Optimal. Leaf size=18 \[ B x+\frac {(A-B) \sinh (x)}{1+\cosh (x)} \]
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Rubi [A]
time = 0.03, 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 = {2814, 2727}
\begin {gather*} \frac {(A-B) \sinh (x)}{\cosh (x)+1}+B x \end {gather*}
Antiderivative was successfully verified.
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Rule 2727
Rule 2814
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 = 0.04, size = 23, normalized size = 1.28 \begin {gather*} \frac {\left (A-B+B x \coth \left (\frac {x}{2}\right )\right ) \sinh (x)}{1+\cosh (x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.35, size = 34, normalized size = 1.89
method | result | size |
risch | \(B x -\frac {2 A}{{\mathrm e}^{x}+1}+\frac {2 B}{{\mathrm e}^{x}+1}\) | \(23\) |
default | \(A \tanh \left (\frac {x}{2}\right )-B \tanh \left (\frac {x}{2}\right )-B \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )+B \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )\) | \(34\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 26, normalized size = 1.44 \begin {gather*} B {\left (x - \frac {2}{e^{\left (-x\right )} + 1}\right )} + \frac {2 \, A}{e^{\left (-x\right )} + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.42, size = 29, normalized size = 1.61 \begin {gather*} \frac {B x \cosh \left (x\right ) + B x \sinh \left (x\right ) + B x - 2 \, A + 2 \, B}{\cosh \left (x\right ) + \sinh \left (x\right ) + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.13, size = 15, normalized size = 0.83 \begin {gather*} A \tanh {\left (\frac {x}{2} \right )} + B x - B \tanh {\left (\frac {x}{2} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 17, normalized size = 0.94 \begin {gather*} B x - \frac {2 \, {\left (A - B\right )}}{e^{x} + 1} \end {gather*}
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 \begin {gather*} B\,x-\frac {2\,A-2\,B}{{\mathrm {e}}^x+1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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