Optimal. Leaf size=5 \[ \log (\cosh (x)+1) \]
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Rubi [A] time = 0.0318504, antiderivative size = 5, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {3160, 2667, 31} \[ \log (\cosh (x)+1) \]
Antiderivative was successfully verified.
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Rule 3160
Rule 2667
Rule 31
Rubi steps
\begin{align*} \int \frac{1}{\coth (x)+\text{csch}(x)} \, dx &=i \int \frac{\sinh (x)}{i+i \cosh (x)} \, dx\\ &=\operatorname{Subst}\left (\int \frac{1}{i+x} \, dx,x,i \cosh (x)\right )\\ &=\log (1+\cosh (x))\\ \end{align*}
Mathematica [A] time = 0.0188026, size = 9, normalized size = 1.8 \[ 2 \log \left (\cosh \left (\frac{x}{2}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.027, size = 20, normalized size = 4. \begin{align*} -\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) -\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 = 1.05773, size = 15, normalized size = 3. \begin{align*} x + 2 \, \log \left (e^{\left (-x\right )} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.03081, size = 47, normalized size = 9.4 \begin{align*} -x + 2 \, \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) \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}{\coth{\left (x \right )} + \operatorname{csch}{\left (x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.12061, size = 15, normalized size = 3. \begin{align*} -x + 2 \, \log \left (e^{x} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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