Optimal. Leaf size=16 \[ \frac{x}{2}-\frac{1}{2 (\coth (x)+1)} \]
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Rubi [A] time = 0.0086758, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {3479, 8} \[ \frac{x}{2}-\frac{1}{2 (\coth (x)+1)} \]
Antiderivative was successfully verified.
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Rule 3479
Rule 8
Rubi steps
\begin{align*} \int \frac{1}{1+\coth (x)} \, dx &=-\frac{1}{2 (1+\coth (x))}+\frac{\int 1 \, dx}{2}\\ &=\frac{x}{2}-\frac{1}{2 (1+\coth (x))}\\ \end{align*}
Mathematica [A] time = 0.0309119, size = 18, normalized size = 1.12 \[ \frac{1}{4} (2 x-\sinh (2 x)+\cosh (2 x)) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.016, size = 24, normalized size = 1.5 \begin{align*} -{\frac{1}{2+2\,{\rm coth} \left (x\right )}}+{\frac{\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 = 1.01401, size = 14, normalized size = 0.88 \begin{align*} \frac{1}{2} \, x + \frac{1}{4} \, e^{\left (-2 \, x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.01175, size = 88, normalized size = 5.5 \begin{align*} \frac{{\left (2 \, x + 1\right )} \cosh \left (x\right ) +{\left (2 \, x - 1\right )} \sinh \left (x\right )}{4 \,{\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.57725, size = 27, normalized size = 1.69 \begin{align*} \frac{x \tanh{\left (x \right )}}{2 \tanh{\left (x \right )} + 2} + \frac{x}{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 = 1.14728, size = 14, normalized size = 0.88 \begin{align*} \frac{1}{2} \, x + \frac{1}{4} \, e^{\left (-2 \, x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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