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 ∫ sn ⁡ ( x , k ) ⁢ d x = k - 1 ⁢ ln ⁡ ( dn ⁡ ( x , k ) - k ⁢ cn ⁡ ( x , k ) ) Jacobi-elliptic-sn 𝑥 𝑘 𝑥 superscript 𝑘 1 Jacobi-elliptic-dn 𝑥 𝑘 𝑘 Jacobi-elliptic-cn 𝑥 𝑘 {\displaystyle{\displaystyle\int\operatorname{sn}\left(x,k\right)\mathrm{d}x=k% ^{-1}\ln\left(\operatorname{dn}\left(x,k\right)-k\operatorname{cn}\left(x,k% \right)\right)}}
\int\Jacobiellsnk@{x}{k}\diff{x} = k^{-1}\ln@{\Jacobielldnk@{x}{k}-k\Jacobiellcnk@{x}{k}}		 

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 ∫ cn ⁡ ( x , k ) ⁢ d x = k - 1 ⁢ Arccos ⁡ ( dn ⁡ ( x , k ) ) Jacobi-elliptic-cn 𝑥 𝑘 𝑥 superscript 𝑘 1 multivalued-inverse-cosine Jacobi-elliptic-dn 𝑥 𝑘 {\displaystyle{\displaystyle\int\operatorname{cn}\left(x,k\right)\mathrm{d}x=k% ^{-1}\operatorname{Arccos}\left(\operatorname{dn}\left(x,k\right)\right)}}
\int\Jacobiellcnk@{x}{k}\diff{x} = k^{-1}\Acos@{\Jacobielldnk@{x}{k}}		 

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 ∫ dn ⁡ ( x , k ) ⁢ d x = Arcsin ⁡ ( sn ⁡ ( x , k ) ) Jacobi-elliptic-dn 𝑥 𝑘 𝑥 multivalued-inverse-sine Jacobi-elliptic-sn 𝑥 𝑘 {\displaystyle{\displaystyle\int\operatorname{dn}\left(x,k\right)\mathrm{d}x=% \operatorname{Arcsin}\left(\operatorname{sn}\left(x,k\right)\right)}}
\int\Jacobielldnk@{x}{k}\diff{x} = \Asin@{\Jacobiellsnk@{x}{k}}		 

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 Arcsin ⁡ ( sn ⁡ ( x , k ) ) = am ⁡ ( x , k ) multivalued-inverse-sine Jacobi-elliptic-sn 𝑥 𝑘 Jacobi-elliptic-amplitude 𝑥 𝑘 {\displaystyle{\displaystyle\operatorname{Arcsin}\left(\operatorname{sn}\left(% x,k\right)\right)=\operatorname{am}\left(x,k\right)}}
\Asin@{\Jacobiellsnk@{x}{k}} = \Jacobiamk@{x}{k}		 

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 ∫ cd ⁡ ( x , k ) ⁢ d x = k - 1 ⁢ ln ⁡ ( nd ⁡ ( x , k ) + k ⁢ sd ⁡ ( x , k ) ) Jacobi-elliptic-cd 𝑥 𝑘 𝑥 superscript 𝑘 1 Jacobi-elliptic-nd 𝑥 𝑘 𝑘 Jacobi-elliptic-sd 𝑥 𝑘 {\displaystyle{\displaystyle\int\operatorname{cd}\left(x,k\right)\mathrm{d}x=k% ^{-1}\ln\left(\operatorname{nd}\left(x,k\right)+k\operatorname{sd}\left(x,k% \right)\right)}}
\int\Jacobiellcdk@{x}{k}\diff{x} = k^{-1}\ln@{\Jacobiellndk@{x}{k}+k\Jacobiellsdk@{x}{k}}		 

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 ∫ sd ⁡ ( x , k ) ⁢ d x = ( k ⁢ k ′ ) - 1 ⁢ Arcsin ⁡ ( - k ⁢ cd ⁡ ( x , k ) ) Jacobi-elliptic-sd 𝑥 𝑘 𝑥 superscript 𝑘 superscript 𝑘 ′ 1 multivalued-inverse-sine 𝑘 Jacobi-elliptic-cd 𝑥 𝑘 {\displaystyle{\displaystyle\int\operatorname{sd}\left(x,k\right)\mathrm{d}x=(% kk^{\prime})^{-1}\operatorname{Arcsin}\left(-k\operatorname{cd}\left(x,k\right% )\right)}}
\int\Jacobiellsdk@{x}{k}\diff{x} = (kk^{\prime})^{-1}\Asin@{-k\Jacobiellcdk@{x}{k}}		 

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 ∫ nd ⁡ ( x , k ) ⁢ d x = k ′ - 1 ⁢ Arccos ⁡ ( cd ⁡ ( x , k ) ) Jacobi-elliptic-nd 𝑥 𝑘 𝑥 superscript superscript 𝑘 ′ 1 multivalued-inverse-cosine Jacobi-elliptic-cd 𝑥 𝑘 {\displaystyle{\displaystyle\int\operatorname{nd}\left(x,k\right)\mathrm{d}x={% k^{\prime}}^{-1}\operatorname{Arccos}\left(\operatorname{cd}\left(x,k\right)% \right)}}
\int\Jacobiellndk@{x}{k}\diff{x} = {k^{\prime}}^{-1}\Acos@{\Jacobiellcdk@{x}{k}}		 

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 ∫ dc ⁡ ( x , k ) ⁢ d x = ln ⁡ ( nc ⁡ ( x , k ) + sc ⁡ ( x , k ) ) Jacobi-elliptic-dc 𝑥 𝑘 𝑥 Jacobi-elliptic-nc 𝑥 𝑘 Jacobi-elliptic-sc 𝑥 𝑘 {\displaystyle{\displaystyle\int\operatorname{dc}\left(x,k\right)\mathrm{d}x=% \ln\left(\operatorname{nc}\left(x,k\right)+\operatorname{sc}\left(x,k\right)% \right)}}
\int\Jacobielldck@{x}{k}\diff{x} = \ln@{\Jacobiellnck@{x}{k}+\Jacobiellsck@{x}{k}}		 

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 ∫ nc ⁡ ( x , k ) ⁢ d x = k ′ - 1 ⁢ ln ⁡ ( dc ⁡ ( x , k ) + k ′ ⁢ sc ⁡ ( x , k ) ) Jacobi-elliptic-nc 𝑥 𝑘 𝑥 superscript superscript 𝑘 ′ 1 Jacobi-elliptic-dc 𝑥 𝑘 superscript 𝑘 ′ Jacobi-elliptic-sc 𝑥 𝑘 {\displaystyle{\displaystyle\int\operatorname{nc}\left(x,k\right)\mathrm{d}x={% k^{\prime}}^{-1}\ln\left(\operatorname{dc}\left(x,k\right)+k^{\prime}% \operatorname{sc}\left(x,k\right)\right)}}
\int\Jacobiellnck@{x}{k}\diff{x} = {k^{\prime}}^{-1}\ln@{\Jacobielldck@{x}{k}+k^{\prime}\Jacobiellsck@{x}{k}}		 

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 ∫ sc ⁡ ( x , k ) ⁢ d x = k ′ - 1 ⁢ ln ⁡ ( dc ⁡ ( x , k ) + k ′ ⁢ nc ⁡ ( x , k ) ) Jacobi-elliptic-sc 𝑥 𝑘 𝑥 superscript superscript 𝑘 ′ 1 Jacobi-elliptic-dc 𝑥 𝑘 superscript 𝑘 ′ Jacobi-elliptic-nc 𝑥 𝑘 {\displaystyle{\displaystyle\int\operatorname{sc}\left(x,k\right)\mathrm{d}x={% k^{\prime}}^{-1}\ln\left(\operatorname{dc}\left(x,k\right)+k^{\prime}% \operatorname{nc}\left(x,k\right)\right)}}
\int\Jacobiellsck@{x}{k}\diff{x} = {k^{\prime}}^{-1}\ln@{\Jacobielldck@{x}{k}+k^{\prime}\Jacobiellnck@{x}{k}}		 

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 ∫ ns ⁡ ( x , k ) ⁢ d x = ln ⁡ ( ds ⁡ ( x , k ) - cs ⁡ ( x , k ) ) Jacobi-elliptic-ns 𝑥 𝑘 𝑥 Jacobi-elliptic-ds 𝑥 𝑘 Jacobi-elliptic-cs 𝑥 𝑘 {\displaystyle{\displaystyle\int\operatorname{ns}\left(x,k\right)\mathrm{d}x=% \ln\left(\operatorname{ds}\left(x,k\right)-\operatorname{cs}\left(x,k\right)% \right)}}
\int\Jacobiellnsk@{x}{k}\diff{x} = \ln@{\Jacobielldsk@{x}{k}-\Jacobiellcsk@{x}{k}}		 

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 ∫ ds ⁡ ( x , k ) ⁢ d x = ln ⁡ ( ns ⁡ ( x , k ) - cs ⁡ ( x , k ) ) Jacobi-elliptic-ds 𝑥 𝑘 𝑥 Jacobi-elliptic-ns 𝑥 𝑘 Jacobi-elliptic-cs 𝑥 𝑘 {\displaystyle{\displaystyle\int\operatorname{ds}\left(x,k\right)\mathrm{d}x=% \ln\left(\operatorname{ns}\left(x,k\right)-\operatorname{cs}\left(x,k\right)% \right)}}
\int\Jacobielldsk@{x}{k}\diff{x} = \ln@{\Jacobiellnsk@{x}{k}-\Jacobiellcsk@{x}{k}}		 

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 ∫ cs ⁡ ( x , k ) ⁢ d x = ln ⁡ ( ns ⁡ ( x , k ) - ds ⁡ ( x , k ) ) Jacobi-elliptic-cs 𝑥 𝑘 𝑥 Jacobi-elliptic-ns 𝑥 𝑘 Jacobi-elliptic-ds 𝑥 𝑘 {\displaystyle{\displaystyle\int\operatorname{cs}\left(x,k\right)\mathrm{d}x=% \ln\left(\operatorname{ns}\left(x,k\right)-\operatorname{ds}\left(x,k\right)% \right)}}
\int\Jacobiellcsk@{x}{k}\diff{x} = \ln@{\Jacobiellnsk@{x}{k}-\Jacobielldsk@{x}{k}}		 

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 ∫ d x sn ⁡ ( x , k ) = ln ⁡ ( sn ⁡ ( x , k ) cn ⁡ ( x , k ) + dn ⁡ ( x , k ) ) 𝑥 Jacobi-elliptic-sn 𝑥 𝑘 Jacobi-elliptic-sn 𝑥 𝑘 Jacobi-elliptic-cn 𝑥 𝑘 Jacobi-elliptic-dn 𝑥 𝑘 {\displaystyle{\displaystyle\int\frac{\mathrm{d}x}{\operatorname{sn}\left(x,k% \right)}=\ln\left(\frac{\operatorname{sn}\left(x,k\right)}{\operatorname{cn}% \left(x,k\right)+\operatorname{dn}\left(x,k\right)}\right)}}
\int\frac{\diff{x}}{\Jacobiellsnk@{x}{k}} = \ln@{\frac{\Jacobiellsnk@{x}{k}}{\Jacobiellcnk@{x}{k}+\Jacobielldnk@{x}{k}}}		 

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 ∫ cn ⁡ ( x , k ) ⁢ d x sn ⁡ ( x , k ) = 1 2 ⁢ ln ⁡ ( 1 - dn ⁡ ( x , k ) 1 + dn ⁡ ( x , k ) ) Jacobi-elliptic-cn 𝑥 𝑘 𝑥 Jacobi-elliptic-sn 𝑥 𝑘 1 2 1 Jacobi-elliptic-dn 𝑥 𝑘 1 Jacobi-elliptic-dn 𝑥 𝑘 {\displaystyle{\displaystyle\int\frac{\operatorname{cn}\left(x,k\right)\mathrm% {d}x}{\operatorname{sn}\left(x,k\right)}=\frac{1}{2}\ln\left(\frac{1-% \operatorname{dn}\left(x,k\right)}{1+\operatorname{dn}\left(x,k\right)}\right)}}
\int\frac{\Jacobiellcnk@{x}{k}\diff{x}}{\Jacobiellsnk@{x}{k}} = \frac{1}{2}\ln@{\frac{1-\Jacobielldnk@{x}{k}}{1+\Jacobielldnk@{x}{k}}}		 

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 ∫ cn ⁡ ( x , k ) ⁢ d x sn 2 ⁡ ( x , k ) = - dn ⁡ ( x , k ) sn ⁡ ( x , k ) Jacobi-elliptic-cn 𝑥 𝑘 𝑥 Jacobi-elliptic-sn 2 𝑥 𝑘 Jacobi-elliptic-dn 𝑥 𝑘 Jacobi-elliptic-sn 𝑥 𝑘 {\displaystyle{\displaystyle\int\frac{\operatorname{cn}\left(x,k\right)\mathrm% {d}x}{{\operatorname{sn}^{2}}\left(x,k\right)}=-\frac{\operatorname{dn}\left(x% ,k\right)}{\operatorname{sn}\left(x,k\right)}}}
\int\frac{\Jacobiellcnk@{x}{k}\diff{x}}{\Jacobiellsnk^{2}@{x}{k}} = -\frac{\Jacobielldnk@{x}{k}}{\Jacobiellsnk@{x}{k}}		 

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 ∫ 0 K ⁡ ( k ) ln ⁡ ( sn ⁡ ( t , k ) ) ⁢ d t = - π 4 ⁢ K ′ ⁡ ( k ) - 1 2 ⁢ K ⁡ ( k ) ⁢ ln ⁡ k superscript subscript 0 complete-elliptic-integral-first-kind-K 𝑘 Jacobi-elliptic-sn 𝑡 𝑘 𝑡 4 complementary-complete-elliptic-integral-first-kind-K 𝑘 1 2 complete-elliptic-integral-first-kind-K 𝑘 𝑘 {\displaystyle{\displaystyle\int_{0}^{K\left(k\right)}\ln\left(\operatorname{% sn}\left(t,k\right)\right)\mathrm{d}t=-\tfrac{\pi}{4}{K^{\prime}}\left(k\right% )-\tfrac{1}{2}K\left(k\right)\ln k}}
\int_{0}^{\compellintKk@{k}}\ln@{\Jacobiellsnk@{t}{k}}\diff{t} = -\tfrac{\cpi}{4}\ccompellintKk@{k}-\tfrac{1}{2}\compellintKk@{k}\ln@@{k}		 

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 ∫ 0 K ⁡ ( k ) ln ⁡ ( cn ⁡ ( t , k ) ) ⁢ d t = - π 4 ⁢ K ′ ⁡ ( k ) + 1 2 ⁢ K ⁡ ( k ) ⁢ ln ⁡ ( k ′ / k ) superscript subscript 0 complete-elliptic-integral-first-kind-K 𝑘 Jacobi-elliptic-cn 𝑡 𝑘 𝑡 4 complementary-complete-elliptic-integral-first-kind-K 𝑘 1 2 complete-elliptic-integral-first-kind-K 𝑘 superscript 𝑘 ′ 𝑘 {\displaystyle{\displaystyle\int_{0}^{K\left(k\right)}\ln\left(\operatorname{% cn}\left(t,k\right)\right)\mathrm{d}t=-\tfrac{\pi}{4}{K^{\prime}}\left(k\right% )+\tfrac{1}{2}K\left(k\right)\ln\left(k^{\prime}/k\right)}}
\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(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 ∫ 0 K ⁡ ( k ) ln ⁡ ( dn ⁡ ( t , k ) ) ⁢ d t = 1 2 ⁢ K ⁡ ( k ) ⁢ ln ⁡ k ′ superscript subscript 0 complete-elliptic-integral-first-kind-K 𝑘 Jacobi-elliptic-dn 𝑡 𝑘 𝑡 1 2 complete-elliptic-integral-first-kind-K 𝑘 superscript 𝑘 ′ {\displaystyle{\displaystyle\int_{0}^{K\left(k\right)}\ln\left(\operatorname{% dn}\left(t,k\right)\right)\mathrm{d}t=\tfrac{1}{2}K\left(k\right)\ln k^{\prime% }}}
\int_{0}^{\compellintKk@{k}}\ln@{\Jacobielldnk@{t}{k}}\diff{t} = \tfrac{1}{2}\compellintKk@{k}\ln@@{k^{\prime}}		 

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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