Optimal. Leaf size=11 \[ \frac{\log (a+b \tanh (x))}{b} \]
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Rubi [A] time = 0.0401233, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {3506, 31} \[ \frac{\log (a+b \tanh (x))}{b} \]
Antiderivative was successfully verified.
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Rule 3506
Rule 31
Rubi steps
\begin{align*} \int \frac{\text{sech}^2(x)}{a+b \tanh (x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{a+x} \, dx,x,b \tanh (x)\right )}{b}\\ &=\frac{\log (a+b \tanh (x))}{b}\\ \end{align*}
Mathematica [A] time = 0.055458, size = 20, normalized size = 1.82 \[ \frac{\log (a \cosh (x)+b \sinh (x))-\log (\cosh (x))}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.023, size = 12, normalized size = 1.1 \begin{align*}{\frac{\ln \left ( a+b\tanh \left ( x \right ) \right ) }{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04408, size = 15, normalized size = 1.36 \begin{align*} \frac{\log \left (b \tanh \left (x\right ) + a\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.06499, size = 126, normalized size = 11.45 \begin{align*} \frac{\log \left (\frac{2 \,{\left (a \cosh \left (x\right ) + b \sinh \left (x\right )\right )}}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) - \log \left (\frac{2 \, \cosh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\right )}{b} \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*} \int \frac{\operatorname{sech}^{2}{\left (x \right )}}{a + b \tanh{\left (x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.17001, size = 61, normalized size = 5.55 \begin{align*} \frac{{\left (a + b\right )} \log \left ({\left | a e^{\left (2 \, x\right )} + b e^{\left (2 \, x\right )} + a - b \right |}\right )}{a b + b^{2}} - \frac{\log \left (e^{\left (2 \, x\right )} + 1\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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