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 = 0.0175013, 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 = 0.068527, 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 = 0.002, size = 43, normalized size = 1.6 \begin{align*} -{\frac{ \left ({\rm coth} \left (bx+a\right ) \right ) ^{2}}{2\,b}}-{\frac{\ln \left ({\rm coth} \left (bx+a\right )-1 \right ) }{2\,b}}-{\frac{\ln \left ({\rm coth} \left (bx+a\right )+1 \right ) }{2\,b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.02922, 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 = 2.38538, 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 [A] time = 4.26156, size = 61, normalized size = 2.26 \begin{align*} \begin{cases} x \coth ^{3}{\left (a \right )} & \text{for}\: b = 0 \\\tilde{\infty } x & \text{for}\: a = \log{\left (- e^{- b x} \right )} \vee a = \log{\left (e^{- b x} \right )} \\x - \frac{\log{\left (\tanh{\left (a + b x \right )} + 1 \right )}}{b} + \frac{\log{\left (\tanh{\left (a + b x \right )} \right )}}{b} - \frac{1}{2 b \tanh ^{2}{\left (a + b x \right )}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.22073, size = 74, normalized size = 2.74 \begin{align*} -\frac{b x + a}{b} + \frac{\log \left ({\left | e^{\left (2 \, b x + 2 \, a\right )} - 1 \right |}\right )}{b} - \frac{2 \, e^{\left (2 \, b x + 2 \, a\right )}}{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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