Integrand size = 22, antiderivative size = 84 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^5 \, dx=-\frac {16 c^5 (1+a x)^7}{7 a}+\frac {4 c^5 (1+a x)^8}{a}-\frac {8 c^5 (1+a x)^9}{3 a}+\frac {4 c^5 (1+a x)^{10}}{5 a}-\frac {c^5 (1+a x)^{11}}{11 a} \]
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Time = 0.08 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {6302, 6275, 45} \[ \int e^{2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^5 \, dx=-\frac {c^5 (a x+1)^{11}}{11 a}+\frac {4 c^5 (a x+1)^{10}}{5 a}-\frac {8 c^5 (a x+1)^9}{3 a}+\frac {4 c^5 (a x+1)^8}{a}-\frac {16 c^5 (a x+1)^7}{7 a} \]
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Rule 45
Rule 6275
Rule 6302
Rubi steps \begin{align*} \text {integral}& = -\int e^{2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^5 \, dx \\ & = -\left (c^5 \int (1-a x)^4 (1+a x)^6 \, dx\right ) \\ & = -\left (c^5 \int \left (16 (1+a x)^6-32 (1+a x)^7+24 (1+a x)^8-8 (1+a x)^9+(1+a x)^{10}\right ) \, dx\right ) \\ & = -\frac {16 c^5 (1+a x)^7}{7 a}+\frac {4 c^5 (1+a x)^8}{a}-\frac {8 c^5 (1+a x)^9}{3 a}+\frac {4 c^5 (1+a x)^{10}}{5 a}-\frac {c^5 (1+a x)^{11}}{11 a} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.56 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^5 \, dx=-\frac {c^5 (1+a x)^7 \left (281-812 a x+938 a^2 x^2-504 a^3 x^3+105 a^4 x^4\right )}{1155 a} \]
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Time = 0.65 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.01
method | result | size |
gosper | \(-\frac {c^{5} x \left (105 a^{10} x^{10}+231 a^{9} x^{9}-385 a^{8} x^{8}-1155 a^{7} x^{7}+330 a^{6} x^{6}+2310 a^{5} x^{5}+462 a^{4} x^{4}-2310 a^{3} x^{3}-1155 a^{2} x^{2}+1155 a x +1155\right )}{1155}\) | \(85\) |
default | \(c^{5} \left (-\frac {1}{11} a^{10} x^{11}-\frac {1}{5} a^{9} x^{10}+\frac {1}{3} a^{8} x^{9}+a^{7} x^{8}-\frac {2}{7} a^{6} x^{7}-2 a^{5} x^{6}-\frac {2}{5} a^{4} x^{5}+2 a^{3} x^{4}+a^{2} x^{3}-a \,x^{2}-x \right )\) | \(85\) |
norman | \(a^{7} c^{5} x^{8}+c^{5} a^{2} x^{3}-c^{5} x -a \,c^{5} x^{2}+2 a^{3} c^{5} x^{4}-\frac {2}{5} a^{4} c^{5} x^{5}-2 a^{5} c^{5} x^{6}-\frac {2}{7} a^{6} c^{5} x^{7}+\frac {1}{3} a^{8} c^{5} x^{9}-\frac {1}{5} a^{9} c^{5} x^{10}-\frac {1}{11} a^{10} c^{5} x^{11}\) | \(114\) |
risch | \(a^{7} c^{5} x^{8}+c^{5} a^{2} x^{3}-c^{5} x -a \,c^{5} x^{2}+2 a^{3} c^{5} x^{4}-\frac {2}{5} a^{4} c^{5} x^{5}-2 a^{5} c^{5} x^{6}-\frac {2}{7} a^{6} c^{5} x^{7}+\frac {1}{3} a^{8} c^{5} x^{9}-\frac {1}{5} a^{9} c^{5} x^{10}-\frac {1}{11} a^{10} c^{5} x^{11}\) | \(114\) |
parallelrisch | \(a^{7} c^{5} x^{8}+c^{5} a^{2} x^{3}-c^{5} x -a \,c^{5} x^{2}+2 a^{3} c^{5} x^{4}-\frac {2}{5} a^{4} c^{5} x^{5}-2 a^{5} c^{5} x^{6}-\frac {2}{7} a^{6} c^{5} x^{7}+\frac {1}{3} a^{8} c^{5} x^{9}-\frac {1}{5} a^{9} c^{5} x^{10}-\frac {1}{11} a^{10} c^{5} x^{11}\) | \(114\) |
meijerg | \(\frac {c^{5} \left (-\frac {x a \left (2520 a^{10} x^{10}+2772 a^{9} x^{9}+3080 a^{8} x^{8}+3465 a^{7} x^{7}+3960 a^{6} x^{6}+4620 a^{5} x^{5}+5544 a^{4} x^{4}+6930 a^{3} x^{3}+9240 a^{2} x^{2}+13860 a x +27720\right )}{27720}-\ln \left (-a x +1\right )\right )}{a}-\frac {5 c^{5} \left (-\frac {a x \left (280 a^{8} x^{8}+315 a^{7} x^{7}+360 a^{6} x^{6}+420 a^{5} x^{5}+504 a^{4} x^{4}+630 a^{3} x^{3}+840 a^{2} x^{2}+1260 a x +2520\right )}{2520}-\ln \left (-a x +1\right )\right )}{a}+\frac {10 c^{5} \left (-\frac {a x \left (120 a^{6} x^{6}+140 a^{5} x^{5}+168 a^{4} x^{4}+210 a^{3} x^{3}+280 a^{2} x^{2}+420 a x +840\right )}{840}-\ln \left (-a x +1\right )\right )}{a}-\frac {10 c^{5} \left (-\frac {a x \left (12 a^{4} x^{4}+15 a^{3} x^{3}+20 a^{2} x^{2}+30 a x +60\right )}{60}-\ln \left (-a x +1\right )\right )}{a}+\frac {5 c^{5} \left (-\frac {a x \left (4 a^{2} x^{2}+6 a x +12\right )}{12}-\ln \left (-a x +1\right )\right )}{a}-\frac {c^{5} \left (-a x -\ln \left (-a x +1\right )\right )}{a}-\frac {c^{5} \left (\frac {x a \left (2772 a^{9} x^{9}+3080 a^{8} x^{8}+3465 a^{7} x^{7}+3960 a^{6} x^{6}+4620 a^{5} x^{5}+5544 a^{4} x^{4}+6930 a^{3} x^{3}+9240 a^{2} x^{2}+13860 a x +27720\right )}{27720}+\ln \left (-a x +1\right )\right )}{a}+\frac {5 c^{5} \left (\frac {a x \left (315 a^{7} x^{7}+360 a^{6} x^{6}+420 a^{5} x^{5}+504 a^{4} x^{4}+630 a^{3} x^{3}+840 a^{2} x^{2}+1260 a x +2520\right )}{2520}+\ln \left (-a x +1\right )\right )}{a}-\frac {10 c^{5} \left (\frac {a x \left (70 a^{5} x^{5}+84 a^{4} x^{4}+105 a^{3} x^{3}+140 a^{2} x^{2}+210 a x +420\right )}{420}+\ln \left (-a x +1\right )\right )}{a}+\frac {10 c^{5} \left (\frac {a x \left (15 a^{3} x^{3}+20 a^{2} x^{2}+30 a x +60\right )}{60}+\ln \left (-a x +1\right )\right )}{a}-\frac {5 c^{5} \left (\frac {a x \left (3 a x +6\right )}{6}+\ln \left (-a x +1\right )\right )}{a}+\frac {c^{5} \ln \left (-a x +1\right )}{a}\) | \(667\) |
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Time = 0.24 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.35 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^5 \, dx=-\frac {1}{11} \, a^{10} c^{5} x^{11} - \frac {1}{5} \, a^{9} c^{5} x^{10} + \frac {1}{3} \, a^{8} c^{5} x^{9} + a^{7} c^{5} x^{8} - \frac {2}{7} \, a^{6} c^{5} x^{7} - 2 \, a^{5} c^{5} x^{6} - \frac {2}{5} \, a^{4} c^{5} x^{5} + 2 \, a^{3} c^{5} x^{4} + a^{2} c^{5} x^{3} - a c^{5} x^{2} - c^{5} x \]
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Time = 0.04 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.42 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^5 \, dx=- \frac {a^{10} c^{5} x^{11}}{11} - \frac {a^{9} c^{5} x^{10}}{5} + \frac {a^{8} c^{5} x^{9}}{3} + a^{7} c^{5} x^{8} - \frac {2 a^{6} c^{5} x^{7}}{7} - 2 a^{5} c^{5} x^{6} - \frac {2 a^{4} c^{5} x^{5}}{5} + 2 a^{3} c^{5} x^{4} + a^{2} c^{5} x^{3} - a c^{5} x^{2} - c^{5} x \]
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Time = 0.20 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.35 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^5 \, dx=-\frac {1}{11} \, a^{10} c^{5} x^{11} - \frac {1}{5} \, a^{9} c^{5} x^{10} + \frac {1}{3} \, a^{8} c^{5} x^{9} + a^{7} c^{5} x^{8} - \frac {2}{7} \, a^{6} c^{5} x^{7} - 2 \, a^{5} c^{5} x^{6} - \frac {2}{5} \, a^{4} c^{5} x^{5} + 2 \, a^{3} c^{5} x^{4} + a^{2} c^{5} x^{3} - a c^{5} x^{2} - c^{5} x \]
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Time = 0.28 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.35 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^5 \, dx=-\frac {1}{11} \, a^{10} c^{5} x^{11} - \frac {1}{5} \, a^{9} c^{5} x^{10} + \frac {1}{3} \, a^{8} c^{5} x^{9} + a^{7} c^{5} x^{8} - \frac {2}{7} \, a^{6} c^{5} x^{7} - 2 \, a^{5} c^{5} x^{6} - \frac {2}{5} \, a^{4} c^{5} x^{5} + 2 \, a^{3} c^{5} x^{4} + a^{2} c^{5} x^{3} - a c^{5} x^{2} - c^{5} x \]
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Time = 3.90 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.35 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^5 \, dx=-\frac {a^{10}\,c^5\,x^{11}}{11}-\frac {a^9\,c^5\,x^{10}}{5}+\frac {a^8\,c^5\,x^9}{3}+a^7\,c^5\,x^8-\frac {2\,a^6\,c^5\,x^7}{7}-2\,a^5\,c^5\,x^6-\frac {2\,a^4\,c^5\,x^5}{5}+2\,a^3\,c^5\,x^4+a^2\,c^5\,x^3-a\,c^5\,x^2-c^5\,x \]
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