Optimal. Leaf size=287 \[ -3 x^2 \cosh (x) \text{PolyLog}\left (2,-e^x\right ) \sqrt{a \text{sech}^2(x)}+3 x^2 \cosh (x) \text{PolyLog}\left (2,e^x\right ) \sqrt{a \text{sech}^2(x)}+6 i x \cosh (x) \text{PolyLog}\left (2,-i e^x\right ) \sqrt{a \text{sech}^2(x)}-6 i x \cosh (x) \text{PolyLog}\left (2,i e^x\right ) \sqrt{a \text{sech}^2(x)}+6 x \cosh (x) \text{PolyLog}\left (3,-e^x\right ) \sqrt{a \text{sech}^2(x)}-6 x \cosh (x) \text{PolyLog}\left (3,e^x\right ) \sqrt{a \text{sech}^2(x)}-6 i \cosh (x) \text{PolyLog}\left (3,-i e^x\right ) \sqrt{a \text{sech}^2(x)}+6 i \cosh (x) \text{PolyLog}\left (3,i e^x\right ) \sqrt{a \text{sech}^2(x)}-6 \cosh (x) \text{PolyLog}\left (4,-e^x\right ) \sqrt{a \text{sech}^2(x)}+6 \cosh (x) \text{PolyLog}\left (4,e^x\right ) \sqrt{a \text{sech}^2(x)}+x^3 \sqrt{a \text{sech}^2(x)}-6 x^2 \tan ^{-1}\left (e^x\right ) \cosh (x) \sqrt{a \text{sech}^2(x)}-2 x^3 \cosh (x) \tanh ^{-1}\left (e^x\right ) \sqrt{a \text{sech}^2(x)} \]
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Rubi [A] time = 0.673415, antiderivative size = 287, normalized size of antiderivative = 1., number of steps used = 21, number of rules used = 13, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.722, Rules used = {6720, 2622, 321, 207, 5462, 14, 6273, 4182, 2531, 6609, 2282, 6589, 4180} \[ -3 x^2 \cosh (x) \text{PolyLog}\left (2,-e^x\right ) \sqrt{a \text{sech}^2(x)}+3 x^2 \cosh (x) \text{PolyLog}\left (2,e^x\right ) \sqrt{a \text{sech}^2(x)}+6 i x \cosh (x) \text{PolyLog}\left (2,-i e^x\right ) \sqrt{a \text{sech}^2(x)}-6 i x \cosh (x) \text{PolyLog}\left (2,i e^x\right ) \sqrt{a \text{sech}^2(x)}+6 x \cosh (x) \text{PolyLog}\left (3,-e^x\right ) \sqrt{a \text{sech}^2(x)}-6 x \cosh (x) \text{PolyLog}\left (3,e^x\right ) \sqrt{a \text{sech}^2(x)}-6 i \cosh (x) \text{PolyLog}\left (3,-i e^x\right ) \sqrt{a \text{sech}^2(x)}+6 i \cosh (x) \text{PolyLog}\left (3,i e^x\right ) \sqrt{a \text{sech}^2(x)}-6 \cosh (x) \text{PolyLog}\left (4,-e^x\right ) \sqrt{a \text{sech}^2(x)}+6 \cosh (x) \text{PolyLog}\left (4,e^x\right ) \sqrt{a \text{sech}^2(x)}+x^3 \sqrt{a \text{sech}^2(x)}-6 x^2 \tan ^{-1}\left (e^x\right ) \cosh (x) \sqrt{a \text{sech}^2(x)}-2 x^3 \cosh (x) \tanh ^{-1}\left (e^x\right ) \sqrt{a \text{sech}^2(x)} \]
Antiderivative was successfully verified.
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Rule 6720
Rule 2622
Rule 321
Rule 207
Rule 5462
Rule 14
Rule 6273
Rule 4182
Rule 2531
Rule 6609
Rule 2282
Rule 6589
Rule 4180
Rubi steps
\begin{align*} \int x^3 \text{csch}(x) \text{sech}(x) \sqrt{a \text{sech}^2(x)} \, dx &=\left (\cosh (x) \sqrt{a \text{sech}^2(x)}\right ) \int x^3 \text{csch}(x) \text{sech}^2(x) \, dx\\ &=x^3 \sqrt{a \text{sech}^2(x)}-x^3 \tanh ^{-1}(\cosh (x)) \cosh (x) \sqrt{a \text{sech}^2(x)}-\left (3 \cosh (x) \sqrt{a \text{sech}^2(x)}\right ) \int x^2 \left (-\tanh ^{-1}(\cosh (x))+\text{sech}(x)\right ) \, dx\\ &=x^3 \sqrt{a \text{sech}^2(x)}-x^3 \tanh ^{-1}(\cosh (x)) \cosh (x) \sqrt{a \text{sech}^2(x)}-\left (3 \cosh (x) \sqrt{a \text{sech}^2(x)}\right ) \int \left (-x^2 \tanh ^{-1}(\cosh (x))+x^2 \text{sech}(x)\right ) \, dx\\ &=x^3 \sqrt{a \text{sech}^2(x)}-x^3 \tanh ^{-1}(\cosh (x)) \cosh (x) \sqrt{a \text{sech}^2(x)}+\left (3 \cosh (x) \sqrt{a \text{sech}^2(x)}\right ) \int x^2 \tanh ^{-1}(\cosh (x)) \, dx-\left (3 \cosh (x) \sqrt{a \text{sech}^2(x)}\right ) \int x^2 \text{sech}(x) \, dx\\ &=x^3 \sqrt{a \text{sech}^2(x)}-6 x^2 \tan ^{-1}\left (e^x\right ) \cosh (x) \sqrt{a \text{sech}^2(x)}+\left (6 i \cosh (x) \sqrt{a \text{sech}^2(x)}\right ) \int x \log \left (1-i e^x\right ) \, dx-\left (6 i \cosh (x) \sqrt{a \text{sech}^2(x)}\right ) \int x \log \left (1+i e^x\right ) \, dx+\left (\cosh (x) \sqrt{a \text{sech}^2(x)}\right ) \int x^3 \text{csch}(x) \, dx\\ &=x^3 \sqrt{a \text{sech}^2(x)}-6 x^2 \tan ^{-1}\left (e^x\right ) \cosh (x) \sqrt{a \text{sech}^2(x)}-2 x^3 \tanh ^{-1}\left (e^x\right ) \cosh (x) \sqrt{a \text{sech}^2(x)}+6 i x \cosh (x) \text{Li}_2\left (-i e^x\right ) \sqrt{a \text{sech}^2(x)}-6 i x \cosh (x) \text{Li}_2\left (i e^x\right ) \sqrt{a \text{sech}^2(x)}-\left (6 i \cosh (x) \sqrt{a \text{sech}^2(x)}\right ) \int \text{Li}_2\left (-i e^x\right ) \, dx+\left (6 i \cosh (x) \sqrt{a \text{sech}^2(x)}\right ) \int \text{Li}_2\left (i e^x\right ) \, dx-\left (3 \cosh (x) \sqrt{a \text{sech}^2(x)}\right ) \int x^2 \log \left (1-e^x\right ) \, dx+\left (3 \cosh (x) \sqrt{a \text{sech}^2(x)}\right ) \int x^2 \log \left (1+e^x\right ) \, dx\\ &=x^3 \sqrt{a \text{sech}^2(x)}-6 x^2 \tan ^{-1}\left (e^x\right ) \cosh (x) \sqrt{a \text{sech}^2(x)}-2 x^3 \tanh ^{-1}\left (e^x\right ) \cosh (x) \sqrt{a \text{sech}^2(x)}-3 x^2 \cosh (x) \text{Li}_2\left (-e^x\right ) \sqrt{a \text{sech}^2(x)}+6 i x \cosh (x) \text{Li}_2\left (-i e^x\right ) \sqrt{a \text{sech}^2(x)}-6 i x \cosh (x) \text{Li}_2\left (i e^x\right ) \sqrt{a \text{sech}^2(x)}+3 x^2 \cosh (x) \text{Li}_2\left (e^x\right ) \sqrt{a \text{sech}^2(x)}-\left (6 i \cosh (x) \sqrt{a \text{sech}^2(x)}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-i x)}{x} \, dx,x,e^x\right )+\left (6 i \cosh (x) \sqrt{a \text{sech}^2(x)}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(i x)}{x} \, dx,x,e^x\right )+\left (6 \cosh (x) \sqrt{a \text{sech}^2(x)}\right ) \int x \text{Li}_2\left (-e^x\right ) \, dx-\left (6 \cosh (x) \sqrt{a \text{sech}^2(x)}\right ) \int x \text{Li}_2\left (e^x\right ) \, dx\\ &=x^3 \sqrt{a \text{sech}^2(x)}-6 x^2 \tan ^{-1}\left (e^x\right ) \cosh (x) \sqrt{a \text{sech}^2(x)}-2 x^3 \tanh ^{-1}\left (e^x\right ) \cosh (x) \sqrt{a \text{sech}^2(x)}-3 x^2 \cosh (x) \text{Li}_2\left (-e^x\right ) \sqrt{a \text{sech}^2(x)}+6 i x \cosh (x) \text{Li}_2\left (-i e^x\right ) \sqrt{a \text{sech}^2(x)}-6 i x \cosh (x) \text{Li}_2\left (i e^x\right ) \sqrt{a \text{sech}^2(x)}+3 x^2 \cosh (x) \text{Li}_2\left (e^x\right ) \sqrt{a \text{sech}^2(x)}+6 x \cosh (x) \text{Li}_3\left (-e^x\right ) \sqrt{a \text{sech}^2(x)}-6 i \cosh (x) \text{Li}_3\left (-i e^x\right ) \sqrt{a \text{sech}^2(x)}+6 i \cosh (x) \text{Li}_3\left (i e^x\right ) \sqrt{a \text{sech}^2(x)}-6 x \cosh (x) \text{Li}_3\left (e^x\right ) \sqrt{a \text{sech}^2(x)}-\left (6 \cosh (x) \sqrt{a \text{sech}^2(x)}\right ) \int \text{Li}_3\left (-e^x\right ) \, dx+\left (6 \cosh (x) \sqrt{a \text{sech}^2(x)}\right ) \int \text{Li}_3\left (e^x\right ) \, dx\\ &=x^3 \sqrt{a \text{sech}^2(x)}-6 x^2 \tan ^{-1}\left (e^x\right ) \cosh (x) \sqrt{a \text{sech}^2(x)}-2 x^3 \tanh ^{-1}\left (e^x\right ) \cosh (x) \sqrt{a \text{sech}^2(x)}-3 x^2 \cosh (x) \text{Li}_2\left (-e^x\right ) \sqrt{a \text{sech}^2(x)}+6 i x \cosh (x) \text{Li}_2\left (-i e^x\right ) \sqrt{a \text{sech}^2(x)}-6 i x \cosh (x) \text{Li}_2\left (i e^x\right ) \sqrt{a \text{sech}^2(x)}+3 x^2 \cosh (x) \text{Li}_2\left (e^x\right ) \sqrt{a \text{sech}^2(x)}+6 x \cosh (x) \text{Li}_3\left (-e^x\right ) \sqrt{a \text{sech}^2(x)}-6 i \cosh (x) \text{Li}_3\left (-i e^x\right ) \sqrt{a \text{sech}^2(x)}+6 i \cosh (x) \text{Li}_3\left (i e^x\right ) \sqrt{a \text{sech}^2(x)}-6 x \cosh (x) \text{Li}_3\left (e^x\right ) \sqrt{a \text{sech}^2(x)}-\left (6 \cosh (x) \sqrt{a \text{sech}^2(x)}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(-x)}{x} \, dx,x,e^x\right )+\left (6 \cosh (x) \sqrt{a \text{sech}^2(x)}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(x)}{x} \, dx,x,e^x\right )\\ &=x^3 \sqrt{a \text{sech}^2(x)}-6 x^2 \tan ^{-1}\left (e^x\right ) \cosh (x) \sqrt{a \text{sech}^2(x)}-2 x^3 \tanh ^{-1}\left (e^x\right ) \cosh (x) \sqrt{a \text{sech}^2(x)}-3 x^2 \cosh (x) \text{Li}_2\left (-e^x\right ) \sqrt{a \text{sech}^2(x)}+6 i x \cosh (x) \text{Li}_2\left (-i e^x\right ) \sqrt{a \text{sech}^2(x)}-6 i x \cosh (x) \text{Li}_2\left (i e^x\right ) \sqrt{a \text{sech}^2(x)}+3 x^2 \cosh (x) \text{Li}_2\left (e^x\right ) \sqrt{a \text{sech}^2(x)}+6 x \cosh (x) \text{Li}_3\left (-e^x\right ) \sqrt{a \text{sech}^2(x)}-6 i \cosh (x) \text{Li}_3\left (-i e^x\right ) \sqrt{a \text{sech}^2(x)}+6 i \cosh (x) \text{Li}_3\left (i e^x\right ) \sqrt{a \text{sech}^2(x)}-6 x \cosh (x) \text{Li}_3\left (e^x\right ) \sqrt{a \text{sech}^2(x)}-6 \cosh (x) \text{Li}_4\left (-e^x\right ) \sqrt{a \text{sech}^2(x)}+6 \cosh (x) \text{Li}_4\left (e^x\right ) \sqrt{a \text{sech}^2(x)}\\ \end{align*}
Mathematica [A] time = 0.631798, size = 249, normalized size = 0.87 \[ \frac{1}{8} \sqrt{a \text{sech}^2(x)} \left (24 x^2 \cosh (x) \text{PolyLog}\left (2,-e^{-x}\right )+24 x^2 \cosh (x) \text{PolyLog}\left (2,e^x\right )+48 i x \cosh (x) \text{PolyLog}\left (2,-i e^{-x}\right )-48 i x \cosh (x) \text{PolyLog}\left (2,i e^{-x}\right )+48 x \cosh (x) \text{PolyLog}\left (3,-e^{-x}\right )-48 x \cosh (x) \text{PolyLog}\left (3,e^x\right )+48 i \cosh (x) \text{PolyLog}\left (3,-i e^{-x}\right )-48 i \cosh (x) \text{PolyLog}\left (3,i e^{-x}\right )+48 \cosh (x) \text{PolyLog}\left (4,-e^{-x}\right )+48 \cosh (x) \text{PolyLog}\left (4,e^x\right )+8 x^3-2 x^4 \cosh (x)-8 x^3 \log \left (e^{-x}+1\right ) \cosh (x)+8 x^3 \log \left (1-e^x\right ) \cosh (x)+24 i x^2 \log \left (1-i e^{-x}\right ) \cosh (x)-24 i x^2 \log \left (1+i e^{-x}\right ) \cosh (x)+\pi ^4 \cosh (x)\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.133, size = 0, normalized size = 0. \begin{align*} \int{x}^{3}{\rm csch} \left (x\right ){\rm sech} \left (x\right )\sqrt{a \left ({\rm sech} \left (x\right ) \right ) ^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{2 \, \sqrt{a} x^{3} e^{x}}{e^{\left (2 \, x\right )} + 1} -{\left (x^{3} \log \left (e^{x} + 1\right ) + 3 \, x^{2}{\rm Li}_2\left (-e^{x}\right ) - 6 \, x{\rm Li}_{3}(-e^{x}) + 6 \,{\rm Li}_{4}(-e^{x})\right )} \sqrt{a} +{\left (x^{3} \log \left (-e^{x} + 1\right ) + 3 \, x^{2}{\rm Li}_2\left (e^{x}\right ) - 6 \, x{\rm Li}_{3}(e^{x}) + 6 \,{\rm Li}_{4}(e^{x})\right )} \sqrt{a} - 12 \, \sqrt{a} \int \frac{x^{2} e^{x}}{2 \,{\left (e^{\left (2 \, x\right )} + 1\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 2.51009, size = 3722, normalized size = 12.97 \begin{align*} \text{result too large to display} \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 x^{3} \sqrt{a \operatorname{sech}^{2}{\left (x \right )}} \operatorname{csch}{\left (x \right )} \operatorname{sech}{\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 \sqrt{a \operatorname{sech}\left (x\right )^{2}} x^{3} \operatorname{csch}\left (x\right ) \operatorname{sech}\left (x\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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