Optimal. Leaf size=20 \[ -\frac {2}{1-\cosh (x)}-\log (1-\cosh (x)) \]
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Rubi [A] time = 0.06, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {4392, 2667, 43} \[ -\frac {2}{1-\cosh (x)}-\log (1-\cosh (x)) \]
Antiderivative was successfully verified.
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Rule 43
Rule 2667
Rule 4392
Rubi steps
\begin {align*} \int \frac {1}{(-\coth (x)+\text {csch}(x))^3} \, dx &=-\left (i \int \frac {\sinh ^3(x)}{(i-i \cosh (x))^3} \, dx\right )\\ &=\operatorname {Subst}\left (\int \frac {i-x}{(i+x)^2} \, dx,x,-i \cosh (x)\right )\\ &=\operatorname {Subst}\left (\int \left (\frac {1}{-i-x}+\frac {2 i}{(i+x)^2}\right ) \, dx,x,-i \cosh (x)\right )\\ &=-\frac {2 i}{i-i \cosh (x)}-\log (1-\cosh (x))\\ \end {align*}
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Mathematica [A] time = 0.03, size = 18, normalized size = 0.90 \[ \text {csch}^2\left (\frac {x}{2}\right )-2 \log \left (\sinh \left (\frac {x}{2}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.44, size = 90, normalized size = 4.50 \[ \frac {x \cosh \relax (x)^{2} + x \sinh \relax (x)^{2} - 2 \, {\left (x - 2\right )} \cosh \relax (x) - 2 \, {\left (\cosh \relax (x)^{2} + 2 \, {\left (\cosh \relax (x) - 1\right )} \sinh \relax (x) + \sinh \relax (x)^{2} - 2 \, \cosh \relax (x) + 1\right )} \log \left (\cosh \relax (x) + \sinh \relax (x) - 1\right ) + 2 \, {\left (x \cosh \relax (x) - x + 2\right )} \sinh \relax (x) + x}{\cosh \relax (x)^{2} + 2 \, {\left (\cosh \relax (x) - 1\right )} \sinh \relax (x) + \sinh \relax (x)^{2} - 2 \, \cosh \relax (x) + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 20, normalized size = 1.00 \[ x + \frac {4 \, e^{x}}{{\left (e^{x} - 1\right )}^{2}} - 2 \, \log \left ({\left | e^{x} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.20, size = 29, normalized size = 1.45 \[ \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )+\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )+\frac {1}{\tanh \left (\frac {x}{2}\right )^{2}}-2 \ln \left (\tanh \left (\frac {x}{2}\right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.40, size = 35, normalized size = 1.75 \[ -x - \frac {4 \, e^{\left (-x\right )}}{2 \, e^{\left (-x\right )} - e^{\left (-2 \, x\right )} - 1} - 2 \, \log \left (e^{\left (-x\right )} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.51, size = 31, normalized size = 1.55 \[ x-2\,\ln \left ({\mathrm {e}}^x-1\right )+\frac {4}{{\mathrm {e}}^{2\,x}-2\,{\mathrm {e}}^x+1}+\frac {4}{{\mathrm {e}}^x-1} \]
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 \[ - \int \frac {1}{\coth ^{3}{\relax (x )} - 3 \coth ^{2}{\relax (x )} \operatorname {csch}{\relax (x )} + 3 \coth {\relax (x )} \operatorname {csch}^{2}{\relax (x )} - \operatorname {csch}^{3}{\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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