Optimal. Leaf size=12 \[ -\frac{\log (a+b \coth (x))}{b} \]
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Rubi [A] time = 0.0484419, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {3506, 31} \[ -\frac{\log (a+b \coth (x))}{b} \]
Antiderivative was successfully verified.
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Rule 3506
Rule 31
Rubi steps
\begin{align*} \int \frac{\text{csch}^2(x)}{a+b \coth (x)} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{1}{a+x} \, dx,x,b \coth (x)\right )}{b}\\ &=-\frac{\log (a+b \coth (x))}{b}\\ \end{align*}
Mathematica [A] time = 0.0536766, size = 20, normalized size = 1.67 \[ \frac{\log (\sinh (x))-\log (a \sinh (x)+b \cosh (x))}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.019, size = 13, normalized size = 1.1 \begin{align*} -{\frac{\ln \left ( a+b{\rm coth} \left (x\right ) \right ) }{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00074, size = 16, normalized size = 1.33 \begin{align*} -\frac{\log \left (b \coth \left (x\right ) + a\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.16145, size = 127, normalized size = 10.58 \begin{align*} -\frac{\log \left (\frac{2 \,{\left (b \cosh \left (x\right ) + a \sinh \left (x\right )\right )}}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) - \log \left (\frac{2 \, \sinh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\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 \frac{\operatorname{csch}^{2}{\left (x \right )}}{a + b \coth{\left (x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.09929, size = 62, normalized size = 5.17 \begin{align*} -\frac{{\left (a + b\right )} \log \left ({\left | a e^{\left (2 \, x\right )} + b e^{\left (2 \, x\right )} - a + b \right |}\right )}{a b + b^{2}} + \frac{\log \left ({\left | e^{\left (2 \, x\right )} - 1 \right |}\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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