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 = 0.04, antiderivative size = 37, normalized size of antiderivative = 1.00, 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 2648
Rule 2750
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*}
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Mathematica [A] time = 0.06, 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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fricas [A] time = 1.20, size = 48, normalized size = 1.30 \[ \frac {2 \, {\left ({\left (A - 5 \, B\right )} \cosh \relax (x) - {\left (A + B\right )} \sinh \relax (x) - 3 \, A + 3 \, B\right )}}{3 \, {\left (\cosh \relax (x)^{2} + 2 \, {\left (\cosh \relax (x) - 1\right )} \sinh \relax (x) + \sinh \relax (x)^{2} - 4 \, \cosh \relax (x) + 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.11, size = 32, normalized size = 0.86 \[ -\frac {2 \, {\left (3 \, B e^{\left (2 \, x\right )} + 3 \, A e^{x} - 3 \, B e^{x} - A + 2 \, B\right )}}{3 \, {\left (e^{x} - 1\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 26, normalized size = 0.70 \[ -\frac {A +B}{6 \tanh \left (\frac {x}{2}\right )^{3}}-\frac {-A +B}{2 \tanh \left (\frac {x}{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.32, size = 131, normalized size = 3.54 \[ -\frac {2}{3} \, B {\left (\frac {3 \, e^{\left (-x\right )}}{3 \, e^{\left (-x\right )} - 3 \, e^{\left (-2 \, x\right )} + e^{\left (-3 \, x\right )} - 1} - \frac {3 \, e^{\left (-2 \, x\right )}}{3 \, e^{\left (-x\right )} - 3 \, e^{\left (-2 \, x\right )} + e^{\left (-3 \, x\right )} - 1} - \frac {2}{3 \, e^{\left (-x\right )} - 3 \, e^{\left (-2 \, x\right )} + e^{\left (-3 \, x\right )} - 1}\right )} + \frac {2}{3} \, A {\left (\frac {3 \, e^{\left (-x\right )}}{3 \, e^{\left (-x\right )} - 3 \, e^{\left (-2 \, x\right )} + e^{\left (-3 \, x\right )} - 1} - \frac {1}{3 \, e^{\left (-x\right )} - 3 \, e^{\left (-2 \, x\right )} + e^{\left (-3 \, x\right )} - 1}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.93, size = 32, normalized size = 0.86 \[ -\frac {2\,\left (2\,B-A+3\,A\,{\mathrm {e}}^x-3\,B\,{\mathrm {e}}^x+3\,B\,{\mathrm {e}}^{2\,x}\right )}{3\,{\left ({\mathrm {e}}^x-1\right )}^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.84, size = 36, normalized size = 0.97 \[ \frac {A}{2 \tanh {\left (\frac {x}{2} \right )}} - \frac {A}{6 \tanh ^{3}{\left (\frac {x}{2} \right )}} - \frac {B}{2 \tanh {\left (\frac {x}{2} \right )}} - \frac {B}{6 \tanh ^{3}{\left (\frac {x}{2} \right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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