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 = 0.05, antiderivative size = 77, normalized size of antiderivative = 1.00, 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*}
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Mathematica [A] time = 0.19, 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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fricas [A] time = 0.44, size = 110, normalized size = 1.43 \[ -\frac {B a b - {\left (2 \, A a b - B b^{2}\right )} x \cosh \relax (x) - {\left (2 \, A a b - B b^{2}\right )} x \sinh \relax (x) - {\left ({\left (B a^{2} - 2 \, A a b + B b^{2}\right )} \cosh \relax (x) + {\left (B a^{2} - 2 \, A a b + B b^{2}\right )} \sinh \relax (x)\right )} \log \left (b \cosh \relax (x) + b \sinh \relax (x) + a\right )}{2 \, {\left (a^{2} b \cosh \relax (x) + a^{2} b \sinh \relax (x)\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 58, normalized size = 0.75 \[ -\frac {B e^{\left (-x\right )}}{2 \, a} + \frac {{\left (2 \, A a - B b\right )} x}{2 \, a^{2}} + \frac {{\left (B a^{2} - 2 \, A a b + B b^{2}\right )} \log \left ({\left | b e^{x} + a \right |}\right )}{2 \, a^{2} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.20, size = 137, normalized size = 1.78 \[ -\frac {B \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{2 b}-\frac {\ln \left (a \tanh \left (\frac {x}{2}\right )-\tanh \left (\frac {x}{2}\right ) b -a -b \right ) A}{a}+\frac {\ln \left (a \tanh \left (\frac {x}{2}\right )-\tanh \left (\frac {x}{2}\right ) b -a -b \right ) B}{2 b}+\frac {b \ln \left (a \tanh \left (\frac {x}{2}\right )-\tanh \left (\frac {x}{2}\right ) b -a -b \right ) B}{2 a^{2}}-\frac {B}{a \left (\tanh \left (\frac {x}{2}\right )+1\right )}+\frac {\ln \left (\tanh \left (\frac {x}{2}\right )+1\right ) A}{a}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )+1\right ) B b}{2 a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 57, normalized size = 0.74 \[ \frac {1}{2} \, B {\left (\frac {x}{b} - \frac {e^{\left (-x\right )}}{a} + \frac {{\left (a^{2} + b^{2}\right )} \log \left (a e^{\left (-x\right )} + b\right )}{a^{2} b}\right )} - \frac {A \log \left (a e^{\left (-x\right )} + b\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.69, size = 57, normalized size = 0.74 \[ \frac {x\,\left (2\,A\,a-B\,b\right )}{2\,a^2}-\frac {B\,{\mathrm {e}}^{-x}}{2\,a}+\frac {\ln \left (a+b\,{\mathrm {e}}^x\right )\,\left (B\,a^2-2\,A\,a\,b+B\,b^2\right )}{2\,a^2\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.15, size = 806, normalized size = 10.47 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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