Elliptic Integrals - 19.29 Reduction of General Elliptic Integrals

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DLMF Formula Constraints Maple Mathematica Symbolic
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Mathematica 		Numeric
Maple 		Numeric
Mathematica
19.29#Ex1 X α = a α + b α ⁢ x subscript 𝑋 𝛼 subscript 𝑎 𝛼 subscript 𝑏 𝛼 𝑥 {\displaystyle{\displaystyle X_{\alpha}=\sqrt{a_{\alpha}+b_{\alpha}x}}}
X_{\alpha} = \sqrt{a_{\alpha}+b_{\alpha}x}		 

X[alpha] = sqrt(a[alpha]+ b[alpha]*x)
 
Subscript[X, \[Alpha]] == Sqrt[Subscript[a, \[Alpha]]+ Subscript[b, \[Alpha]]*x]
 Skipped - no semantic math Skipped - no semantic math - -
19.29#Ex2 Y α = a α + b α ⁢ y subscript 𝑌 𝛼 subscript 𝑎 𝛼 subscript 𝑏 𝛼 𝑦 {\displaystyle{\displaystyle Y_{\alpha}=\sqrt{a_{\alpha}+b_{\alpha}y}}}
Y_{\alpha} = \sqrt{a_{\alpha}+b_{\alpha}y}		 x > y , 1 ≤ α , α ≤ 5 formulae-sequence 𝑥 𝑦 formulae-sequence 1 𝛼 𝛼 5 {\displaystyle{\displaystyle x>y,1\leq\alpha,\alpha\leq 5}} 
Y[alpha] = sqrt(a[alpha]+ b[alpha]*y)
 
Subscript[Y, \[Alpha]] == Sqrt[Subscript[a, \[Alpha]]+ Subscript[b, \[Alpha]]*y]
 Skipped - no semantic math Skipped - no semantic math - -
19.29.E2 d α ⁢ β = a α ⁢ b β - a β ⁢ b α subscript 𝑑 𝛼 𝛽 subscript 𝑎 𝛼 subscript 𝑏 𝛽 subscript 𝑎 𝛽 subscript 𝑏 𝛼 {\displaystyle{\displaystyle d_{\alpha\beta}=a_{\alpha}b_{\beta}-a_{\beta}b_{% \alpha}}}
d_{\alpha\beta} = a_{\alpha}b_{\beta}-a_{\beta}b_{\alpha}		 d α ⁢ β ≠ 0 , α ≠ β formulae-sequence subscript 𝑑 𝛼 𝛽 0 𝛼 𝛽 {\displaystyle{\displaystyle d_{\alpha\beta}\neq 0,\alpha\neq\beta}} 
d[alpha*beta] = a[alpha]*b[beta]- a[beta]*b[alpha]
 
Subscript[d, \[Alpha]*\[Beta]] == Subscript[a, \[Alpha]]*Subscript[b, \[Beta]]- Subscript[a, \[Beta]]*Subscript[b, \[Alpha]]
 Skipped - no semantic math Skipped - no semantic math - -
19.29.E3 s ⁢ ( t ) = ∏ α = 1 4 a α + b α ⁢ t 𝑠 𝑡 superscript subscript product 𝛼 1 4 subscript 𝑎 𝛼 subscript 𝑏 𝛼 𝑡 {\displaystyle{\displaystyle s(t)=\prod_{\alpha=1}^{4}\sqrt{a_{\alpha}+b_{% \alpha}t}}}
s(t) = \prod_{\alpha=1}^{4}\sqrt{a_{\alpha}+b_{\alpha}t}		 

s(t) = product(sqrt(a[alpha]+ b[alpha]*t), alpha = 1..4)
 
s[t] == Product[Sqrt[Subscript[a, \[Alpha]]+ Subscript[b, \[Alpha]]*t], {\[Alpha], 1, 4}, GenerateConditions->None]
 Skipped - no semantic math Skipped - no semantic math - -
19.29.E4 ∫ y x d t s ⁢ ( t ) = 2 ⁢ R F ⁡ ( U 12 2 , U 13 2 , U 23 2 ) superscript subscript 𝑦 𝑥 𝑡 𝑠 𝑡 2 Carlson-integral-RF superscript subscript 𝑈 12 2 superscript subscript 𝑈 13 2 superscript subscript 𝑈 23 2 {\displaystyle{\displaystyle\int_{y}^{x}\frac{\mathrm{d}t}{s(t)}=2R_{F}\left(U% _{12}^{2},U_{13}^{2},U_{23}^{2}\right)}}
\int_{y}^{x}\frac{\diff{t}}{s(t)} = 2\CarlsonsymellintRF@{U_{12}^{2}}{U_{13}^{2}}{U_{23}^{2}}		 

int((1)/(s(t)), t = y..x) = 2*0.5*int(1/(sqrt(t+(U[12])^(2))*sqrt(t+(U[13])^(2))*sqrt(t+(U[23])^(2))), t = 0..infinity)

Integrate[Divide[1,s[t]], {t, y, x}, GenerateConditions->None] == 2*EllipticF[ArcCos[Sqrt[(Subscript[U, 12])^(2)/(Subscript[U, 23])^(2)]],((Subscript[U, 23])^(2)-(Subscript[U, 13])^(2))/((Subscript[U, 23])^(2)-(Subscript[U, 12])^(2))]/Sqrt[(Subscript[U, 23])^(2)-(Subscript[U, 12])^(2)]
Aborted Aborted Skipped - Because timed out Skipped - Because timed out
19.29#Ex3 U α ⁢ β = ( X α ⁢ X β ⁢ Y γ ⁢ Y δ + Y α ⁢ Y β ⁢ X γ ⁢ X δ ) / ( x - y ) subscript 𝑈 𝛼 𝛽 subscript 𝑋 𝛼 subscript 𝑋 𝛽 subscript 𝑌 𝛾 subscript 𝑌 𝛿 subscript 𝑌 𝛼 subscript 𝑌 𝛽 subscript 𝑋 𝛾 subscript 𝑋 𝛿 𝑥 𝑦 {\displaystyle{\displaystyle U_{\alpha\beta}=(X_{\alpha}X_{\beta}Y_{\gamma}Y_{% \delta}+Y_{\alpha}Y_{\beta}X_{\gamma}X_{\delta})/(x-y)}}
U_{\alpha\beta} = (X_{\alpha}X_{\beta}Y_{\gamma}Y_{\delta}+Y_{\alpha}Y_{\beta}X_{\gamma}X_{\delta})/(x-y)		 

U[alpha*beta] = (X[alpha]*X[beta]*Y[gamma]*Y[delta]+ Y[alpha]*Y[beta]*X[gamma]*X[delta])/(x - y)
 
Subscript[U, \[Alpha]*\[Beta]] == (Subscript[X, \[Alpha]]*Subscript[X, \[Beta]]*Subscript[Y, \[Gamma]]*Subscript[Y, \[Delta]]+ Subscript[Y, \[Alpha]]*Subscript[Y, \[Beta]]*Subscript[X, \[Gamma]]*Subscript[X, \[Delta]])/(x - y)
 Skipped - no semantic math Skipped - no semantic math - -
19.29#Ex4 U α ⁢ β = U β ⁢ α subscript 𝑈 𝛼 𝛽 subscript 𝑈 𝛽 𝛼 {\displaystyle{\displaystyle U_{\alpha\beta}=U_{\beta\alpha}}}
U_{\alpha\beta} = U_{\beta\alpha}		 

U[alpha*beta] = U[beta*alpha]
 
Subscript[U, \[Alpha]*\[Beta]] == Subscript[U, \[Beta]*\[Alpha]]
 Skipped - no semantic math Skipped - no semantic math - -
19.29#Ex5 U α ⁢ β 2 - U α ⁢ γ 2 = d α ⁢ δ ⁢ d β ⁢ γ superscript subscript 𝑈 𝛼 𝛽 2 superscript subscript 𝑈 𝛼 𝛾 2 subscript 𝑑 𝛼 𝛿 subscript 𝑑 𝛽 𝛾 {\displaystyle{\displaystyle U_{\alpha\beta}^{2}-U_{\alpha\gamma}^{2}=d_{% \alpha\delta}d_{\beta\gamma}}}
U_{\alpha\beta}^{2}-U_{\alpha\gamma}^{2} = d_{\alpha\delta}d_{\beta\gamma}		 

(U[alpha*beta])^(2)- (U[alpha*gamma])^(2) = d[alpha*delta]*d[beta*gamma]
 
(Subscript[U, \[Alpha]*\[Beta]])^(2)- (Subscript[U, \[Alpha]*\[Gamma]])^(2) == Subscript[d, \[Alpha]*\[Delta]]*Subscript[d, \[Beta]*\[Gamma]]
 Skipped - no semantic math Skipped - no semantic math - -
19.29#Ex6 U α ⁢ β = b α ⁢ b β ⁢ Y γ ⁢ Y δ + Y α ⁢ Y β ⁢ b γ ⁢ b δ , subscript 𝑈 𝛼 𝛽 subscript 𝑏 𝛼 subscript 𝑏 𝛽 subscript 𝑌 𝛾 subscript 𝑌 𝛿 subscript 𝑌 𝛼 subscript 𝑌 𝛽 subscript 𝑏 𝛾 subscript 𝑏 𝛿 {\displaystyle{\displaystyle U_{\alpha\beta}=\sqrt{b_{\alpha}}\sqrt{b_{\beta}}% Y_{\gamma}Y_{\delta}+Y_{\alpha}Y_{\beta}\sqrt{b_{\gamma}}\sqrt{b_{\delta}},}}
U_{\alpha\beta} = \sqrt{b_{\alpha}}\sqrt{b_{\beta}}Y_{\gamma}Y_{\delta}+Y_{\alpha}Y_{\beta}\sqrt{b_{\gamma}}\sqrt{b_{\delta}},		 x = ∞ 𝑥 {\displaystyle{\displaystyle x=\infty}} 
U[alpha*beta] = sqrt(b[alpha])*sqrt(b[beta])*Y[gamma]*Y[delta]+ Y[alpha]*Y[beta]*sqrt(b[gamma])*sqrt(b[delta])
 
Subscript[U, \[Alpha]*\[Beta]] == Sqrt[Subscript[b, \[Alpha]]]*Sqrt[Subscript[b, \[Beta]]]*Subscript[Y, \[Gamma]]*Subscript[Y, \[Delta]]+ Subscript[Y, \[Alpha]]*Subscript[Y, \[Beta]]*Sqrt[Subscript[b, \[Gamma]]]*Sqrt[Subscript[b, \[Delta]]]
 Skipped - no semantic math Skipped - no semantic math - -
19.29#Ex7 U α ⁢ β = X α ⁢ X β ⁢ - b γ ⁢ - b δ + - b α ⁢ - b β ⁢ X γ ⁢ X δ subscript 𝑈 𝛼 𝛽 subscript 𝑋 𝛼 subscript 𝑋 𝛽 subscript 𝑏 𝛾 subscript 𝑏 𝛿 subscript 𝑏 𝛼 subscript 𝑏 𝛽 subscript 𝑋 𝛾 subscript 𝑋 𝛿 {\displaystyle{\displaystyle U_{\alpha\beta}=X_{\alpha}X_{\beta}\sqrt{-b_{% \gamma}}\sqrt{-b_{\delta}}+\sqrt{-b_{\alpha}}\sqrt{-b_{\beta}}X_{\gamma}X_{% \delta}}}
U_{\alpha\beta} = X_{\alpha}X_{\beta}\sqrt{-b_{\gamma}}\sqrt{-b_{\delta}}+\sqrt{-b_{\alpha}}\sqrt{-b_{\beta}}X_{\gamma}X_{\delta}		 y = - ∞ 𝑦 {\displaystyle{\displaystyle y=-\infty}} 
U[alpha*beta] = X[alpha]*X[beta]*sqrt(- b[gamma])*sqrt(- b[delta])+sqrt(- b[alpha])*sqrt(- b[beta])*X[gamma]*X[delta]
 
Subscript[U, \[Alpha]*\[Beta]] == Subscript[X, \[Alpha]]*Subscript[X, \[Beta]]*Sqrt[- Subscript[b, \[Gamma]]]*Sqrt[- Subscript[b, \[Delta]]]+Sqrt[- Subscript[b, \[Alpha]]]*Sqrt[- Subscript[b, \[Beta]]]*Subscript[X, \[Gamma]]*Subscript[X, \[Delta]]
 Skipped - no semantic math Skipped - no semantic math - -
19.29.E7 ∫ y x a α + b α ⁢ t a δ + b δ ⁢ t ⁢ d t s ⁢ ( t ) = 2 3 ⁢ d α ⁢ β ⁢ d α ⁢ γ ⁢ R D ⁡ ( U α ⁢ β 2 , U α ⁢ γ 2 , U α ⁢ δ 2 ) + 2 ⁢ X α ⁢ Y α X δ ⁢ Y δ ⁢ U α ⁢ δ superscript subscript 𝑦 𝑥 subscript 𝑎 𝛼 subscript 𝑏 𝛼 𝑡 subscript 𝑎 𝛿 subscript 𝑏 𝛿 𝑡 𝑡 𝑠 𝑡 2 3 subscript 𝑑 𝛼 𝛽 subscript 𝑑 𝛼 𝛾 Carlson-integral-RD superscript subscript 𝑈 𝛼 𝛽 2 superscript subscript 𝑈 𝛼 𝛾 2 superscript subscript 𝑈 𝛼 𝛿 2 2 subscript 𝑋 𝛼 subscript 𝑌 𝛼 subscript 𝑋 𝛿 subscript 𝑌 𝛿 subscript 𝑈 𝛼 𝛿 {\displaystyle{\displaystyle\int_{y}^{x}\frac{a_{\alpha}+b_{\alpha}t}{a_{% \delta}+b_{\delta}t}\frac{\mathrm{d}t}{s(t)}=\tfrac{2}{3}d_{\alpha\beta}d_{% \alpha\gamma}R_{D}\left(U_{\alpha\beta}^{2},U_{\alpha\gamma}^{2},U_{\alpha% \delta}^{2}\right)+\frac{2X_{\alpha}Y_{\alpha}}{X_{\delta}Y_{\delta}U_{\alpha% \delta}}}}
\int_{y}^{x}\frac{a_{\alpha}+b_{\alpha}t}{a_{\delta}+b_{\delta}t}\frac{\diff{t}}{s(t)} = \tfrac{2}{3}d_{\alpha\beta}d_{\alpha\gamma}\CarlsonsymellintRD@{U_{\alpha\beta}^{2}}{U_{\alpha\gamma}^{2}}{U_{\alpha\delta}^{2}}+\frac{2X_{\alpha}Y_{\alpha}}{X_{\delta}Y_{\delta}U_{\alpha\delta}}		 U α ⁢ δ ≠ 0 subscript 𝑈 𝛼 𝛿 0 {\displaystyle{\displaystyle U_{\alpha\delta}\neq 0}} 
Error

Integrate[Divide[Subscript[a, \[Alpha]]+ Subscript[b, \[Alpha]]*t,Subscript[a, \[Delta]]+ Subscript[b, \[Delta]]*t]*Divide[1,s[t]], {t, y, x}, GenerateConditions->None] == Divide[2,3]*Subscript[d, \[Alpha]*\[Beta]]*Subscript[d, \[Alpha]*\[Gamma]]*3*(EllipticF[ArcCos[Sqrt[(Subscript[U, \[Alpha]*\[Beta]])^(2)/(Subscript[U, \[Alpha]*\[Delta]])^(2)]],((Subscript[U, \[Alpha]*\[Delta]])^(2)-(Subscript[U, \[Alpha]*\[Gamma]])^(2))/((Subscript[U, \[Alpha]*\[Delta]])^(2)-(Subscript[U, \[Alpha]*\[Beta]])^(2))]-EllipticE[ArcCos[Sqrt[(Subscript[U, \[Alpha]*\[Beta]])^(2)/(Subscript[U, \[Alpha]*\[Delta]])^(2)]],((Subscript[U, \[Alpha]*\[Delta]])^(2)-(Subscript[U, \[Alpha]*\[Gamma]])^(2))/((Subscript[U, \[Alpha]*\[Delta]])^(2)-(Subscript[U, \[Alpha]*\[Beta]])^(2))])/(((Subscript[U, \[Alpha]*\[Delta]])^(2)-(Subscript[U, \[Alpha]*\[Gamma]])^(2))*((Subscript[U, \[Alpha]*\[Delta]])^(2)-(Subscript[U, \[Alpha]*\[Beta]])^(2))^(1/2))+Divide[2*Subscript[X, \[Alpha]]*Subscript[Y, \[Alpha]],Subscript[X, \[Delta]]*Subscript[Y, \[Delta]]*Subscript[U, \[Alpha]*\[Delta]]]
Missing Macro Error Aborted - Skipped - Because timed out
19.29.E8 ∫ y x a α + b α ⁢ t a 5 + b 5 ⁢ t ⁢ d t s ⁢ ( t ) = 2 3 ⁢ d α ⁢ β ⁢ d α ⁢ γ ⁢ d α ⁢ δ d α ⁢ 5 ⁢ R J ⁡ ( U 12 2 , U 13 2 , U 23 2 , U α ⁢ 5 2 ) + 2 ⁢ R C ⁡ ( S α ⁢ 5 2 , Q α ⁢ 5 2 ) superscript subscript 𝑦 𝑥 subscript 𝑎 𝛼 subscript 𝑏 𝛼 𝑡 subscript 𝑎 5 subscript 𝑏 5 𝑡 𝑡 𝑠 𝑡 2 3 subscript 𝑑 𝛼 𝛽 subscript 𝑑 𝛼 𝛾 subscript 𝑑 𝛼 𝛿 subscript 𝑑 𝛼 5 Carlson-integral-RJ superscript subscript 𝑈 12 2 superscript subscript 𝑈 13 2 superscript subscript 𝑈 23 2 superscript subscript 𝑈 𝛼 5 2 2 Carlson-integral-RC superscript subscript 𝑆 𝛼 5 2 superscript subscript 𝑄 𝛼 5 2 {\displaystyle{\displaystyle\int_{y}^{x}\frac{a_{\alpha}+b_{\alpha}t}{a_{5}+b_% {5}t}\frac{\mathrm{d}t}{s(t)}=\frac{2}{3}\frac{d_{\alpha\beta}d_{\alpha\gamma}% d_{\alpha\delta}}{d_{\alpha 5}}R_{J}\left(U_{12}^{2},U_{13}^{2},U_{23}^{2},U_{% \alpha 5}^{2}\right)+2R_{C}\left(S_{\alpha 5}^{2},Q_{\alpha 5}^{2}\right)}}
\int_{y}^{x}\frac{a_{\alpha}+b_{\alpha}t}{a_{5}+b_{5}t}\frac{\diff{t}}{s(t)} = \frac{2}{3}\frac{d_{\alpha\beta}d_{\alpha\gamma}d_{\alpha\delta}}{d_{\alpha 5}}\CarlsonsymellintRJ@{U_{12}^{2}}{U_{13}^{2}}{U_{23}^{2}}{U_{\alpha 5}^{2}}+2\CarlsonellintRC@{S_{\alpha 5}^{2}}{Q_{\alpha 5}^{2}}		 

Error

Integrate[Divide[Subscript[a, \[Alpha]]+ Subscript[b, \[Alpha]]*t,Subscript[a, 5]+ Subscript[b, 5]*t]*Divide[1,s[t]], {t, y, x}, GenerateConditions->None] == Divide[2,3]*Divide[Subscript[d, \[Alpha]*\[Beta]]*Subscript[d, \[Alpha]*\[Gamma]]*Subscript[d, \[Alpha]*\[Delta]],Subscript[d, \[Alpha]*5]]*3*((Subscript[U, 23])^(2)-(Subscript[U, 12])^(2))/((Subscript[U, 23])^(2)-(Subscript[U, \[Alpha]*5])^(2))*(EllipticPi[((Subscript[U, 23])^(2)-(Subscript[U, \[Alpha]*5])^(2))/((Subscript[U, 23])^(2)-(Subscript[U, 12])^(2)),ArcCos[Sqrt[(Subscript[U, 12])^(2)/(Subscript[U, 23])^(2)]],((Subscript[U, 23])^(2)-(Subscript[U, 13])^(2))/((Subscript[U, 23])^(2)-(Subscript[U, 12])^(2))]-EllipticF[ArcCos[Sqrt[(Subscript[U, 12])^(2)/(Subscript[U, 23])^(2)]],((Subscript[U, 23])^(2)-(Subscript[U, 13])^(2))/((Subscript[U, 23])^(2)-(Subscript[U, 12])^(2))])/Sqrt[(Subscript[U, 23])^(2)-(Subscript[U, 12])^(2)]+ 2*1/Sqrt[(Subscript[Q, \[Alpha]*5])^(2)]*Hypergeometric2F1[1/2,1/2,3/2,1-((Subscript[S, \[Alpha]*5])^(2))/((Subscript[Q, \[Alpha]*5])^(2))]
Missing Macro Error Aborted - Skipped - Because timed out
19.29#Ex8 U α ⁢ 5 2 = U α ⁢ β 2 - d α ⁢ γ ⁢ d α ⁢ δ ⁢ d β ⁢ 5 d α ⁢ 5 superscript subscript 𝑈 𝛼 5 2 superscript subscript 𝑈 𝛼 𝛽 2 subscript 𝑑 𝛼 𝛾 subscript 𝑑 𝛼 𝛿 subscript 𝑑 𝛽 5 subscript 𝑑 𝛼 5 {\displaystyle{\displaystyle U_{\alpha 5}^{2}=U_{\alpha\beta}^{2}-\frac{d_{% \alpha\gamma}d_{\alpha\delta}d_{\beta 5}}{d_{\alpha 5}}}}
U_{\alpha 5}^{2} = U_{\alpha\beta}^{2}-\frac{d_{\alpha\gamma}d_{\alpha\delta}d_{\beta 5}}{d_{\alpha 5}}		 

(U[alpha*5])^(2) = (U[alpha*beta])^(2)-(d[alpha*gamma]*d[alpha*delta]*d[beta*5])/(d[alpha*5])
 
(Subscript[U, \[Alpha]*5])^(2) == (Subscript[U, \[Alpha]*\[Beta]])^(2)-Divide[Subscript[d, \[Alpha]*\[Gamma]]*Subscript[d, \[Alpha]*\[Delta]]*Subscript[d, \[Beta]*5],Subscript[d, \[Alpha]*5]]
 Skipped - no semantic math Skipped - no semantic math - -
19.29#Ex9 S α ⁢ 5 = 1 x - y ⁢ ( X β ⁢ X γ ⁢ X δ X α ⁢ Y 5 2 + Y β ⁢ Y γ ⁢ Y δ Y α ⁢ X 5 2 ) subscript 𝑆 𝛼 5 1 𝑥 𝑦 subscript 𝑋 𝛽 subscript 𝑋 𝛾 subscript 𝑋 𝛿 subscript 𝑋 𝛼 superscript subscript 𝑌 5 2 subscript 𝑌 𝛽 subscript 𝑌 𝛾 subscript 𝑌 𝛿 subscript 𝑌 𝛼 superscript subscript 𝑋 5 2 {\displaystyle{\displaystyle S_{\alpha 5}=\frac{1}{x-y}\left(\frac{X_{\beta}X_% {\gamma}X_{\delta}}{X_{\alpha}}Y_{5}^{2}+\frac{Y_{\beta}Y_{\gamma}Y_{\delta}}{% Y_{\alpha}}X_{5}^{2}\right)}}
S_{\alpha 5} = \frac{1}{x-y}\left(\frac{X_{\beta}X_{\gamma}X_{\delta}}{X_{\alpha}}Y_{5}^{2}+\frac{Y_{\beta}Y_{\gamma}Y_{\delta}}{Y_{\alpha}}X_{5}^{2}\right)		 

S[alpha*5] = (1)/(x - y)*((X[beta]*X[gamma]*X[delta])/(X[alpha])*(Y[5])^(2)+(Y[beta]*Y[gamma]*Y[delta])/(Y[alpha])*(X[5])^(2))
 
Subscript[S, \[Alpha]*5] == Divide[1,x - y]*(Divide[Subscript[X, \[Beta]]*Subscript[X, \[Gamma]]*Subscript[X, \[Delta]],Subscript[X, \[Alpha]]]*(Subscript[Y, 5])^(2)+Divide[Subscript[Y, \[Beta]]*Subscript[Y, \[Gamma]]*Subscript[Y, \[Delta]],Subscript[Y, \[Alpha]]]*(Subscript[X, 5])^(2))
 Skipped - no semantic math Skipped - no semantic math - -
19.29#Ex10 Q α ⁢ 5 = X 5 ⁢ Y 5 X α ⁢ Y α ⁢ U α ⁢ 5 subscript 𝑄 𝛼 5 subscript 𝑋 5 subscript 𝑌 5 subscript 𝑋 𝛼 subscript 𝑌 𝛼 subscript 𝑈 𝛼 5 {\displaystyle{\displaystyle Q_{\alpha 5}=\frac{X_{5}Y_{5}}{X_{\alpha}Y_{% \alpha}}U_{\alpha 5}}}
Q_{\alpha 5} = \frac{X_{5}Y_{5}}{X_{\alpha}Y_{\alpha}}U_{\alpha 5}		 

Q[alpha*5] = (X[5]*Y[5])/(X[alpha]*Y[alpha])*U[alpha*5]
 
Subscript[Q, \[Alpha]*5] == Divide[Subscript[X, 5]*Subscript[Y, 5],Subscript[X, \[Alpha]]*Subscript[Y, \[Alpha]]]*Subscript[U, \[Alpha]*5]
 Skipped - no semantic math Skipped - no semantic math - -
19.29#Ex11 S α ⁢ 5 2 - Q α ⁢ 5 2 = d β ⁢ 5 ⁢ d γ ⁢ 5 ⁢ d δ ⁢ 5 d α ⁢ 5 superscript subscript 𝑆 𝛼 5 2 superscript subscript 𝑄 𝛼 5 2 subscript 𝑑 𝛽 5 subscript 𝑑 𝛾 5 subscript 𝑑 𝛿 5 subscript 𝑑 𝛼 5 {\displaystyle{\displaystyle S_{\alpha 5}^{2}-Q_{\alpha 5}^{2}=\frac{d_{\beta 5% }d_{\gamma 5}d_{\delta 5}}{d_{\alpha 5}}}}
S_{\alpha 5}^{2}-Q_{\alpha 5}^{2} = \frac{d_{\beta 5}d_{\gamma 5}d_{\delta 5}}{d_{\alpha 5}}		 

(S[alpha*5])^(2)- (Q[alpha*5])^(2) = (d[beta*5]*d[gamma*5]*d[delta*5])/(d[alpha*5])
 
(Subscript[S, \[Alpha]*5])^(2)- (Subscript[Q, \[Alpha]*5])^(2) == Divide[Subscript[d, \[Beta]*5]*Subscript[d, \[Gamma]*5]*Subscript[d, \[Delta]*5],Subscript[d, \[Alpha]*5]]
 Skipped - no semantic math Skipped - no semantic math - -
19.29.E10 ∫ u b a - t ( b - t ) ⁢ ( t - c ) 3 ⁢ d t = - 2 3 ⁢ ( a - b ) ⁢ ( b - u ) 3 / 2 ⁢ R D ⁡ + 2 b - c ⁢ ( a - u ) ⁢ ( b - u ) u - c superscript subscript 𝑢 𝑏 𝑎 𝑡 𝑏 𝑡 superscript 𝑡 𝑐 3 𝑡 2 3 𝑎 𝑏 superscript 𝑏 𝑢 3 2 Carlson-integral-RD 𝑎 𝑏 𝑢 𝑐 𝑏 𝑐 𝑎 𝑢 𝑎 𝑏 𝑏 𝑐 2 𝑏 𝑐 𝑎 𝑢 𝑏 𝑢 𝑢 𝑐 {\displaystyle{\displaystyle\int_{u}^{b}\sqrt{\frac{a-t}{(b-t)(t-c)^{3}}}% \mathrm{d}t=-\tfrac{2}{3}{(a-b)}{(b-u)}^{3/2}R_{D}+\frac{2}{b-c}\sqrt{\frac{(a% -u)(b-u)}{u-c}}}}
\int_{u}^{b}\sqrt{\frac{a-t}{(b-t)(t-c)^{3}}}\diff{t} = -\tfrac{2}{3}{(a-b)}{(b-u)}^{3/2}\CarlsonsymellintRD@@{(a-b)(u-c)}{(b-c)(a-u)}{(a-b)(b-c)}+\frac{2}{b-c}\sqrt{\frac{(a-u)(b-u)}{u-c}}		 a > b , b > u , u > c formulae-sequence 𝑎 𝑏 formulae-sequence 𝑏 𝑢 𝑢 𝑐 {\displaystyle{\displaystyle a>b,b>u,u>c}} 
Error

Integrate[Sqrt[Divide[a - t,(b - t)*(t - c)^(3)]], {t, u, b}, GenerateConditions->None] == -Divide[2,3]*(a - b)*(b - u)^(3/2)* 3*(EllipticF[ArcCos[Sqrt[(a - b)*(u - c)/(a - b)*(b - c)]],((a - b)*(b - c)-(b - c)*(a - u))/((a - b)*(b - c)-(a - b)*(u - c))]-EllipticE[ArcCos[Sqrt[(a - b)*(u - c)/(a - b)*(b - c)]],((a - b)*(b - c)-(b - c)*(a - u))/((a - b)*(b - c)-(a - b)*(u - c))])/(((a - b)*(b - c)-(b - c)*(a - u))*((a - b)*(b - c)-(a - b)*(u - c))^(1/2))+Divide[2,b - c]*Sqrt[Divide[(a - u)*(b - u),u - c]]
Missing Macro Error Aborted - Skipped - Because timed out
19.29.E11 I ⁢ ( 𝐦 ) = ∫ y x ∏ α = 1 h ( a α + b α ⁢ t ) - 1 / 2 ⁢ ∏ j = 1 n ( a j + b j ⁢ t ) m j ⁢ d t 𝐼 𝐦 superscript subscript 𝑦 𝑥 superscript subscript product 𝛼 1 ℎ superscript subscript 𝑎 𝛼 subscript 𝑏 𝛼 𝑡 1 2 superscript subscript product 𝑗 1 𝑛 superscript subscript 𝑎 𝑗 subscript 𝑏 𝑗 𝑡 subscript 𝑚 𝑗 𝑡 {\displaystyle{\displaystyle I(\mathbf{m})=\int_{y}^{x}\prod_{\alpha=1}^{h}(a_% {\alpha}+b_{\alpha}t)^{-1/2}\prod_{j=1}^{n}(a_{j}+b_{j}t)^{m_{j}}\mathrm{d}t}}
I(\mathbf{m}) = \int_{y}^{x}\prod_{\alpha=1}^{h}(a_{\alpha}+b_{\alpha}t)^{-1/2}\prod_{j=1}^{n}(a_{j}+b_{j}t)^{m_{j}}\diff{t}		 

I(m) = int(product((a[alpha]+ b[alpha]*t)^(- 1/2)* product((a[j]+ b[j]*t)^(m[j]), j = 1..n), alpha = 1..h), t = y..x)

I[m] == Integrate[Product[(Subscript[a, \[Alpha]]+ Subscript[b, \[Alpha]]*t)^(- 1/2)* Product[(Subscript[a, j]+ Subscript[b, j]*t)^(Subscript[m, j]), {j, 1, n}, GenerateConditions->None], {\[Alpha], 1, h}, GenerateConditions->None], {t, y, x}, GenerateConditions->None]
Aborted Aborted Error Skipped - Because timed out
19.29.E15 b j ⁢ I ⁢ ( 𝐞 l - 𝐞 j ) = d l ⁢ j ⁢ I ⁢ ( - 𝐞 j ) + b l ⁢ I ⁢ ( 𝟎 ) subscript 𝑏 𝑗 𝐼 subscript 𝐞 𝑙 subscript 𝐞 𝑗 subscript 𝑑 𝑙 𝑗 𝐼 subscript 𝐞 𝑗 subscript 𝑏 𝑙 𝐼 0 {\displaystyle{\displaystyle b_{j}I(\mathbf{e}_{l}-\mathbf{e}_{j})=d_{lj}I(-% \mathbf{e}_{j})+b_{l}I(\boldsymbol{{0}})}}
b_{j}I(\mathbf{e}_{l}-\mathbf{e}_{j}) = d_{lj}I(-\mathbf{e}_{j})+b_{l}I(\boldsymbol{{0}})		 j = 1 , l = 1 formulae-sequence 𝑗 1 𝑙 1 {\displaystyle{\displaystyle j=1,l=1}} 
b[j]*I(e[l]- e[j]) = d[l, j]*I(- e[j])+ b[l]*I(0)
 
Subscript[b, j]*I[Subscript[e, l]- Subscript[e, j]] == Subscript[d, l, j]*I[- Subscript[e, j]]+ Subscript[b, l]*I[0]
 Skipped - no semantic math Skipped - no semantic math - -
19.29.E16 b β ⁢ b γ ⁢ I ⁢ ( 𝐞 α ) = d α ⁢ β ⁢ d α ⁢ γ ⁢ I ⁢ ( - 𝐞 α ) + 2 ⁢ b α ⁢ ( s ⁢ ( x ) a α + b α ⁢ x - s ⁢ ( y ) a α + b α ⁢ y ) subscript 𝑏 𝛽 subscript 𝑏 𝛾 𝐼 subscript 𝐞 𝛼 subscript 𝑑 𝛼 𝛽 subscript 𝑑 𝛼 𝛾 𝐼 subscript 𝐞 𝛼 2 subscript 𝑏 𝛼 𝑠 𝑥 subscript 𝑎 𝛼 subscript 𝑏 𝛼 𝑥 𝑠 𝑦 subscript 𝑎 𝛼 subscript 𝑏 𝛼 𝑦 {\displaystyle{\displaystyle b_{\beta}b_{\gamma}I(\mathbf{e}_{\alpha})=d_{% \alpha\beta}d_{\alpha\gamma}I(-\mathbf{e}_{\alpha})+2b_{\alpha}\left(\frac{s(x% )}{a_{\alpha}+b_{\alpha}x}-\frac{s(y)}{a_{\alpha}+b_{\alpha}y}\right)}}
b_{\beta}b_{\gamma}I(\mathbf{e}_{\alpha}) = d_{\alpha\beta}d_{\alpha\gamma}I(-\mathbf{e}_{\alpha})+2b_{\alpha}\left(\frac{s(x)}{a_{\alpha}+b_{\alpha}x}-\frac{s(y)}{a_{\alpha}+b_{\alpha}y}\right)		 

b[beta]*b[gamma]*I(e[alpha]) = d[alpha*beta]*d[alpha*gamma]*I(- e[alpha])+ 2*b[alpha]*((s(x))/(a[alpha]+ b[alpha]*x)-(s(y))/(a[alpha]+ b[alpha]*y))
 
Subscript[b, \[Beta]]*Subscript[b, \[Gamma]]*I[Subscript[e, \[Alpha]]] == Subscript[d, \[Alpha]*\[Beta]]*Subscript[d, \[Alpha]*\[Gamma]]*I[- Subscript[e, \[Alpha]]]+ 2*Subscript[b, \[Alpha]]*(Divide[s[x],Subscript[a, \[Alpha]]+ Subscript[b, \[Alpha]]*x]-Divide[s[y],Subscript[a, \[Alpha]]+ Subscript[b, \[Alpha]]*y])
 Skipped - no semantic math Skipped - no semantic math - -
19.29.E17 s ⁢ ( t ) = ∏ α = 1 3 a α + b α ⁢ t 𝑠 𝑡 superscript subscript product 𝛼 1 3 subscript 𝑎 𝛼 subscript 𝑏 𝛼 𝑡 {\displaystyle{\displaystyle s(t)=\prod_{\alpha=1}^{3}\sqrt{a_{\alpha}+b_{% \alpha}t}}}
s(t) = \prod_{\alpha=1}^{3}\sqrt{a_{\alpha}+b_{\alpha}t}		 

s(t) = product(sqrt(a[alpha]+ b[alpha]*t), alpha = 1..3)
 
s[t] == Product[Sqrt[Subscript[a, \[Alpha]]+ Subscript[b, \[Alpha]]*t], {\[Alpha], 1, 3}, GenerateConditions->None]
 Skipped - no semantic math Skipped - no semantic math - -
19.29.E18 b j q ⁢ I ⁢ ( q ⁢ 𝐞 l ) = ∑ r = 0 q ( q r ) ⁢ b l r ⁢ d l ⁢ j q - r ⁢ I ⁢ ( r ⁢ 𝐞 j ) superscript subscript 𝑏 𝑗 𝑞 𝐼 𝑞 subscript 𝐞 𝑙 superscript subscript 𝑟 0 𝑞 binomial 𝑞 𝑟 superscript subscript 𝑏 𝑙 𝑟 superscript subscript 𝑑 𝑙 𝑗 𝑞 𝑟 𝐼 𝑟 subscript 𝐞 𝑗 {\displaystyle{\displaystyle b_{j}^{q}I(q\mathbf{e}_{l})=\sum_{r=0}^{q}% \genfrac{(}{)}{0.0pt}{}{q}{r}b_{l}^{r}d_{lj}^{q-r}I(r\mathbf{e}_{j})}}
b_{j}^{q}I(q\mathbf{e}_{l}) = \sum_{r=0}^{q}\binom{q}{r}b_{l}^{r}d_{lj}^{q-r}I(r\mathbf{e}_{j})		 j = 1 , l = 1 formulae-sequence 𝑗 1 𝑙 1 {\displaystyle{\displaystyle j=1,l=1}} 
(b[j])^(q)*I(q*e[l]) = sum(binomial(q,r)*(b[l])^(r)*(d[l, j])^(q - r)*I(r*e[j]), r = 0..q)

(Subscript[b, j])^(q)*I[q*Subscript[e, l]] == Sum[Binomial[q,r]*(Subscript[b, l])^(r)*(Subscript[d, l, j])^(q - r)*I[r*Subscript[e, j]], {r, 0, q}, GenerateConditions->None]
Failure Failure Error Skipped - Because timed out
19.29.E19 ∫ y x d t Q 1 ⁢ ( t ) ⁢ Q 2 ⁢ ( t ) = R F ⁡ ( U 2 + a 1 ⁢ b 2 , U 2 + a 2 ⁢ b 1 , U 2 ) superscript subscript 𝑦 𝑥 𝑡 subscript 𝑄 1 𝑡 subscript 𝑄 2 𝑡 Carlson-integral-RF superscript 𝑈 2 subscript 𝑎 1 subscript 𝑏 2 superscript 𝑈 2 subscript 𝑎 2 subscript 𝑏 1 superscript 𝑈 2 {\displaystyle{\displaystyle\int_{y}^{x}\frac{\mathrm{d}t}{\sqrt{Q_{1}(t)Q_{2}% (t)}}=R_{F}\left(U^{2}+a_{1}b_{2},U^{2}+a_{2}b_{1},U^{2}\right)}}
\int_{y}^{x}\frac{\diff{t}}{\sqrt{Q_{1}(t)Q_{2}(t)}} = \CarlsonsymellintRF@{U^{2}+a_{1}b_{2}}{U^{2}+a_{2}b_{1}}{U^{2}}		 

int((1)/(sqrt(Q[1](t)* Q[2](t))), t = y..x) = 0.5*int(1/(sqrt(t+(U)^(2)+ a[1]*b[2])*sqrt(t+(U)^(2)+ a[2]*b[1])*sqrt(t+(U)^(2))), t = 0..infinity)

Integrate[Divide[1,Sqrt[Subscript[Q, 1][t]* Subscript[Q, 2][t]]], {t, y, x}, GenerateConditions->None] == EllipticF[ArcCos[Sqrt[(U)^(2)+ Subscript[a, 1]*Subscript[b, 2]/(U)^(2)]],((U)^(2)-(U)^(2)+ Subscript[a, 2]*Subscript[b, 1])/((U)^(2)-(U)^(2)+ Subscript[a, 1]*Subscript[b, 2])]/Sqrt[(U)^(2)-(U)^(2)+ Subscript[a, 1]*Subscript[b, 2]]
Aborted Aborted Manual Skip! Skipped - Because timed out
19.29.E20 ∫ y x t 2 ⁢ d t Q 1 ⁢ ( t ) ⁢ Q 2 ⁢ ( t ) = 1 3 ⁢ a 1 ⁢ a 2 ⁢ R D ⁡ ( U 2 + a 1 ⁢ b 2 , U 2 + a 2 ⁢ b 1 , U 2 ) + ( x ⁢ y / U ) superscript subscript 𝑦 𝑥 superscript 𝑡 2 𝑡 subscript 𝑄 1 𝑡 subscript 𝑄 2 𝑡 1 3 subscript 𝑎 1 subscript 𝑎 2 Carlson-integral-RD superscript 𝑈 2 subscript 𝑎 1 subscript 𝑏 2 superscript 𝑈 2 subscript 𝑎 2 subscript 𝑏 1 superscript 𝑈 2 𝑥 𝑦 𝑈 {\displaystyle{\displaystyle\int_{y}^{x}\frac{t^{2}\mathrm{d}t}{\sqrt{Q_{1}(t)% Q_{2}(t)}}=\tfrac{1}{3}a_{1}a_{2}R_{D}\left(U^{2}+a_{1}b_{2},U^{2}+a_{2}b_{1},% U^{2}\right)+(xy/U)}}
\int_{y}^{x}\frac{t^{2}\diff{t}}{\sqrt{Q_{1}(t)Q_{2}(t)}} = \tfrac{1}{3}a_{1}a_{2}\CarlsonsymellintRD@{U^{2}+a_{1}b_{2}}{U^{2}+a_{2}b_{1}}{U^{2}}+(xy/U)		 

Error

Integrate[Divide[(t)^(2),Sqrt[Subscript[Q, 1][t]* Subscript[Q, 2][t]]], {t, y, x}, GenerateConditions->None] == Divide[1,3]*Subscript[a, 1]*Subscript[a, 2]*3*(EllipticF[ArcCos[Sqrt[(U)^(2)+ Subscript[a, 1]*Subscript[b, 2]/(U)^(2)]],((U)^(2)-(U)^(2)+ Subscript[a, 2]*Subscript[b, 1])/((U)^(2)-(U)^(2)+ Subscript[a, 1]*Subscript[b, 2])]-EllipticE[ArcCos[Sqrt[(U)^(2)+ Subscript[a, 1]*Subscript[b, 2]/(U)^(2)]],((U)^(2)-(U)^(2)+ Subscript[a, 2]*Subscript[b, 1])/((U)^(2)-(U)^(2)+ Subscript[a, 1]*Subscript[b, 2])])/(((U)^(2)-(U)^(2)+ Subscript[a, 2]*Subscript[b, 1])*((U)^(2)-(U)^(2)+ Subscript[a, 1]*Subscript[b, 2])^(1/2))+(x*y/U)
Missing Macro Error Aborted - Skipped - Because timed out
19.29.E21 ∫ y x d t t 2 ⁢ Q 1 ⁢ ( t ) ⁢ Q 2 ⁢ ( t ) = 1 3 ⁢ b 1 ⁢ b 2 ⁢ R D ⁡ ( U 2 + a 1 ⁢ b 2 , U 2 + a 2 ⁢ b 1 , U 2 ) + ( x ⁢ y ⁢ U ) - 1 superscript subscript 𝑦 𝑥 𝑡 superscript 𝑡 2 subscript 𝑄 1 𝑡 subscript 𝑄 2 𝑡 1 3 subscript 𝑏 1 subscript 𝑏 2 Carlson-integral-RD superscript 𝑈 2 subscript 𝑎 1 subscript 𝑏 2 superscript 𝑈 2 subscript 𝑎 2 subscript 𝑏 1 superscript 𝑈 2 superscript 𝑥 𝑦 𝑈 1 {\displaystyle{\displaystyle\int_{y}^{x}\frac{\mathrm{d}t}{t^{2}\sqrt{Q_{1}(t)% Q_{2}(t)}}=\tfrac{1}{3}b_{1}b_{2}R_{D}\left(U^{2}+a_{1}b_{2},U^{2}+a_{2}b_{1},% U^{2}\right)+(xyU)^{-1}}}
\int_{y}^{x}\frac{\diff{t}}{t^{2}\sqrt{Q_{1}(t)Q_{2}(t)}} = \tfrac{1}{3}b_{1}b_{2}\CarlsonsymellintRD@{U^{2}+a_{1}b_{2}}{U^{2}+a_{2}b_{1}}{U^{2}}+(xyU)^{-1}		 

Error

Integrate[Divide[1,(t)^(2)*Sqrt[Subscript[Q, 1][t]* Subscript[Q, 2][t]]], {t, y, x}, GenerateConditions->None] == Divide[1,3]*Subscript[b, 1]*Subscript[b, 2]*3*(EllipticF[ArcCos[Sqrt[(U)^(2)+ Subscript[a, 1]*Subscript[b, 2]/(U)^(2)]],((U)^(2)-(U)^(2)+ Subscript[a, 2]*Subscript[b, 1])/((U)^(2)-(U)^(2)+ Subscript[a, 1]*Subscript[b, 2])]-EllipticE[ArcCos[Sqrt[(U)^(2)+ Subscript[a, 1]*Subscript[b, 2]/(U)^(2)]],((U)^(2)-(U)^(2)+ Subscript[a, 2]*Subscript[b, 1])/((U)^(2)-(U)^(2)+ Subscript[a, 1]*Subscript[b, 2])])/(((U)^(2)-(U)^(2)+ Subscript[a, 2]*Subscript[b, 1])*((U)^(2)-(U)^(2)+ Subscript[a, 1]*Subscript[b, 2])^(1/2))+(x*y*U)^(- 1)
Missing Macro Error Aborted - Skipped - Because timed out
19.29.E22 ( x 2 - y 2 ) ⁢ U = x ⁢ Q 1 ⁢ ( y ) ⁢ Q 2 ⁢ ( y ) + y ⁢ Q 1 ⁢ ( x ) ⁢ Q 2 ⁢ ( x ) superscript 𝑥 2 superscript 𝑦 2 𝑈 𝑥 subscript 𝑄 1 𝑦 subscript 𝑄 2 𝑦 𝑦 subscript 𝑄 1 𝑥 subscript 𝑄 2 𝑥 {\displaystyle{\displaystyle(x^{2}-y^{2})U=x\sqrt{Q_{1}(y)Q_{2}(y)}+y\sqrt{Q_{% 1}(x)Q_{2}(x)}}}
(x^{2}-y^{2})U = x\sqrt{Q_{1}(y)Q_{2}(y)}+y\sqrt{Q_{1}(x)Q_{2}(x)}		 

((x)^(2)- (y)^(2))*U = x*sqrt(Q[1](y)* Q[2](y))+ y*sqrt(Q[1](x)* Q[2](x))
 
((x)^(2)- (y)^(2))*U == x*Sqrt[Subscript[Q, 1][y]* Subscript[Q, 2][y]]+ y*Sqrt[Subscript[Q, 1][x]* Subscript[Q, 2][x]]
 Skipped - no semantic math Skipped - no semantic math - -
19.29.E23 ∫ y ∞ d t ( t 2 + a 2 ) ⁢ ( t 2 - b 2 ) = R F ⁡ ( y 2 + a 2 , y 2 - b 2 , y 2 ) superscript subscript 𝑦 𝑡 superscript 𝑡 2 superscript 𝑎 2 superscript 𝑡 2 superscript 𝑏 2 Carlson-integral-RF superscript 𝑦 2 superscript 𝑎 2 superscript 𝑦 2 superscript 𝑏 2 superscript 𝑦 2 {\displaystyle{\displaystyle\int_{y}^{\infty}\frac{\mathrm{d}t}{\sqrt{(t^{2}+a% ^{2})(t^{2}-b^{2})}}=R_{F}\left(y^{2}+a^{2},y^{2}-b^{2},y^{2}\right)}}
\int_{y}^{\infty}\frac{\diff{t}}{\sqrt{(t^{2}+a^{2})(t^{2}-b^{2})}} = \CarlsonsymellintRF@{y^{2}+a^{2}}{y^{2}-b^{2}}{y^{2}}		 

int((1)/(sqrt(((t)^(2)+ (a)^(2))*((t)^(2)- (b)^(2)))), t = y..infinity) = 0.5*int(1/(sqrt(t+(y)^(2)+ (a)^(2))*sqrt(t+(y)^(2)- (b)^(2))*sqrt(t+(y)^(2))), t = 0..infinity)

Integrate[Divide[1,Sqrt[((t)^(2)+ (a)^(2))*((t)^(2)- (b)^(2))]], {t, y, Infinity}, GenerateConditions->None] == EllipticF[ArcCos[Sqrt[(y)^(2)+ (a)^(2)/(y)^(2)]],((y)^(2)-(y)^(2)- (b)^(2))/((y)^(2)-(y)^(2)+ (a)^(2))]/Sqrt[(y)^(2)-(y)^(2)+ (a)^(2)]
Aborted Aborted Skipped - Because timed out Skipped - Because timed out
19.29.E24 ∫ y x d t Q 1 ⁢ ( t ) ⁢ Q 2 ⁢ ( t ) = 4 ⁢ R F ⁡ ( U , U + D 12 + V , U + D 12 - V ) superscript subscript 𝑦 𝑥 𝑡 subscript 𝑄 1 𝑡 subscript 𝑄 2 𝑡 4 Carlson-integral-RF 𝑈 𝑈 subscript 𝐷 12 𝑉 𝑈 subscript 𝐷 12 𝑉 {\displaystyle{\displaystyle\int_{y}^{x}\frac{\mathrm{d}t}{\sqrt{Q_{1}(t)Q_{2}% (t)}}=4R_{F}\left(U,U+D_{12}+V,U+D_{12}-V\right)}}
\int_{y}^{x}\frac{\diff{t}}{\sqrt{Q_{1}(t)Q_{2}(t)}} = 4\CarlsonsymellintRF@{U}{U+D_{12}+V}{U+D_{12}-V}		 

int((1)/(sqrt(Q[1](t)* Q[2](t))), t = y..x) = 4*0.5*int(1/(sqrt(t+U)*sqrt(t+U + D[12]+ V)*sqrt(t+U + D[12]- V)), t = 0..infinity)

Integrate[Divide[1,Sqrt[Subscript[Q, 1][t]* Subscript[Q, 2][t]]], {t, y, x}, GenerateConditions->None] == 4*EllipticF[ArcCos[Sqrt[U/U + Subscript[D, 12]- V]],(U + Subscript[D, 12]- V-U + Subscript[D, 12]+ V)/(U + Subscript[D, 12]- V-U)]/Sqrt[U + Subscript[D, 12]- V-U]
Aborted Aborted Skipped - Because timed out Skipped - Because timed out
19.29#Ex17 ( x - y ) 2 ⁢ U = S 1 ⁢ S 2 superscript 𝑥 𝑦 2 𝑈 subscript 𝑆 1 subscript 𝑆 2 {\displaystyle{\displaystyle(x-y)^{2}U=S_{1}S_{2}}}
(x-y)^{2}U = S_{1}S_{2}		 

(x - y)^(2)* U = S[1]*S[2]
 
(x - y)^(2)* U == Subscript[S, 1]*Subscript[S, 2]
 Skipped - no semantic math Skipped - no semantic math - -
19.29#Ex18 S j = ( Q j ⁢ ( x ) + Q j ⁢ ( y ) ) 2 - h j ⁢ ( x - y ) 2 subscript 𝑆 𝑗 superscript subscript 𝑄 𝑗 𝑥 subscript 𝑄 𝑗 𝑦 2 subscript ℎ 𝑗 superscript 𝑥 𝑦 2 {\displaystyle{\displaystyle S_{j}=\left(\sqrt{Q_{j}(x)}+\sqrt{Q_{j}(y)}\right% )^{2}-h_{j}(x-y)^{2}}}
S_{j} = \left(\sqrt{Q_{j}(x)}+\sqrt{Q_{j}(y)}\right)^{2}-h_{j}(x-y)^{2}		 

S[j] = (sqrt(Q[j](x))+sqrt(Q[j](y)))^(2)- h[j]*(x - y)^(2)
 
Subscript[S, j] == (Sqrt[Subscript[Q, j][x]]+Sqrt[Subscript[Q, j][y]])^(2)- Subscript[h, j]*(x - y)^(2)
 Skipped - no semantic math Skipped - no semantic math - -
19.29#Ex19 D j ⁢ l = 2 ⁢ f j ⁢ h l + 2 ⁢ h j ⁢ f l - g j ⁢ g l subscript 𝐷 𝑗 𝑙 2 subscript 𝑓 𝑗 subscript ℎ 𝑙 2 subscript ℎ 𝑗 subscript 𝑓 𝑙 subscript 𝑔 𝑗 subscript 𝑔 𝑙 {\displaystyle{\displaystyle D_{jl}=2f_{j}h_{l}+2h_{j}f_{l}-g_{j}g_{l}}}
D_{jl} = 2f_{j}h_{l}+2h_{j}f_{l}-g_{j}g_{l}		 

D[j, l] = 2*f[j]*h[l]+ 2*h[j]*f[l]- g[j]*g[l]
 
Subscript[D, j, l] == 2*Subscript[f, j]*Subscript[h, l]+ 2*Subscript[h, j]*Subscript[f, l]- Subscript[g, j]*Subscript[g, l]
 Skipped - no semantic math Skipped - no semantic math - -
19.29#Ex20 V = D 12 2 - D 11 ⁢ D 22 𝑉 superscript subscript 𝐷 12 2 subscript 𝐷 11 subscript 𝐷 22 {\displaystyle{\displaystyle V=\sqrt{D_{12}^{2}-D_{11}D_{22}}}}
V = \sqrt{D_{12}^{2}-D_{11}D_{22}}		 

V = sqrt((D[12])^(2)- D[11]*D[22])
 
V == Sqrt[(Subscript[D, 12])^(2)- Subscript[D, 11]*Subscript[D, 22]]
 Skipped - no semantic math Skipped - no semantic math - -
19.29#Ex21 S 1 = ( X 1 ⁢ Y 2 + Y 1 ⁢ X 2 ) 2 subscript 𝑆 1 superscript subscript 𝑋 1 subscript 𝑌 2 subscript 𝑌 1 subscript 𝑋 2 2 {\displaystyle{\displaystyle S_{1}=(X_{1}Y_{2}+Y_{1}X_{2})^{2}}}
S_{1} = (X_{1}Y_{2}+Y_{1}X_{2})^{2}		 

S[1] = (X[1]*Y[2]+ Y[1]*X[2])^(2)
 
Subscript[S, 1] == (Subscript[X, 1]*Subscript[Y, 2]+ Subscript[Y, 1]*Subscript[X, 2])^(2)
 Skipped - no semantic math Skipped - no semantic math - -
19.29#Ex22 X j = a j + b j ⁢ x subscript 𝑋 𝑗 subscript 𝑎 𝑗 subscript 𝑏 𝑗 𝑥 {\displaystyle{\displaystyle X_{j}=\sqrt{a_{j}+b_{j}x}}}
X_{j} = \sqrt{a_{j}+b_{j}x}		 

X[j] = sqrt(a[j]+ b[j]*x)
 
Subscript[X, j] == Sqrt[Subscript[a, j]+ Subscript[b, j]*x]
 Skipped - no semantic math Skipped - no semantic math - -
19.29#Ex23 Y j = a j + b j ⁢ y subscript 𝑌 𝑗 subscript 𝑎 𝑗 subscript 𝑏 𝑗 𝑦 {\displaystyle{\displaystyle Y_{j}=\sqrt{a_{j}+b_{j}y}}}
Y_{j} = \sqrt{a_{j}+b_{j}y}		 

Y[j] = sqrt(a[j]+ b[j]*y)
 
Subscript[Y, j] == Sqrt[Subscript[a, j]+ Subscript[b, j]*y]
 Skipped - no semantic math Skipped - no semantic math - -
19.29#Ex24 D 12 = 2 ⁢ a 1 ⁢ a 2 ⁢ h 2 + 2 ⁢ b 1 ⁢ b 2 ⁢ f 2 - ( a 1 ⁢ b 2 + a 2 ⁢ b 1 ) ⁢ g 2 subscript 𝐷 12 2 subscript 𝑎 1 subscript 𝑎 2 subscript ℎ 2 2 subscript 𝑏 1 subscript 𝑏 2 subscript 𝑓 2 subscript 𝑎 1 subscript 𝑏 2 subscript 𝑎 2 subscript 𝑏 1 subscript 𝑔 2 {\displaystyle{\displaystyle D_{12}=2a_{1}a_{2}h_{2}+2b_{1}b_{2}f_{2}-(a_{1}b_% {2}+a_{2}b_{1})g_{2}}}
D_{12} = 2a_{1}a_{2}h_{2}+2b_{1}b_{2}f_{2}-(a_{1}b_{2}+a_{2}b_{1})g_{2}		 

D[12] = 2*a[1]*a[2]*h[2]+ 2*b[1]*b[2]*f[2]-(a[1]*b[2]+ a[2]*b[1])*g[2]
 
Subscript[D, 12] == 2*Subscript[a, 1]*Subscript[a, 2]*Subscript[h, 2]+ 2*Subscript[b, 1]*Subscript[b, 2]*Subscript[f, 2]-(Subscript[a, 1]*Subscript[b, 2]+ Subscript[a, 2]*Subscript[b, 1])*Subscript[g, 2]
 Skipped - no semantic math Skipped - no semantic math - -
19.29#Ex25 D 11 = - ( a 1 ⁢ b 2 - a 2 ⁢ b 1 ) 2 subscript 𝐷 11 superscript subscript 𝑎 1 subscript 𝑏 2 subscript 𝑎 2 subscript 𝑏 1 2 {\displaystyle{\displaystyle D_{11}=-(a_{1}b_{2}-a_{2}b_{1})^{2}}}
D_{11} = -(a_{1}b_{2}-a_{2}b_{1})^{2}		 

D[11] = -(a[1]*b[2]- a[2]*b[1])^(2)
 
Subscript[D, 11] == -(Subscript[a, 1]*Subscript[b, 2]- Subscript[a, 2]*Subscript[b, 1])^(2)
 Skipped - no semantic math Skipped - no semantic math - -
19.29#Ex26 S 1 = ( X 1 + Y 1 ) 2 subscript 𝑆 1 superscript subscript 𝑋 1 subscript 𝑌 1 2 {\displaystyle{\displaystyle S_{1}=(X_{1}+Y_{1})^{2}}}
S_{1} = (X_{1}+Y_{1})^{2}		 

S[1] = (X[1]+ Y[1])^(2)
 
Subscript[S, 1] == (Subscript[X, 1]+ Subscript[Y, 1])^(2)
 Skipped - no semantic math Skipped - no semantic math - -
19.29#Ex27 D 12 = 2 ⁢ a 1 ⁢ h 2 - b 1 ⁢ g 2 subscript 𝐷 12 2 subscript 𝑎 1 subscript ℎ 2 subscript 𝑏 1 subscript 𝑔 2 {\displaystyle{\displaystyle D_{12}=2a_{1}h_{2}-b_{1}g_{2}}}
D_{12} = 2a_{1}h_{2}-b_{1}g_{2}		 

D[12] = 2*a[1]*h[2]- b[1]*g[2]
 
Subscript[D, 12] == 2*Subscript[a, 1]*Subscript[h, 2]- Subscript[b, 1]*Subscript[g, 2]
 Skipped - no semantic math Skipped - no semantic math - -
19.29#Ex28 D 11 = - b 1 2 subscript 𝐷 11 superscript subscript 𝑏 1 2 {\displaystyle{\displaystyle D_{11}=-b_{1}^{2}}}
D_{11} = -b_{1}^{2}		 

D[11] = - (b[1])^(2)
 
Subscript[D, 11] == - (Subscript[b, 1])^(2)
 Skipped - no semantic math Skipped - no semantic math - -
19.29.E28 ∫ y x d t t 3 - a 3 = 4 ⁢ R F ⁡ ( U , U - 3 ⁢ a + 2 ⁢ 3 ⁢ a , U - 3 ⁢ a - 2 ⁢ 3 ⁢ a ) superscript subscript 𝑦 𝑥 𝑡 superscript 𝑡 3 superscript 𝑎 3 4 Carlson-integral-RF 𝑈 𝑈 3 𝑎 2 3 𝑎 𝑈 3 𝑎 2 3 𝑎 {\displaystyle{\displaystyle\int_{y}^{x}\frac{\mathrm{d}t}{\sqrt{t^{3}-a^{3}}}% =4R_{F}\left(U,U-3a+2\sqrt{3}a,U-3a-2\sqrt{3}a\right)}}
\int_{y}^{x}\frac{\diff{t}}{\sqrt{t^{3}-a^{3}}} = 4\CarlsonsymellintRF@{U}{U-3a+2\sqrt{3}a}{U-3a-2\sqrt{3}a}		 

int((1)/(sqrt((t)^(3)- (a)^(3))), t = y..x) = 4*0.5*int(1/(sqrt(t+U)*sqrt(t+U - 3*a + 2*sqrt(3)*a)*sqrt(t+U - 3*a - 2*sqrt(3)*a)), t = 0..infinity)

Integrate[Divide[1,Sqrt[(t)^(3)- (a)^(3)]], {t, y, x}, GenerateConditions->None] == 4*EllipticF[ArcCos[Sqrt[U/U - 3*a - 2*Sqrt[3]*a]],(U - 3*a - 2*Sqrt[3]*a-U - 3*a + 2*Sqrt[3]*a)/(U - 3*a - 2*Sqrt[3]*a-U)]/Sqrt[U - 3*a - 2*Sqrt[3]*a-U]
Aborted Aborted Skipped - Because timed out Skipped - Because timed out
19.29#Ex29 ( x - y ) 2 ⁢ U = ( x - a + y - a ) 2 ⁢ ( ( ξ + η ) 2 - ( x - y ) 2 ) superscript 𝑥 𝑦 2 𝑈 superscript 𝑥 𝑎 𝑦 𝑎 2 superscript 𝜉 𝜂 2 superscript 𝑥 𝑦 2 {\displaystyle{\displaystyle(x-y)^{2}U=(\sqrt{x-a}+\sqrt{y-a})^{2}\left((\xi+% \eta)^{2}-(x-y)^{2}\right)}}
(x-y)^{2}U = (\sqrt{x-a}+\sqrt{y-a})^{2}\left((\xi+\eta)^{2}-(x-y)^{2}\right)		 

(x - y)^(2)* U = (sqrt(x - a)+sqrt(y - a))^(2)*((xi + eta)^(2)-(x - y)^(2))
 
(x - y)^(2)* U == (Sqrt[x - a]+Sqrt[y - a])^(2)*((\[Xi]+ \[Eta])^(2)-(x - y)^(2))
 Skipped - no semantic math Skipped - no semantic math - -
19.29#Ex30 ξ = x 2 + a ⁢ x + a 2 𝜉 superscript 𝑥 2 𝑎 𝑥 superscript 𝑎 2 {\displaystyle{\displaystyle\xi=\sqrt{x^{2}+ax+a^{2}}}}
\xi = \sqrt{x^{2}+ax+a^{2}}		 

xi = sqrt((x)^(2)+ a*x + (a)^(2))
 
\[Xi] == Sqrt[(x)^(2)+ a*x + (a)^(2)]
 Skipped - no semantic math Skipped - no semantic math - -
19.29#Ex31 η = y 2 + a ⁢ y + a 2 𝜂 superscript 𝑦 2 𝑎 𝑦 superscript 𝑎 2 {\displaystyle{\displaystyle\eta=\sqrt{y^{2}+ay+a^{2}}}}
\eta = \sqrt{y^{2}+ay+a^{2}}		 

eta = sqrt((y)^(2)+ a*y + (a)^(2))
 
\[Eta] == Sqrt[(y)^(2)+ a*y + (a)^(2)]
 Skipped - no semantic math Skipped - no semantic math - -
19.29.E30 ∫ y x d t Q ⁢ ( t 2 ) = 2 ⁢ R F ⁡ ( U , U - g + 2 ⁢ f ⁢ h , U - g - 2 ⁢ f ⁢ h ) superscript subscript 𝑦 𝑥 𝑡 𝑄 superscript 𝑡 2 2 Carlson-integral-RF 𝑈 𝑈 𝑔 2 𝑓 ℎ 𝑈 𝑔 2 𝑓 ℎ {\displaystyle{\displaystyle\int_{y}^{x}\frac{\mathrm{d}t}{\sqrt{Q(t^{2})}}=2R% _{F}\left(U,U-g+2\sqrt{fh},U-g-2\sqrt{fh}\right)}}
\int_{y}^{x}\frac{\diff{t}}{\sqrt{Q(t^{2})}} = 2\CarlsonsymellintRF@{U}{U-g+2\sqrt{fh}}{U-g-2\sqrt{fh}}		 

int((1)/(sqrt(Q((t)^(2)))), t = y..x) = 2*0.5*int(1/(sqrt(t+U)*sqrt(t+U - g + 2*sqrt(f*h))*sqrt(t+U - g - 2*sqrt(f*h))), t = 0..infinity)

Integrate[Divide[1,Sqrt[Q[(t)^(2)]]], {t, y, x}, GenerateConditions->None] == 2*EllipticF[ArcCos[Sqrt[U/U - g - 2*Sqrt[f*h]]],(U - g - 2*Sqrt[f*h]-U - g + 2*Sqrt[f*h])/(U - g - 2*Sqrt[f*h]-U)]/Sqrt[U - g - 2*Sqrt[f*h]-U]
Aborted Aborted Skipped - Because timed out Skipped - Because timed out
19.29.E31 ( x - y ) 2 ⁢ U = ( Q ⁢ ( x 2 ) + Q ⁢ ( y 2 ) ) 2 - h ⁢ ( x 2 - y 2 ) 2 superscript 𝑥 𝑦 2 𝑈 superscript 𝑄 superscript 𝑥 2 𝑄 superscript 𝑦 2 2 ℎ superscript superscript 𝑥 2 superscript 𝑦 2 2 {\displaystyle{\displaystyle(x-y)^{2}U=\left(\sqrt{Q(x^{2})}+\sqrt{Q(y^{2})}% \right)^{2}-h(x^{2}-y^{2})^{2}}}
(x-y)^{2}U = \left(\sqrt{Q(x^{2})}+\sqrt{Q(y^{2})}\right)^{2}-h(x^{2}-y^{2})^{2}		 

(x - y)^(2)* U = (sqrt(Q((x)^(2)))+sqrt(Q((y)^(2))))^(2)- h*((x)^(2)- (y)^(2))^(2)
 
(x - y)^(2)* U == (Sqrt[Q[(x)^(2)]]+Sqrt[Q[(y)^(2)]])^(2)- h*((x)^(2)- (y)^(2))^(2)
 Skipped - no semantic math Skipped - no semantic math - -
19.29.E32 ∫ y x d t t 4 + a 4 = 2 ⁢ R F ⁡ ( U , U + 2 ⁢ a 2 , U - 2 ⁢ a 2 ) superscript subscript 𝑦 𝑥 𝑡 superscript 𝑡 4 superscript 𝑎 4 2 Carlson-integral-RF 𝑈 𝑈 2 superscript 𝑎 2 𝑈 2 superscript 𝑎 2 {\displaystyle{\displaystyle\int_{y}^{x}\frac{\mathrm{d}t}{\sqrt{t^{4}+a^{4}}}% =2R_{F}\left(U,U+2a^{2},U-2a^{2}\right)}}
\int_{y}^{x}\frac{\diff{t}}{\sqrt{t^{4}+a^{4}}} = 2\CarlsonsymellintRF@{U}{U+2a^{2}}{U-2a^{2}}		 

int((1)/(sqrt((t)^(4)+ (a)^(4))), t = y..x) = 2*0.5*int(1/(sqrt(t+U)*sqrt(t+U + 2*(a)^(2))*sqrt(t+U - 2*(a)^(2))), t = 0..infinity)

Integrate[Divide[1,Sqrt[(t)^(4)+ (a)^(4)]], {t, y, x}, GenerateConditions->None] == 2*EllipticF[ArcCos[Sqrt[U/U - 2*(a)^(2)]],(U - 2*(a)^(2)-U + 2*(a)^(2))/(U - 2*(a)^(2)-U)]/Sqrt[U - 2*(a)^(2)-U]
Aborted Failure Skipped - Because timed out
Failed [300 / 300]
Result: Complex[0.06910876495694751, 1.480960979386122]
Test Values: {Rule[a, -1.5], Rule[U, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[x, 1.5], Rule[y, -1.5]}


Result: Complex[1.3051585498245286, 1.480960979386122]
Test Values: {Rule[a, -1.5], Rule[U, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[x, 1.5], Rule[y, 1.5]}

... skip entries to safe data
19.29.E33 ( x - y ) 2 ⁢ U = ( x 4 + a 4 + y 4 + a 4 ) 2 - ( x 2 - y 2 ) 2 superscript 𝑥 𝑦 2 𝑈 superscript superscript 𝑥 4 superscript 𝑎 4 superscript 𝑦 4 superscript 𝑎 4 2 superscript superscript 𝑥 2 superscript 𝑦 2 2 {\displaystyle{\displaystyle(x-y)^{2}U=\left(\sqrt{x^{4}+a^{4}}+\sqrt{y^{4}+a^% {4}}\right)^{2}-(x^{2}-y^{2})^{2}}}
(x-y)^{2}U = \left(\sqrt{x^{4}+a^{4}}+\sqrt{y^{4}+a^{4}}\right)^{2}-(x^{2}-y^{2})^{2}		 

(x - y)^(2)* U = (sqrt((x)^(4)+ (a)^(4))+sqrt((y)^(4)+ (a)^(4)))^(2)-((x)^(2)- (y)^(2))^(2)
 
(x - y)^(2)* U == (Sqrt[(x)^(4)+ (a)^(4)]+Sqrt[(y)^(4)+ (a)^(4)])^(2)-((x)^(2)- (y)^(2))^(2)
 Skipped - no semantic math Skipped - no semantic math - -
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