Optimal. Leaf size=9 \[ -\log (1-\cosh (x)) \]
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Rubi [A] time = 0.03, antiderivative size = 9, normalized size of antiderivative = 1.00, 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 31
Rule 2667
Rule 3160
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*}
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Mathematica [A] time = 0.02, size = 9, normalized size = 1.00 \[ -2 \log \left (\sinh \left (\frac {x}{2}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 11, normalized size = 1.22 \[ x - 2 \, \log \left (\cosh \relax (x) + \sinh \relax (x) - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 10, normalized size = 1.11 \[ x - 2 \, \log \left ({\left | e^{x} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.19, size = 23, normalized size = 2.56 \[ \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )+\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )-2 \ln \left (\tanh \left (\frac {x}{2}\right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.37, size = 13, normalized size = 1.44 \[ -x - 2 \, \log \left (e^{\left (-x\right )} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 9, normalized size = 1.00 \[ x-2\,\ln \left ({\mathrm {e}}^x-1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ - \int \frac {1}{\coth {\relax (x )} - \operatorname {csch}{\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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