Optimal. Leaf size=31 \[ 8 x-\frac {1}{3} (\coth (x)+1)^3-(\coth (x)+1)^2-4 \coth (x)+8 \log (\sinh (x)) \]
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Rubi [A] time = 0.03, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3478, 3477, 3475} \[ 8 x-\frac {1}{3} (\coth (x)+1)^3-(\coth (x)+1)^2-4 \coth (x)+8 \log (\sinh (x)) \]
Antiderivative was successfully verified.
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Rule 3475
Rule 3477
Rule 3478
Rubi steps
\begin {align*} \int (1+\coth (x))^4 \, dx &=-\frac {1}{3} (1+\coth (x))^3+2 \int (1+\coth (x))^3 \, dx\\ &=-(1+\coth (x))^2-\frac {1}{3} (1+\coth (x))^3+4 \int (1+\coth (x))^2 \, dx\\ &=8 x-4 \coth (x)-(1+\coth (x))^2-\frac {1}{3} (1+\coth (x))^3+8 \int \coth (x) \, dx\\ &=8 x-4 \coth (x)-(1+\coth (x))^2-\frac {1}{3} (1+\coth (x))^3+8 \log (\sinh (x))\\ \end {align*}
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Mathematica [C] time = 0.19, size = 84, normalized size = 2.71 \[ \frac {\sinh (x) (\coth (x)+1)^4 \left (3 \sinh (x) \left (-6 \sinh (x) \cosh (x) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\tanh ^2(x)\right )-2 \cosh ^2(x)+\sinh ^2(x) (x+8 \log (\tanh (x))+8 \log (\cosh (x)))\right )-\cosh ^3(x) \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};\tanh ^2(x)\right )\right )}{3 (\sinh (x)+\cosh (x))^4} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.40, size = 273, normalized size = 8.81 \[ -\frac {4 \, {\left (18 \, \cosh \relax (x)^{4} + 72 \, \cosh \relax (x) \sinh \relax (x)^{3} + 18 \, \sinh \relax (x)^{4} + 27 \, {\left (4 \, \cosh \relax (x)^{2} - 1\right )} \sinh \relax (x)^{2} - 27 \, \cosh \relax (x)^{2} - 6 \, {\left (\cosh \relax (x)^{6} + 6 \, \cosh \relax (x) \sinh \relax (x)^{5} + \sinh \relax (x)^{6} + 3 \, {\left (5 \, \cosh \relax (x)^{2} - 1\right )} \sinh \relax (x)^{4} - 3 \, \cosh \relax (x)^{4} + 4 \, {\left (5 \, \cosh \relax (x)^{3} - 3 \, \cosh \relax (x)\right )} \sinh \relax (x)^{3} + 3 \, {\left (5 \, \cosh \relax (x)^{4} - 6 \, \cosh \relax (x)^{2} + 1\right )} \sinh \relax (x)^{2} + 3 \, \cosh \relax (x)^{2} + 6 \, {\left (\cosh \relax (x)^{5} - 2 \, \cosh \relax (x)^{3} + \cosh \relax (x)\right )} \sinh \relax (x) - 1\right )} \log \left (\frac {2 \, \sinh \relax (x)}{\cosh \relax (x) - \sinh \relax (x)}\right ) + 18 \, {\left (4 \, \cosh \relax (x)^{3} - 3 \, \cosh \relax (x)\right )} \sinh \relax (x) + 11\right )}}{3 \, {\left (\cosh \relax (x)^{6} + 6 \, \cosh \relax (x) \sinh \relax (x)^{5} + \sinh \relax (x)^{6} + 3 \, {\left (5 \, \cosh \relax (x)^{2} - 1\right )} \sinh \relax (x)^{4} - 3 \, \cosh \relax (x)^{4} + 4 \, {\left (5 \, \cosh \relax (x)^{3} - 3 \, \cosh \relax (x)\right )} \sinh \relax (x)^{3} + 3 \, {\left (5 \, \cosh \relax (x)^{4} - 6 \, \cosh \relax (x)^{2} + 1\right )} \sinh \relax (x)^{2} + 3 \, \cosh \relax (x)^{2} + 6 \, {\left (\cosh \relax (x)^{5} - 2 \, \cosh \relax (x)^{3} + \cosh \relax (x)\right )} \sinh \relax (x) - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.11, size = 35, normalized size = 1.13 \[ -\frac {4 \, {\left (18 \, e^{\left (4 \, x\right )} - 27 \, e^{\left (2 \, x\right )} + 11\right )}}{3 \, {\left (e^{\left (2 \, x\right )} - 1\right )}^{3}} + 8 \, \log \left ({\left | e^{\left (2 \, x\right )} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 25, normalized size = 0.81 \[ -\frac {\left (\coth ^{3}\relax (x )\right )}{3}-2 \left (\coth ^{2}\relax (x )\right )-7 \coth \relax (x )-8 \ln \left (\coth \relax (x )-1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.32, size = 95, normalized size = 3.06 \[ 12 \, x - \frac {4 \, {\left (3 \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-4 \, x\right )} - 2\right )}}{3 \, {\left (3 \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} - 1\right )}} + \frac {8 \, e^{\left (-2 \, x\right )}}{2 \, e^{\left (-2 \, x\right )} - e^{\left (-4 \, x\right )} - 1} + \frac {12}{e^{\left (-2 \, x\right )} - 1} + 4 \, \log \left (e^{\left (-x\right )} + 1\right ) + 4 \, \log \left (e^{\left (-x\right )} - 1\right ) + 4 \, \log \left (\sinh \relax (x)\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.14, size = 60, normalized size = 1.94 \[ 8\,\ln \left ({\mathrm {e}}^{2\,x}-1\right )-\frac {8}{3\,\left (3\,{\mathrm {e}}^{2\,x}-3\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}-1\right )}-\frac {12}{{\mathrm {e}}^{4\,x}-2\,{\mathrm {e}}^{2\,x}+1}-\frac {24}{{\mathrm {e}}^{2\,x}-1} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.92, size = 37, normalized size = 1.19 \[ 16 x - 8 \log {\left (\tanh {\relax (x )} + 1 \right )} + 8 \log {\left (\tanh {\relax (x )} \right )} - \frac {7}{\tanh {\relax (x )}} - \frac {2}{\tanh ^{2}{\relax (x )}} - \frac {1}{3 \tanh ^{3}{\relax (x )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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