Optimal. Leaf size=132 \[ \frac {1}{2} x \cosh ^2(x) \sqrt {a \text {sech}^4(x)}-2 x \tanh ^{-1}\left (e^{2 x}\right ) \cosh ^2(x) \sqrt {a \text {sech}^4(x)}-\frac {1}{2} \cosh ^2(x) \text {PolyLog}\left (2,-e^{2 x}\right ) \sqrt {a \text {sech}^4(x)}+\frac {1}{2} \cosh ^2(x) \text {PolyLog}\left (2,e^{2 x}\right ) \sqrt {a \text {sech}^4(x)}-\frac {1}{2} \cosh (x) \sqrt {a \text {sech}^4(x)} \sinh (x)-\frac {1}{2} x \sqrt {a \text {sech}^4(x)} \sinh ^2(x) \]
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Rubi [A]
time = 0.31, antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 11, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.688, Rules used = {6852, 2700,
14, 5570, 2628, 5569, 4267, 2317, 2438, 3554, 8} \begin {gather*} -\frac {1}{2} \text {Li}_2\left (-e^{2 x}\right ) \cosh ^2(x) \sqrt {a \text {sech}^4(x)}+\frac {1}{2} \text {Li}_2\left (e^{2 x}\right ) \cosh ^2(x) \sqrt {a \text {sech}^4(x)}+\frac {1}{2} x \cosh ^2(x) \sqrt {a \text {sech}^4(x)}-\frac {1}{2} x \sinh ^2(x) \sqrt {a \text {sech}^4(x)}-2 x \cosh ^2(x) \tanh ^{-1}\left (e^{2 x}\right ) \sqrt {a \text {sech}^4(x)}-\frac {1}{2} \sinh (x) \cosh (x) \sqrt {a \text {sech}^4(x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 14
Rule 2317
Rule 2438
Rule 2628
Rule 2700
Rule 3554
Rule 4267
Rule 5569
Rule 5570
Rule 6852
Rubi steps
\begin {align*} \int x \text {csch}(x) \text {sech}(x) \sqrt {a \text {sech}^4(x)} \, dx &=\left (\cosh ^2(x) \sqrt {a \text {sech}^4(x)}\right ) \int x \text {csch}(x) \text {sech}^3(x) \, dx\\ &=x \cosh ^2(x) \log (\tanh (x)) \sqrt {a \text {sech}^4(x)}-\frac {1}{2} x \sqrt {a \text {sech}^4(x)} \sinh ^2(x)-\left (\cosh ^2(x) \sqrt {a \text {sech}^4(x)}\right ) \int \left (\log (\tanh (x))-\frac {\tanh ^2(x)}{2}\right ) \, dx\\ &=x \cosh ^2(x) \log (\tanh (x)) \sqrt {a \text {sech}^4(x)}-\frac {1}{2} x \sqrt {a \text {sech}^4(x)} \sinh ^2(x)+\frac {1}{2} \left (\cosh ^2(x) \sqrt {a \text {sech}^4(x)}\right ) \int \tanh ^2(x) \, dx-\left (\cosh ^2(x) \sqrt {a \text {sech}^4(x)}\right ) \int \log (\tanh (x)) \, dx\\ &=-\frac {1}{2} \cosh (x) \sqrt {a \text {sech}^4(x)} \sinh (x)-\frac {1}{2} x \sqrt {a \text {sech}^4(x)} \sinh ^2(x)+\frac {1}{2} \left (\cosh ^2(x) \sqrt {a \text {sech}^4(x)}\right ) \int 1 \, dx+\left (\cosh ^2(x) \sqrt {a \text {sech}^4(x)}\right ) \int x \text {csch}(x) \text {sech}(x) \, dx\\ &=\frac {1}{2} x \cosh ^2(x) \sqrt {a \text {sech}^4(x)}-\frac {1}{2} \cosh (x) \sqrt {a \text {sech}^4(x)} \sinh (x)-\frac {1}{2} x \sqrt {a \text {sech}^4(x)} \sinh ^2(x)+\left (2 \cosh ^2(x) \sqrt {a \text {sech}^4(x)}\right ) \int x \text {csch}(2 x) \, dx\\ &=\frac {1}{2} x \cosh ^2(x) \sqrt {a \text {sech}^4(x)}-2 x \tanh ^{-1}\left (e^{2 x}\right ) \cosh ^2(x) \sqrt {a \text {sech}^4(x)}-\frac {1}{2} \cosh (x) \sqrt {a \text {sech}^4(x)} \sinh (x)-\frac {1}{2} x \sqrt {a \text {sech}^4(x)} \sinh ^2(x)-\left (\cosh ^2(x) \sqrt {a \text {sech}^4(x)}\right ) \int \log \left (1-e^{2 x}\right ) \, dx+\left (\cosh ^2(x) \sqrt {a \text {sech}^4(x)}\right ) \int \log \left (1+e^{2 x}\right ) \, dx\\ &=\frac {1}{2} x \cosh ^2(x) \sqrt {a \text {sech}^4(x)}-2 x \tanh ^{-1}\left (e^{2 x}\right ) \cosh ^2(x) \sqrt {a \text {sech}^4(x)}-\frac {1}{2} \cosh (x) \sqrt {a \text {sech}^4(x)} \sinh (x)-\frac {1}{2} x \sqrt {a \text {sech}^4(x)} \sinh ^2(x)-\frac {1}{2} \left (\cosh ^2(x) \sqrt {a \text {sech}^4(x)}\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 x}\right )+\frac {1}{2} \left (\cosh ^2(x) \sqrt {a \text {sech}^4(x)}\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 x}\right )\\ &=\frac {1}{2} x \cosh ^2(x) \sqrt {a \text {sech}^4(x)}-2 x \tanh ^{-1}\left (e^{2 x}\right ) \cosh ^2(x) \sqrt {a \text {sech}^4(x)}-\frac {1}{2} \cosh ^2(x) \text {Li}_2\left (-e^{2 x}\right ) \sqrt {a \text {sech}^4(x)}+\frac {1}{2} \cosh ^2(x) \text {Li}_2\left (e^{2 x}\right ) \sqrt {a \text {sech}^4(x)}-\frac {1}{2} \cosh (x) \sqrt {a \text {sech}^4(x)} \sinh (x)-\frac {1}{2} x \sqrt {a \text {sech}^4(x)} \sinh ^2(x)\\ \end {align*}
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Mathematica [A]
time = 0.15, size = 71, normalized size = 0.54 \begin {gather*} \frac {1}{2} \cosh ^2(x) \sqrt {a \text {sech}^4(x)} \left (2 x \log \left (1-e^{-2 x}\right )-2 x \log \left (1+e^{-2 x}\right )+\text {PolyLog}\left (2,-e^{-2 x}\right )-\text {PolyLog}\left (2,e^{-2 x}\right )+x \text {sech}^2(x)-\tanh (x)\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(251\) vs.
\(2(107)=214\).
time = 1.48, size = 252, normalized size = 1.91
method | result | size |
risch | \(\sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left (1+{\mathrm e}^{2 x}\right )^{4}}}\, {\mathrm e}^{-2 x} \left (2 x \,{\mathrm e}^{2 x}+{\mathrm e}^{2 x}+1\right )+\sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left (1+{\mathrm e}^{2 x}\right )^{4}}}\, {\mathrm e}^{-2 x} \left (1+{\mathrm e}^{2 x}\right )^{2} x \ln \left ({\mathrm e}^{x}+1\right )+\sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left (1+{\mathrm e}^{2 x}\right )^{4}}}\, {\mathrm e}^{-2 x} \left (1+{\mathrm e}^{2 x}\right )^{2} \polylog \left (2, -{\mathrm e}^{x}\right )-\sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left (1+{\mathrm e}^{2 x}\right )^{4}}}\, {\mathrm e}^{-2 x} \left (1+{\mathrm e}^{2 x}\right )^{2} x \ln \left (1+{\mathrm e}^{2 x}\right )-\frac {\sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left (1+{\mathrm e}^{2 x}\right )^{4}}}\, {\mathrm e}^{-2 x} \left (1+{\mathrm e}^{2 x}\right )^{2} \polylog \left (2, -{\mathrm e}^{2 x}\right )}{2}+\sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left (1+{\mathrm e}^{2 x}\right )^{4}}}\, {\mathrm e}^{-2 x} \left (1+{\mathrm e}^{2 x}\right )^{2} x \ln \left (1-{\mathrm e}^{x}\right )+\sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left (1+{\mathrm e}^{2 x}\right )^{4}}}\, {\mathrm e}^{-2 x} \left (1+{\mathrm e}^{2 x}\right )^{2} \polylog \left (2, {\mathrm e}^{x}\right )\) | \(252\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 92, normalized size = 0.70 \begin {gather*} -\frac {1}{2} \, {\left (2 \, x \log \left (e^{\left (2 \, x\right )} + 1\right ) + {\rm Li}_2\left (-e^{\left (2 \, x\right )}\right )\right )} \sqrt {a} + {\left (x \log \left (e^{x} + 1\right ) + {\rm Li}_2\left (-e^{x}\right )\right )} \sqrt {a} + {\left (x \log \left (-e^{x} + 1\right ) + {\rm Li}_2\left (e^{x}\right )\right )} \sqrt {a} + \frac {{\left (2 \, \sqrt {a} x + \sqrt {a}\right )} e^{\left (2 \, x\right )} + \sqrt {a}}{e^{\left (4 \, x\right )} + 2 \, e^{\left (2 \, x\right )} + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains complex when optimal does not.
time = 0.50, size = 1757, normalized size = 13.31 \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 \sqrt {a \operatorname {sech}^{4}{\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 = -0.01 \begin {gather*} \int \frac {x\,\sqrt {\frac {a}{{\mathrm {cosh}\left (x\right )}^4}}}{\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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