Optimal. Leaf size=20 \[ -\frac{(A+B) \sinh (x)}{1-\cosh (x)}-B x \]
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Rubi [A] time = 0.0395878, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {2735, 2648} \[ -\frac{(A+B) \sinh (x)}{1-\cosh (x)}-B x \]
Antiderivative was successfully verified.
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Rule 2735
Rule 2648
Rubi steps
\begin{align*} \int \frac{A+B \cosh (x)}{1-\cosh (x)} \, dx &=-B x-(-A-B) \int \frac{1}{1-\cosh (x)} \, dx\\ &=-B x-\frac{(A+B) \sinh (x)}{1-\cosh (x)}\\ \end{align*}
Mathematica [A] time = 0.0540247, size = 35, normalized size = 1.75 \[ \frac{2 \sinh \left (\frac{x}{2}\right ) \left ((A+B) \cosh \left (\frac{x}{2}\right )-B x \sinh \left (\frac{x}{2}\right )\right )}{\cosh (x)-1} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.016, size = 37, normalized size = 1.9 \begin{align*} -B\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) +{A \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-1}}+{B \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-1}}+B\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.03264, size = 36, normalized size = 1.8 \begin{align*} -B{\left (x + \frac{2}{e^{\left (-x\right )} - 1}\right )} - \frac{2 \, A}{e^{\left (-x\right )} - 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.17154, size = 97, normalized size = 4.85 \begin{align*} -\frac{B x \cosh \left (x\right ) + B x \sinh \left (x\right ) - B x - 2 \, A - 2 \, B}{\cosh \left (x\right ) + \sinh \left (x\right ) - 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.66039, size = 15, normalized size = 0.75 \begin{align*} \frac{A}{\tanh{\left (\frac{x}{2} \right )}} - B x + \frac{B}{\tanh{\left (\frac{x}{2} \right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1627, size = 22, normalized size = 1.1 \begin{align*} -B x + \frac{2 \,{\left (A + B\right )}}{e^{x} - 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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