Elliptic Integrals - 19.30 Lengths of Plane Curves

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
19.30#Ex1 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle x = a\sin@@{\phi}}
x = a\sin@@{\phi}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
x = a*sin(phi)

x == a*Sin[\[Phi]]
Failure Failure
Failed [180 / 180]
Result: 2.788470502+.5063946946*I
Test Values: {a = -3/2, phi = 1/2*3^(1/2)+1/2*I, x = 3/2}


Result: 1.788470502+.5063946946*I
Test Values: {a = -3/2, phi = 1/2*3^(1/2)+1/2*I, x = 1/2}

... skip entries to safe data
Failed [180 / 180]
Result: Complex[2.1491827752870476, 0.34394646701016035]
Test Values: {Rule[a, -1.5], Rule[x, 1.5], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}


Result: Complex[1.093555858156998, 0.6491787480429551]
Test Values: {Rule[a, -1.5], Rule[x, 1.5], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]}

... skip entries to safe data
19.30#Ex2 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle y = b\cos@@{\phi}}
y = b\cos@@{\phi}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle 0 \leq \phi, \phi \leq 2\pi} 
y = b*cos(phi)

y == b*Cos[\[Phi]]
Failure Failure
Failed [108 / 108]
Result: -1.393894198
Test Values: {b = -3/2, phi = 3/2, y = -3/2}


Result: 1.606105802
Test Values: {b = -3/2, phi = 3/2, y = 3/2}

... skip entries to safe data
Failed [108 / 108]
Result: -1.3938941974984456
Test Values: {Rule[b, -1.5], Rule[y, -1.5], Rule[ϕ, 1.5]}


Result: -0.18362615716444086
Test Values: {Rule[b, -1.5], Rule[y, -1.5], Rule[ϕ, 0.5]}

... skip entries to safe data
19.30.E2 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle s = a\int_{0}^{\phi}\sqrt{1-k^{2}\sin^{2}@@{\theta}}\diff{\theta}}
s = a\int_{0}^{\phi}\sqrt{1-k^{2}\sin^{2}@@{\theta}}\diff{\theta}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
s = a*int(sqrt(1 - (k)^(2)* (sin(theta))^(2)), theta = 0..phi)

s == a*Integrate[Sqrt[1 - (k)^(2)* (Sin[\[Theta]])^(2)], {\[Theta], 0, \[Phi]}, GenerateConditions->None]
Aborted Aborted Skipped - Because timed out Skipped - Because timed out
19.30.E3 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle s/a = \incellintEk@{\phi}{k}}
s/a = \incellintEk@{\phi}{k}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
s/a = EllipticE(sin(phi), k)

s/a == EllipticE[\[Phi], (k)^2]
Failure Failure
Failed [300 / 300]
Result: .1410196655-.3375964631*I
Test Values: {a = -3/2, phi = 1/2*3^(1/2)+1/2*I, s = -3/2, k = 1}


Result: -.36391978e-1+.5433649104e-1*I
Test Values: {a = -3/2, phi = 1/2*3^(1/2)+1/2*I, s = -3/2, k = 2}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[0.5672114831419685, -0.22929764467344024]
Test Values: {Rule[a, -1.5], Rule[k, 1], Rule[s, -1.5], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}


Result: Complex[0.5579190406370536, -0.16535187593702125]
Test Values: {Rule[a, -1.5], Rule[k, 2], Rule[s, -1.5], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}

... skip entries to safe data
19.30.E3 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \incellintEk@{\phi}{k} = {\CarlsonsymellintRF@{c-1}{c-k^{2}}{c}-\tfrac{1}{3}k^{2}\CarlsonsymellintRD@{c-1}{c-k^{2}}{c}}}
\incellintEk@{\phi}{k} = {\CarlsonsymellintRF@{c-1}{c-k^{2}}{c}-\tfrac{1}{3}k^{2}\CarlsonsymellintRD@{c-1}{c-k^{2}}{c}}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
Error

EllipticE[\[Phi], (k)^2] == EllipticF[ArcCos[Sqrt[c - 1/c]],(c-c - (k)^(2))/(c-c - 1)]/Sqrt[c-c - 1]-Divide[1,3]*(k)^(2)* 3*(EllipticF[ArcCos[Sqrt[c - 1/c]],(c-c - (k)^(2))/(c-c - 1)]-EllipticE[ArcCos[Sqrt[c - 1/c]],(c-c - (k)^(2))/(c-c - 1)])/((c-c - (k)^(2))*(c-c - 1)^(1/2))
Missing Macro Error Failure Skip - symbolical successful subtest
Failed [180 / 180]
Result: Complex[3.5743811704478246, 0.7698502565730785]
Test Values: {Rule[c, -1.5], Rule[k, 1], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}


Result: Complex[3.9424508382496875, -1.017653751864599]
Test Values: {Rule[c, -1.5], Rule[k, 2], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}

... skip entries to safe data
19.30#Ex3 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle k^{2} = 1-(b^{2}/a^{2})}
k^{2} = 1-(b^{2}/a^{2})		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
(k)^(2) = 1 -((b)^(2)/(a)^(2))
 
(k)^(2) == 1 -((b)^(2)/(a)^(2))
 Skipped - no semantic math Skipped - no semantic math - -
19.30#Ex4 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle c = \csc^{2}@@{\phi}}
c = \csc^{2}@@{\phi}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
c = (csc(phi))^(2)

c == (Csc[\[Phi]])^(2)
Failure Failure
Failed [60 / 60]
Result: -2.359812877+.7993130071*I
Test Values: {c = -3/2, phi = 1/2*3^(1/2)+1/2*I}


Result: -1.296085040-.8173084059*I
Test Values: {c = -3/2, phi = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [60 / 60]
Result: Complex[-3.841312467237177, 3.4490957612740374]
Test Values: {Rule[c, -1.5], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}


Result: Complex[0.17530792640393877, -3.4502399957777015]
Test Values: {Rule[c, -1.5], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]}

... skip entries to safe data
19.30.E5 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle L(a,b) = 4a\compellintEk@{k}}
L(a,b) = 4a\compellintEk@{k}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
L(a , b) = 4*a*EllipticE(k)

L[a , b] == 4*a*EllipticE[(k)^2]
Failure Failure
Failed [300 / 300]
Result: (.8660254040+.5000000000*I)*(-1.500000000, -1.500000000)+6.000000000
Test Values: {L = 1/2*3^(1/2)+1/2*I, a = -3/2, b = -3/2, k = 1}


Result: (.8660254040+.5000000000*I)*(-1.500000000, -1.500000000)+2.437793319+8.063125386*I
Test Values: {L = 1/2*3^(1/2)+1/2*I, a = -3/2, b = -3/2, k = 2}

... skip entries to safe data
 Error
19.30.E5 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle 4a\compellintEk@{k} = 8a\CarlsonsymellintRG@{0}{b^{2}/a^{2}}{1}}
4a\compellintEk@{k} = 8a\CarlsonsymellintRG@{0}{b^{2}/a^{2}}{1}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
Error

4*a*EllipticE[(k)^2] == 8*a*Sqrt[1-0]*(EllipticE[ArcCos[Sqrt[0/1]],(1-(b)^(2)/(a)^(2))/(1-0)]+(Cot[ArcCos[Sqrt[0/1]]])^2*EllipticF[ArcCos[Sqrt[0/1]],(1-(b)^(2)/(a)^(2))/(1-0)]+Cot[ArcCos[Sqrt[0/1]]]*Sqrt[1-k^2*Sin[ArcCos[Sqrt[0/1]]]^2])
Missing Macro Error Failure Skip - symbolical successful subtest
Failed [108 / 108]
Result: 12.849555921538759
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[k, 1]}


Result: Complex[16.411762602778996, -8.063125388322588]
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[k, 2]}

... skip entries to safe data
19.30.E5 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle 8a\CarlsonsymellintRG@{0}{b^{2}/a^{2}}{1} = 8\CarlsonsymellintRG@{0}{a^{2}}{b^{2}}}
8a\CarlsonsymellintRG@{0}{b^{2}/a^{2}}{1} = 8\CarlsonsymellintRG@{0}{a^{2}}{b^{2}}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
Error

8*a*Sqrt[1-0]*(EllipticE[ArcCos[Sqrt[0/1]],(1-(b)^(2)/(a)^(2))/(1-0)]+(Cot[ArcCos[Sqrt[0/1]]])^2*EllipticF[ArcCos[Sqrt[0/1]],(1-(b)^(2)/(a)^(2))/(1-0)]+Cot[ArcCos[Sqrt[0/1]]]*Sqrt[1-k^2*Sin[ArcCos[Sqrt[0/1]]]^2]) == 8*Sqrt[(b)^(2)-0]*(EllipticE[ArcCos[Sqrt[0/(b)^(2)]],((b)^(2)-(a)^(2))/((b)^(2)-0)]+(Cot[ArcCos[Sqrt[0/(b)^(2)]]])^2*EllipticF[ArcCos[Sqrt[0/(b)^(2)]],((b)^(2)-(a)^(2))/((b)^(2)-0)]+Cot[ArcCos[Sqrt[0/(b)^(2)]]]*Sqrt[1-k^2*Sin[ArcCos[Sqrt[0/(b)^(2)]]]^2])
Missing Macro Error Failure Skip - symbolical successful subtest
Failed [18 / 36]
Result: -37.69911184307752
Test Values: {Rule[a, -1.5], Rule[b, -1.5]}


Result: -37.69911184307752
Test Values: {Rule[a, -1.5], Rule[b, 1.5]}

... skip entries to safe data
19.30.E5 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle 8\CarlsonsymellintRG@{0}{a^{2}}{b^{2}} = 8ab\CarlsonsymellintRG@{0}{a^{-2}}{b^{-2}}}
8\CarlsonsymellintRG@{0}{a^{2}}{b^{2}} = 8ab\CarlsonsymellintRG@{0}{a^{-2}}{b^{-2}}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
Error

8*Sqrt[(b)^(2)-0]*(EllipticE[ArcCos[Sqrt[0/(b)^(2)]],((b)^(2)-(a)^(2))/((b)^(2)-0)]+(Cot[ArcCos[Sqrt[0/(b)^(2)]]])^2*EllipticF[ArcCos[Sqrt[0/(b)^(2)]],((b)^(2)-(a)^(2))/((b)^(2)-0)]+Cot[ArcCos[Sqrt[0/(b)^(2)]]]*Sqrt[1-k^2*Sin[ArcCos[Sqrt[0/(b)^(2)]]]^2]) == 8*a*b*Sqrt[(b)^(- 2)-0]*(EllipticE[ArcCos[Sqrt[0/(b)^(- 2)]],((b)^(- 2)-(a)^(- 2))/((b)^(- 2)-0)]+(Cot[ArcCos[Sqrt[0/(b)^(- 2)]]])^2*EllipticF[ArcCos[Sqrt[0/(b)^(- 2)]],((b)^(- 2)-(a)^(- 2))/((b)^(- 2)-0)]+Cot[ArcCos[Sqrt[0/(b)^(- 2)]]]*Sqrt[1-k^2*Sin[ArcCos[Sqrt[0/(b)^(- 2)]]]^2])
Missing Macro Error Failure Skip - symbolical successful subtest
Failed [18 / 36]
Result: 37.69911184307752
Test Values: {Rule[a, -1.5], Rule[b, 1.5]}


Result: 26.729786441110512
Test Values: {Rule[a, -1.5], Rule[b, 0.5]}

... skip entries to safe data
19.30.E6 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \pderiv{s}{(1/k)} = \sqrt{a^{2}-b^{2}}\incellintFk@{\phi}{k}}
\pderiv{s}{(1/k)} = \sqrt{a^{2}-b^{2}}\incellintFk@{\phi}{k}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle k^{2} = (a^{2}-b^{2})/(a^{2}+\lambda), c = \csc^{2}@@{\phi}} 
subs( temp=(1/k), diff( s, temp$(1) ) ) = sqrt((a)^(2)- (b)^(2))*EllipticF(sin(phi), k)

(D[s, {temp, 1}]/.temp-> (1/k)) == Sqrt[(a)^(2)- (b)^(2)]*EllipticF[\[Phi], (k)^2]
Failure Failure Successful [Tested: 300]
Failed [20 / 300]
Result: Indeterminate
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[k, 1], Rule[s, -1.5], Rule[ϕ, -2]}


Result: Indeterminate
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[k, 1], Rule[s, -1.5], Rule[ϕ, 2]}

... skip entries to safe data
19.30.E6 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sqrt{a^{2}-b^{2}}\incellintFk@{\phi}{k} = \sqrt{a^{2}-b^{2}}\CarlsonsymellintRF@{c-1}{c-k^{2}}{c}}
\sqrt{a^{2}-b^{2}}\incellintFk@{\phi}{k} = \sqrt{a^{2}-b^{2}}\CarlsonsymellintRF@{c-1}{c-k^{2}}{c}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle k^{2} = (a^{2}-b^{2})/(a^{2}+\lambda), c = \csc^{2}@@{\phi}} 
sqrt((a)^(2)- (b)^(2))*EllipticF(sin(phi), k) = sqrt((a)^(2)- (b)^(2))*0.5*int(1/(sqrt(t+c - 1)*sqrt(t+c - (k)^(2))*sqrt(t+c)), t = 0..infinity)

Sqrt[(a)^(2)- (b)^(2)]*EllipticF[\[Phi], (k)^2] == Sqrt[(a)^(2)- (b)^(2)]*EllipticF[ArcCos[Sqrt[c - 1/c]],(c-c - (k)^(2))/(c-c - 1)]/Sqrt[c-c - 1]
Error Failure Skip - symbolical successful subtest Skip - No test values generated
19.30#Ex5 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle x = a\sqrt{t+1}}
x = a\sqrt{t+1}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
x = a*sqrt(t + 1)
 
x == a*Sqrt[t + 1]
 Skipped - no semantic math Skipped - no semantic math - -
19.30#Ex6 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle y = b\sqrt{t}}
y = b\sqrt{t}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
y = b*sqrt(t)
 
y == b*Sqrt[t]
 Skipped - no semantic math Skipped - no semantic math - -
19.30.E8 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle s = \frac{1}{2}\int_{0}^{y^{2}/b^{2}}\sqrt{\frac{(a^{2}+b^{2})t+b^{2}}{t(t+1)}}\diff{t}}
s = \frac{1}{2}\int_{0}^{y^{2}/b^{2}}\sqrt{\frac{(a^{2}+b^{2})t+b^{2}}{t(t+1)}}\diff{t}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
s = (1)/(2)*int(sqrt((((a)^(2)+ (b)^(2))*t + (b)^(2))/(t*(t + 1))), t = 0..(y)^(2)/(b)^(2))

s == Divide[1,2]*Integrate[Sqrt[Divide[((a)^(2)+ (b)^(2))*t + (b)^(2),t*(t + 1)]], {t, 0, (y)^(2)/(b)^(2)}, GenerateConditions->None]
Failure Aborted
Failed [300 / 300]
Result: -3.149531120
Test Values: {a = -3/2, b = -3/2, s = -3/2, y = -3/2}


Result: -3.149531120
Test Values: {a = -3/2, b = -3/2, s = -3/2, y = 3/2}

... skip entries to safe data
 Skipped - Because timed out
19.30.E9 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle s = \tfrac{1}{2}I(\mathbf{e}_{1})}
s = \tfrac{1}{2}I(\mathbf{e}_{1})		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle r = b^{4}/y^{2}} 
s = (1)/(2)*I(e[1])

s == Divide[1,2]*I[Subscript[e, 1]]
Failure Failure
Failed [298 / 300]
Result: -1.750000000-.4330127020*I
Test Values: {I = 1/2*3^(1/2)+1/2*I, s = -3/2, e[1] = 1/2*3^(1/2)+1/2*I}


Result: -1.066987298-.2500000002*I
Test Values: {I = 1/2*3^(1/2)+1/2*I, s = -3/2, e[1] = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [180 / 180]
Result: Complex[-1.375, -0.21650635094610968]
Test Values: {Rule[Complex[0, 1], 1], Rule[s, -1.5], Rule[Subscript[e, 1], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}


Result: Complex[-1.375, -0.21650635094610968]
Test Values: {Rule[Complex[0, 1], 2], Rule[s, -1.5], Rule[Subscript[e, 1], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}

... skip entries to safe data
19.30.E9 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \tfrac{1}{2}I(\mathbf{e}_{1}) = -\tfrac{1}{3}a^{2}b^{2}\CarlsonsymellintRD@{r}{r+b^{2}+a^{2}}{r+b^{2}}+y\sqrt{\frac{r+b^{2}+a^{2}}{r+b^{2}}}}
\tfrac{1}{2}I(\mathbf{e}_{1}) = -\tfrac{1}{3}a^{2}b^{2}\CarlsonsymellintRD@{r}{r+b^{2}+a^{2}}{r+b^{2}}+y\sqrt{\frac{r+b^{2}+a^{2}}{r+b^{2}}}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle r = b^{4}/y^{2}} 
Error

Divide[1,2]*I[Subscript[e, 1]] == -Divide[1,3]*(a)^(2)* (b)^(2)* 3*(EllipticF[ArcCos[Sqrt[r/r + (b)^(2)]],(r + (b)^(2)-r + (b)^(2)+ (a)^(2))/(r + (b)^(2)-r)]-EllipticE[ArcCos[Sqrt[r/r + (b)^(2)]],(r + (b)^(2)-r + (b)^(2)+ (a)^(2))/(r + (b)^(2)-r)])/((r + (b)^(2)-r + (b)^(2)+ (a)^(2))*(r + (b)^(2)-r)^(1/2))+ y*Sqrt[Divide[r + (b)^(2)+ (a)^(2),r + (b)^(2)]]
Missing Macro Error Failure Skip - symbolical successful subtest Skip - No test values generated
19.30.E10 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle r^{2} = 2a^{2}\cos@{2\theta}}
r^{2} = 2a^{2}\cos@{2\theta}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle 0 \leq \theta, \theta \leq 2\pi} 
(r)^(2) = 2*(a)^(2)* cos(2*theta)

(r)^(2) == 2*(a)^(2)* Cos[2*\[Theta]]
Failure Failure
Failed [108 / 108]
Result: 6.704966234
Test Values: {a = -3/2, r = -3/2, theta = 3/2}


Result: -.181360376
Test Values: {a = -3/2, r = -3/2, theta = 1/2}

... skip entries to safe data
Failed [108 / 108]
Result: 6.704966234702004
Test Values: {Rule[a, -1.5], Rule[r, -1.5], Rule[θ, 1.5]}


Result: -0.18136037640662916
Test Values: {Rule[a, -1.5], Rule[r, -1.5], Rule[θ, 0.5]}

... skip entries to safe data
19.30.E11 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle s = 2a^{2}\int_{0}^{r}\frac{\diff{t}}{\sqrt{4a^{4}-t^{4}}}}
s = 2a^{2}\int_{0}^{r}\frac{\diff{t}}{\sqrt{4a^{4}-t^{4}}}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle q = 2a^{2}/r^{2}, 2a^{2}/r^{2} = \sec@{2\theta}} 
s = 2*(a)^(2)* int((1)/(sqrt(4*(a)^(4)- (t)^(4))), t = 0..r)

s == 2*(a)^(2)* Integrate[Divide[1,Sqrt[4*(a)^(4)- (t)^(4)]], {t, 0, r}, GenerateConditions->None]
Error Failure -
Failed [208 / 216]
Result: 0.042085201578189846
Test Values: {Rule[a, -1.5], Rule[r, -1.5], Rule[s, -1.5]}


Result: 3.04208520157819
Test Values: {Rule[a, -1.5], Rule[r, -1.5], Rule[s, 1.5]}

... skip entries to safe data
19.30.E11 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle 2a^{2}\int_{0}^{r}\frac{\diff{t}}{\sqrt{4a^{4}-t^{4}}} = \sqrt{2a^{2}}\CarlsonsymellintRF@{q-1}{q}{q+1}}
2a^{2}\int_{0}^{r}\frac{\diff{t}}{\sqrt{4a^{4}-t^{4}}} = \sqrt{2a^{2}}\CarlsonsymellintRF@{q-1}{q}{q+1}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle q = 2a^{2}/r^{2}, 2a^{2}/r^{2} = \sec@{2\theta}} 
2*(a)^(2)* int((1)/(sqrt(4*(a)^(4)- (t)^(4))), t = 0..r) = sqrt(2*(a)^(2))*0.5*int(1/(sqrt(t+q - 1)*sqrt(t+q)*sqrt(t+q + 1)), t = 0..infinity)

2*(a)^(2)* Integrate[Divide[1,Sqrt[4*(a)^(4)- (t)^(4)]], {t, 0, r}, GenerateConditions->None] == Sqrt[2*(a)^(2)]*EllipticF[ArcCos[Sqrt[q - 1/q + 1]],(q + 1-q)/(q + 1-q - 1)]/Sqrt[q + 1-q - 1]
Error Failure -
Failed [12 / 12]
Result: Indeterminate
Test Values: {Rule[a, -1.5], Rule[q, 2], Rule[r, -1.5]}


Result: Indeterminate
Test Values: {Rule[a, -1.5], Rule[q, 2], Rule[r, 1.5]}

... skip entries to safe data
19.30.E12 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle s = a\incellintFk@{\phi}{1/\sqrt{2}}}
s = a\incellintFk@{\phi}{1/\sqrt{2}}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \phi = \asin@@{\sqrt{2/(q+1)}}, \asin@@{\sqrt{2/(q+1)}} = \acos@{\tan@@{\theta}}} 
s = a*EllipticF(sin(phi), 1/(sqrt(2)))
 
s == a*EllipticF[\[Phi], (1/(Sqrt[2]))^2]
 Failure Failure
Failed [300 / 300]
Result: -.201379324+.8785912788*I
Test Values: {a = -3/2, phi = 1/2*3^(1/2)+1/2*I, s = -3/2}

Result: 2.798620676+.8785912788*I
Test Values: {a = -3/2, phi = 1/2*3^(1/2)+1/2*I, s = 3/2}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[-0.8505476575870029, 0.390685462269601]
Test Values: {Rule[a, -1.5], Rule[s, -1.5], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}

Result: Complex[-1.859414812385125, 0.6494166239344216]
Test Values: {Rule[a, -1.5], Rule[s, -1.5], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]}

... skip entries to safe data
19.30.E13 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle P = 4\sqrt{2a^{2}}\CarlsonsymellintRF@{0}{1}{2}}
P = 4\sqrt{2a^{2}}\CarlsonsymellintRF@{0}{1}{2}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
P = 4*sqrt(2*(a)^(2))*0.5*int(1/(sqrt(t+0)*sqrt(t+1)*sqrt(t+2)), t = 0..infinity)
 
P == 4*Sqrt[2*(a)^(2)]*EllipticF[ArcCos[Sqrt[0/2]],(2-1)/(2-0)]/Sqrt[2-0]
 Failure Failure
Failed [60 / 60]
Result: -10.25842266+.5000000000*I
Test Values: {P = 1/2*3^(1/2)+1/2*I, a = -3/2}

Result: -10.25842266+.5000000000*I
Test Values: {P = 1/2*3^(1/2)+1/2*I, a = 3/2}

... skip entries to safe data
Failed [60 / 60]
Result: Complex[-10.691435361916012, 0.24999999999999997]
Test Values: {Rule[a, -1.5], Rule[P, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}

Result: Complex[-11.37444806380823, 0.43301270189221935]
Test Values: {Rule[a, -1.5], Rule[P, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]}

... skip entries to safe data
19.30.E13 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle 4\sqrt{2a^{2}}\CarlsonsymellintRF@{0}{1}{2} = \sqrt{2a^{2}}\times 5.24411\;51\ldots}
4\sqrt{2a^{2}}\CarlsonsymellintRF@{0}{1}{2} = \sqrt{2a^{2}}\times 5.24411\;51\ldots		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
4*sqrt(2*(a)^(2))*0.5*int(1/(sqrt(t+0)*sqrt(t+1)*sqrt(t+2)), t = 0..infinity) = sqrt(2*(a)^(2)) * 5.2441151
 
4*Sqrt[2*(a)^(2)]*EllipticF[ArcCos[Sqrt[0/2]],(2-1)/(2-0)]/Sqrt[2-0] == Sqrt[2*(a)^(2)] * 5.2441151
 Translation Error Translation Error Skip - symbolical successful subtest Skip - symbolical successful subtest
19.30.E13 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sqrt{2a^{2}}\times 5.24411\;51\ldots = 4a\compellintKk@{1/\sqrt{2}}}
\sqrt{2a^{2}}\times 5.24411\;51\ldots = 4a\compellintKk@{1/\sqrt{2}}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
sqrt(2*(a)^(2)) * 5.2441151 = 4*a*EllipticK(1/(sqrt(2)))
 
Sqrt[2*(a)^(2)] * 5.2441151 == 4*a*EllipticK[(1/(Sqrt[2]))^2]
 Translation Error Translation Error Skip - symbolical successful subtest Skip - symbolical successful subtest
19.30.E13 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle 4a\compellintKk@{1/\sqrt{2}} = a\times 7.41629\;87\dots}
4a\compellintKk@{1/\sqrt{2}} = a\times 7.41629\;87\dots		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
4*a*EllipticK(1/(sqrt(2))) = a * 7.4162987
 
4*a*EllipticK[(1/(Sqrt[2]))^2] == a * 7.4162987
 Translation Error Translation Error Skip - symbolical successful subtest Skip - symbolical successful subtest
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