Optimal. Leaf size=18 \[ \frac {(A-B) \sinh (x)}{\cosh (x)+1}+B x \]
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Rubi [A] time = 0.04, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2735, 2648} \[ \frac {(A-B) \sinh (x)}{\cosh (x)+1}+B x \]
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 = 23, normalized size = 1.28 \[ \frac {\sinh (x) \left (A+B x \coth \left (\frac {x}{2}\right )-B\right )}{\cosh (x)+1} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.84, size = 29, normalized size = 1.61 \[ \frac {B x \cosh \relax (x) + B x \sinh \relax (x) + B x - 2 \, A + 2 \, B}{\cosh \relax (x) + \sinh \relax (x) + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 17, normalized size = 0.94 \[ B x - \frac {2 \, {\left (A - B\right )}}{e^{x} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 34, normalized size = 1.89 \[ A \tanh \left (\frac {x}{2}\right )-B \tanh \left (\frac {x}{2}\right )-B \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )+B \ln \left (\tanh \left (\frac {x}{2}\right )+1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 26, normalized size = 1.44 \[ B {\left (x - \frac {2}{e^{\left (-x\right )} + 1}\right )} + \frac {2 \, A}{e^{\left (-x\right )} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 19, normalized size = 1.06 \[ B\,x-\frac {2\,A-2\,B}{{\mathrm {e}}^x+1} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.34, size = 15, normalized size = 0.83 \[ A \tanh {\left (\frac {x}{2} \right )} + B x - B \tanh {\left (\frac {x}{2} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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