Optimal. Leaf size=25 \[ -\frac {x}{a}+\frac {\sinh (x)}{a}+\frac {\sinh (x)}{a (\cosh (x)+1)} \]
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Rubi [A] time = 0.07, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {2746, 12, 2735, 2648} \[ -\frac {x}{a}+\frac {\sinh (x)}{a}+\frac {\sinh (x)}{a (\cosh (x)+1)} \]
Antiderivative was successfully verified.
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Rule 12
Rule 2648
Rule 2735
Rule 2746
Rubi steps
\begin {align*} \int \frac {\cosh ^2(x)}{a+a \cosh (x)} \, dx &=\frac {\sinh (x)}{a}-\frac {\int \frac {a \cosh (x)}{a+a \cosh (x)} \, dx}{a}\\ &=\frac {\sinh (x)}{a}-\int \frac {\cosh (x)}{a+a \cosh (x)} \, dx\\ &=-\frac {x}{a}+\frac {\sinh (x)}{a}+\int \frac {1}{a+a \cosh (x)} \, dx\\ &=-\frac {x}{a}+\frac {\sinh (x)}{a}+\frac {\sinh (x)}{a+a \cosh (x)}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 32, normalized size = 1.28 \[ \frac {-2 x+3 \tanh \left (\frac {x}{2}\right )+\sinh \left (\frac {3 x}{2}\right ) \text {sech}\left (\frac {x}{2}\right )}{2 a} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.46, size = 47, normalized size = 1.88 \[ -\frac {2 \, x \cosh \relax (x) - \cosh \relax (x)^{2} + 2 \, {\left (x - \cosh \relax (x) - 1\right )} \sinh \relax (x) - \sinh \relax (x)^{2} + 2 \, x + 5}{2 \, {\left (a \cosh \relax (x) + a \sinh \relax (x) + a\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 35, normalized size = 1.40 \[ -\frac {x}{a} - \frac {{\left (5 \, e^{x} + 1\right )} e^{\left (-x\right )}}{2 \, a {\left (e^{x} + 1\right )}} + \frac {e^{x}}{2 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.07, size = 59, normalized size = 2.36 \[ \frac {\tanh \left (\frac {x}{2}\right )}{a}-\frac {1}{a \left (\tanh \left (\frac {x}{2}\right )-1\right )}+\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{a}-\frac {1}{a \left (\tanh \left (\frac {x}{2}\right )+1\right )}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.37, size = 41, normalized size = 1.64 \[ -\frac {x}{a} + \frac {5 \, e^{\left (-x\right )} + 1}{2 \, {\left (a e^{\left (-x\right )} + a e^{\left (-2 \, x\right )}\right )}} - \frac {e^{\left (-x\right )}}{2 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.90, size = 34, normalized size = 1.36 \[ \frac {{\mathrm {e}}^x}{2\,a}-\frac {x}{a}-\frac {2}{a\,\left ({\mathrm {e}}^x+1\right )}-\frac {{\mathrm {e}}^{-x}}{2\,a} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.60, size = 63, normalized size = 2.52 \[ - \frac {x \tanh ^{2}{\left (\frac {x}{2} \right )}}{a \tanh ^{2}{\left (\frac {x}{2} \right )} - a} + \frac {x}{a \tanh ^{2}{\left (\frac {x}{2} \right )} - a} + \frac {\tanh ^{3}{\left (\frac {x}{2} \right )}}{a \tanh ^{2}{\left (\frac {x}{2} \right )} - a} - \frac {3 \tanh {\left (\frac {x}{2} \right )}}{a \tanh ^{2}{\left (\frac {x}{2} \right )} - a} \]
Verification of antiderivative is not currently implemented for this CAS.
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