Optimal. Leaf size=29 \[ -\frac{b \log (\tanh (x))}{a^2}+\frac{b \log (a+b \tanh (x))}{a^2}-\frac{\coth (x)}{a} \]
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Rubi [A] time = 0.0541423, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {3516, 44} \[ -\frac{b \log (\tanh (x))}{a^2}+\frac{b \log (a+b \tanh (x))}{a^2}-\frac{\coth (x)}{a} \]
Antiderivative was successfully verified.
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Rule 3516
Rule 44
Rubi steps
\begin{align*} \int \frac{\text{csch}^2(x)}{a+b \tanh (x)} \, dx &=b \operatorname{Subst}\left (\int \frac{1}{x^2 (a+x)} \, dx,x,b \tanh (x)\right )\\ &=b \operatorname{Subst}\left (\int \left (\frac{1}{a x^2}-\frac{1}{a^2 x}+\frac{1}{a^2 (a+x)}\right ) \, dx,x,b \tanh (x)\right )\\ &=-\frac{\coth (x)}{a}-\frac{b \log (\tanh (x))}{a^2}+\frac{b \log (a+b \tanh (x))}{a^2}\\ \end{align*}
Mathematica [A] time = 0.0834297, size = 28, normalized size = 0.97 \[ -\frac{-b \log (a \cosh (x)+b \sinh (x))+a \coth (x)+b \log (\sinh (x))}{a^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.039, size = 56, normalized size = 1.9 \begin{align*} -{\frac{1}{2\,a}\tanh \left ({\frac{x}{2}} \right ) }+{\frac{b}{{a}^{2}}\ln \left ( a \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}+2\,\tanh \left ( x/2 \right ) b+a \right ) }-{\frac{1}{2\,a} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-1}}-{\frac{b}{{a}^{2}}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.04015, size = 88, normalized size = 3.03 \begin{align*} \frac{b \log \left (-{\left (a - b\right )} e^{\left (-2 \, x\right )} - a - b\right )}{a^{2}} - \frac{b \log \left (e^{\left (-x\right )} + 1\right )}{a^{2}} - \frac{b \log \left (e^{\left (-x\right )} - 1\right )}{a^{2}} + \frac{2}{a e^{\left (-2 \, x\right )} - a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.25994, size = 360, normalized size = 12.41 \begin{align*} \frac{{\left (b \cosh \left (x\right )^{2} + 2 \, b \cosh \left (x\right ) \sinh \left (x\right ) + b \sinh \left (x\right )^{2} - b\right )} \log \left (\frac{2 \,{\left (a \cosh \left (x\right ) + b \sinh \left (x\right )\right )}}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) -{\left (b \cosh \left (x\right )^{2} + 2 \, b \cosh \left (x\right ) \sinh \left (x\right ) + b \sinh \left (x\right )^{2} - b\right )} \log \left (\frac{2 \, \sinh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) - 2 \, a}{a^{2} \cosh \left (x\right )^{2} + 2 \, a^{2} \cosh \left (x\right ) \sinh \left (x\right ) + a^{2} \sinh \left (x\right )^{2} - a^{2}} \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 \tanh{\left (x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.19888, size = 105, normalized size = 3.62 \begin{align*} \frac{{\left (a b + b^{2}\right )} \log \left ({\left | a e^{\left (2 \, x\right )} + b e^{\left (2 \, x\right )} + a - b \right |}\right )}{a^{3} + a^{2} b} - \frac{b \log \left ({\left | e^{\left (2 \, x\right )} - 1 \right |}\right )}{a^{2}} + \frac{b e^{\left (2 \, x\right )} - 2 \, a - b}{a^{2}{\left (e^{\left (2 \, x\right )} - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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