Optimal. Leaf size=35 \[ \frac {(A+2 B) \sinh (x)}{3 (\cosh (x)+1)}+\frac {(A-B) \sinh (x)}{3 (\cosh (x)+1)^2} \]
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Rubi [A] time = 0.04, antiderivative size = 35, 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 = {2750, 2648} \[ \frac {(A+2 B) \sinh (x)}{3 (\cosh (x)+1)}+\frac {(A-B) \sinh (x)}{3 (\cosh (x)+1)^2} \]
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 = 0.71 \[ \frac {\sinh (x) ((A+2 B) \cosh (x)+2 A+B)}{3 (\cosh (x)+1)^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.85, size = 50, normalized size = 1.43 \[ -\frac {2 \, {\left ({\left (A + 5 \, B\right )} \cosh \relax (x) - {\left (A - B\right )} \sinh \relax (x) + 3 \, A + 3 \, B\right )}}{3 \, {\left (\cosh \relax (x)^{2} + 2 \, {\left (\cosh \relax (x) + 1\right )} \sinh \relax (x) + \sinh \relax (x)^{2} + 4 \, \cosh \relax (x) + 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 30, 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 = 0.04, size = 34, normalized size = 0.97 \[ -\frac {A \left (\tanh ^{3}\left (\frac {x}{2}\right )\right )}{6}+\frac {B \left (\tanh ^{3}\left (\frac {x}{2}\right )\right )}{6}+\frac {A \tanh \left (\frac {x}{2}\right )}{2}+\frac {B \tanh \left (\frac {x}{2}\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.32, size = 129, normalized size = 3.69 \[ \frac {2}{3} \, B {\left (\frac {3 \, e^{\left (-x\right )}}{3 \, e^{\left (-x\right )} + 3 \, e^{\left (-2 \, x\right )} + e^{\left (-3 \, x\right )} + 1} + \frac {3 \, e^{\left (-2 \, x\right )}}{3 \, e^{\left (-x\right )} + 3 \, e^{\left (-2 \, x\right )} + e^{\left (-3 \, x\right )} + 1} + \frac {2}{3 \, e^{\left (-x\right )} + 3 \, e^{\left (-2 \, x\right )} + e^{\left (-3 \, x\right )} + 1}\right )} + \frac {2}{3} \, A {\left (\frac {3 \, e^{\left (-x\right )}}{3 \, e^{\left (-x\right )} + 3 \, e^{\left (-2 \, x\right )} + e^{\left (-3 \, x\right )} + 1} + \frac {1}{3 \, e^{\left (-x\right )} + 3 \, e^{\left (-2 \, x\right )} + e^{\left (-3 \, x\right )} + 1}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 30, normalized size = 0.86 \[ -\frac {2\,\left (A+2\,B+3\,A\,{\mathrm {e}}^x+3\,B\,{\mathrm {e}}^x+3\,B\,{\mathrm {e}}^{2\,x}\right )}{3\,{\left ({\mathrm {e}}^x+1\right )}^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.61, size = 36, normalized size = 1.03 \[ - \frac {A \tanh ^{3}{\left (\frac {x}{2} \right )}}{6} + \frac {A \tanh {\left (\frac {x}{2} \right )}}{2} + \frac {B \tanh ^{3}{\left (\frac {x}{2} \right )}}{6} + \frac {B \tanh {\left (\frac {x}{2} \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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