22.14

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
22.14.E1	${\displaystyle{\displaystyle\int\operatorname{sn}\left(x,k\right)\mathrm{d}x=k^{-1}\ln\left(\operatorname{dn}\left(x,k\right)-k\operatorname{cn}\left(x,k\right)\right)}}$ \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	${\displaystyle{\displaystyle\int\operatorname{cn}\left(x,k\right)\mathrm{d}x=k^{-1}\operatorname{Arccos}\left(\operatorname{dn}\left(x,k\right)\right)}}$ \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	${\displaystyle{\displaystyle\int\operatorname{dn}\left(x,k\right)\mathrm{d}x=\operatorname{Arcsin}\left(\operatorname{sn}\left(x,k\right)\right)}}$ \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	${\displaystyle{\displaystyle\operatorname{Arcsin}\left(\operatorname{sn}\left(x,k\right)\right)=\operatorname{am}\left(x,k\right)}}$ \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	${\displaystyle{\displaystyle\int\operatorname{cd}\left(x,k\right)\mathrm{d}x=k^{-1}\ln\left(\operatorname{nd}\left(x,k\right)+k\operatorname{sd}\left(x,k\right)\right)}}$ \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	${\displaystyle{\displaystyle\int\operatorname{sd}\left(x,k\right)\mathrm{d}x=(kk^{\prime})^{-1}\operatorname{Arcsin}\left(-k\operatorname{cd}\left(x,k\right)\right)}}$ \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	${\displaystyle{\displaystyle\int\operatorname{nd}\left(x,k\right)\mathrm{d}x={k^{\prime}}^{-1}\operatorname{Arccos}\left(\operatorname{cd}\left(x,k\right)\right)}}$ \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	${\displaystyle{\displaystyle\int\operatorname{dc}\left(x,k\right)\mathrm{d}x=\ln\left(\operatorname{nc}\left(x,k\right)+\operatorname{sc}\left(x,k\right)\right)}}$ \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	${\displaystyle{\displaystyle\int\operatorname{nc}\left(x,k\right)\mathrm{d}x={k^{\prime}}^{-1}\ln\left(\operatorname{dc}\left(x,k\right)+k^{\prime}\operatorname{sc}\left(x,k\right)\right)}}$ \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	${\displaystyle{\displaystyle\int\operatorname{sc}\left(x,k\right)\mathrm{d}x={k^{\prime}}^{-1}\ln\left(\operatorname{dc}\left(x,k\right)+k^{\prime}\operatorname{nc}\left(x,k\right)\right)}}$ \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	${\displaystyle{\displaystyle\int\operatorname{ns}\left(x,k\right)\mathrm{d}x=\ln\left(\operatorname{ds}\left(x,k\right)-\operatorname{cs}\left(x,k\right)\right)}}$ \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	${\displaystyle{\displaystyle\int\operatorname{ds}\left(x,k\right)\mathrm{d}x=\ln\left(\operatorname{ns}\left(x,k\right)-\operatorname{cs}\left(x,k\right)\right)}}$ \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	${\displaystyle{\displaystyle\int\operatorname{cs}\left(x,k\right)\mathrm{d}x=\ln\left(\operatorname{ns}\left(x,k\right)-\operatorname{ds}\left(x,k\right)\right)}}$ \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	${\displaystyle{\displaystyle\int\frac{\mathrm{d}x}{\operatorname{sn}\left(x,k\right)}=\ln\left(\frac{\operatorname{sn}\left(x,k\right)}{\operatorname{cn}\left(x,k\right)+\operatorname{dn}\left(x,k\right)}\right)}}$ \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	${\displaystyle{\displaystyle\int\frac{\operatorname{cn}\left(x,k\right)\mathrm{d}x}{\operatorname{sn}\left(x,k\right)}=\frac{1}{2}\ln\left(\frac{1-\operatorname{dn}\left(x,k\right)}{1+\operatorname{dn}\left(x,k\right)}\right)}}$ \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	${\displaystyle{\displaystyle\int\frac{\operatorname{cn}\left(x,k\right)\mathrm{d}x}{{\operatorname{sn}^{2}}\left(x,k\right)}=-\frac{\operatorname{dn}\left(x,k\right)}{\operatorname{sn}\left(x,k\right)}}}$ \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	${\displaystyle{\displaystyle\int_{0}^{K\left(k\right)}\ln\left(\operatorname{sn}\left(t,k\right)\right)\mathrm{d}t=-\tfrac{\pi}{4}{K^{\prime}}\left(k\right)-\tfrac{1}{2}K\left(k\right)\ln k}}$ \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	${\displaystyle{\displaystyle\int_{0}^{K\left(k\right)}\ln\left(\operatorname{cn}\left(t,k\right)\right)\mathrm{d}t=-\tfrac{\pi}{4}{K^{\prime}}\left(k\right)+\tfrac{1}{2}K\left(k\right)\ln\left(k^{\prime}/k\right)}}$ \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	${\displaystyle{\displaystyle\int_{0}^{K\left(k\right)}\ln\left(\operatorname{dn}\left(t,k\right)\right)\mathrm{d}t=\tfrac{1}{2}K\left(k\right)\ln k^{\prime}}}$ \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