Optimal. Leaf size=77 \[ \frac{\left (a^2 C+2 a A b-b^2 C\right ) \log (a-b \sinh (x)+b \cosh (x))}{2 a^2 b}+\frac{x (2 a A-b C)}{2 a^2}+\frac{C \sinh (x)}{2 a}+\frac{C \cosh (x)}{2 a} \]
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Rubi [A] time = 0.0512851, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {3131} \[ \frac{\left (a^2 C+2 a A b-b^2 C\right ) \log (a-b \sinh (x)+b \cosh (x))}{2 a^2 b}+\frac{x (2 a A-b C)}{2 a^2}+\frac{C \sinh (x)}{2 a}+\frac{C \cosh (x)}{2 a} \]
Antiderivative was successfully verified.
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Rule 3131
Rubi steps
\begin{align*} \int \frac{A+C \sinh (x)}{a+b \cosh (x)-b \sinh (x)} \, dx &=\frac{(2 a A-b C) x}{2 a^2}+\frac{C \cosh (x)}{2 a}+\frac{\left (2 a A b+a^2 C-b^2 C\right ) \log (a+b \cosh (x)-b \sinh (x))}{2 a^2 b}+\frac{C \sinh (x)}{2 a}\\ \end{align*}
Mathematica [A] time = 0.233741, size = 86, normalized size = 1.12 \[ \frac{\frac{2 \left (a^2 C+2 a A b-b^2 C\right ) \log \left ((a-b) \sinh \left (\frac{x}{2}\right )+(a+b) \cosh \left (\frac{x}{2}\right )\right )}{b}+x \left (-\frac{a^2 C}{b}+2 a A-b C\right )+2 a C \sinh (x)+2 a C \cosh (x)}{4 a^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.048, size = 125, normalized size = 1.6 \begin{align*} -{\frac{C}{2\,b}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) }-{\frac{C}{a} \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-1}}-{\frac{A}{a}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) }+{\frac{bC}{2\,{a}^{2}}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) }+{\frac{A}{a}\ln \left ( a\tanh \left ({\frac{x}{2}} \right ) -\tanh \left ({\frac{x}{2}} \right ) b+a+b \right ) }+{\frac{C}{2\,b}\ln \left ( a\tanh \left ({\frac{x}{2}} \right ) -\tanh \left ({\frac{x}{2}} \right ) b+a+b \right ) }-{\frac{bC}{2\,{a}^{2}}\ln \left ( a\tanh \left ({\frac{x}{2}} \right ) -\tanh \left ({\frac{x}{2}} \right ) b+a+b \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.12609, size = 88, normalized size = 1.14 \begin{align*} A{\left (\frac{x}{a} + \frac{\log \left (b e^{\left (-x\right )} + a\right )}{a}\right )} - \frac{1}{2} \, C{\left (\frac{b x}{a^{2}} - \frac{e^{x}}{a} - \frac{{\left (a^{2} - b^{2}\right )} \log \left (b e^{\left (-x\right )} + a\right )}{a^{2} b}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.3295, size = 155, normalized size = 2.01 \begin{align*} -\frac{C a^{2} x - C a b \cosh \left (x\right ) - C a b \sinh \left (x\right ) -{\left (C a^{2} + 2 \, A a b - C b^{2}\right )} \log \left (a \cosh \left (x\right ) + a \sinh \left (x\right ) + b\right )}{2 \, a^{2} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13059, size = 66, normalized size = 0.86 \begin{align*} -\frac{C x}{2 \, b} + \frac{C e^{x}}{2 \, a} + \frac{{\left (C a^{2} + 2 \, A a b - C b^{2}\right )} \log \left ({\left | a e^{x} + b \right |}\right )}{2 \, a^{2} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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