Optimal. Leaf size=88 \[ -\text {Li}_2\left (-e^x\right ) \cosh (x) \sqrt {a \text {sech}^2(x)}+\text {Li}_2\left (e^x\right ) \cosh (x) \sqrt {a \text {sech}^2(x)}+x \sqrt {a \text {sech}^2(x)}-2 x \cosh (x) \tanh ^{-1}\left (e^x\right ) \sqrt {a \text {sech}^2(x)}-\cosh (x) \sqrt {a \text {sech}^2(x)} \tan ^{-1}(\sinh (x)) \]
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Rubi [A] time = 0.36, 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 = {6720, 2622, 321, 207, 5462, 6271, 4182, 2279, 2391, 3770} \[ -\cosh (x) \text {PolyLog}\left (2,-e^x\right ) \sqrt {a \text {sech}^2(x)}+\cosh (x) \text {PolyLog}\left (2,e^x\right ) \sqrt {a \text {sech}^2(x)}+x \sqrt {a \text {sech}^2(x)}-2 x \cosh (x) \tanh ^{-1}\left (e^x\right ) \sqrt {a \text {sech}^2(x)}-\cosh (x) \sqrt {a \text {sech}^2(x)} \tan ^{-1}(\sinh (x)) \]
Antiderivative was successfully verified.
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Rule 207
Rule 321
Rule 2279
Rule 2391
Rule 2622
Rule 3770
Rule 4182
Rule 5462
Rule 6271
Rule 6720
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 ) \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^x\right )+\left (\cosh (x) \sqrt {a \text {sech}^2(x)}\right ) \operatorname {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 = 0.06, size = 74, normalized size = 0.84 \[ \sqrt {a \text {sech}^2(x)} \left (\text {Li}_2\left (-e^{-x}\right ) \cosh (x)-\text {Li}_2\left (e^{-x}\right ) \cosh (x)+x+x \log \left (1-e^{-x}\right ) \cosh (x)-x \log \left (e^{-x}+1\right ) \cosh (x)-2 \cosh (x) \tan ^{-1}\left (\tanh \left (\frac {x}{2}\right )\right )\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.49, size = 351, normalized size = 3.99 \[ \frac {{\left (2 \, x \cosh \relax (x) e^{\left (2 \, x\right )} - 2 \, {\left ({\left (e^{\left (2 \, x\right )} + 1\right )} \sinh \relax (x)^{2} + \cosh \relax (x)^{2} + {\left (\cosh \relax (x)^{2} + 1\right )} e^{\left (2 \, x\right )} + 2 \, {\left (\cosh \relax (x) e^{\left (2 \, x\right )} + \cosh \relax (x)\right )} \sinh \relax (x) + 1\right )} \arctan \left (\cosh \relax (x) + \sinh \relax (x)\right ) + 2 \, x \cosh \relax (x) + {\left ({\left (e^{\left (2 \, x\right )} + 1\right )} \sinh \relax (x)^{2} + \cosh \relax (x)^{2} + {\left (\cosh \relax (x)^{2} + 1\right )} e^{\left (2 \, x\right )} + 2 \, {\left (\cosh \relax (x) e^{\left (2 \, x\right )} + \cosh \relax (x)\right )} \sinh \relax (x) + 1\right )} {\rm Li}_2\left (\cosh \relax (x) + \sinh \relax (x)\right ) - {\left ({\left (e^{\left (2 \, x\right )} + 1\right )} \sinh \relax (x)^{2} + \cosh \relax (x)^{2} + {\left (\cosh \relax (x)^{2} + 1\right )} e^{\left (2 \, x\right )} + 2 \, {\left (\cosh \relax (x) e^{\left (2 \, x\right )} + \cosh \relax (x)\right )} \sinh \relax (x) + 1\right )} {\rm Li}_2\left (-\cosh \relax (x) - \sinh \relax (x)\right ) - {\left (x \cosh \relax (x)^{2} + {\left (x e^{\left (2 \, x\right )} + x\right )} \sinh \relax (x)^{2} + {\left (x \cosh \relax (x)^{2} + x\right )} e^{\left (2 \, x\right )} + 2 \, {\left (x \cosh \relax (x) e^{\left (2 \, x\right )} + x \cosh \relax (x)\right )} \sinh \relax (x) + x\right )} \log \left (\cosh \relax (x) + \sinh \relax (x) + 1\right ) + {\left (x \cosh \relax (x)^{2} + {\left (x e^{\left (2 \, x\right )} + x\right )} \sinh \relax (x)^{2} + {\left (x \cosh \relax (x)^{2} + x\right )} e^{\left (2 \, x\right )} + 2 \, {\left (x \cosh \relax (x) e^{\left (2 \, x\right )} + x \cosh \relax (x)\right )} \sinh \relax (x) + x\right )} \log \left (-\cosh \relax (x) - \sinh \relax (x) + 1\right ) + 2 \, {\left (x e^{\left (2 \, x\right )} + x\right )} \sinh \relax (x)\right )} \sqrt {\frac {a}{e^{\left (4 \, x\right )} + 2 \, e^{\left (2 \, x\right )} + 1}} e^{x}}{2 \, \cosh \relax (x) e^{x} \sinh \relax (x) + e^{x} \sinh \relax (x)^{2} + {\left (\cosh \relax (x)^{2} + 1\right )} e^{x}} \]
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 \[ \int \sqrt {a \operatorname {sech}\relax (x)^{2}} x \operatorname {csch}\relax (x) \operatorname {sech}\relax (x)\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.38, size = 150, normalized size = 1.70 \[ 2 \sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}\, x -2 \sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}\, {\mathrm e}^{-x} \left (1+{\mathrm e}^{2 x}\right ) \arctan \left ({\mathrm e}^{x}\right )-\sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}\, {\mathrm e}^{-x} \left (1+{\mathrm e}^{2 x}\right ) \dilog \left ({\mathrm e}^{x}\right )-\sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}\, {\mathrm e}^{-x} \left (1+{\mathrm e}^{2 x}\right ) \dilog \left ({\mathrm e}^{x}+1\right )-\sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}\, {\mathrm e}^{-x} \left (1+{\mathrm e}^{2 x}\right ) x \ln \left ({\mathrm e}^{x}+1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 60, normalized size = 0.68 \[ -{\left (x \log \left (e^{x} + 1\right ) + {\rm Li}_2\left (-e^{x}\right )\right )} \sqrt {a} + {\left (x \log \left (-e^{x} + 1\right ) + {\rm Li}_2\left (e^{x}\right )\right )} \sqrt {a} - 2 \, \sqrt {a} \arctan \left (e^{x}\right ) + \frac {2 \, \sqrt {a} x e^{x}}{e^{\left (2 \, x\right )} + 1} \]
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 \[ \int \frac {x\,\sqrt {\frac {a}{{\mathrm {cosh}\relax (x)}^2}}}{\mathrm {cosh}\relax (x)\,\mathrm {sinh}\relax (x)} \,d x \]
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 \[ \int x \sqrt {a \operatorname {sech}^{2}{\relax (x )}} \operatorname {csch}{\relax (x )} \operatorname {sech}{\relax (x )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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