Optimal. Leaf size=11 \[ \frac{\log (a+b \sinh (x))}{b} \]
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Rubi [A] time = 0.0385105, antiderivative size = 11, normalized size of antiderivative = 1., 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 = {3159, 2668, 31} \[ \frac{\log (a+b \sinh (x))}{b} \]
Antiderivative was successfully verified.
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Rule 3159
Rule 2668
Rule 31
Rubi steps
\begin{align*} \int \frac{1}{a \text{sech}(x)+b \tanh (x)} \, dx &=\int \frac{\cosh (x)}{a+b \sinh (x)} \, dx\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{a+x} \, dx,x,b \sinh (x)\right )}{b}\\ &=\frac{\log (a+b \sinh (x))}{b}\\ \end{align*}
Mathematica [A] time = 0.0058458, size = 11, normalized size = 1. \[ \frac{\log (a+b \sinh (x))}{b} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.046, size = 50, normalized size = 4.6 \begin{align*} -{\frac{1}{b}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) }+{\frac{1}{b}\ln \left ( a \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}-2\,\tanh \left ( x/2 \right ) b-a \right ) }-{\frac{1}{b}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.02182, size = 38, normalized size = 3.45 \begin{align*} \frac{x}{b} + \frac{\log \left (-2 \, a e^{\left (-x\right )} + b e^{\left (-2 \, x\right )} - b\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.28765, size = 72, normalized size = 6.55 \begin{align*} -\frac{x - \log \left (\frac{2 \,{\left (b \sinh \left (x\right ) + a\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 [A] time = 0.654301, size = 32, normalized size = 2.91 \begin{align*} \begin{cases} \frac{x}{b} + \frac{\log{\left (\frac{a \operatorname{sech}{\left (x \right )}}{b} + \tanh{\left (x \right )} \right )}}{b} - \frac{\log{\left (\tanh{\left (x \right )} + 1 \right )}}{b} & \text{for}\: b \neq 0 \\\frac{\tanh{\left (x \right )}}{a \operatorname{sech}{\left (x \right )}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15566, size = 30, normalized size = 2.73 \begin{align*} \frac{\log \left ({\left | -b{\left (e^{\left (-x\right )} - e^{x}\right )} + 2 \, a \right |}\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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