Optimal. Leaf size=16 \[ \frac {x}{2}-\frac {1}{2 (\coth (x)+1)} \]
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Rubi [A] time = 0.01, antiderivative size = 16, normalized size of antiderivative = 1.00, 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 8
Rule 3479
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*}
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Mathematica [A] time = 0.03, 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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fricas [B] time = 0.41, size = 26, normalized size = 1.62 \[ \frac {{\left (2 \, x + 1\right )} \cosh \relax (x) + {\left (2 \, x - 1\right )} \sinh \relax (x)}{4 \, {\left (\cosh \relax (x) + \sinh \relax (x)\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 10, normalized size = 0.62 \[ \frac {1}{2} \, x + \frac {1}{4} \, e^{\left (-2 \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 24, normalized size = 1.50 \[ -\frac {\ln \left (\coth \relax (x )-1\right )}{4}-\frac {1}{2 \left (1+\coth \relax (x )\right )}+\frac {\ln \left (1+\coth \relax (x )\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 10, normalized size = 0.62 \[ \frac {1}{2} \, x + \frac {1}{4} \, e^{\left (-2 \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.00, size = 14, normalized size = 0.88 \[ \frac {x}{2}-\frac {1}{2\,\left (\mathrm {coth}\relax (x)+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.45, size = 27, normalized size = 1.69 \[ \frac {x \tanh {\relax (x )}}{2 \tanh {\relax (x )} + 2} + \frac {x}{2 \tanh {\relax (x )} + 2} + \frac {1}{2 \tanh {\relax (x )} + 2} \]
Verification of antiderivative is not currently implemented for this CAS.
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