Optimal. Leaf size=19 \[ -\frac{x}{2}-\frac{1}{2 (\coth (x)+1)}+\log (\sinh (x)) \]
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Rubi [A] time = 0.0382739, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {3540, 3475} \[ -\frac{x}{2}-\frac{1}{2 (\coth (x)+1)}+\log (\sinh (x)) \]
Antiderivative was successfully verified.
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Rule 3540
Rule 3475
Rubi steps
\begin{align*} \int \frac{\coth ^2(x)}{1+\coth (x)} \, dx &=-\frac{1}{2 (1+\coth (x))}-\frac{1}{2} \int (1-2 \coth (x)) \, dx\\ &=-\frac{x}{2}-\frac{1}{2 (1+\coth (x))}+\int \coth (x) \, dx\\ &=-\frac{x}{2}-\frac{1}{2 (1+\coth (x))}+\log (\sinh (x))\\ \end{align*}
Mathematica [A] time = 0.0285523, size = 23, normalized size = 1.21 \[ \frac{1}{4} (-2 x-\sinh (2 x)+\cosh (2 x)+4 \log (\sinh (x))) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 24, normalized size = 1.3 \begin{align*} -{\frac{1}{2+2\,{\rm coth} \left (x\right )}}-{\frac{3\,\ln \left ( 1+{\rm coth} \left (x\right ) \right ) }{4}}-{\frac{\ln \left ({\rm coth} \left (x\right )-1 \right ) }{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.10925, size = 32, normalized size = 1.68 \begin{align*} \frac{1}{2} \, x + \frac{1}{4} \, e^{\left (-2 \, x\right )} + \log \left (e^{\left (-x\right )} + 1\right ) + \log \left (e^{\left (-x\right )} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 3.09236, size = 259, normalized size = 13.63 \begin{align*} -\frac{6 \, x \cosh \left (x\right )^{2} + 12 \, x \cosh \left (x\right ) \sinh \left (x\right ) + 6 \, x \sinh \left (x\right )^{2} - 4 \,{\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}\right )} \log \left (\frac{2 \, \sinh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) - 1}{4 \,{\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.905729, size = 92, normalized size = 4.84 \begin{align*} \frac{x \tanh{\left (x \right )}}{2 \tanh{\left (x \right )} + 2} + \frac{x}{2 \tanh{\left (x \right )} + 2} - \frac{2 \log{\left (\tanh{\left (x \right )} + 1 \right )} \tanh{\left (x \right )}}{2 \tanh{\left (x \right )} + 2} - \frac{2 \log{\left (\tanh{\left (x \right )} + 1 \right )}}{2 \tanh{\left (x \right )} + 2} + \frac{2 \log{\left (\tanh{\left (x \right )} \right )} \tanh{\left (x \right )}}{2 \tanh{\left (x \right )} + 2} + \frac{2 \log{\left (\tanh{\left (x \right )} \right )}}{2 \tanh{\left (x \right )} + 2} + \frac{1}{2 \tanh{\left (x \right )} + 2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12442, size = 24, normalized size = 1.26 \begin{align*} -\frac{3}{2} \, x + \frac{1}{4} \, e^{\left (-2 \, x\right )} + \log \left ({\left | e^{\left (2 \, x\right )} - 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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