Optimal. Leaf size=4 \[ x+\coth (x) \]
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Rubi [A] time = 0.0618662, antiderivative size = 4, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {453, 206} \[ x+\coth (x) \]
Antiderivative was successfully verified.
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Rule 453
Rule 206
Rubi steps
\begin{align*} \int \left (-1-\frac{1}{1-\coth ^2(x)}\right ) \text{csch}^2(x) \, dx &=-\operatorname{Subst}\left (\int \frac{1-2 x^2}{x^2 \left (1-x^2\right )} \, dx,x,\tanh (x)\right )\\ &=\coth (x)+\operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\tanh (x)\right )\\ &=x+\coth (x)\\ \end{align*}
Mathematica [A] time = 0.0057326, size = 4, normalized size = 1. \[ x+\coth (x) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.03, size = 32, normalized size = 8. \begin{align*}{\frac{1}{2}\tanh \left ({\frac{x}{2}} \right ) }+\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) +{\frac{1}{2} \left ( \tanh \left ({\frac{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 [B] time = 1.06225, size = 16, normalized size = 4. \begin{align*} x - \frac{2}{e^{\left (-2 \, x\right )} - 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.97203, size = 50, normalized size = 12.5 \begin{align*} \frac{{\left (x - 1\right )} \sinh \left (x\right ) + \cosh \left (x\right )}{\sinh \left (x\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{2 \operatorname{csch}^{2}{\left (x \right )}}{\coth ^{2}{\left (x \right )} - 1}\, dx - \int \frac{\coth ^{2}{\left (x \right )} \operatorname{csch}^{2}{\left (x \right )}}{\coth ^{2}{\left (x \right )} - 1}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.11284, size = 16, normalized size = 4. \begin{align*} x + \frac{2}{e^{\left (2 \, x\right )} - 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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