Optimal. Leaf size=77 \[ -\frac{\left (a^2 (-B)+2 a A b-b^2 B\right ) \log (a+b \sinh (x)+b \cosh (x))}{2 a^2 b}+\frac{x (2 a A-b B)}{2 a^2}+\frac{B \sinh (x)}{2 a}-\frac{B \cosh (x)}{2 a} \]
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Rubi [A] time = 0.049617, antiderivative size = 77, 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 = {3132} \[ -\frac{\left (a^2 (-B)+2 a A b-b^2 B\right ) \log (a+b \sinh (x)+b \cosh (x))}{2 a^2 b}+\frac{x (2 a A-b B)}{2 a^2}+\frac{B \sinh (x)}{2 a}-\frac{B \cosh (x)}{2 a} \]
Antiderivative was successfully verified.
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Rule 3132
Rubi steps
\begin{align*} \int \frac{A+B \cosh (x)}{a+b \cosh (x)+b \sinh (x)} \, dx &=\frac{(2 a A-b B) x}{2 a^2}-\frac{B \cosh (x)}{2 a}-\frac{\left (2 a A b-a^2 B-b^2 B\right ) \log (a+b \cosh (x)+b \sinh (x))}{2 a^2 b}+\frac{B \sinh (x)}{2 a}\\ \end{align*}
Mathematica [A] time = 0.168832, size = 84, normalized size = 1.09 \[ \frac{\frac{2 \left (a^2 B-2 a A b+b^2 B\right ) \log \left ((b-a) \sinh \left (\frac{x}{2}\right )+(a+b) \cosh \left (\frac{x}{2}\right )\right )}{b}+x \left (\frac{a^2 B}{b}+2 a A-b B\right )+2 a B \sinh (x)-2 a B \cosh (x)}{4 a^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.052, size = 137, normalized size = 1.8 \begin{align*} -{\frac{B}{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{Bb}{2\,{a}^{2}}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) }-{\frac{B}{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{B}{2\,b}\ln \left ( a\tanh \left ({\frac{x}{2}} \right ) -\tanh \left ({\frac{x}{2}} \right ) b-a-b \right ) }+{\frac{Bb}{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 = 1.13735, size = 77, normalized size = 1. \begin{align*} \frac{1}{2} \, B{\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 = 2.25281, size = 285, normalized size = 3.7 \begin{align*} -\frac{B a b -{\left (2 \, A a b - B b^{2}\right )} x \cosh \left (x\right ) -{\left (2 \, A a b - B b^{2}\right )} x \sinh \left (x\right ) -{\left ({\left (B a^{2} - 2 \, A a b + B b^{2}\right )} \cosh \left (x\right ) +{\left (B a^{2} - 2 \, A a b + B 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 = 1.13103, size = 78, normalized size = 1.01 \begin{align*} -\frac{B e^{\left (-x\right )}}{2 \, a} + \frac{{\left (2 \, A a - B b\right )} x}{2 \, a^{2}} + \frac{{\left (B a^{2} - 2 \, A a b + B 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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