Optimal. Leaf size=24 \[ -\frac {A \sinh (x)}{1-\cosh (x)}-B \log (1-\cosh (x)) \]
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Rubi [A] time = 0.09, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {4401, 2648, 2667, 31} \[ -\frac {A \sinh (x)}{1-\cosh (x)}-B \log (1-\cosh (x)) \]
Antiderivative was successfully verified.
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Rule 31
Rule 2648
Rule 2667
Rule 4401
Rubi steps
\begin {align*} \int \frac {A+B \sinh (x)}{1-\cosh (x)} \, dx &=\int \left (-\frac {A}{-1+\cosh (x)}-\frac {B \sinh (x)}{-1+\cosh (x)}\right ) \, dx\\ &=-\left (A \int \frac {1}{-1+\cosh (x)} \, dx\right )-B \int \frac {\sinh (x)}{-1+\cosh (x)} \, dx\\ &=-\frac {A \sinh (x)}{1-\cosh (x)}-B \operatorname {Subst}\left (\int \frac {1}{-1+x} \, dx,x,\cosh (x)\right )\\ &=-B \log (1-\cosh (x))-\frac {A \sinh (x)}{1-\cosh (x)}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 19, normalized size = 0.79 \[ A \coth \left (\frac {x}{2}\right )-2 B \log \left (\sinh \left (\frac {x}{2}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.48, size = 48, normalized size = 2.00 \[ \frac {B x \cosh \relax (x) + B x \sinh \relax (x) - B x - 2 \, {\left (B \cosh \relax (x) + B \sinh \relax (x) - B\right )} \log \left (\cosh \relax (x) + \sinh \relax (x) - 1\right ) + 2 \, A}{\cosh \relax (x) + \sinh \relax (x) - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 22, normalized size = 0.92 \[ B x - 2 \, B \log \left ({\left | e^{x} - 1 \right |}\right ) + \frac {2 \, A}{e^{x} - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 36, normalized size = 1.50 \[ B \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )+B \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )+\frac {A}{\tanh \left (\frac {x}{2}\right )}-2 B \ln \left (\tanh \left (\frac {x}{2}\right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.37, size = 20, normalized size = 0.83 \[ -B \log \left (\cosh \relax (x) - 1\right ) - \frac {2 \, A}{e^{\left (-x\right )} - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.91, size = 21, normalized size = 0.88 \[ B\,x+\frac {2\,A}{{\mathrm {e}}^x-1}-2\,B\,\ln \left ({\mathrm {e}}^x-1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.51, size = 31, normalized size = 1.29 \[ \frac {A}{\tanh {\left (\frac {x}{2} \right )}} - B x + 2 B \log {\left (\tanh {\left (\frac {x}{2} \right )} + 1 \right )} - 2 B \log {\left (\tanh {\left (\frac {x}{2} \right )} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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