Integrand size = 18, antiderivative size = 37 \[ \int e^{2 \coth ^{-1}(a x)} (c-a c x)^5 \, dx=\frac {2 c^5 (1-a x)^5}{5 a}-\frac {c^5 (1-a x)^6}{6 a} \]
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Time = 0.04 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6302, 6264, 45} \[ \int e^{2 \coth ^{-1}(a x)} (c-a c x)^5 \, dx=\frac {2 c^5 (1-a x)^5}{5 a}-\frac {c^5 (1-a x)^6}{6 a} \]
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Rule 45
Rule 6264
Rule 6302
Rubi steps \begin{align*} \text {integral}& = -\int e^{2 \text {arctanh}(a x)} (c-a c x)^5 \, dx \\ & = -\left (c^5 \int (1-a x)^4 (1+a x) \, dx\right ) \\ & = -\left (c^5 \int \left (2 (1-a x)^4-(1-a x)^5\right ) \, dx\right ) \\ & = \frac {2 c^5 (1-a x)^5}{5 a}-\frac {c^5 (1-a x)^6}{6 a} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.62 \[ \int e^{2 \coth ^{-1}(a x)} (c-a c x)^5 \, dx=-\frac {c^5 (-1+a x)^5 (7+5 a x)}{30 a} \]
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Time = 0.46 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.22
method | result | size |
gosper | \(-\frac {\left (5 a^{5} x^{5}-18 a^{4} x^{4}+15 a^{3} x^{3}+20 a^{2} x^{2}-45 a x +30\right ) c^{5} x}{30}\) | \(45\) |
default | \(c^{5} \left (-\frac {1}{6} a^{5} x^{6}+\frac {3}{5} a^{4} x^{5}-\frac {1}{2} a^{3} x^{4}-\frac {2}{3} a^{2} x^{3}+\frac {3}{2} a \,x^{2}-x \right )\) | \(47\) |
norman | \(-c^{5} x +\frac {3}{2} a \,c^{5} x^{2}-\frac {1}{2} a^{3} c^{5} x^{4}+\frac {3}{5} a^{4} c^{5} x^{5}-\frac {1}{6} a^{5} c^{5} x^{6}-\frac {2}{3} c^{5} a^{2} x^{3}\) | \(61\) |
risch | \(-c^{5} x +\frac {3}{2} a \,c^{5} x^{2}-\frac {1}{2} a^{3} c^{5} x^{4}+\frac {3}{5} a^{4} c^{5} x^{5}-\frac {1}{6} a^{5} c^{5} x^{6}-\frac {2}{3} c^{5} a^{2} x^{3}\) | \(61\) |
parallelrisch | \(-c^{5} x +\frac {3}{2} a \,c^{5} x^{2}-\frac {1}{2} a^{3} c^{5} x^{4}+\frac {3}{5} a^{4} c^{5} x^{5}-\frac {1}{6} a^{5} c^{5} x^{6}-\frac {2}{3} c^{5} a^{2} x^{3}\) | \(61\) |
meijerg | \(-\frac {c^{5} \left (\frac {a x \left (70 a^{5} x^{5}+84 a^{4} x^{4}+105 a^{3} x^{3}+140 a^{2} x^{2}+210 a x +420\right )}{420}+\ln \left (-a x +1\right )\right )}{a}-\frac {4 c^{5} \left (-\frac {a x \left (12 a^{4} x^{4}+15 a^{3} x^{3}+20 a^{2} x^{2}+30 a x +60\right )}{60}-\ln \left (-a x +1\right )\right )}{a}-\frac {5 c^{5} \left (\frac {a x \left (15 a^{3} x^{3}+20 a^{2} x^{2}+30 a x +60\right )}{60}+\ln \left (-a x +1\right )\right )}{a}+\frac {5 c^{5} \left (\frac {a x \left (3 a x +6\right )}{6}+\ln \left (-a x +1\right )\right )}{a}+\frac {4 c^{5} \left (-a x -\ln \left (-a x +1\right )\right )}{a}+\frac {c^{5} \ln \left (-a x +1\right )}{a}\) | \(216\) |
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Time = 0.24 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.62 \[ \int e^{2 \coth ^{-1}(a x)} (c-a c x)^5 \, dx=-\frac {1}{6} \, a^{5} c^{5} x^{6} + \frac {3}{5} \, a^{4} c^{5} x^{5} - \frac {1}{2} \, a^{3} c^{5} x^{4} - \frac {2}{3} \, a^{2} c^{5} x^{3} + \frac {3}{2} \, a c^{5} x^{2} - c^{5} x \]
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Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (27) = 54\).
Time = 0.06 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.78 \[ \int e^{2 \coth ^{-1}(a x)} (c-a c x)^5 \, dx=- \frac {a^{5} c^{5} x^{6}}{6} + \frac {3 a^{4} c^{5} x^{5}}{5} - \frac {a^{3} c^{5} x^{4}}{2} - \frac {2 a^{2} c^{5} x^{3}}{3} + \frac {3 a c^{5} x^{2}}{2} - c^{5} x \]
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Time = 0.19 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.62 \[ \int e^{2 \coth ^{-1}(a x)} (c-a c x)^5 \, dx=-\frac {1}{6} \, a^{5} c^{5} x^{6} + \frac {3}{5} \, a^{4} c^{5} x^{5} - \frac {1}{2} \, a^{3} c^{5} x^{4} - \frac {2}{3} \, a^{2} c^{5} x^{3} + \frac {3}{2} \, a c^{5} x^{2} - c^{5} x \]
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Time = 0.28 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.62 \[ \int e^{2 \coth ^{-1}(a x)} (c-a c x)^5 \, dx=-\frac {1}{6} \, a^{5} c^{5} x^{6} + \frac {3}{5} \, a^{4} c^{5} x^{5} - \frac {1}{2} \, a^{3} c^{5} x^{4} - \frac {2}{3} \, a^{2} c^{5} x^{3} + \frac {3}{2} \, a c^{5} x^{2} - c^{5} x \]
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Time = 0.04 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.62 \[ \int e^{2 \coth ^{-1}(a x)} (c-a c x)^5 \, dx=-\frac {a^5\,c^5\,x^6}{6}+\frac {3\,a^4\,c^5\,x^5}{5}-\frac {a^3\,c^5\,x^4}{2}-\frac {2\,a^2\,c^5\,x^3}{3}+\frac {3\,a\,c^5\,x^2}{2}-c^5\,x \]
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