Optimal. Leaf size=11 \[ \frac {\tanh (x)}{a \text {sech}(x)+a} \]
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Rubi [A] time = 0.02, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {3794} \[ \frac {\tanh (x)}{a \text {sech}(x)+a} \]
Antiderivative was successfully verified.
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Rule 3794
Rubi steps
\begin {align*} \int \frac {\text {sech}(x)}{a+a \text {sech}(x)} \, dx &=\frac {\tanh (x)}{a+a \text {sech}(x)}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 10, normalized size = 0.91 \[ \frac {\tanh \left (\frac {x}{2}\right )}{a} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.37, size = 14, normalized size = 1.27 \[ -\frac {2}{a \cosh \relax (x) + a \sinh \relax (x) + a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 11, normalized size = 1.00 \[ -\frac {2}{a {\left (e^{x} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 9, normalized size = 0.82 \[ \frac {\tanh \left (\frac {x}{2}\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 12, normalized size = 1.09 \[ \frac {2}{a e^{\left (-x\right )} + a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.31, size = 11, normalized size = 1.00 \[ -\frac {2}{a\,\left ({\mathrm {e}}^x+1\right )} \]
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 \[ \frac {\int \frac {\operatorname {sech}{\relax (x )}}{\operatorname {sech}{\relax (x )} + 1}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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