Optimal. Leaf size=20 \[ \frac{\sinh (x)}{a}-\frac{b \log (a \sinh (x)+b)}{a^2} \]
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Rubi [A] time = 0.0800283, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {3872, 2833, 12, 43} \[ \frac{\sinh (x)}{a}-\frac{b \log (a \sinh (x)+b)}{a^2} \]
Antiderivative was successfully verified.
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Rule 3872
Rule 2833
Rule 12
Rule 43
Rubi steps
\begin{align*} \int \frac{\cosh (x)}{a+b \text{csch}(x)} \, dx &=i \int \frac{\cosh (x) \sinh (x)}{i b+i a \sinh (x)} \, dx\\ &=-\frac{i \operatorname{Subst}\left (\int \frac{x}{a (i b+x)} \, dx,x,i a \sinh (x)\right )}{a}\\ &=-\frac{i \operatorname{Subst}\left (\int \frac{x}{i b+x} \, dx,x,i a \sinh (x)\right )}{a^2}\\ &=-\frac{i \operatorname{Subst}\left (\int \left (1-\frac{b}{b-i x}\right ) \, dx,x,i a \sinh (x)\right )}{a^2}\\ &=-\frac{b \log (b+a \sinh (x))}{a^2}+\frac{\sinh (x)}{a}\\ \end{align*}
Mathematica [A] time = 0.0124498, size = 19, normalized size = 0.95 \[ \frac{a \sinh (x)-b \log (a \sinh (x)+b)}{a^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.022, size = 31, normalized size = 1.6 \begin{align*}{\frac{1}{a{\rm csch} \left (x\right )}}+{\frac{b\ln \left ({\rm csch} \left (x\right ) \right ) }{{a}^{2}}}-{\frac{b\ln \left ( a+b{\rm csch} \left (x\right ) \right ) }{{a}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.01637, size = 65, normalized size = 3.25 \begin{align*} -\frac{b x}{a^{2}} - \frac{e^{\left (-x\right )}}{2 \, a} + \frac{e^{x}}{2 \, a} - \frac{b \log \left (-2 \, b e^{\left (-x\right )} + a e^{\left (-2 \, x\right )} - a\right )}{a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.58189, size = 246, normalized size = 12.3 \begin{align*} \frac{2 \, b x \cosh \left (x\right ) + a \cosh \left (x\right )^{2} + a \sinh \left (x\right )^{2} - 2 \,{\left (b \cosh \left (x\right ) + b \sinh \left (x\right )\right )} \log \left (\frac{2 \,{\left (a \sinh \left (x\right ) + b\right )}}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) + 2 \,{\left (b x + a \cosh \left (x\right )\right )} \sinh \left (x\right ) - a}{2 \,{\left (a^{2} \cosh \left (x\right ) + a^{2} \sinh \left (x\right )\right )}} \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{\cosh{\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 = 1.16308, size = 53, normalized size = 2.65 \begin{align*} -\frac{e^{\left (-x\right )} - e^{x}}{2 \, a} - \frac{b \log \left ({\left | -a{\left (e^{\left (-x\right )} - e^{x}\right )} + 2 \, b \right |}\right )}{a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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