Optimal. Leaf size=11 \[ \frac{\tanh (x)}{a \text{sech}(x)+a} \]
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Rubi [A] time = 0.0242042, antiderivative size = 11, normalized size of antiderivative = 1., 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*}
Mathematica [A] time = 0.0073096, size = 10, normalized size = 0.91 \[ \frac{\tanh \left (\frac{x}{2}\right )}{a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 9, normalized size = 0.8 \begin{align*}{\frac{1}{a}\tanh \left ({\frac{x}{2}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.13703, size = 16, normalized size = 1.45 \begin{align*} \frac{2}{a e^{\left (-x\right )} + a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.27118, size = 43, normalized size = 3.91 \begin{align*} -\frac{2}{a \cosh \left (x\right ) + a \sinh \left (x\right ) + a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{\operatorname{sech}{\left (x \right )}}{\operatorname{sech}{\left (x \right )} + 1}\, dx}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14796, size = 15, normalized size = 1.36 \begin{align*} -\frac{2}{a{\left (e^{x} + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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