Optimal. Leaf size=11 \[ \frac{\log (a \cosh (x)+b)}{a} \]
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Rubi [A] time = 0.0462933, 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 = {3160, 2668, 31} \[ \frac{\log (a \cosh (x)+b)}{a} \]
Antiderivative was successfully verified.
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Rule 3160
Rule 2668
Rule 31
Rubi steps
\begin{align*} \int \frac{1}{a \coth (x)+b \text{csch}(x)} \, dx &=i \int \frac{\sinh (x)}{i b+i a \cosh (x)} \, dx\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{i b+x} \, dx,x,i a \cosh (x)\right )}{a}\\ &=\frac{\log (b+a \cosh (x))}{a}\\ \end{align*}
Mathematica [A] time = 0.0179501, size = 11, normalized size = 1. \[ \frac{\log (a \cosh (x)+b)}{a} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.034, size = 51, normalized size = 4.6 \begin{align*} -{\frac{1}{a}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) }+{\frac{1}{a}\ln \left ( a \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}- \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}b+a+b \right ) }-{\frac{1}{a}\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 = 1.03671, size = 35, normalized size = 3.18 \begin{align*} \frac{x}{a} + \frac{\log \left (2 \, b e^{\left (-x\right )} + a e^{\left (-2 \, x\right )} + a\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.06924, size = 72, normalized size = 6.55 \begin{align*} -\frac{x - \log \left (\frac{2 \,{\left (a \cosh \left (x\right ) + b\right )}}{\cosh \left (x\right ) - \sinh \left (x\right )}\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*} \int \frac{1}{a \coth{\left (x \right )} + b \operatorname{csch}{\left (x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12907, size = 26, normalized size = 2.36 \begin{align*} \frac{\log \left ({\left | a{\left (e^{\left (-x\right )} + e^{x}\right )} + 2 \, b \right |}\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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