Optimal. Leaf size=34 \[ \frac{\sinh (c+d x)}{a d}-\frac{b \log (a \sinh (c+d x)+b)}{a^2 d} \]
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Rubi [A] time = 0.0890107, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {3872, 2833, 12, 43} \[ \frac{\sinh (c+d x)}{a d}-\frac{b \log (a \sinh (c+d x)+b)}{a^2 d} \]
Antiderivative was successfully verified.
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Rule 3872
Rule 2833
Rule 12
Rule 43
Rubi steps
\begin{align*} \int \frac{\cosh (c+d x)}{a+b \text{csch}(c+d x)} \, dx &=i \int \frac{\cosh (c+d x) \sinh (c+d x)}{i b+i a \sinh (c+d x)} \, dx\\ &=-\frac{i \operatorname{Subst}\left (\int \frac{x}{a (i b+x)} \, dx,x,i a \sinh (c+d x)\right )}{a d}\\ &=-\frac{i \operatorname{Subst}\left (\int \frac{x}{i b+x} \, dx,x,i a \sinh (c+d x)\right )}{a^2 d}\\ &=-\frac{i \operatorname{Subst}\left (\int \left (1-\frac{b}{b-i x}\right ) \, dx,x,i a \sinh (c+d x)\right )}{a^2 d}\\ &=-\frac{b \log (b+a \sinh (c+d x))}{a^2 d}+\frac{\sinh (c+d x)}{a d}\\ \end{align*}
Mathematica [A] time = 0.0217856, size = 30, normalized size = 0.88 \[ \frac{a \sinh (c+d x)-b \log (a \sinh (c+d x)+b)}{a^2 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 52, normalized size = 1.5 \begin{align*}{\frac{1}{da{\rm csch} \left (dx+c\right )}}+{\frac{b\ln \left ({\rm csch} \left (dx+c\right ) \right ) }{d{a}^{2}}}-{\frac{b\ln \left ( a+b{\rm csch} \left (dx+c\right ) \right ) }{d{a}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.23365, size = 112, normalized size = 3.29 \begin{align*} -\frac{{\left (d x + c\right )} b}{a^{2} d} + \frac{e^{\left (d x + c\right )}}{2 \, a d} - \frac{e^{\left (-d x - c\right )}}{2 \, a d} - \frac{b \log \left (-2 \, b e^{\left (-d x - c\right )} + a e^{\left (-2 \, d x - 2 \, c\right )} - a\right )}{a^{2} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.65089, size = 354, normalized size = 10.41 \begin{align*} \frac{2 \, b d x \cosh \left (d x + c\right ) + a \cosh \left (d x + c\right )^{2} + a \sinh \left (d x + c\right )^{2} - 2 \,{\left (b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right )\right )} \log \left (\frac{2 \,{\left (a \sinh \left (d x + c\right ) + b\right )}}{\cosh \left (d x + c\right ) - \sinh \left (d x + c\right )}\right ) + 2 \,{\left (b d x + a \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) - a}{2 \,{\left (a^{2} d \cosh \left (d x + c\right ) + a^{2} d \sinh \left (d x + c\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 (c + d x \right )}}{a + b \operatorname{csch}{\left (c + d x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1507, size = 84, normalized size = 2.47 \begin{align*} \frac{e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}}{2 \, a d} - \frac{b \log \left ({\left | a{\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 2 \, b \right |}\right )}{a^{2} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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