Optimal. Leaf size=20 \[ \frac {\tan ^{-1}(\sinh (x))}{a}-\frac {\sinh (x)}{a \cosh (x)+a} \]
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Rubi [A] time = 0.04, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {2747, 3770, 2648} \[ \frac {\tan ^{-1}(\sinh (x))}{a}-\frac {\sinh (x)}{a \cosh (x)+a} \]
Antiderivative was successfully verified.
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Rule 2648
Rule 2747
Rule 3770
Rubi steps
\begin {align*} \int \frac {\text {sech}(x)}{a+a \cosh (x)} \, dx &=\frac {\int \text {sech}(x) \, dx}{a}-\int \frac {1}{a+a \cosh (x)} \, dx\\ &=\frac {\tan ^{-1}(\sinh (x))}{a}-\frac {\sinh (x)}{a+a \cosh (x)}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 22, normalized size = 1.10 \[ \frac {2 \tan ^{-1}\left (\tanh \left (\frac {x}{2}\right )\right )-\tanh \left (\frac {x}{2}\right )}{a} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.61, size = 29, normalized size = 1.45 \[ \frac {2 \, {\left ({\left (\cosh \relax (x) + \sinh \relax (x) + 1\right )} \arctan \left (\cosh \relax (x) + \sinh \relax (x)\right ) + 1\right )}}{a \cosh \relax (x) + a \sinh \relax (x) + a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 20, normalized size = 1.00 \[ \frac {2 \, \arctan \left (e^{x}\right )}{a} + \frac {2}{a {\left (e^{x} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 21, normalized size = 1.05 \[ -\frac {\tanh \left (\frac {x}{2}\right )}{a}+\frac {2 \arctan \left (\tanh \left (\frac {x}{2}\right )\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.40, size = 23, normalized size = 1.15 \[ -\frac {2 \, \arctan \left (e^{\left (-x\right )}\right )}{a} - \frac {2}{a e^{\left (-x\right )} + a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.88, size = 31, normalized size = 1.55 \[ \frac {2}{a\,\left ({\mathrm {e}}^x+1\right )}+\frac {2\,\mathrm {atan}\left (\frac {{\mathrm {e}}^x\,\sqrt {a^2}}{a}\right )}{\sqrt {a^2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {\operatorname {sech}{\relax (x )}}{\cosh {\relax (x )} + 1}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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