Optimal. Leaf size=79 \[ \frac{i \text{PolyLog}\left (2,-i e^{a+b x}\right )}{b^2}-\frac{i \text{PolyLog}\left (2,i e^{a+b x}\right )}{b^2}-\frac{\tanh ^{-1}(\cosh (a+b x))}{b^2}-\frac{2 x \tan ^{-1}\left (e^{a+b x}\right )}{b}-\frac{x \text{csch}(a+b x)}{b} \]
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Rubi [A] time = 0.11245, antiderivative size = 79, normalized size of antiderivative = 1., 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 = {2621, 321, 207, 5462, 5203, 12, 4180, 2279, 2391, 3770} \[ \frac{i \text{PolyLog}\left (2,-i e^{a+b x}\right )}{b^2}-\frac{i \text{PolyLog}\left (2,i e^{a+b x}\right )}{b^2}-\frac{\tanh ^{-1}(\cosh (a+b x))}{b^2}-\frac{2 x \tan ^{-1}\left (e^{a+b x}\right )}{b}-\frac{x \text{csch}(a+b x)}{b} \]
Antiderivative was successfully verified.
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Rule 2621
Rule 321
Rule 207
Rule 5462
Rule 5203
Rule 12
Rule 4180
Rule 2279
Rule 2391
Rule 3770
Rubi steps
\begin{align*} \int x \text{csch}^2(a+b x) \text{sech}(a+b x) \, dx &=-\frac{x \tan ^{-1}(\sinh (a+b x))}{b}-\frac{x \text{csch}(a+b x)}{b}-\int \left (-\frac{\tan ^{-1}(\sinh (a+b x))}{b}-\frac{\text{csch}(a+b x)}{b}\right ) \, dx\\ &=-\frac{x \tan ^{-1}(\sinh (a+b x))}{b}-\frac{x \text{csch}(a+b x)}{b}+\frac{\int \tan ^{-1}(\sinh (a+b x)) \, dx}{b}+\frac{\int \text{csch}(a+b x) \, dx}{b}\\ &=-\frac{\tanh ^{-1}(\cosh (a+b x))}{b^2}-\frac{x \text{csch}(a+b x)}{b}-\frac{\int b x \text{sech}(a+b x) \, dx}{b}\\ &=-\frac{\tanh ^{-1}(\cosh (a+b x))}{b^2}-\frac{x \text{csch}(a+b x)}{b}-\int x \text{sech}(a+b x) \, dx\\ &=-\frac{2 x \tan ^{-1}\left (e^{a+b x}\right )}{b}-\frac{\tanh ^{-1}(\cosh (a+b x))}{b^2}-\frac{x \text{csch}(a+b x)}{b}+\frac{i \int \log \left (1-i e^{a+b x}\right ) \, dx}{b}-\frac{i \int \log \left (1+i e^{a+b x}\right ) \, dx}{b}\\ &=-\frac{2 x \tan ^{-1}\left (e^{a+b x}\right )}{b}-\frac{\tanh ^{-1}(\cosh (a+b x))}{b^2}-\frac{x \text{csch}(a+b x)}{b}+\frac{i \operatorname{Subst}\left (\int \frac{\log (1-i x)}{x} \, dx,x,e^{a+b x}\right )}{b^2}-\frac{i \operatorname{Subst}\left (\int \frac{\log (1+i x)}{x} \, dx,x,e^{a+b x}\right )}{b^2}\\ &=-\frac{2 x \tan ^{-1}\left (e^{a+b x}\right )}{b}-\frac{\tanh ^{-1}(\cosh (a+b x))}{b^2}-\frac{x \text{csch}(a+b x)}{b}+\frac{i \text{Li}_2\left (-i e^{a+b x}\right )}{b^2}-\frac{i \text{Li}_2\left (i e^{a+b x}\right )}{b^2}\\ \end{align*}
Mathematica [A] time = 0.805017, size = 112, normalized size = 1.42 \[ \frac{2 i \text{PolyLog}(2,-i (\sinh (a+b x)+\cosh (a+b x)))-2 i \text{PolyLog}(2,i (\sinh (a+b x)+\cosh (a+b x)))+b x \tanh \left (\frac{1}{2} (a+b x)\right )-b x \coth \left (\frac{1}{2} (a+b x)\right )+2 \log \left (\tanh \left (\frac{1}{2} (a+b x)\right )\right )-4 b x \tan ^{-1}(\sinh (a+b x)+\cosh (a+b x))}{2 b^2} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.041, size = 179, normalized size = 2.3 \begin{align*} -2\,{\frac{x{{\rm e}^{bx+a}}}{b \left ({{\rm e}^{2\,bx+2\,a}}-1 \right ) }}+2\,{\frac{a\arctan \left ({{\rm e}^{bx+a}} \right ) }{{b}^{2}}}-{\frac{i{\it dilog} \left ( 1-i{{\rm e}^{bx+a}} \right ) }{{b}^{2}}}+{\frac{i{\it dilog} \left ( 1+i{{\rm e}^{bx+a}} \right ) }{{b}^{2}}}+{\frac{i\ln \left ( 1+i{{\rm e}^{bx+a}} \right ) x}{b}}+{\frac{i\ln \left ( 1+i{{\rm e}^{bx+a}} \right ) a}{{b}^{2}}}-{\frac{i\ln \left ( 1-i{{\rm e}^{bx+a}} \right ) x}{b}}-{\frac{i\ln \left ( 1-i{{\rm e}^{bx+a}} \right ) a}{{b}^{2}}}+{\frac{\ln \left ({{\rm e}^{bx+a}}-1 \right ) }{{b}^{2}}}-{\frac{\ln \left ( 1+{{\rm e}^{bx+a}} \right ) }{{b}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{2 \, x e^{\left (b x + a\right )}}{b e^{\left (2 \, b x + 2 \, a\right )} - b} - \frac{\log \left ({\left (e^{\left (b x + a\right )} + 1\right )} e^{\left (-a\right )}\right )}{b^{2}} + \frac{\log \left ({\left (e^{\left (b x + a\right )} - 1\right )} e^{\left (-a\right )}\right )}{b^{2}} - 8 \, \int \frac{x e^{\left (b x + a\right )}}{4 \,{\left (e^{\left (2 \, b x + 2 \, a\right )} + 1\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.67801, size = 1624, normalized size = 20.56 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x \operatorname{csch}^{2}{\left (a + b x \right )} \operatorname{sech}{\left (a + b x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x \operatorname{csch}\left (b x + a\right )^{2} \operatorname{sech}\left (b x + a\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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