Jacobian Elliptic Functions - 22.14 Integrals

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
22.14.E1 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int\Jacobiellsnk@{x}{k}\diff{x} = k^{-1}\ln@{\Jacobielldnk@{x}{k}-k\Jacobiellcnk@{x}{k}}}
\int\Jacobiellsnk@{x}{k}\diff{x} = k^{-1}\ln@{\Jacobielldnk@{x}{k}-k\Jacobiellcnk@{x}{k}}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
int(JacobiSN(x, k), x) = (k)^(- 1)* ln(JacobiDN(x, k)- k*JacobiCN(x, k))

Integrate[JacobiSN[x, (k)^2], x, GenerateConditions->None] == (k)^(- 1)* Log[JacobiDN[x, (k)^2]- k*JacobiCN[x, (k)^2]]
Successful Failure -
Failed [3 / 9]
Result: Indeterminate
Test Values: {Rule[k, 1], Rule[x, 1.5]}


Result: Indeterminate
Test Values: {Rule[k, 1], Rule[x, 0.5]}

... skip entries to safe data
22.14.E2 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int\Jacobiellcnk@{x}{k}\diff{x} = k^{-1}\Acos@{\Jacobielldnk@{x}{k}}}
\int\Jacobiellcnk@{x}{k}\diff{x} = k^{-1}\Acos@{\Jacobielldnk@{x}{k}}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
Error

Integrate[JacobiCN[x, (k)^2], x, GenerateConditions->None] == (k)^(- 1)* ArcCos[JacobiDN[x, (k)^2]]
Missing Macro Error Failure -
Failed [3 / 9]
Result: -1.2690416691147375
Test Values: {Rule[k, 3], Rule[x, 1.5]}


Result: -2.5226182800392123
Test Values: {Rule[k, 2], Rule[x, 2]}

... skip entries to safe data
22.14.E3 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int\Jacobielldnk@{x}{k}\diff{x} = \Asin@{\Jacobiellsnk@{x}{k}}}
\int\Jacobielldnk@{x}{k}\diff{x} = \Asin@{\Jacobiellsnk@{x}{k}}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
Error

Integrate[JacobiDN[x, (k)^2], x, GenerateConditions->None] == ArcSin[JacobiSN[x, (k)^2]]
Missing Macro Error Failure -
Failed [3 / 9]
Result: 6.283185307179586
Test Values: {Rule[k, 3], Rule[x, 1.5]}


Result: 6.283185307179586
Test Values: {Rule[k, 2], Rule[x, 2]}

... skip entries to safe data
22.14.E3 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Asin@{\Jacobiellsnk@{x}{k}} = \Jacobiamk@{x}{k}}
\Asin@{\Jacobiellsnk@{x}{k}} = \Jacobiamk@{x}{k}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
Error

ArcSin[JacobiSN[x, (k)^2]] == JacobiAmplitude[x, Power[k, 2]]
Missing Macro Error Failure -
Failed [1 / 3]
Result: -6.283185307179586
Test Values: {Rule[k, 3], Rule[x, Rational[3, 2]]}

22.14.E4 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int\Jacobiellcdk@{x}{k}\diff{x} = k^{-1}\ln@{\Jacobiellndk@{x}{k}+k\Jacobiellsdk@{x}{k}}}
\int\Jacobiellcdk@{x}{k}\diff{x} = k^{-1}\ln@{\Jacobiellndk@{x}{k}+k\Jacobiellsdk@{x}{k}}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
int(JacobiCD(x, k), x) = (k)^(- 1)* ln(JacobiND(x, k)+ k*JacobiSD(x, k))

Integrate[JacobiCD[x, (k)^2], x, GenerateConditions->None] == (k)^(- 1)* Log[JacobiND[x, (k)^2]+ k*JacobiSD[x, (k)^2]]
Successful Failure - Successful [Tested: 9]
22.14.E5 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int\Jacobiellsdk@{x}{k}\diff{x} = (kk^{\prime})^{-1}\Asin@{-k\Jacobiellcdk@{x}{k}}}
\int\Jacobiellsdk@{x}{k}\diff{x} = (kk^{\prime})^{-1}\Asin@{-k\Jacobiellcdk@{x}{k}}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
Error

Integrate[JacobiSD[x, (k)^2], x, GenerateConditions->None] == (k*Sqrt[1 - (k)^(2)])^(- 1)* ArcSin[- k*JacobiCD[x, (k)^2]]
Missing Macro Error Aborted -
Failed [6 / 9]
Result: Indeterminate
Test Values: {Rule[k, 1], Rule[x, 1.5]}


Result: Complex[0.7955664885698261, 0.9068996821171089]
Test Values: {Rule[k, 2], Rule[x, 1.5]}

... skip entries to safe data
22.14.E6 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int\Jacobiellndk@{x}{k}\diff{x} = {k^{\prime}}^{-1}\Acos@{\Jacobiellcdk@{x}{k}}}
\int\Jacobiellndk@{x}{k}\diff{x} = {k^{\prime}}^{-1}\Acos@{\Jacobiellcdk@{x}{k}}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
Error

Integrate[JacobiND[x, (k)^2], x, GenerateConditions->None] == (Sqrt[1 - (k)^(2)])^(- 1)* ArcCos[JacobiCD[x, (k)^2]]
Missing Macro Error Failure -
Failed [3 / 3]
Result: Indeterminate
Test Values: {Rule[k, 1]}


Result: Plus[Complex[0.0, -1.7320508075688772], Times[-0.3333333333333333, ArcCos[JacobiCD[x, 4.0]], Power[Plus[1.0, Times[-1.0, Power[JacobiCD[x, 4.0], 2]]], Rational[1, 2]], JacobiDN[x, 4.0], Power[JacobiSN[x, 4.0], -1]]]
Test Values: {Rule[k, 2]}

... skip entries to safe data
22.14.E7 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int\Jacobielldck@{x}{k}\diff{x} = \ln@{\Jacobiellnck@{x}{k}+\Jacobiellsck@{x}{k}}}
\int\Jacobielldck@{x}{k}\diff{x} = \ln@{\Jacobiellnck@{x}{k}+\Jacobiellsck@{x}{k}}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
int(JacobiDC(x, k), x) = ln(JacobiNC(x, k)+ JacobiSC(x, k))

Integrate[JacobiDC[x, (k)^2], x, GenerateConditions->None] == Log[JacobiNC[x, (k)^2]+ JacobiSC[x, (k)^2]]
Successful Successful - Successful [Tested: 9]
22.14.E8 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int\Jacobiellnck@{x}{k}\diff{x} = {k^{\prime}}^{-1}\ln@{\Jacobielldck@{x}{k}+k^{\prime}\Jacobiellsck@{x}{k}}}
\int\Jacobiellnck@{x}{k}\diff{x} = {k^{\prime}}^{-1}\ln@{\Jacobielldck@{x}{k}+k^{\prime}\Jacobiellsck@{x}{k}}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
int(JacobiNC(x, k), x) = (sqrt(1 - (k)^(2)))^(- 1)* ln(JacobiDC(x, k)+sqrt(1 - (k)^(2))*JacobiSC(x, k))

Integrate[JacobiNC[x, (k)^2], x, GenerateConditions->None] == (Sqrt[1 - (k)^(2)])^(- 1)* Log[JacobiDC[x, (k)^2]+Sqrt[1 - (k)^(2)]*JacobiSC[x, (k)^2]]
Successful Successful - Successful [Tested: 9]
22.14.E9 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int\Jacobiellsck@{x}{k}\diff{x} = {k^{\prime}}^{-1}\ln@{\Jacobielldck@{x}{k}+k^{\prime}\Jacobiellnck@{x}{k}}}
\int\Jacobiellsck@{x}{k}\diff{x} = {k^{\prime}}^{-1}\ln@{\Jacobielldck@{x}{k}+k^{\prime}\Jacobiellnck@{x}{k}}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
int(JacobiSC(x, k), x) = (sqrt(1 - (k)^(2)))^(- 1)* ln(JacobiDC(x, k)+sqrt(1 - (k)^(2))*JacobiNC(x, k))

Integrate[JacobiSC[x, (k)^2], x, GenerateConditions->None] == (Sqrt[1 - (k)^(2)])^(- 1)* Log[JacobiDC[x, (k)^2]+Sqrt[1 - (k)^(2)]*JacobiNC[x, (k)^2]]
Successful Successful - Successful [Tested: 9]
22.14.E10 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int\Jacobiellnsk@{x}{k}\diff{x} = \ln@{\Jacobielldsk@{x}{k}-\Jacobiellcsk@{x}{k}}}
\int\Jacobiellnsk@{x}{k}\diff{x} = \ln@{\Jacobielldsk@{x}{k}-\Jacobiellcsk@{x}{k}}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
int(JacobiNS(x, k), x) = ln(JacobiDS(x, k)- JacobiCS(x, k))

Integrate[JacobiNS[x, (k)^2], x, GenerateConditions->None] == Log[JacobiDS[x, (k)^2]- JacobiCS[x, (k)^2]]
Successful Successful - Successful [Tested: 9]
22.14.E11 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int\Jacobielldsk@{x}{k}\diff{x} = \ln@{\Jacobiellnsk@{x}{k}-\Jacobiellcsk@{x}{k}}}
\int\Jacobielldsk@{x}{k}\diff{x} = \ln@{\Jacobiellnsk@{x}{k}-\Jacobiellcsk@{x}{k}}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
int(JacobiDS(x, k), x) = ln(JacobiNS(x, k)- JacobiCS(x, k))

Integrate[JacobiDS[x, (k)^2], x, GenerateConditions->None] == Log[JacobiNS[x, (k)^2]- JacobiCS[x, (k)^2]]
Successful Successful - Successful [Tested: 9]
22.14.E12 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int\Jacobiellcsk@{x}{k}\diff{x} = \ln@{\Jacobiellnsk@{x}{k}-\Jacobielldsk@{x}{k}}}
\int\Jacobiellcsk@{x}{k}\diff{x} = \ln@{\Jacobiellnsk@{x}{k}-\Jacobielldsk@{x}{k}}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
int(JacobiCS(x, k), x) = ln(JacobiNS(x, k)- JacobiDS(x, k))

Integrate[JacobiCS[x, (k)^2], x, GenerateConditions->None] == Log[JacobiNS[x, (k)^2]- JacobiDS[x, (k)^2]]
Successful Successful - Successful [Tested: 9]
22.14.E13 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int\frac{\diff{x}}{\Jacobiellsnk@{x}{k}} = \ln@{\frac{\Jacobiellsnk@{x}{k}}{\Jacobiellcnk@{x}{k}+\Jacobielldnk@{x}{k}}}}
\int\frac{\diff{x}}{\Jacobiellsnk@{x}{k}} = \ln@{\frac{\Jacobiellsnk@{x}{k}}{\Jacobiellcnk@{x}{k}+\Jacobielldnk@{x}{k}}}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
int((1)/(JacobiSN(x, k)), x) = ln((JacobiSN(x, k))/(JacobiCN(x, k)+ JacobiDN(x, k)))

Integrate[Divide[1,JacobiSN[x, (k)^2]], x, GenerateConditions->None] == Log[Divide[JacobiSN[x, (k)^2],JacobiCN[x, (k)^2]+ JacobiDN[x, (k)^2]]]
Successful Successful - Successful [Tested: 9]
22.14.E14 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int\frac{\Jacobiellcnk@{x}{k}\diff{x}}{\Jacobiellsnk@{x}{k}} = \frac{1}{2}\ln@{\frac{1-\Jacobielldnk@{x}{k}}{1+\Jacobielldnk@{x}{k}}}}
\int\frac{\Jacobiellcnk@{x}{k}\diff{x}}{\Jacobiellsnk@{x}{k}} = \frac{1}{2}\ln@{\frac{1-\Jacobielldnk@{x}{k}}{1+\Jacobielldnk@{x}{k}}}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
int((JacobiCN(x, k))/(JacobiSN(x, k)), x) = (1)/(2)*ln((1 - JacobiDN(x, k))/(1 + JacobiDN(x, k)))

Integrate[Divide[JacobiCN[x, (k)^2],JacobiSN[x, (k)^2]], x, GenerateConditions->None] == Divide[1,2]*Log[Divide[1 - JacobiDN[x, (k)^2],1 + JacobiDN[x, (k)^2]]]
Failure Failure Successful [Tested: 9]
Failed [6 / 9]
Result: 0.6931471805599452
Test Values: {Rule[k, 2], Rule[x, 1.5]}


Result: Complex[1.0986122886681102, 3.141592653589793]
Test Values: {Rule[k, 3], Rule[x, 1.5]}

... skip entries to safe data
22.14.E15 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int\frac{\Jacobiellcnk@{x}{k}\diff{x}}{\Jacobiellsnk^{2}@{x}{k}} = -\frac{\Jacobielldnk@{x}{k}}{\Jacobiellsnk@{x}{k}}}
\int\frac{\Jacobiellcnk@{x}{k}\diff{x}}{\Jacobiellsnk^{2}@{x}{k}} = -\frac{\Jacobielldnk@{x}{k}}{\Jacobiellsnk@{x}{k}}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
int((JacobiCN(x, k))/((JacobiSN(x, k))^(2)), x) = -(JacobiDN(x, k))/(JacobiSN(x, k))

Integrate[Divide[JacobiCN[x, (k)^2],(JacobiSN[x, (k)^2])^(2)], x, GenerateConditions->None] == -Divide[JacobiDN[x, (k)^2],JacobiSN[x, (k)^2]]
Successful Successful - Successful [Tested: 9]
22.14.E16 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{0}^{\compellintKk@{k}}\ln@{\Jacobiellsnk@{t}{k}}\diff{t} = -\tfrac{\cpi}{4}\ccompellintKk@{k}-\tfrac{1}{2}\compellintKk@{k}\ln@@{k}}
\int_{0}^{\compellintKk@{k}}\ln@{\Jacobiellsnk@{t}{k}}\diff{t} = -\tfrac{\cpi}{4}\ccompellintKk@{k}-\tfrac{1}{2}\compellintKk@{k}\ln@@{k}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
int(ln(JacobiSN(t, k)), t = 0..EllipticK(k)) = -(Pi)/(4)*EllipticCK(k)-(1)/(2)*EllipticK(k)*ln(k)

Integrate[Log[JacobiSN[t, (k)^2]], {t, 0, EllipticK[(k)^2]}, GenerateConditions->None] == -Divide[Pi,4]*EllipticK[1-(k)^2]-Divide[1,2]*EllipticK[(k)^2]*Log[k]
Failure Failure Error Skipped - Because timed out
22.14.E17 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{0}^{\compellintKk@{k}}\ln@{\Jacobiellcnk@{t}{k}}\diff{t} = -\tfrac{\cpi}{4}\ccompellintKk@{k}+\tfrac{1}{2}\compellintKk@{k}\ln@{k^{\prime}/k}}
\int_{0}^{\compellintKk@{k}}\ln@{\Jacobiellcnk@{t}{k}}\diff{t} = -\tfrac{\cpi}{4}\ccompellintKk@{k}+\tfrac{1}{2}\compellintKk@{k}\ln@{k^{\prime}/k}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
int(ln(JacobiCN(t, k)), t = 0..EllipticK(k)) = -(Pi)/(4)*EllipticCK(k)+(1)/(2)*EllipticK(k)*ln(sqrt(1 - (k)^(2))/k)

Integrate[Log[JacobiCN[t, (k)^2]], {t, 0, EllipticK[(k)^2]}, GenerateConditions->None] == -Divide[Pi,4]*EllipticK[1-(k)^2]+Divide[1,2]*EllipticK[(k)^2]*Log[Sqrt[1 - (k)^(2)]/k]
Failure Failure Error Skipped - Because timed out
22.14.E18 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{0}^{\compellintKk@{k}}\ln@{\Jacobielldnk@{t}{k}}\diff{t} = \tfrac{1}{2}\compellintKk@{k}\ln@@{k^{\prime}}}
\int_{0}^{\compellintKk@{k}}\ln@{\Jacobielldnk@{t}{k}}\diff{t} = \tfrac{1}{2}\compellintKk@{k}\ln@@{k^{\prime}}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
int(ln(JacobiDN(t, k)), t = 0..EllipticK(k)) = (1)/(2)*EllipticK(k)*ln(sqrt(1 - (k)^(2)))

Integrate[Log[JacobiDN[t, (k)^2]], {t, 0, EllipticK[(k)^2]}, GenerateConditions->None] == Divide[1,2]*EllipticK[(k)^2]*Log[Sqrt[1 - (k)^(2)]]
Failure Failure Error Skipped - Because timed out
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