Optimal. Leaf size=88 \[ x \sqrt {a \text {sech}^2(x)}-\text {ArcTan}(\sinh (x)) \cosh (x) \sqrt {a \text {sech}^2(x)}-2 x \tanh ^{-1}\left (e^x\right ) \cosh (x) \sqrt {a \text {sech}^2(x)}-\cosh (x) \text {PolyLog}\left (2,-e^x\right ) \sqrt {a \text {sech}^2(x)}+\cosh (x) \text {PolyLog}\left (2,e^x\right ) \sqrt {a \text {sech}^2(x)} \]
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Rubi [A]
time = 0.24, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 10, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {6852, 2702,
327, 213, 5570, 6406, 4267, 2317, 2438, 3855} \begin {gather*} -\cosh (x) \sqrt {a \text {sech}^2(x)} \text {ArcTan}(\sinh (x))-\text {Li}_2\left (-e^x\right ) \cosh (x) \sqrt {a \text {sech}^2(x)}+\text {Li}_2\left (e^x\right ) \cosh (x) \sqrt {a \text {sech}^2(x)}+x \sqrt {a \text {sech}^2(x)}-2 x \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 213
Rule 327
Rule 2317
Rule 2438
Rule 2702
Rule 3855
Rule 4267
Rule 5570
Rule 6406
Rule 6852
Rubi steps
\begin {align*} \int x \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 \text {csch}(x) \text {sech}^2(x) \, dx\\ &=x \sqrt {a \text {sech}^2(x)}-x \tanh ^{-1}(\cosh (x)) \cosh (x) \sqrt {a \text {sech}^2(x)}-\left (\cosh (x) \sqrt {a \text {sech}^2(x)}\right ) \int \left (-\tanh ^{-1}(\cosh (x))+\text {sech}(x)\right ) \, dx\\ &=x \sqrt {a \text {sech}^2(x)}-x \tanh ^{-1}(\cosh (x)) \cosh (x) \sqrt {a \text {sech}^2(x)}+\left (\cosh (x) \sqrt {a \text {sech}^2(x)}\right ) \int \tanh ^{-1}(\cosh (x)) \, dx-\left (\cosh (x) \sqrt {a \text {sech}^2(x)}\right ) \int \text {sech}(x) \, dx\\ &=x \sqrt {a \text {sech}^2(x)}-\tan ^{-1}(\sinh (x)) \cosh (x) \sqrt {a \text {sech}^2(x)}+\left (\cosh (x) \sqrt {a \text {sech}^2(x)}\right ) \int x \text {csch}(x) \, dx\\ &=x \sqrt {a \text {sech}^2(x)}-\tan ^{-1}(\sinh (x)) \cosh (x) \sqrt {a \text {sech}^2(x)}-2 x \tanh ^{-1}\left (e^x\right ) \cosh (x) \sqrt {a \text {sech}^2(x)}-\left (\cosh (x) \sqrt {a \text {sech}^2(x)}\right ) \int \log \left (1-e^x\right ) \, dx+\left (\cosh (x) \sqrt {a \text {sech}^2(x)}\right ) \int \log \left (1+e^x\right ) \, dx\\ &=x \sqrt {a \text {sech}^2(x)}-\tan ^{-1}(\sinh (x)) \cosh (x) \sqrt {a \text {sech}^2(x)}-2 x \tanh ^{-1}\left (e^x\right ) \cosh (x) \sqrt {a \text {sech}^2(x)}-\left (\cosh (x) \sqrt {a \text {sech}^2(x)}\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^x\right )+\left (\cosh (x) \sqrt {a \text {sech}^2(x)}\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^x\right )\\ &=x \sqrt {a \text {sech}^2(x)}-\tan ^{-1}(\sinh (x)) \cosh (x) \sqrt {a \text {sech}^2(x)}-2 x \tanh ^{-1}\left (e^x\right ) \cosh (x) \sqrt {a \text {sech}^2(x)}-\cosh (x) \text {Li}_2\left (-e^x\right ) \sqrt {a \text {sech}^2(x)}+\cosh (x) \text {Li}_2\left (e^x\right ) \sqrt {a \text {sech}^2(x)}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 74, normalized size = 0.84 \begin {gather*} \left (x-2 \text {ArcTan}\left (\tanh \left (\frac {x}{2}\right )\right ) \cosh (x)+x \cosh (x) \log \left (1-e^{-x}\right )-x \cosh (x) \log \left (1+e^{-x}\right )+\cosh (x) \text {PolyLog}\left (2,-e^{-x}\right )-\cosh (x) \text {PolyLog}\left (2,e^{-x}\right )\right ) \sqrt {a \text {sech}^2(x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.50, size = 150, normalized size = 1.70
method | result | size |
risch | \(2 \sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}\, x -2 \sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}\, {\mathrm e}^{-x} \left (1+{\mathrm e}^{2 x}\right ) \arctan \left ({\mathrm e}^{x}\right )-\sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}\, {\mathrm e}^{-x} \left (1+{\mathrm e}^{2 x}\right ) \dilog \left ({\mathrm e}^{x}\right )-\sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}\, {\mathrm e}^{-x} \left (1+{\mathrm e}^{2 x}\right ) \dilog \left ({\mathrm e}^{x}+1\right )-\sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}\, {\mathrm e}^{-x} \left (1+{\mathrm e}^{2 x}\right ) x \ln \left ({\mathrm e}^{x}+1\right )\) | \(150\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 60, normalized size = 0.68 \begin {gather*} -{\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} - 2 \, \sqrt {a} \arctan \left (e^{x}\right ) + \frac {2 \, \sqrt {a} x e^{x}}{e^{\left (2 \, x\right )} + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 351 vs.
\(2 (73) = 146\).
time = 0.36, size = 351, normalized size = 3.99 \begin {gather*} \frac {{\left (2 \, x \cosh \left (x\right ) e^{\left (2 \, x\right )} - 2 \, {\left ({\left (e^{\left (2 \, x\right )} + 1\right )} \sinh \left (x\right )^{2} + \cosh \left (x\right )^{2} + {\left (\cosh \left (x\right )^{2} + 1\right )} e^{\left (2 \, x\right )} + 2 \, {\left (\cosh \left (x\right ) e^{\left (2 \, x\right )} + \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1\right )} \arctan \left (\cosh \left (x\right ) + \sinh \left (x\right )\right ) + 2 \, x \cosh \left (x\right ) + {\left ({\left (e^{\left (2 \, x\right )} + 1\right )} \sinh \left (x\right )^{2} + \cosh \left (x\right )^{2} + {\left (\cosh \left (x\right )^{2} + 1\right )} e^{\left (2 \, x\right )} + 2 \, {\left (\cosh \left (x\right ) e^{\left (2 \, x\right )} + \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1\right )} {\rm Li}_2\left (\cosh \left (x\right ) + \sinh \left (x\right )\right ) - {\left ({\left (e^{\left (2 \, x\right )} + 1\right )} \sinh \left (x\right )^{2} + \cosh \left (x\right )^{2} + {\left (\cosh \left (x\right )^{2} + 1\right )} e^{\left (2 \, x\right )} + 2 \, {\left (\cosh \left (x\right ) e^{\left (2 \, x\right )} + \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1\right )} {\rm Li}_2\left (-\cosh \left (x\right ) - \sinh \left (x\right )\right ) - {\left (x \cosh \left (x\right )^{2} + {\left (x e^{\left (2 \, x\right )} + x\right )} \sinh \left (x\right )^{2} + {\left (x \cosh \left (x\right )^{2} + x\right )} e^{\left (2 \, x\right )} + 2 \, {\left (x \cosh \left (x\right ) e^{\left (2 \, x\right )} + x \cosh \left (x\right )\right )} \sinh \left (x\right ) + x\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) + {\left (x \cosh \left (x\right )^{2} + {\left (x e^{\left (2 \, x\right )} + x\right )} \sinh \left (x\right )^{2} + {\left (x \cosh \left (x\right )^{2} + x\right )} e^{\left (2 \, x\right )} + 2 \, {\left (x \cosh \left (x\right ) e^{\left (2 \, x\right )} + x \cosh \left (x\right )\right )} \sinh \left (x\right ) + x\right )} \log \left (-\cosh \left (x\right ) - \sinh \left (x\right ) + 1\right ) + 2 \, {\left (x e^{\left (2 \, x\right )} + x\right )} \sinh \left (x\right )\right )} \sqrt {\frac {a}{e^{\left (4 \, x\right )} + 2 \, e^{\left (2 \, x\right )} + 1}} e^{x}}{2 \, \cosh \left (x\right ) e^{x} \sinh \left (x\right ) + e^{x} \sinh \left (x\right )^{2} + {\left (\cosh \left (x\right )^{2} + 1\right )} e^{x}} \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}^{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 = -0.01 \begin {gather*} \int \frac {x\,\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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