Optimal. Leaf size=287 \[ x^3 \sqrt {a \text {sech}^2(x)}-6 x^2 \text {ArcTan}\left (e^x\right ) \cosh (x) \sqrt {a \text {sech}^2(x)}-2 x^3 \tanh ^{-1}\left (e^x\right ) \cosh (x) \sqrt {a \text {sech}^2(x)}-3 x^2 \cosh (x) \text {PolyLog}\left (2,-e^x\right ) \sqrt {a \text {sech}^2(x)}+6 i x \cosh (x) \text {PolyLog}\left (2,-i e^x\right ) \sqrt {a \text {sech}^2(x)}-6 i x \cosh (x) \text {PolyLog}\left (2,i e^x\right ) \sqrt {a \text {sech}^2(x)}+3 x^2 \cosh (x) \text {PolyLog}\left (2,e^x\right ) \sqrt {a \text {sech}^2(x)}+6 x \cosh (x) \text {PolyLog}\left (3,-e^x\right ) \sqrt {a \text {sech}^2(x)}-6 i \cosh (x) \text {PolyLog}\left (3,-i e^x\right ) \sqrt {a \text {sech}^2(x)}+6 i \cosh (x) \text {PolyLog}\left (3,i e^x\right ) \sqrt {a \text {sech}^2(x)}-6 x \cosh (x) \text {PolyLog}\left (3,e^x\right ) \sqrt {a \text {sech}^2(x)}-6 \cosh (x) \text {PolyLog}\left (4,-e^x\right ) \sqrt {a \text {sech}^2(x)}+6 \cosh (x) \text {PolyLog}\left (4,e^x\right ) \sqrt {a \text {sech}^2(x)} \]
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Rubi [A]
time = 0.44, 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 = {6852, 2702, 327, 213, 5570, 14, 6408, 4267, 2611, 6744, 2320, 6724, 4265}
\begin {gather*} -6 x^2 \text {ArcTan}\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)}+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)} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 213
Rule 327
Rule 2320
Rule 2611
Rule 2702
Rule 4265
Rule 4267
Rule 5570
Rule 6408
Rule 6724
Rule 6744
Rule 6852
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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 = 0.40, size = 249, normalized size = 0.87 \begin {gather*} \frac {1}{8} \left (8 x^3+\pi ^4 \cosh (x)-2 x^4 \cosh (x)+24 i x^2 \cosh (x) \log \left (1-i e^{-x}\right )-24 i x^2 \cosh (x) \log \left (1+i e^{-x}\right )-8 x^3 \cosh (x) \log \left (1+e^{-x}\right )+8 x^3 \cosh (x) \log \left (1-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 )+24 x^2 \cosh (x) \text {PolyLog}\left (2,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 x \cosh (x) \text {PolyLog}\left (3,e^x\right )+48 \cosh (x) \text {PolyLog}\left (4,-e^{-x}\right )+48 \cosh (x) \text {PolyLog}\left (4,e^x\right )\right ) \sqrt {a \text {sech}^2(x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 1.47, size = 0, normalized size = 0.00 \[\int x^{3} \mathrm {csch}\left (x \right ) \mathrm {sech}\left (x \right ) \sqrt {a \mathrm {sech}\left (x \right )^{2}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 1202 vs. \(2 (229) = 458\).
time = 0.41, size = 1202, normalized size = 4.19 \begin {gather*} \text {Too large to display} \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 x^{3} \sqrt {a \operatorname {sech}^{2}{\left (x \right )}} \operatorname {csch}{\left (x \right )} \operatorname {sech}{\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 = -0.00 \begin {gather*} \int \frac {x^3\,\sqrt {\frac {a}{{\mathrm {cosh}\left (x\right )}^2}}}{\mathrm {cosh}\left (x\right )\,\mathrm {sinh}\left (x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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