Optimal. Leaf size=37 \[ -\frac{(A-2 B) \sinh (x)}{3 (1-\cosh (x))}-\frac{(A+B) \sinh (x)}{3 (1-\cosh (x))^2} \]
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Rubi [A] time = 0.0408369, antiderivative size = 37, 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 = {2750, 2648} \[ -\frac{(A-2 B) \sinh (x)}{3 (1-\cosh (x))}-\frac{(A+B) \sinh (x)}{3 (1-\cosh (x))^2} \]
Antiderivative was successfully verified.
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Rule 2750
Rule 2648
Rubi steps
\begin{align*} \int \frac{A+B \cosh (x)}{(1-\cosh (x))^2} \, dx &=-\frac{(A+B) \sinh (x)}{3 (1-\cosh (x))^2}+\frac{1}{3} (A-2 B) \int \frac{1}{1-\cosh (x)} \, dx\\ &=-\frac{(A+B) \sinh (x)}{3 (1-\cosh (x))^2}-\frac{(A-2 B) \sinh (x)}{3 (1-\cosh (x))}\\ \end{align*}
Mathematica [A] time = 0.0485829, size = 25, normalized size = 0.68 \[ \frac{\sinh (x) ((A-2 B) \cosh (x)-2 A+B)}{3 (\cosh (x)-1)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 26, normalized size = 0.7 \begin{align*} -{\frac{-A+B}{2} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-1}}-{\frac{A+B}{6} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.04776, size = 177, normalized size = 4.78 \begin{align*} -\frac{2}{3} \, B{\left (\frac{3 \, e^{\left (-x\right )}}{3 \, e^{\left (-x\right )} - 3 \, e^{\left (-2 \, x\right )} + e^{\left (-3 \, x\right )} - 1} - \frac{3 \, e^{\left (-2 \, x\right )}}{3 \, e^{\left (-x\right )} - 3 \, e^{\left (-2 \, x\right )} + e^{\left (-3 \, x\right )} - 1} - \frac{2}{3 \, e^{\left (-x\right )} - 3 \, e^{\left (-2 \, x\right )} + e^{\left (-3 \, x\right )} - 1}\right )} + \frac{2}{3} \, A{\left (\frac{3 \, e^{\left (-x\right )}}{3 \, e^{\left (-x\right )} - 3 \, e^{\left (-2 \, x\right )} + e^{\left (-3 \, x\right )} - 1} - \frac{1}{3 \, e^{\left (-x\right )} - 3 \, e^{\left (-2 \, x\right )} + e^{\left (-3 \, x\right )} - 1}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.06134, size = 163, normalized size = 4.41 \begin{align*} \frac{2 \,{\left ({\left (A - 5 \, B\right )} \cosh \left (x\right ) -{\left (A + B\right )} \sinh \left (x\right ) - 3 \, A + 3 \, B\right )}}{3 \,{\left (\cosh \left (x\right )^{2} + 2 \,{\left (\cosh \left (x\right ) - 1\right )} \sinh \left (x\right ) + \sinh \left (x\right )^{2} - 4 \, \cosh \left (x\right ) + 3\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.20393, size = 36, normalized size = 0.97 \begin{align*} \frac{A}{2 \tanh{\left (\frac{x}{2} \right )}} - \frac{A}{6 \tanh ^{3}{\left (\frac{x}{2} \right )}} - \frac{B}{2 \tanh{\left (\frac{x}{2} \right )}} - \frac{B}{6 \tanh ^{3}{\left (\frac{x}{2} \right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16434, size = 43, normalized size = 1.16 \begin{align*} -\frac{2 \,{\left (3 \, B e^{\left (2 \, x\right )} + 3 \, A e^{x} - 3 \, B e^{x} - A + 2 \, B\right )}}{3 \,{\left (e^{x} - 1\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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