Optimal. Leaf size=41 \[ 16 x-\frac {1}{4} (\coth (x)+1)^4-\frac {2}{3} (\coth (x)+1)^3-2 (\coth (x)+1)^2-8 \coth (x)+16 \log (\sinh (x)) \]
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Rubi [A] time = 0.04, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3478, 3477, 3475} \[ 16 x-\frac {1}{4} (\coth (x)+1)^4-\frac {2}{3} (\coth (x)+1)^3-2 (\coth (x)+1)^2-8 \coth (x)+16 \log (\sinh (x)) \]
Antiderivative was successfully verified.
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Rule 3475
Rule 3477
Rule 3478
Rubi steps
\begin {align*} \int (1+\coth (x))^5 \, dx &=-\frac {1}{4} (1+\coth (x))^4+2 \int (1+\coth (x))^4 \, dx\\ &=-\frac {2}{3} (1+\coth (x))^3-\frac {1}{4} (1+\coth (x))^4+4 \int (1+\coth (x))^3 \, dx\\ &=-2 (1+\coth (x))^2-\frac {2}{3} (1+\coth (x))^3-\frac {1}{4} (1+\coth (x))^4+8 \int (1+\coth (x))^2 \, dx\\ &=16 x-8 \coth (x)-2 (1+\coth (x))^2-\frac {2}{3} (1+\coth (x))^3-\frac {1}{4} (1+\coth (x))^4+16 \int \coth (x) \, dx\\ &=16 x-8 \coth (x)-2 (1+\coth (x))^2-\frac {2}{3} (1+\coth (x))^3-\frac {1}{4} (1+\coth (x))^4+16 \log (\sinh (x))\\ \end {align*}
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Mathematica [C] time = 0.25, size = 94, normalized size = 2.29 \[ \frac {\sinh (x) (\coth (x)+1)^5 \left (-20 \sinh (x) \cosh ^3(x) \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};\tanh ^2(x)\right )-120 \sinh ^3(x) \cosh (x) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\tanh ^2(x)\right )-3 \cosh ^4(x)-66 \sinh ^2(x) \cosh ^2(x)+12 \sinh ^4(x) (x+16 \log (\tanh (x))+16 \log (\cosh (x)))\right )}{12 (\sinh (x)+\cosh (x))^5} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.39, size = 448, normalized size = 10.93 \[ -\frac {4 \, {\left (48 \, \cosh \relax (x)^{6} + 288 \, \cosh \relax (x) \sinh \relax (x)^{5} + 48 \, \sinh \relax (x)^{6} + 36 \, {\left (20 \, \cosh \relax (x)^{2} - 3\right )} \sinh \relax (x)^{4} - 108 \, \cosh \relax (x)^{4} + 48 \, {\left (20 \, \cosh \relax (x)^{3} - 9 \, \cosh \relax (x)\right )} \sinh \relax (x)^{3} + 8 \, {\left (90 \, \cosh \relax (x)^{4} - 81 \, \cosh \relax (x)^{2} + 11\right )} \sinh \relax (x)^{2} + 88 \, \cosh \relax (x)^{2} - 12 \, {\left (\cosh \relax (x)^{8} + 8 \, \cosh \relax (x) \sinh \relax (x)^{7} + \sinh \relax (x)^{8} + 4 \, {\left (7 \, \cosh \relax (x)^{2} - 1\right )} \sinh \relax (x)^{6} - 4 \, \cosh \relax (x)^{6} + 8 \, {\left (7 \, \cosh \relax (x)^{3} - 3 \, \cosh \relax (x)\right )} \sinh \relax (x)^{5} + 2 \, {\left (35 \, \cosh \relax (x)^{4} - 30 \, \cosh \relax (x)^{2} + 3\right )} \sinh \relax (x)^{4} + 6 \, \cosh \relax (x)^{4} + 8 \, {\left (7 \, \cosh \relax (x)^{5} - 10 \, \cosh \relax (x)^{3} + 3 \, \cosh \relax (x)\right )} \sinh \relax (x)^{3} + 4 \, {\left (7 \, \cosh \relax (x)^{6} - 15 \, \cosh \relax (x)^{4} + 9 \, \cosh \relax (x)^{2} - 1\right )} \sinh \relax (x)^{2} - 4 \, \cosh \relax (x)^{2} + 8 \, {\left (\cosh \relax (x)^{7} - 3 \, \cosh \relax (x)^{5} + 3 \, \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 ) + 16 \, {\left (18 \, \cosh \relax (x)^{5} - 27 \, \cosh \relax (x)^{3} + 11 \, \cosh \relax (x)\right )} \sinh \relax (x) - 25\right )}}{3 \, {\left (\cosh \relax (x)^{8} + 8 \, \cosh \relax (x) \sinh \relax (x)^{7} + \sinh \relax (x)^{8} + 4 \, {\left (7 \, \cosh \relax (x)^{2} - 1\right )} \sinh \relax (x)^{6} - 4 \, \cosh \relax (x)^{6} + 8 \, {\left (7 \, \cosh \relax (x)^{3} - 3 \, \cosh \relax (x)\right )} \sinh \relax (x)^{5} + 2 \, {\left (35 \, \cosh \relax (x)^{4} - 30 \, \cosh \relax (x)^{2} + 3\right )} \sinh \relax (x)^{4} + 6 \, \cosh \relax (x)^{4} + 8 \, {\left (7 \, \cosh \relax (x)^{5} - 10 \, \cosh \relax (x)^{3} + 3 \, \cosh \relax (x)\right )} \sinh \relax (x)^{3} + 4 \, {\left (7 \, \cosh \relax (x)^{6} - 15 \, \cosh \relax (x)^{4} + 9 \, \cosh \relax (x)^{2} - 1\right )} \sinh \relax (x)^{2} - 4 \, \cosh \relax (x)^{2} + 8 \, {\left (\cosh \relax (x)^{7} - 3 \, \cosh \relax (x)^{5} + 3 \, \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 = 41, normalized size = 1.00 \[ -\frac {4 \, {\left (48 \, e^{\left (6 \, x\right )} - 108 \, e^{\left (4 \, x\right )} + 88 \, e^{\left (2 \, x\right )} - 25\right )}}{3 \, {\left (e^{\left (2 \, x\right )} - 1\right )}^{4}} + 16 \, \log \left ({\left | e^{\left (2 \, x\right )} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 31, normalized size = 0.76 \[ -\frac {\left (\coth ^{4}\relax (x )\right )}{4}-\frac {5 \left (\coth ^{3}\relax (x )\right )}{3}-\frac {11 \left (\coth ^{2}\relax (x )\right )}{2}-15 \coth \relax (x )-16 \ln \left (\coth \relax (x )-1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.33, size = 140, normalized size = 3.41 \[ 27 \, x - \frac {20 \, {\left (3 \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-4 \, x\right )} - 2\right )}}{3 \, {\left (3 \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} - 1\right )}} + \frac {4 \, {\left (e^{\left (-2 \, x\right )} - e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )}\right )}}{4 \, e^{\left (-2 \, x\right )} - 6 \, e^{\left (-4 \, x\right )} + 4 \, e^{\left (-6 \, x\right )} - e^{\left (-8 \, x\right )} - 1} + \frac {20 \, e^{\left (-2 \, x\right )}}{2 \, e^{\left (-2 \, x\right )} - e^{\left (-4 \, x\right )} - 1} + \frac {20}{e^{\left (-2 \, x\right )} - 1} + 11 \, \log \left (e^{\left (-x\right )} + 1\right ) + 11 \, \log \left (e^{\left (-x\right )} - 1\right ) + 5 \, \log \left (\sinh \relax (x)\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.13, size = 88, normalized size = 2.15 \[ 16\,\ln \left ({\mathrm {e}}^{2\,x}-1\right )-\frac {64}{3\,\left (3\,{\mathrm {e}}^{2\,x}-3\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}-1\right )}-\frac {48}{{\mathrm {e}}^{4\,x}-2\,{\mathrm {e}}^{2\,x}+1}-\frac {4}{6\,{\mathrm {e}}^{4\,x}-4\,{\mathrm {e}}^{2\,x}-4\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1}-\frac {64}{{\mathrm {e}}^{2\,x}-1} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.22, size = 48, normalized size = 1.17 \[ 32 x - 16 \log {\left (\tanh {\relax (x )} + 1 \right )} + 16 \log {\left (\tanh {\relax (x )} \right )} - \frac {15}{\tanh {\relax (x )}} - \frac {11}{2 \tanh ^{2}{\relax (x )}} - \frac {5}{3 \tanh ^{3}{\relax (x )}} - \frac {1}{4 \tanh ^{4}{\relax (x )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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