Optimal. Leaf size=25 \[ -\frac{\tanh ^{-1}(\cosh (a+b x))}{b^2}-\frac{x \text{csch}(a+b x)}{b} \]
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Rubi [A] time = 0.018915, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {5419, 3770} \[ -\frac{\tanh ^{-1}(\cosh (a+b x))}{b^2}-\frac{x \text{csch}(a+b x)}{b} \]
Antiderivative was successfully verified.
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Rule 5419
Rule 3770
Rubi steps
\begin{align*} \int x \coth (a+b x) \text{csch}(a+b x) \, dx &=-\frac{x \text{csch}(a+b x)}{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}\\ \end{align*}
Mathematica [B] time = 0.0483079, size = 114, normalized size = 4.56 \[ \frac{\log \left (\sinh \left (\frac{a}{2}+\frac{b x}{2}\right )\right )}{b^2}-\frac{\log \left (\cosh \left (\frac{a}{2}+\frac{b x}{2}\right )\right )}{b^2}-\frac{x \text{csch}(a)}{b}+\frac{x \text{csch}\left (\frac{a}{2}\right ) \sinh \left (\frac{b x}{2}\right ) \text{csch}\left (\frac{a}{2}+\frac{b x}{2}\right )}{2 b}+\frac{x \text{sech}\left (\frac{a}{2}\right ) \sinh \left (\frac{b x}{2}\right ) \text{sech}\left (\frac{a}{2}+\frac{b x}{2}\right )}{2 b} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.029, size = 54, normalized size = 2.2 \begin{align*} -2\,{\frac{x{{\rm e}^{bx+a}}}{b \left ({{\rm e}^{2\,bx+2\,a}}-1 \right ) }}+{\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 [B] time = 1.31727, size = 86, normalized size = 3.44 \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.05509, size = 483, normalized size = 19.32 \begin{align*} -\frac{2 \, b x \cosh \left (b x + a\right ) + 2 \, b x \sinh \left (b x + a\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 )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1\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 )} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.27432, size = 126, normalized size = 5.04 \begin{align*} -\frac{2 \, b x e^{\left (b x + a\right )} + e^{\left (2 \, b x + 2 \, a\right )} \log \left (e^{\left (b x + a\right )} + 1\right ) - e^{\left (2 \, b x + 2 \, a\right )} \log \left (e^{\left (b x + a\right )} - 1\right ) - \log \left (e^{\left (b x + a\right )} + 1\right ) + \log \left (e^{\left (b x + a\right )} - 1\right )}{b^{2} e^{\left (2 \, b x + 2 \, a\right )} - b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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