Optimal. Leaf size=9 \[ -\log (1-\cosh (x)) \]
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Rubi [A] time = 0.0342147, antiderivative size = 9, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {3160, 2667, 31} \[ -\log (1-\cosh (x)) \]
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 = 0.0212703, size = 9, normalized size = 1. \[ -2 \log \left (\sinh \left (\frac{x}{2}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.03, size = 23, normalized size = 2.6 \begin{align*} \ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) -2\,\ln \left ( \tanh \left ( x/2 \right ) \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 [A] time = 1.22438, size = 18, normalized size = 2. \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 [A] time = 2.12802, size = 46, normalized size = 5.11 \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 [A] time = 1.1381, size = 14, normalized size = 1.56 \begin{align*} x - 2 \, \log \left ({\left | e^{x} - 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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