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 = 0.02, antiderivative size = 26, normalized size of antiderivative = 1.00, 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 8
Rule 3479
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*}
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Mathematica [A] time = 0.06, 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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fricas [B] time = 0.38, size = 52, normalized size = 2.00 \[ \frac {{\left (4 \, x - 1\right )} \cosh \relax (x)^{2} + 2 \, {\left (4 \, x + 1\right )} \cosh \relax (x) \sinh \relax (x) + {\left (4 \, x - 1\right )} \sinh \relax (x)^{2} + 4}{16 \, {\left (\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.11, size = 18, normalized size = 0.69 \[ \frac {1}{16} \, {\left (4 \, e^{\left (2 \, x\right )} - 1\right )} e^{\left (-4 \, x\right )} + \frac {1}{4} \, x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 32, normalized size = 1.23 \[ -\frac {\ln \left (\coth \relax (x )-1\right )}{8}-\frac {1}{4 \left (1+\coth \relax (x )\right )^{2}}-\frac {1}{4 \left (1+\coth \relax (x )\right )}+\frac {\ln \left (1+\coth \relax (x )\right )}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 16, normalized size = 0.62 \[ \frac {1}{4} \, x + \frac {1}{4} \, e^{\left (-2 \, x\right )} - \frac {1}{16} \, e^{\left (-4 \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 16, normalized size = 0.62 \[ \frac {x}{4}+\frac {{\mathrm {e}}^{-2\,x}}{4}-\frac {{\mathrm {e}}^{-4\,x}}{16} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.81, size = 88, normalized size = 3.38 \[ \frac {x \tanh ^{2}{\relax (x )}}{4 \tanh ^{2}{\relax (x )} + 8 \tanh {\relax (x )} + 4} + \frac {2 x \tanh {\relax (x )}}{4 \tanh ^{2}{\relax (x )} + 8 \tanh {\relax (x )} + 4} + \frac {x}{4 \tanh ^{2}{\relax (x )} + 8 \tanh {\relax (x )} + 4} + \frac {3 \tanh {\relax (x )}}{4 \tanh ^{2}{\relax (x )} + 8 \tanh {\relax (x )} + 4} + \frac {2}{4 \tanh ^{2}{\relax (x )} + 8 \tanh {\relax (x )} + 4} \]
Verification of antiderivative is not currently implemented for this CAS.
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