Optimal. Leaf size=14 \[ x+\frac {2 \sinh (x)}{1-\cosh (x)} \]
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Rubi [A] time = 0.05, antiderivative size = 14, 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 = {4392, 2680, 8} \[ x+\frac {2 \sinh (x)}{1-\cosh (x)} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2680
Rule 4392
Rubi steps
\begin {align*} \int \frac {1}{(-\coth (x)+\text {csch}(x))^2} \, dx &=-\int \frac {\sinh ^2(x)}{(i-i \cosh (x))^2} \, dx\\ &=\frac {2 \sinh (x)}{1-\cosh (x)}+\int 1 \, dx\\ &=x+\frac {2 \sinh (x)}{1-\cosh (x)}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 24, normalized size = 1.71 \[ -2 \coth \left (\frac {x}{2}\right ) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\tanh ^2\left (\frac {x}{2}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.39, size = 22, normalized size = 1.57 \[ \frac {x \cosh \relax (x) + x \sinh \relax (x) - x - 4}{\cosh \relax (x) + \sinh \relax (x) - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 10, normalized size = 0.71 \[ x - \frac {4}{e^{x} - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.20, size = 26, normalized size = 1.86 \[ -\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )+\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )-\frac {2}{\tanh \left (\frac {x}{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.37, size = 12, normalized size = 0.86 \[ x + \frac {4}{e^{\left (-x\right )} - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.67, size = 10, normalized size = 0.71 \[ x-\frac {4}{{\mathrm {e}}^x-1} \]
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 \[ \int \frac {1}{\left (- \coth {\relax (x )} + \operatorname {csch}{\relax (x )}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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