Optimal. Leaf size=59 \[ -\frac{\text{sech}(x) \text{PolyLog}\left (2,-e^x\right )}{\sqrt{a \text{sech}^2(x)}}+\frac{\text{sech}(x) \text{PolyLog}\left (2,e^x\right )}{\sqrt{a \text{sech}^2(x)}}-\frac{2 x \tanh ^{-1}\left (e^x\right ) \text{sech}(x)}{\sqrt{a \text{sech}^2(x)}} \]
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Rubi [A] time = 0.701356, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {6720, 4182, 2279, 2391} \[ -\frac{\text{sech}(x) \text{PolyLog}\left (2,-e^x\right )}{\sqrt{a \text{sech}^2(x)}}+\frac{\text{sech}(x) \text{PolyLog}\left (2,e^x\right )}{\sqrt{a \text{sech}^2(x)}}-\frac{2 x \tanh ^{-1}\left (e^x\right ) \text{sech}(x)}{\sqrt{a \text{sech}^2(x)}} \]
Antiderivative was successfully verified.
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Rule 6720
Rule 4182
Rule 2279
Rule 2391
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) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^x\right )}{\sqrt{a \text{sech}^2(x)}}+\frac{\text{sech}(x) \operatorname{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*}
Mathematica [A] time = 0.0542851, size = 55, normalized size = 0.93 \[ \frac{\text{sech}(x) \left (\text{PolyLog}\left (2,-e^{-x}\right )-\text{PolyLog}\left (2,e^{-x}\right )+x \left (\log \left (1-e^{-x}\right )-\log \left (e^{-x}+1\right )\right )\right )}{\sqrt{a \text{sech}^2(x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.079, size = 136, normalized size = 2.3 \begin{align*} -{\frac{x{{\rm e}^{x}}\ln \left ({{\rm e}^{x}}+1 \right ) }{{{\rm e}^{2\,x}}+1}{\frac{1}{\sqrt{{\frac{a{{\rm e}^{2\,x}}}{ \left ({{\rm e}^{2\,x}}+1 \right ) ^{2}}}}}}}-{\frac{{{\rm e}^{x}}{\it polylog} \left ( 2,-{{\rm e}^{x}} \right ) }{{{\rm e}^{2\,x}}+1}{\frac{1}{\sqrt{{\frac{a{{\rm e}^{2\,x}}}{ \left ({{\rm e}^{2\,x}}+1 \right ) ^{2}}}}}}}+{\frac{x{{\rm e}^{x}}\ln \left ( 1-{{\rm e}^{x}} \right ) }{{{\rm e}^{2\,x}}+1}{\frac{1}{\sqrt{{\frac{a{{\rm e}^{2\,x}}}{ \left ({{\rm e}^{2\,x}}+1 \right ) ^{2}}}}}}}+{\frac{{{\rm e}^{x}}{\it polylog} \left ( 2,{{\rm e}^{x}} \right ) }{{{\rm e}^{2\,x}}+1}{\frac{1}{\sqrt{{\frac{a{{\rm e}^{2\,x}}}{ \left ({{\rm e}^{2\,x}}+1 \right ) ^{2}}}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.81995, size = 49, normalized size = 0.83 \begin{align*} -\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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.09782, size = 285, normalized size = 4.83 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x \operatorname{csch}{\left (x \right )} \operatorname{sech}{\left (x \right )}}{\sqrt{a \operatorname{sech}^{2}{\left (x \right )}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x \operatorname{csch}\left (x\right ) \operatorname{sech}\left (x\right )}{\sqrt{a \operatorname{sech}\left (x\right )^{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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