Optimal. Leaf size=287 \[ -3 x^2 \text {Li}_2\left (-e^x\right ) \cosh (x) \sqrt {a \text {sech}^2(x)}+3 x^2 \text {Li}_2\left (e^x\right ) \cosh (x) \sqrt {a \text {sech}^2(x)}+6 i x \text {Li}_2\left (-i e^x\right ) \cosh (x) \sqrt {a \text {sech}^2(x)}-6 i x \text {Li}_2\left (i e^x\right ) \cosh (x) \sqrt {a \text {sech}^2(x)}+6 x \text {Li}_3\left (-e^x\right ) \cosh (x) \sqrt {a \text {sech}^2(x)}-6 x \text {Li}_3\left (e^x\right ) \cosh (x) \sqrt {a \text {sech}^2(x)}-6 i \text {Li}_3\left (-i e^x\right ) \cosh (x) \sqrt {a \text {sech}^2(x)}+6 i \text {Li}_3\left (i e^x\right ) \cosh (x) \sqrt {a \text {sech}^2(x)}-6 \text {Li}_4\left (-e^x\right ) \cosh (x) \sqrt {a \text {sech}^2(x)}+6 \text {Li}_4\left (e^x\right ) \cosh (x) \sqrt {a \text {sech}^2(x)}+x^3 \sqrt {a \text {sech}^2(x)}-2 x^3 \cosh (x) \tanh ^{-1}\left (e^x\right ) \sqrt {a \text {sech}^2(x)}-6 x^2 \tan ^{-1}\left (e^x\right ) \cosh (x) \sqrt {a \text {sech}^2(x)} \]
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Rubi [A] time = 0.67, antiderivative size = 287, normalized size of antiderivative = 1.00, 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 14
Rule 207
Rule 321
Rule 2282
Rule 2531
Rule 2622
Rule 4180
Rule 4182
Rule 5462
Rule 6273
Rule 6589
Rule 6609
Rule 6720
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*}
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Mathematica [A] time = 0.66, size = 249, normalized size = 0.87 \[ \frac {1}{8} \sqrt {a \text {sech}^2(x)} \left (24 x^2 \text {Li}_2\left (-e^{-x}\right ) \cosh (x)+24 x^2 \text {Li}_2\left (e^x\right ) \cosh (x)+48 i x \text {Li}_2\left (-i e^{-x}\right ) \cosh (x)-48 i x \text {Li}_2\left (i e^{-x}\right ) \cosh (x)+48 x \text {Li}_3\left (-e^{-x}\right ) \cosh (x)-48 x \text {Li}_3\left (e^x\right ) \cosh (x)+48 i \text {Li}_3\left (-i e^{-x}\right ) \cosh (x)-48 i \text {Li}_3\left (i e^{-x}\right ) \cosh (x)+48 \text {Li}_4\left (-e^{-x}\right ) \cosh (x)+48 \text {Li}_4\left (e^x\right ) \cosh (x)-2 x^4 \cosh (x)+8 x^3-8 x^3 \log \left (e^{-x}+1\right ) \cosh (x)+8 x^3 \log \left (1-e^x\right ) \cosh (x)+24 i x^2 \log \left (1-i e^{-x}\right ) \cosh (x)-24 i x^2 \log \left (1+i e^{-x}\right ) \cosh (x)+\pi ^4 \cosh (x)\right ) \]
Antiderivative was successfully verified.
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fricas [C] time = 0.54, size = 1190, normalized size = 4.15 \[ \text {result too large to display} \]
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 \[ \int \sqrt {a \operatorname {sech}\relax (x)^{2}} x^{3} \operatorname {csch}\relax (x) \operatorname {sech}\relax (x)\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.53, size = 0, normalized size = 0.00 \[ \int x^{3} \mathrm {csch}\relax (x ) \mathrm {sech}\relax (x ) \sqrt {a \mathrm {sech}\relax (x )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {2 \, \sqrt {a} x^{3} e^{x}}{e^{\left (2 \, x\right )} + 1} - {\left (x^{3} \log \left (e^{x} + 1\right ) + 3 \, x^{2} {\rm Li}_2\left (-e^{x}\right ) - 6 \, x {\rm Li}_{3}(-e^{x}) + 6 \, {\rm Li}_{4}(-e^{x})\right )} \sqrt {a} + {\left (x^{3} \log \left (-e^{x} + 1\right ) + 3 \, x^{2} {\rm Li}_2\left (e^{x}\right ) - 6 \, x {\rm Li}_{3}(e^{x}) + 6 \, {\rm Li}_{4}(e^{x})\right )} \sqrt {a} - 12 \, \sqrt {a} \int \frac {x^{2} e^{x}}{2 \, {\left (e^{\left (2 \, x\right )} + 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^3\,\sqrt {\frac {a}{{\mathrm {cosh}\relax (x)}^2}}}{\mathrm {cosh}\relax (x)\,\mathrm {sinh}\relax (x)} \,d x \]
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 \[ \int x^{3} \sqrt {a \operatorname {sech}^{2}{\relax (x )}} \operatorname {csch}{\relax (x )} \operatorname {sech}{\relax (x )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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