Optimal. Leaf size=30 \[ \frac{\log (\sinh (2 a+2 b x))}{b^2}-\frac{2 x \coth (2 a+2 b x)}{b} \]
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Rubi [A] time = 0.0568041, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {5461, 4184, 3475} \[ \frac{\log (\sinh (2 a+2 b x))}{b^2}-\frac{2 x \coth (2 a+2 b x)}{b} \]
Antiderivative was successfully verified.
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Rule 5461
Rule 4184
Rule 3475
Rubi steps
\begin{align*} \int x \text{csch}^2(a+b x) \text{sech}^2(a+b x) \, dx &=4 \int x \text{csch}^2(2 a+2 b x) \, dx\\ &=-\frac{2 x \coth (2 a+2 b x)}{b}+\frac{2 \int \coth (2 a+2 b x) \, dx}{b}\\ &=-\frac{2 x \coth (2 a+2 b x)}{b}+\frac{\log (\sinh (2 a+2 b x))}{b^2}\\ \end{align*}
Mathematica [A] time = 0.145231, size = 26, normalized size = 0.87 \[ \frac{\log (\sinh (2 (a+b x)))-2 b x \coth (2 (a+b x))}{b^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.037, size = 62, normalized size = 2.1 \begin{align*} -4\,{\frac{x}{b}}-4\,{\frac{a}{{b}^{2}}}-4\,{\frac{x}{b \left ( 1+{{\rm e}^{2\,bx+2\,a}} \right ) \left ({{\rm e}^{2\,bx+2\,a}}-1 \right ) }}+{\frac{\ln \left ({{\rm e}^{4\,bx+4\,a}}-1 \right ) }{{b}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.11937, size = 117, normalized size = 3.9 \begin{align*} -\frac{4 \, x e^{\left (4 \, b x + 4 \, a\right )}}{b e^{\left (4 \, b x + 4 \, 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}} + \frac{\log \left ({\left (e^{\left (2 \, b x + 2 \, a\right )} + 1\right )} e^{\left (-2 \, a\right )}\right )}{b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.2761, size = 788, normalized size = 26.27 \begin{align*} -\frac{4 \, b x \cosh \left (b x + a\right )^{4} + 16 \, b x \cosh \left (b x + a\right )^{3} \sinh \left (b x + a\right ) + 24 \, b x \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right )^{2} + 16 \, b x \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + 4 \, b x \sinh \left (b x + a\right )^{4} -{\left (\cosh \left (b x + a\right )^{4} + 4 \, \cosh \left (b x + a\right )^{3} \sinh \left (b x + a\right ) + 6 \, \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right )^{2} + 4 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + \sinh \left (b x + a\right )^{4} - 1\right )} \log \left (\frac{4 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )}{\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}}\right )}{b^{2} \cosh \left (b x + a\right )^{4} + 4 \, b^{2} \cosh \left (b x + a\right )^{3} \sinh \left (b x + a\right ) + 6 \, b^{2} \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right )^{2} + 4 \, b^{2} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + b^{2} \sinh \left (b x + a\right )^{4} - b^{2}} \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}^{2}{\left (a + b x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.16235, size = 97, normalized size = 3.23 \begin{align*} -\frac{4 \, b x e^{\left (4 \, b x + 4 \, a\right )} - e^{\left (4 \, b x + 4 \, a\right )} \log \left (e^{\left (4 \, b x + 4 \, a\right )} - 1\right ) + \log \left (e^{\left (4 \, b x + 4 \, a\right )} - 1\right )}{b^{2} e^{\left (4 \, b x + 4 \, a\right )} - b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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