Optimal. Leaf size=14 \[ x+\frac{2 \sinh (x)}{1-\cosh (x)} \]
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Rubi [A] time = 0.0513362, antiderivative size = 14, 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 = {4392, 2680, 8} \[ x+\frac{2 \sinh (x)}{1-\cosh (x)} \]
Antiderivative was successfully verified.
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Rule 4392
Rule 2680
Rule 8
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*}
Mathematica [C] time = 0.0086892, 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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Maple [A] time = 0.036, size = 26, normalized size = 1.9 \begin{align*} \ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) -2\, \left ( \tanh \left ( x/2 \right ) \right ) ^{-1}-\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 = 1.23996, size = 16, normalized size = 1.14 \begin{align*} x + \frac{4}{e^{\left (-x\right )} - 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.21153, size = 77, normalized size = 5.5 \begin{align*} \frac{x \cosh \left (x\right ) + x \sinh \left (x\right ) - x - 4}{\cosh \left (x\right ) + \sinh \left (x\right ) - 1} \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}{\left (- \coth{\left (x \right )} + \operatorname{csch}{\left (x \right )}\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13126, size = 14, normalized size = 1. \begin{align*} x - \frac{4}{e^{x} - 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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