Integrand size = 6, antiderivative size = 41 \[ \int (1+\coth (x))^5 \, dx=16 x-8 \coth (x)-2 (1+\coth (x))^2-\frac {2}{3} (1+\coth (x))^3-\frac {1}{4} (1+\coth (x))^4+16 \log (\sinh (x)) \]
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Time = 0.03 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3559, 3558, 3556} \[ \int (1+\coth (x))^5 \, dx=16 x-\frac {1}{4} (\coth (x)+1)^4-\frac {2}{3} (\coth (x)+1)^3-2 (\coth (x)+1)^2-8 \coth (x)+16 \log (\sinh (x)) \]
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Rule 3556
Rule 3558
Rule 3559
Rubi steps \begin{align*} \text {integral}& = -\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*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.27 (sec) , antiderivative size = 94, normalized size of antiderivative = 2.29 \[ \int (1+\coth (x))^5 \, dx=\frac {(1+\coth (x))^5 \sinh (x) \left (-3 \cosh ^4(x)-20 \cosh ^3(x) \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},\tanh ^2(x)\right ) \sinh (x)-66 \cosh ^2(x) \sinh ^2(x)-120 \cosh (x) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},\tanh ^2(x)\right ) \sinh ^3(x)+12 (x+16 \log (\cosh (x))+16 \log (\tanh (x))) \sinh ^4(x)\right )}{12 (\cosh (x)+\sinh (x))^5} \]
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Time = 0.10 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.76
method | result | size |
derivativedivides | \(-\frac {\coth \left (x \right )^{4}}{4}-\frac {5 \coth \left (x \right )^{3}}{3}-\frac {11 \coth \left (x \right )^{2}}{2}-15 \coth \left (x \right )-16 \ln \left (\coth \left (x \right )-1\right )\) | \(31\) |
default | \(-\frac {\coth \left (x \right )^{4}}{4}-\frac {5 \coth \left (x \right )^{3}}{3}-\frac {11 \coth \left (x \right )^{2}}{2}-15 \coth \left (x \right )-16 \ln \left (\coth \left (x \right )-1\right )\) | \(31\) |
parallelrisch | \(-\frac {\coth \left (x \right )^{4}}{4}+16 \ln \left (\tanh \left (x \right )\right )-16 \ln \left (1-\tanh \left (x \right )\right )-15 \coth \left (x \right )-\frac {11 \coth \left (x \right )^{2}}{2}-\frac {5 \coth \left (x \right )^{3}}{3}\) | \(38\) |
risch | \(-\frac {4 \left (48 \,{\mathrm e}^{6 x}-108 \,{\mathrm e}^{4 x}+88 \,{\mathrm e}^{2 x}-25\right )}{3 \left ({\mathrm e}^{2 x}-1\right )^{4}}+16 \ln \left ({\mathrm e}^{2 x}-1\right )\) | \(41\) |
parts | \(x -\frac {\coth \left (x \right )^{4}}{4}-\frac {11 \coth \left (x \right )^{2}}{2}-13 \ln \left (\coth \left (x \right )-1\right )+2 \ln \left (1+\coth \left (x \right )\right )-15 \coth \left (x \right )-\frac {5 \coth \left (x \right )^{3}}{3}+5 \ln \left (\sinh \left (x \right )\right )\) | \(44\) |
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Leaf count of result is larger than twice the leaf count of optimal. 448 vs. \(2 (37) = 74\).
Time = 0.26 (sec) , antiderivative size = 448, normalized size of antiderivative = 10.93 \[ \int (1+\coth (x))^5 \, dx=-\frac {4 \, {\left (48 \, \cosh \left (x\right )^{6} + 288 \, \cosh \left (x\right ) \sinh \left (x\right )^{5} + 48 \, \sinh \left (x\right )^{6} + 36 \, {\left (20 \, \cosh \left (x\right )^{2} - 3\right )} \sinh \left (x\right )^{4} - 108 \, \cosh \left (x\right )^{4} + 48 \, {\left (20 \, \cosh \left (x\right )^{3} - 9 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 8 \, {\left (90 \, \cosh \left (x\right )^{4} - 81 \, \cosh \left (x\right )^{2} + 11\right )} \sinh \left (x\right )^{2} + 88 \, \cosh \left (x\right )^{2} - 12 \, {\left (\cosh \left (x\right )^{8} + 8 \, \cosh \left (x\right ) \sinh \left (x\right )^{7} + \sinh \left (x\right )^{8} + 4 \, {\left (7 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right )^{6} - 4 \, \cosh \left (x\right )^{6} + 8 \, {\left (7 \, \cosh \left (x\right )^{3} - 3 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{5} + 2 \, {\left (35 \, \cosh \left (x\right )^{4} - 30 \, \cosh \left (x\right )^{2} + 3\right )} \sinh \left (x\right )^{4} + 6 \, \cosh \left (x\right )^{4} + 8 \, {\left (7 \, \cosh \left (x\right )^{5} - 10 \, \cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 4 \, {\left (7 \, \cosh \left (x\right )^{6} - 15 \, \cosh \left (x\right )^{4} + 9 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right )^{2} - 4 \, \cosh \left (x\right )^{2} + 8 \, {\left (\cosh \left (x\right )^{7} - 3 \, \cosh \left (x\right )^{5} + 3 \, \cosh \left (x\right )^{3} - \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1\right )} \log \left (\frac {2 \, \sinh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) + 16 \, {\left (18 \, \cosh \left (x\right )^{5} - 27 \, \cosh \left (x\right )^{3} + 11 \, \cosh \left (x\right )\right )} \sinh \left (x\right ) - 25\right )}}{3 \, {\left (\cosh \left (x\right )^{8} + 8 \, \cosh \left (x\right ) \sinh \left (x\right )^{7} + \sinh \left (x\right )^{8} + 4 \, {\left (7 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right )^{6} - 4 \, \cosh \left (x\right )^{6} + 8 \, {\left (7 \, \cosh \left (x\right )^{3} - 3 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{5} + 2 \, {\left (35 \, \cosh \left (x\right )^{4} - 30 \, \cosh \left (x\right )^{2} + 3\right )} \sinh \left (x\right )^{4} + 6 \, \cosh \left (x\right )^{4} + 8 \, {\left (7 \, \cosh \left (x\right )^{5} - 10 \, \cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 4 \, {\left (7 \, \cosh \left (x\right )^{6} - 15 \, \cosh \left (x\right )^{4} + 9 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right )^{2} - 4 \, \cosh \left (x\right )^{2} + 8 \, {\left (\cosh \left (x\right )^{7} - 3 \, \cosh \left (x\right )^{5} + 3 \, \cosh \left (x\right )^{3} - \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1\right )}} \]
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Time = 0.67 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.17 \[ \int (1+\coth (x))^5 \, dx=32 x - 16 \log {\left (\tanh {\left (x \right )} + 1 \right )} + 16 \log {\left (\tanh {\left (x \right )} \right )} - \frac {15}{\tanh {\left (x \right )}} - \frac {11}{2 \tanh ^{2}{\left (x \right )}} - \frac {5}{3 \tanh ^{3}{\left (x \right )}} - \frac {1}{4 \tanh ^{4}{\left (x \right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 140 vs. \(2 (37) = 74\).
Time = 0.20 (sec) , antiderivative size = 140, normalized size of antiderivative = 3.41 \[ \int (1+\coth (x))^5 \, dx=27 \, x - \frac {20 \, {\left (3 \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-4 \, x\right )} - 2\right )}}{3 \, {\left (3 \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} - 1\right )}} + \frac {4 \, {\left (e^{\left (-2 \, x\right )} - e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )}\right )}}{4 \, e^{\left (-2 \, x\right )} - 6 \, e^{\left (-4 \, x\right )} + 4 \, e^{\left (-6 \, x\right )} - e^{\left (-8 \, x\right )} - 1} + \frac {20 \, e^{\left (-2 \, x\right )}}{2 \, e^{\left (-2 \, x\right )} - e^{\left (-4 \, x\right )} - 1} + \frac {20}{e^{\left (-2 \, x\right )} - 1} + 11 \, \log \left (e^{\left (-x\right )} + 1\right ) + 11 \, \log \left (e^{\left (-x\right )} - 1\right ) + 5 \, \log \left (\sinh \left (x\right )\right ) \]
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Time = 0.28 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00 \[ \int (1+\coth (x))^5 \, dx=-\frac {4 \, {\left (48 \, e^{\left (6 \, x\right )} - 108 \, e^{\left (4 \, x\right )} + 88 \, e^{\left (2 \, x\right )} - 25\right )}}{3 \, {\left (e^{\left (2 \, x\right )} - 1\right )}^{4}} + 16 \, \log \left ({\left | e^{\left (2 \, x\right )} - 1 \right |}\right ) \]
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Time = 1.87 (sec) , antiderivative size = 88, normalized size of antiderivative = 2.15 \[ \int (1+\coth (x))^5 \, dx=16\,\ln \left ({\mathrm {e}}^{2\,x}-1\right )-\frac {64}{3\,\left (3\,{\mathrm {e}}^{2\,x}-3\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}-1\right )}-\frac {48}{{\mathrm {e}}^{4\,x}-2\,{\mathrm {e}}^{2\,x}+1}-\frac {4}{6\,{\mathrm {e}}^{4\,x}-4\,{\mathrm {e}}^{2\,x}-4\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1}-\frac {64}{{\mathrm {e}}^{2\,x}-1} \]
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