Optimal. Leaf size=18 \[ \frac {x}{a}-\frac {\sinh (x)}{a \cosh (x)+a} \]
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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 = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2735, 2648} \[ \frac {x}{a}-\frac {\sinh (x)}{a \cosh (x)+a} \]
Antiderivative was successfully verified.
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Rule 2648
Rule 2735
Rubi steps
\begin {align*} \int \frac {\cosh (x)}{a+a \cosh (x)} \, dx &=\frac {x}{a}-\int \frac {1}{a+a \cosh (x)} \, dx\\ &=\frac {x}{a}-\frac {\sinh (x)}{a+a \cosh (x)}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 14, normalized size = 0.78 \[ \frac {x-\tanh \left (\frac {x}{2}\right )}{a} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.38, size = 24, normalized size = 1.33 \[ \frac {x \cosh \relax (x) + x \sinh \relax (x) + x + 2}{a \cosh \relax (x) + a \sinh \relax (x) + a} \]
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 \[ \frac {x}{a} + \frac {2}{a {\left (e^{x} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 34, normalized size = 1.89 \[ -\frac {\tanh \left (\frac {x}{2}\right )}{a}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{a}+\frac {\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 18, normalized size = 1.00 \[ \frac {x}{a} - \frac {2}{a e^{\left (-x\right )} + a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.87, size = 17, normalized size = 0.94 \[ \frac {x}{a}+\frac {2}{a\,\left ({\mathrm {e}}^x+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.33, size = 8, normalized size = 0.44 \[ \frac {x}{a} - \frac {\tanh {\left (\frac {x}{2} \right )}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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