Optimal. Leaf size=11 \[ \frac{\log (\sinh (a+b x))}{b} \]
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Rubi [A] time = 0.0058973, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3475} \[ \frac{\log (\sinh (a+b x))}{b} \]
Antiderivative was successfully verified.
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Rule 3475
Rubi steps
\begin{align*} \int \coth (a+b x) \, dx &=\frac{\log (\sinh (a+b x))}{b}\\ \end{align*}
Mathematica [A] time = 0.0127889, size = 19, normalized size = 1.73 \[ \frac{\log (\tanh (a+b x))+\log (\cosh (a+b x))}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 13, normalized size = 1.2 \begin{align*} -{\frac{\ln \left ({\rm csch} \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 = 1.09975, size = 31, normalized size = 2.82 \begin{align*} \frac{\log \left (e^{\left (b x + a\right )} - e^{\left (-b x - a\right )}\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.07281, size = 88, normalized size = 8. \begin{align*} -\frac{b x - \log \left (\frac{2 \, \sinh \left (b x + a\right )}{\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )}\right )}{b} \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 \cosh{\left (a + b x \right )} \operatorname{csch}{\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 = 1.21458, size = 38, normalized size = 3.45 \begin{align*} -\frac{b x + a}{b} + \frac{\log \left ({\left | e^{\left (2 \, b x + 2 \, a\right )} - 1 \right |}\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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