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 = 0.0394656, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, 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 3478
Rule 3477
Rule 3475
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*}
Mathematica [C] time = 0.238116, 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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Maple [A] time = 0.007, size = 31, normalized size = 0.8 \begin{align*} -{\frac{ \left ({\rm coth} \left (x\right ) \right ) ^{4}}{4}}-{\frac{5\, \left ({\rm coth} \left (x\right ) \right ) ^{3}}{3}}-{\frac{11\, \left ({\rm coth} \left (x\right ) \right ) ^{2}}{2}}-15\,{\rm coth} \left (x\right )-16\,\ln \left ({\rm coth} \left (x\right )-1 \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.0613, size = 189, normalized size = 4.61 \begin{align*} 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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.04947, size = 1481, normalized size = 36.12 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.58923, size = 48, normalized size = 1.17 \begin{align*} 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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18271, size = 55, normalized size = 1.34 \begin{align*} -\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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