Optimal. Leaf size=26 \[ \frac{x}{4}-\frac{1}{4 (\coth (x)+1)}-\frac{1}{4 (\coth (x)+1)^2} \]
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Rubi [A] time = 0.0169901, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {3479, 8} \[ \frac{x}{4}-\frac{1}{4 (\coth (x)+1)}-\frac{1}{4 (\coth (x)+1)^2} \]
Antiderivative was successfully verified.
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Rule 3479
Rule 8
Rubi steps
\begin{align*} \int \frac{1}{(1+\coth (x))^2} \, dx &=-\frac{1}{4 (1+\coth (x))^2}+\frac{1}{2} \int \frac{1}{1+\coth (x)} \, dx\\ &=-\frac{1}{4 (1+\coth (x))^2}-\frac{1}{4 (1+\coth (x))}+\frac{\int 1 \, dx}{4}\\ &=\frac{x}{4}-\frac{1}{4 (1+\coth (x))^2}-\frac{1}{4 (1+\coth (x))}\\ \end{align*}
Mathematica [A] time = 0.0576583, size = 30, normalized size = 1.15 \[ \frac{1}{16} (4 x-4 \sinh (2 x)+\sinh (4 x)+4 \cosh (2 x)-\cosh (4 x)) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 32, normalized size = 1.2 \begin{align*} -{\frac{1}{4\, \left ( 1+{\rm coth} \left (x\right ) \right ) ^{2}}}-{\frac{1}{4+4\,{\rm coth} \left (x\right )}}+{\frac{\ln \left ( 1+{\rm coth} \left (x\right ) \right ) }{8}}-{\frac{\ln \left ({\rm coth} \left (x\right )-1 \right ) }{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02873, size = 22, normalized size = 0.85 \begin{align*} \frac{1}{4} \, x + \frac{1}{4} \, e^{\left (-2 \, x\right )} - \frac{1}{16} \, e^{\left (-4 \, x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.39471, size = 173, normalized size = 6.65 \begin{align*} \frac{{\left (4 \, x - 1\right )} \cosh \left (x\right )^{2} + 2 \,{\left (4 \, x + 1\right )} \cosh \left (x\right ) \sinh \left (x\right ) +{\left (4 \, x - 1\right )} \sinh \left (x\right )^{2} + 4}{16 \,{\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 = 0.990182, size = 88, normalized size = 3.38 \begin{align*} \frac{x \tanh ^{2}{\left (x \right )}}{4 \tanh ^{2}{\left (x \right )} + 8 \tanh{\left (x \right )} + 4} + \frac{2 x \tanh{\left (x \right )}}{4 \tanh ^{2}{\left (x \right )} + 8 \tanh{\left (x \right )} + 4} + \frac{x}{4 \tanh ^{2}{\left (x \right )} + 8 \tanh{\left (x \right )} + 4} + \frac{3 \tanh{\left (x \right )}}{4 \tanh ^{2}{\left (x \right )} + 8 \tanh{\left (x \right )} + 4} + \frac{2}{4 \tanh ^{2}{\left (x \right )} + 8 \tanh{\left (x \right )} + 4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15641, size = 24, normalized size = 0.92 \begin{align*} \frac{1}{16} \,{\left (4 \, e^{\left (2 \, x\right )} - 1\right )} e^{\left (-4 \, x\right )} + \frac{1}{4} \, x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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