Optimal. Leaf size=9 \[ \frac{\log (\cosh (x)+1)}{a} \]
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Rubi [A] time = 0.0266844, antiderivative size = 9, normalized size of antiderivative = 1., 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 = {3879, 31} \[ \frac{\log (\cosh (x)+1)}{a} \]
Antiderivative was successfully verified.
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Rule 3879
Rule 31
Rubi steps
\begin{align*} \int \frac{\tanh (x)}{a+a \text{sech}(x)} \, dx &=\operatorname{Subst}\left (\int \frac{1}{a+a x} \, dx,x,\cosh (x)\right )\\ &=\frac{\log (1+\cosh (x))}{a}\\ \end{align*}
Mathematica [A] time = 0.0069035, size = 12, normalized size = 1.33 \[ \frac{2 \log \left (\cosh \left (\frac{x}{2}\right )\right )}{a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.016, size = 19, normalized size = 2.1 \begin{align*}{\frac{\ln \left ( 1+{\rm sech} \left (x\right ) \right ) }{a}}-{\frac{\ln \left ({\rm sech} \left (x\right ) \right ) }{a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.13244, size = 24, normalized size = 2.67 \begin{align*} \frac{x}{a} + \frac{2 \, \log \left (e^{\left (-x\right )} + 1\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.4193, size = 53, normalized size = 5.89 \begin{align*} -\frac{x - 2 \, \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.20549, size = 19, normalized size = 2.11 \begin{align*} \frac{x}{a} - \frac{\log{\left (\tanh{\left (x \right )} + 1 \right )}}{a} + \frac{\log{\left (\operatorname{sech}{\left (x \right )} + 1 \right )}}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1518, size = 23, normalized size = 2.56 \begin{align*} -\frac{x}{a} + \frac{2 \, \log \left (e^{x} + 1\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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