Optimal. Leaf size=36 \[ \frac {x}{8}-\frac {1}{8 (\coth (x)+1)}-\frac {1}{8 (\coth (x)+1)^2}-\frac {1}{6 (\coth (x)+1)^3} \]
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Rubi [A] time = 0.03, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3479, 8} \[ \frac {x}{8}-\frac {1}{8 (\coth (x)+1)}-\frac {1}{8 (\coth (x)+1)^2}-\frac {1}{6 (\coth (x)+1)^3} \]
Antiderivative was successfully verified.
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Rule 8
Rule 3479
Rubi steps
\begin {align*} \int \frac {1}{(1+\coth (x))^3} \, dx &=-\frac {1}{6 (1+\coth (x))^3}+\frac {1}{2} \int \frac {1}{(1+\coth (x))^2} \, dx\\ &=-\frac {1}{6 (1+\coth (x))^3}-\frac {1}{8 (1+\coth (x))^2}+\frac {1}{4} \int \frac {1}{1+\coth (x)} \, dx\\ &=-\frac {1}{6 (1+\coth (x))^3}-\frac {1}{8 (1+\coth (x))^2}-\frac {1}{8 (1+\coth (x))}+\frac {\int 1 \, dx}{8}\\ &=\frac {x}{8}-\frac {1}{6 (1+\coth (x))^3}-\frac {1}{8 (1+\coth (x))^2}-\frac {1}{8 (1+\coth (x))}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 44, normalized size = 1.22 \[ \frac {1}{96} (12 x-18 \sinh (2 x)+9 \sinh (4 x)-2 \sinh (6 x)+18 \cosh (2 x)-9 \cosh (4 x)+2 \cosh (6 x)) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.38, size = 86, normalized size = 2.39 \[ \frac {2 \, {\left (6 \, x + 1\right )} \cosh \relax (x)^{3} + 6 \, {\left (6 \, x + 1\right )} \cosh \relax (x) \sinh \relax (x)^{2} + 2 \, {\left (6 \, x - 1\right )} \sinh \relax (x)^{3} + 3 \, {\left (2 \, {\left (6 \, x - 1\right )} \cosh \relax (x)^{2} + 9\right )} \sinh \relax (x) + 9 \, \cosh \relax (x)}{96 \, {\left (\cosh \relax (x)^{3} + 3 \, \cosh \relax (x)^{2} \sinh \relax (x) + 3 \, \cosh \relax (x) \sinh \relax (x)^{2} + \sinh \relax (x)^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 24, normalized size = 0.67 \[ \frac {1}{96} \, {\left (18 \, e^{\left (4 \, x\right )} - 9 \, e^{\left (2 \, x\right )} + 2\right )} e^{\left (-6 \, x\right )} + \frac {1}{8} \, x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 40, normalized size = 1.11 \[ -\frac {\ln \left (\coth \relax (x )-1\right )}{16}-\frac {1}{6 \left (1+\coth \relax (x )\right )^{3}}-\frac {1}{8 \left (1+\coth \relax (x )\right )^{2}}-\frac {1}{8 \left (1+\coth \relax (x )\right )}+\frac {\ln \left (1+\coth \relax (x )\right )}{16} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 22, normalized size = 0.61 \[ \frac {1}{8} \, x + \frac {3}{16} \, e^{\left (-2 \, x\right )} - \frac {3}{32} \, e^{\left (-4 \, x\right )} + \frac {1}{48} \, e^{\left (-6 \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 22, normalized size = 0.61 \[ \frac {x}{8}+\frac {3\,{\mathrm {e}}^{-2\,x}}{16}-\frac {3\,{\mathrm {e}}^{-4\,x}}{32}+\frac {{\mathrm {e}}^{-6\,x}}{48} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.05, size = 182, normalized size = 5.06 \[ \frac {3 x \tanh ^{3}{\relax (x )}}{24 \tanh ^{3}{\relax (x )} + 72 \tanh ^{2}{\relax (x )} + 72 \tanh {\relax (x )} + 24} + \frac {9 x \tanh ^{2}{\relax (x )}}{24 \tanh ^{3}{\relax (x )} + 72 \tanh ^{2}{\relax (x )} + 72 \tanh {\relax (x )} + 24} + \frac {9 x \tanh {\relax (x )}}{24 \tanh ^{3}{\relax (x )} + 72 \tanh ^{2}{\relax (x )} + 72 \tanh {\relax (x )} + 24} + \frac {3 x}{24 \tanh ^{3}{\relax (x )} + 72 \tanh ^{2}{\relax (x )} + 72 \tanh {\relax (x )} + 24} + \frac {21 \tanh ^{2}{\relax (x )}}{24 \tanh ^{3}{\relax (x )} + 72 \tanh ^{2}{\relax (x )} + 72 \tanh {\relax (x )} + 24} + \frac {27 \tanh {\relax (x )}}{24 \tanh ^{3}{\relax (x )} + 72 \tanh ^{2}{\relax (x )} + 72 \tanh {\relax (x )} + 24} + \frac {10}{24 \tanh ^{3}{\relax (x )} + 72 \tanh ^{2}{\relax (x )} + 72 \tanh {\relax (x )} + 24} \]
Verification of antiderivative is not currently implemented for this CAS.
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