Optimal. Leaf size=19 \[ \frac{\log (a+b \text{csch}(x))}{a}+\frac{\log (\sinh (x))}{a} \]
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Rubi [A] time = 0.0316641, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {3885, 36, 29, 31} \[ \frac{\log (a+b \text{csch}(x))}{a}+\frac{\log (\sinh (x))}{a} \]
Antiderivative was successfully verified.
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Rule 3885
Rule 36
Rule 29
Rule 31
Rubi steps
\begin{align*} \int \frac{\coth (x)}{a+b \text{csch}(x)} \, dx &=-\operatorname{Subst}\left (\int \frac{1}{x (a+x)} \, dx,x,b \text{csch}(x)\right )\\ &=-\frac{\operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,b \text{csch}(x)\right )}{a}+\frac{\operatorname{Subst}\left (\int \frac{1}{a+x} \, dx,x,b \text{csch}(x)\right )}{a}\\ &=\frac{\log (a+b \text{csch}(x))}{a}+\frac{\log (\sinh (x))}{a}\\ \end{align*}
Mathematica [A] time = 0.0079351, size = 11, normalized size = 0.58 \[ \frac{\log (a \sinh (x)+b)}{a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.021, size = 21, normalized size = 1.1 \begin{align*} -{\frac{\ln \left ({\rm csch} \left (x\right ) \right ) }{a}}+{\frac{\ln \left ( a+b{\rm csch} \left (x\right ) \right ) }{a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01121, size = 38, normalized size = 2. \begin{align*} \frac{x}{a} + \frac{\log \left (-2 \, b e^{\left (-x\right )} + a e^{\left (-2 \, x\right )} - a\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.54861, size = 72, normalized size = 3.79 \begin{align*} -\frac{x - \log \left (\frac{2 \,{\left (a \sinh \left (x\right ) + b\right )}}{\cosh \left (x\right ) - \sinh \left (x\right )}\right )}{a} \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 \frac{\coth{\left (x \right )}}{a + b \operatorname{csch}{\left (x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10651, size = 30, normalized size = 1.58 \begin{align*} \frac{\log \left ({\left | -a{\left (e^{\left (-x\right )} - e^{x}\right )} + 2 \, b \right |}\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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