Optimal. Leaf size=59 \[ -\frac {2 x \tanh ^{-1}\left (e^x\right ) \text {sech}(x)}{\sqrt {a \text {sech}^2(x)}}-\frac {\text {PolyLog}\left (2,-e^x\right ) \text {sech}(x)}{\sqrt {a \text {sech}^2(x)}}+\frac {\text {PolyLog}\left (2,e^x\right ) \text {sech}(x)}{\sqrt {a \text {sech}^2(x)}} \]
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Rubi [A]
time = 0.40, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6852, 4267,
2317, 2438} \begin {gather*} -\frac {\text {Li}_2\left (-e^x\right ) \text {sech}(x)}{\sqrt {a \text {sech}^2(x)}}+\frac {\text {Li}_2\left (e^x\right ) \text {sech}(x)}{\sqrt {a \text {sech}^2(x)}}-\frac {2 x \tanh ^{-1}\left (e^x\right ) \text {sech}(x)}{\sqrt {a \text {sech}^2(x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2317
Rule 2438
Rule 4267
Rule 6852
Rubi steps
\begin {align*} \int \frac {x \text {csch}(x) \text {sech}(x)}{\sqrt {a \text {sech}^2(x)}} \, dx &=\frac {\text {sech}(x) \int x \text {csch}(x) \, dx}{\sqrt {a \text {sech}^2(x)}}\\ &=-\frac {2 x \tanh ^{-1}\left (e^x\right ) \text {sech}(x)}{\sqrt {a \text {sech}^2(x)}}-\frac {\text {sech}(x) \int \log \left (1-e^x\right ) \, dx}{\sqrt {a \text {sech}^2(x)}}+\frac {\text {sech}(x) \int \log \left (1+e^x\right ) \, dx}{\sqrt {a \text {sech}^2(x)}}\\ &=-\frac {2 x \tanh ^{-1}\left (e^x\right ) \text {sech}(x)}{\sqrt {a \text {sech}^2(x)}}-\frac {\text {sech}(x) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^x\right )}{\sqrt {a \text {sech}^2(x)}}+\frac {\text {sech}(x) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^x\right )}{\sqrt {a \text {sech}^2(x)}}\\ &=-\frac {2 x \tanh ^{-1}\left (e^x\right ) \text {sech}(x)}{\sqrt {a \text {sech}^2(x)}}-\frac {\text {Li}_2\left (-e^x\right ) \text {sech}(x)}{\sqrt {a \text {sech}^2(x)}}+\frac {\text {Li}_2\left (e^x\right ) \text {sech}(x)}{\sqrt {a \text {sech}^2(x)}}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 55, normalized size = 0.93 \begin {gather*} \frac {\left (x \left (\log \left (1-e^{-x}\right )-\log \left (1+e^{-x}\right )\right )+\text {PolyLog}\left (2,-e^{-x}\right )-\text {PolyLog}\left (2,e^{-x}\right )\right ) \text {sech}(x)}{\sqrt {a \text {sech}^2(x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(135\) vs.
\(2(50)=100\).
time = 1.70, size = 136, normalized size = 2.31
method | result | size |
risch | \(-\frac {{\mathrm e}^{x} x \ln \left ({\mathrm e}^{x}+1\right )}{\sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}\, \left (1+{\mathrm e}^{2 x}\right )}-\frac {{\mathrm e}^{x} \polylog \left (2, -{\mathrm e}^{x}\right )}{\sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}\, \left (1+{\mathrm e}^{2 x}\right )}+\frac {{\mathrm e}^{x} x \ln \left (1-{\mathrm e}^{x}\right )}{\sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}\, \left (1+{\mathrm e}^{2 x}\right )}+\frac {{\mathrm e}^{x} \polylog \left (2, {\mathrm e}^{x}\right )}{\sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}\, \left (1+{\mathrm e}^{2 x}\right )}\) | \(136\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 36, normalized size = 0.61 \begin {gather*} -\frac {x \log \left (e^{x} + 1\right ) + {\rm Li}_2\left (-e^{x}\right )}{\sqrt {a}} + \frac {x \log \left (-e^{x} + 1\right ) + {\rm Li}_2\left (e^{x}\right )}{\sqrt {a}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 91, normalized size = 1.54 \begin {gather*} \frac {{\left ({\left (e^{\left (2 \, x\right )} + 1\right )} {\rm Li}_2\left (\cosh \left (x\right ) + \sinh \left (x\right )\right ) - {\left (e^{\left (2 \, x\right )} + 1\right )} {\rm Li}_2\left (-\cosh \left (x\right ) - \sinh \left (x\right )\right ) - {\left (x e^{\left (2 \, x\right )} + x\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) + {\left (x e^{\left (2 \, x\right )} + x\right )} \log \left (-\cosh \left (x\right ) - \sinh \left (x\right ) + 1\right )\right )} \sqrt {\frac {a}{e^{\left (4 \, x\right )} + 2 \, e^{\left (2 \, x\right )} + 1}}}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x \operatorname {csch}{\left (x \right )} \operatorname {sech}{\left (x \right )}}{\sqrt {a \operatorname {sech}^{2}{\left (x \right )}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {x}{\mathrm {cosh}\left (x\right )\,\mathrm {sinh}\left (x\right )\,\sqrt {\frac {a}{{\mathrm {cosh}\left (x\right )}^2}}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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