Optimal. Leaf size=27 \[ \frac{\log (\sinh (a+b x))}{b}-\frac{\coth ^2(a+b x)}{2 b} \]
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Rubi [A] time = 0.0183687, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3473, 3475} \[ \frac{\log (\sinh (a+b x))}{b}-\frac{\coth ^2(a+b x)}{2 b} \]
Antiderivative was successfully verified.
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Rule 3473
Rule 3475
Rubi steps
\begin{align*} \int \coth ^3(a+b x) \, dx &=-\frac{\coth ^2(a+b x)}{2 b}+\int \coth (a+b x) \, dx\\ &=-\frac{\coth ^2(a+b x)}{2 b}+\frac{\log (\sinh (a+b x))}{b}\\ \end{align*}
Mathematica [A] time = 0.0710364, size = 34, normalized size = 1.26 \[ -\frac{\coth ^2(a+b x)-2 \log (\tanh (a+b x))-2 \log (\cosh (a+b x))}{2 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.019, size = 26, normalized size = 1. \begin{align*} -{\frac{ \left ({\rm coth} \left (bx+a\right ) \right ) ^{2}}{2\,b}}+{\frac{\ln \left ( \sinh \left ( bx+a \right ) \right ) }{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.07929, size = 107, normalized size = 3.96 \begin{align*} x + \frac{a}{b} + \frac{\log \left (e^{\left (-b x - a\right )} + 1\right )}{b} + \frac{\log \left (e^{\left (-b x - a\right )} - 1\right )}{b} + \frac{2 \, e^{\left (-2 \, b x - 2 \, a\right )}}{b{\left (2 \, e^{\left (-2 \, b x - 2 \, a\right )} - e^{\left (-4 \, b x - 4 \, a\right )} - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.53651, size = 930, normalized size = 34.44 \begin{align*} -\frac{b x \cosh \left (b x + a\right )^{4} + 4 \, b x \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + b x \sinh \left (b x + a\right )^{4} - 2 \,{\left (b x - 1\right )} \cosh \left (b x + a\right )^{2} + 2 \,{\left (3 \, b x \cosh \left (b x + a\right )^{2} - b x + 1\right )} \sinh \left (b x + a\right )^{2} + b x -{\left (\cosh \left (b x + a\right )^{4} + 4 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + \sinh \left (b x + a\right )^{4} + 2 \,{\left (3 \, \cosh \left (b x + a\right )^{2} - 1\right )} \sinh \left (b x + a\right )^{2} - 2 \, \cosh \left (b x + a\right )^{2} + 4 \,{\left (\cosh \left (b x + a\right )^{3} - \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + 1\right )} \log \left (\frac{2 \, \sinh \left (b x + a\right )}{\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )}\right ) + 4 \,{\left (b x \cosh \left (b x + a\right )^{3} -{\left (b x - 1\right )} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )}{b \cosh \left (b x + a\right )^{4} + 4 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + b \sinh \left (b x + a\right )^{4} - 2 \, b \cosh \left (b x + a\right )^{2} + 2 \,{\left (3 \, b \cosh \left (b x + a\right )^{2} - b\right )} \sinh \left (b x + a\right )^{2} + 4 \,{\left (b \cosh \left (b x + a\right )^{3} - b \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + b} \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 = 1.16715, size = 95, normalized size = 3.52 \begin{align*} -\frac{b x + a}{b} + \frac{\log \left ({\left | e^{\left (2 \, b x + 2 \, a\right )} - 1 \right |}\right )}{b} - \frac{3 \, e^{\left (4 \, b x + 4 \, a\right )} - 2 \, e^{\left (2 \, b x + 2 \, a\right )} + 3}{2 \, b{\left (e^{\left (2 \, b x + 2 \, a\right )} - 1\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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