Optimal. Leaf size=104 \[ -\frac{2 x \text{sech}(x) \text{PolyLog}\left (2,-e^x\right )}{\sqrt{a \text{sech}^2(x)}}+\frac{2 x \text{sech}(x) \text{PolyLog}\left (2,e^x\right )}{\sqrt{a \text{sech}^2(x)}}+\frac{2 \text{sech}(x) \text{PolyLog}\left (3,-e^x\right )}{\sqrt{a \text{sech}^2(x)}}-\frac{2 \text{sech}(x) \text{PolyLog}\left (3,e^x\right )}{\sqrt{a \text{sech}^2(x)}}-\frac{2 x^2 \tanh ^{-1}\left (e^x\right ) \text{sech}(x)}{\sqrt{a \text{sech}^2(x)}} \]
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Rubi [A] time = 0.813223, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278, Rules used = {6720, 4182, 2531, 2282, 6589} \[ -\frac{2 x \text{sech}(x) \text{PolyLog}\left (2,-e^x\right )}{\sqrt{a \text{sech}^2(x)}}+\frac{2 x \text{sech}(x) \text{PolyLog}\left (2,e^x\right )}{\sqrt{a \text{sech}^2(x)}}+\frac{2 \text{sech}(x) \text{PolyLog}\left (3,-e^x\right )}{\sqrt{a \text{sech}^2(x)}}-\frac{2 \text{sech}(x) \text{PolyLog}\left (3,e^x\right )}{\sqrt{a \text{sech}^2(x)}}-\frac{2 x^2 \tanh ^{-1}\left (e^x\right ) \text{sech}(x)}{\sqrt{a \text{sech}^2(x)}} \]
Antiderivative was successfully verified.
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Rule 6720
Rule 4182
Rule 2531
Rule 2282
Rule 6589
Rubi steps
\begin{align*} \int \frac{x^2 \text{csch}(x) \text{sech}(x)}{\sqrt{a \text{sech}^2(x)}} \, dx &=\frac{\text{sech}(x) \int x^2 \text{csch}(x) \, dx}{\sqrt{a \text{sech}^2(x)}}\\ &=-\frac{2 x^2 \tanh ^{-1}\left (e^x\right ) \text{sech}(x)}{\sqrt{a \text{sech}^2(x)}}-\frac{(2 \text{sech}(x)) \int x \log \left (1-e^x\right ) \, dx}{\sqrt{a \text{sech}^2(x)}}+\frac{(2 \text{sech}(x)) \int x \log \left (1+e^x\right ) \, dx}{\sqrt{a \text{sech}^2(x)}}\\ &=-\frac{2 x^2 \tanh ^{-1}\left (e^x\right ) \text{sech}(x)}{\sqrt{a \text{sech}^2(x)}}-\frac{2 x \text{Li}_2\left (-e^x\right ) \text{sech}(x)}{\sqrt{a \text{sech}^2(x)}}+\frac{2 x \text{Li}_2\left (e^x\right ) \text{sech}(x)}{\sqrt{a \text{sech}^2(x)}}+\frac{(2 \text{sech}(x)) \int \text{Li}_2\left (-e^x\right ) \, dx}{\sqrt{a \text{sech}^2(x)}}-\frac{(2 \text{sech}(x)) \int \text{Li}_2\left (e^x\right ) \, dx}{\sqrt{a \text{sech}^2(x)}}\\ &=-\frac{2 x^2 \tanh ^{-1}\left (e^x\right ) \text{sech}(x)}{\sqrt{a \text{sech}^2(x)}}-\frac{2 x \text{Li}_2\left (-e^x\right ) \text{sech}(x)}{\sqrt{a \text{sech}^2(x)}}+\frac{2 x \text{Li}_2\left (e^x\right ) \text{sech}(x)}{\sqrt{a \text{sech}^2(x)}}+\frac{(2 \text{sech}(x)) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-x)}{x} \, dx,x,e^x\right )}{\sqrt{a \text{sech}^2(x)}}-\frac{(2 \text{sech}(x)) \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,e^x\right )}{\sqrt{a \text{sech}^2(x)}}\\ &=-\frac{2 x^2 \tanh ^{-1}\left (e^x\right ) \text{sech}(x)}{\sqrt{a \text{sech}^2(x)}}-\frac{2 x \text{Li}_2\left (-e^x\right ) \text{sech}(x)}{\sqrt{a \text{sech}^2(x)}}+\frac{2 x \text{Li}_2\left (e^x\right ) \text{sech}(x)}{\sqrt{a \text{sech}^2(x)}}+\frac{2 \text{Li}_3\left (-e^x\right ) \text{sech}(x)}{\sqrt{a \text{sech}^2(x)}}-\frac{2 \text{Li}_3\left (e^x\right ) \text{sech}(x)}{\sqrt{a \text{sech}^2(x)}}\\ \end{align*}
Mathematica [A] time = 0.0659689, size = 83, normalized size = 0.8 \[ \frac{\text{sech}(x) \left (2 x \text{PolyLog}\left (2,-e^{-x}\right )-2 x \text{PolyLog}\left (2,e^{-x}\right )+2 \text{PolyLog}\left (3,-e^{-x}\right )-2 \text{PolyLog}\left (3,e^{-x}\right )+x^2 \log \left (1-e^{-x}\right )-x^2 \log \left (e^{-x}+1\right )\right )}{\sqrt{a \text{sech}^2(x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.069, size = 209, normalized size = 2. \begin{align*} -{\frac{{x}^{2}{{\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}}}}}}}-2\,{\frac{x{{\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}}}}}}}+2\,{\frac{{{\rm e}^{x}}{\it polylog} \left ( 3,-{{\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}^{2}{{\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}}}}}}}+2\,{\frac{x{{\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}}}}}}}-2\,{\frac{{{\rm e}^{x}}{\it polylog} \left ( 3,{{\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 = 1.65807, size = 81, normalized size = 0.78 \begin{align*} -\frac{x^{2} \log \left (e^{x} + 1\right ) + 2 \, x{\rm Li}_2\left (-e^{x}\right ) - 2 \,{\rm Li}_{3}(-e^{x})}{\sqrt{a}} + \frac{x^{2} \log \left (-e^{x} + 1\right ) + 2 \, x{\rm Li}_2\left (e^{x}\right ) - 2 \,{\rm Li}_{3}(e^{x})}{\sqrt{a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 2.09906, size = 559, normalized size = 5.38 \begin{align*} -\frac{{\left (2 \, \sqrt{\frac{a}{e^{\left (4 \, x\right )} + 2 \, e^{\left (2 \, x\right )} + 1}}{\left (e^{\left (2 \, x\right )} + 1\right )} e^{x}{\rm polylog}\left (3, \cosh \left (x\right ) + \sinh \left (x\right )\right ) - 2 \, \sqrt{\frac{a}{e^{\left (4 \, x\right )} + 2 \, e^{\left (2 \, x\right )} + 1}}{\left (e^{\left (2 \, x\right )} + 1\right )} e^{x}{\rm polylog}\left (3, -\cosh \left (x\right ) - \sinh \left (x\right )\right ) -{\left (2 \,{\left (x e^{\left (2 \, x\right )} + x\right )}{\rm Li}_2\left (\cosh \left (x\right ) + \sinh \left (x\right )\right ) - 2 \,{\left (x e^{\left (2 \, x\right )} + x\right )}{\rm Li}_2\left (-\cosh \left (x\right ) - \sinh \left (x\right )\right ) -{\left (x^{2} e^{\left (2 \, x\right )} + x^{2}\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) +{\left (x^{2} e^{\left (2 \, x\right )} + x^{2}\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}} e^{x}\right )} e^{\left (-x\right )}}{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^{2} \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^{2} \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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