Optimal. Leaf size=20 \[ -\frac {(A+B) \sinh (x)}{1-\cosh (x)}-B x \]
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Rubi [A] time = 0.04, antiderivative size = 20, 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 = {2735, 2648} \[ -\frac {(A+B) \sinh (x)}{1-\cosh (x)}-B x \]
Antiderivative was successfully verified.
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Rule 2648
Rule 2735
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*}
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Mathematica [A] time = 0.06, 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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fricas [A] time = 0.72, size = 31, normalized size = 1.55 \[ -\frac {B x \cosh \relax (x) + B x \sinh \relax (x) - B x - 2 \, A - 2 \, B}{\cosh \relax (x) + \sinh \relax (x) - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 16, normalized size = 0.80 \[ -B x + \frac {2 \, {\left (A + B\right )}}{e^{x} - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 37, normalized size = 1.85 \[ B \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )-B \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )+\frac {A}{\tanh \left (\frac {x}{2}\right )}+\frac {B}{\tanh \left (\frac {x}{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 27, normalized size = 1.35 \[ -B {\left (x + \frac {2}{e^{\left (-x\right )} - 1}\right )} - \frac {2 \, A}{e^{\left (-x\right )} - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 19, normalized size = 0.95 \[ \frac {2\,A+2\,B}{{\mathrm {e}}^x-1}-B\,x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.49, size = 15, normalized size = 0.75 \[ \frac {A}{\tanh {\left (\frac {x}{2} \right )}} - B x + \frac {B}{\tanh {\left (\frac {x}{2} \right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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