Optimal. Leaf size=67 \[ -\frac {\text {Li}_2\left (-e^{a+b x}\right )}{b^2}+\frac {\text {Li}_2\left (e^{a+b x}\right )}{b^2}-\frac {\tan ^{-1}(\sinh (a+b x))}{b^2}-\frac {2 x \tanh ^{-1}\left (e^{a+b x}\right )}{b}+\frac {x \text {sech}(a+b x)}{b} \]
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Rubi [A] time = 0.11, antiderivative size = 67, 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 = {2622, 321, 207, 5462, 6271, 12, 4182, 2279, 2391, 3770} \[ -\frac {\text {PolyLog}\left (2,-e^{a+b x}\right )}{b^2}+\frac {\text {PolyLog}\left (2,e^{a+b x}\right )}{b^2}-\frac {\tan ^{-1}(\sinh (a+b x))}{b^2}-\frac {2 x \tanh ^{-1}\left (e^{a+b x}\right )}{b}+\frac {x \text {sech}(a+b x)}{b} \]
Antiderivative was successfully verified.
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Rule 12
Rule 207
Rule 321
Rule 2279
Rule 2391
Rule 2622
Rule 3770
Rule 4182
Rule 5462
Rule 6271
Rubi steps
\begin {align*} \int x \text {csch}(a+b x) \text {sech}^2(a+b x) \, dx &=-\frac {x \tanh ^{-1}(\cosh (a+b x))}{b}+\frac {x \text {sech}(a+b x)}{b}-\int \left (-\frac {\tanh ^{-1}(\cosh (a+b x))}{b}+\frac {\text {sech}(a+b x)}{b}\right ) \, dx\\ &=-\frac {x \tanh ^{-1}(\cosh (a+b x))}{b}+\frac {x \text {sech}(a+b x)}{b}+\frac {\int \tanh ^{-1}(\cosh (a+b x)) \, dx}{b}-\frac {\int \text {sech}(a+b x) \, dx}{b}\\ &=-\frac {\tan ^{-1}(\sinh (a+b x))}{b^2}+\frac {x \text {sech}(a+b x)}{b}+\frac {\int b x \text {csch}(a+b x) \, dx}{b}\\ &=-\frac {\tan ^{-1}(\sinh (a+b x))}{b^2}+\frac {x \text {sech}(a+b x)}{b}+\int x \text {csch}(a+b x) \, dx\\ &=-\frac {\tan ^{-1}(\sinh (a+b x))}{b^2}-\frac {2 x \tanh ^{-1}\left (e^{a+b x}\right )}{b}+\frac {x \text {sech}(a+b x)}{b}-\frac {\int \log \left (1-e^{a+b x}\right ) \, dx}{b}+\frac {\int \log \left (1+e^{a+b x}\right ) \, dx}{b}\\ &=-\frac {\tan ^{-1}(\sinh (a+b x))}{b^2}-\frac {2 x \tanh ^{-1}\left (e^{a+b x}\right )}{b}+\frac {x \text {sech}(a+b x)}{b}-\frac {\operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{a+b x}\right )}{b^2}+\frac {\operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{a+b x}\right )}{b^2}\\ &=-\frac {\tan ^{-1}(\sinh (a+b x))}{b^2}-\frac {2 x \tanh ^{-1}\left (e^{a+b x}\right )}{b}-\frac {\text {Li}_2\left (-e^{a+b x}\right )}{b^2}+\frac {\text {Li}_2\left (e^{a+b x}\right )}{b^2}+\frac {x \text {sech}(a+b x)}{b}\\ \end {align*}
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Mathematica [A] time = 0.29, size = 106, normalized size = 1.58 \[ \frac {\text {Li}_2\left (-e^{-a-b x}\right )-\text {Li}_2\left (e^{-a-b x}\right )+(a+b x) \left (\log \left (1-e^{-a-b x}\right )-\log \left (e^{-a-b x}+1\right )\right )+b x \text {sech}(a+b x)-a \log \left (\tanh \left (\frac {1}{2} (a+b x)\right )\right )-2 \tan ^{-1}\left (\tanh \left (\frac {1}{2} (a+b x)\right )\right )}{b^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.41, size = 401, normalized size = 5.99 \[ \frac {2 \, b x \cosh \left (b x + a\right ) + 2 \, b x \sinh \left (b x + a\right ) - 2 \, {\left (\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} + 1\right )} \arctan \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) + {\left (\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} + 1\right )} {\rm Li}_2\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) - {\left (\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} + 1\right )} {\rm Li}_2\left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )\right ) - {\left (b x \cosh \left (b x + a\right )^{2} + 2 \, b x \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b x \sinh \left (b x + a\right )^{2} + b x\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1\right ) - {\left (a \cosh \left (b x + a\right )^{2} + 2 \, a \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + a \sinh \left (b x + a\right )^{2} + a\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1\right ) + {\left ({\left (b x + a\right )} \cosh \left (b x + a\right )^{2} + 2 \, {\left (b x + a\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + {\left (b x + a\right )} \sinh \left (b x + a\right )^{2} + b x + a\right )} \log \left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right ) + 1\right )}{b^{2} \cosh \left (b x + a\right )^{2} + 2 \, b^{2} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b^{2} \sinh \left (b x + a\right )^{2} + b^{2}} \]
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 \[ \int x \operatorname {csch}\left (b x + a\right ) \operatorname {sech}\left (b x + a\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.52, size = 95, normalized size = 1.42 \[ \frac {2 x \,{\mathrm e}^{b x +a}}{b \left (1+{\mathrm e}^{2 b x +2 a}\right )}-\frac {2 \arctan \left ({\mathrm e}^{b x +a}\right )}{b^{2}}-\frac {\dilog \left (1+{\mathrm e}^{b x +a}\right )}{b^{2}}-\frac {\ln \left (1+{\mathrm e}^{b x +a}\right ) x}{b}-\frac {\dilog \left ({\mathrm e}^{b x +a}\right )}{b^{2}}-\frac {a \ln \left ({\mathrm e}^{b x +a}-1\right )}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.46, size = 90, normalized size = 1.34 \[ \frac {2 \, x e^{\left (b x + a\right )}}{b e^{\left (2 \, b x + 2 \, a\right )} + b} - \frac {b x \log \left (e^{\left (b x + a\right )} + 1\right ) + {\rm Li}_2\left (-e^{\left (b x + a\right )}\right )}{b^{2}} + \frac {b x \log \left (-e^{\left (b x + a\right )} + 1\right ) + {\rm Li}_2\left (e^{\left (b x + a\right )}\right )}{b^{2}} - \frac {2 \, \arctan \left (e^{\left (b x + a\right )}\right )}{b^{2}} \]
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 \[ \int \frac {x}{{\mathrm {cosh}\left (a+b\,x\right )}^2\,\mathrm {sinh}\left (a+b\,x\right )} \,d x \]
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 \[ \int x \operatorname {csch}{\left (a + b x \right )} \operatorname {sech}^{2}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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