Optimal. Leaf size=71 \[ \frac{x (2 a A+b C)}{2 a^2}-\frac{1}{2} \left (\frac{b C}{a^2}+\frac{2 A}{a}-\frac{C}{b}\right ) \log (a+b \sinh (x)+b \cosh (x))-\frac{C \sinh (x)}{2 a}+\frac{C \cosh (x)}{2 a} \]
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Rubi [A] time = 0.0560363, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {3131} \[ \frac{x (2 a A+b C)}{2 a^2}-\frac{1}{2} \left (\frac{b C}{a^2}+\frac{2 A}{a}-\frac{C}{b}\right ) \log (a+b \sinh (x)+b \cosh (x))-\frac{C \sinh (x)}{2 a}+\frac{C \cosh (x)}{2 a} \]
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{1}{2} \left (\frac{2 A}{a}-\frac{C}{b}+\frac{b C}{a^2}\right ) \log (a+b \cosh (x)+b \sinh (x))-\frac{C \sinh (x)}{2 a}\\ \end{align*}
Mathematica [A] time = 0.210844, size = 86, normalized size = 1.21 \[ \frac{x \left (a^2 C+2 a A b+b^2 C\right )+2 \left (a^2 C-2 a A b-b^2 C\right ) \log \left ((b-a) \sinh \left (\frac{x}{2}\right )+(a+b) \cosh \left (\frac{x}{2}\right )\right )-2 a b C \sinh (x)+2 a b C \cosh (x)}{4 a^2 b} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.053, size = 136, normalized size = 1.9 \begin{align*}{\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{C}{2\,b}\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 = 1.12256, size = 78, normalized size = 1.1 \begin{align*} \frac{1}{2} \, C{\left (\frac{x}{b} + \frac{e^{\left (-x\right )}}{a} + \frac{{\left (a^{2} - b^{2}\right )} \log \left (a e^{\left (-x\right )} + b\right )}{a^{2} b}\right )} - \frac{A \log \left (a e^{\left (-x\right )} + b\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.29658, size = 284, normalized size = 4. \begin{align*} \frac{C a b +{\left (2 \, A a b + C b^{2}\right )} x \cosh \left (x\right ) +{\left (2 \, A a b + C b^{2}\right )} x \sinh \left (x\right ) +{\left ({\left (C a^{2} - 2 \, A a b - C b^{2}\right )} \cosh \left (x\right ) +{\left (C a^{2} - 2 \, A a b - C b^{2}\right )} \sinh \left (x\right )\right )} \log \left (b \cosh \left (x\right ) + b \sinh \left (x\right ) + a\right )}{2 \,{\left (a^{2} b \cosh \left (x\right ) + a^{2} b \sinh \left (x\right )\right )}} \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 = 1.13367, size = 78, normalized size = 1.1 \begin{align*} \frac{C e^{\left (-x\right )}}{2 \, a} + \frac{{\left (2 \, A a + C b\right )} x}{2 \, a^{2}} + \frac{{\left (C a^{2} - 2 \, A a b - C b^{2}\right )} \log \left ({\left | b e^{x} + a \right |}\right )}{2 \, a^{2} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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