Integrand size = 22, antiderivative size = 69 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^4 \, dx=-\frac {4 c^4 (1+a x)^6}{3 a}+\frac {12 c^4 (1+a x)^7}{7 a}-\frac {3 c^4 (1+a x)^8}{4 a}+\frac {c^4 (1+a x)^9}{9 a} \]
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Time = 0.07 (sec) , antiderivative size = 69, 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} \[ \int e^{2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^4 \, dx=\frac {c^4 (a x+1)^9}{9 a}-\frac {3 c^4 (a x+1)^8}{4 a}+\frac {12 c^4 (a x+1)^7}{7 a}-\frac {4 c^4 (a x+1)^6}{3 a} \]
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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 )^4 \, dx \\ & = -\left (c^4 \int (1-a x)^3 (1+a x)^5 \, dx\right ) \\ & = -\left (c^4 \int \left (8 (1+a x)^5-12 (1+a x)^6+6 (1+a x)^7-(1+a x)^8\right ) \, dx\right ) \\ & = -\frac {4 c^4 (1+a x)^6}{3 a}+\frac {12 c^4 (1+a x)^7}{7 a}-\frac {3 c^4 (1+a x)^8}{4 a}+\frac {c^4 (1+a x)^9}{9 a} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.57 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^4 \, dx=\frac {c^4 (1+a x)^6 \left (-65+138 a x-105 a^2 x^2+28 a^3 x^3\right )}{252 a} \]
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Time = 0.62 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.88
method | result | size |
gosper | \(\frac {c^{4} x \left (28 a^{8} x^{8}+63 a^{7} x^{7}-72 a^{6} x^{6}-252 a^{5} x^{5}+378 a^{3} x^{3}+168 a^{2} x^{2}-252 a x -252\right )}{252}\) | \(61\) |
default | \(c^{4} \left (\frac {1}{9} a^{8} x^{9}+\frac {1}{4} a^{7} x^{8}-\frac {2}{7} a^{6} x^{7}-a^{5} x^{6}+\frac {3}{2} a^{3} x^{4}+\frac {2}{3} a^{2} x^{3}-a \,x^{2}-x \right )\) | \(63\) |
norman | \(-c^{4} x -a \,c^{4} x^{2}+\frac {2}{3} a^{2} c^{4} x^{3}+\frac {3}{2} a^{3} c^{4} x^{4}-a^{5} c^{4} x^{6}-\frac {2}{7} a^{6} c^{4} x^{7}+\frac {1}{4} a^{7} c^{4} x^{8}+\frac {1}{9} a^{8} c^{4} x^{9}\) | \(83\) |
risch | \(-c^{4} x -a \,c^{4} x^{2}+\frac {2}{3} a^{2} c^{4} x^{3}+\frac {3}{2} a^{3} c^{4} x^{4}-a^{5} c^{4} x^{6}-\frac {2}{7} a^{6} c^{4} x^{7}+\frac {1}{4} a^{7} c^{4} x^{8}+\frac {1}{9} a^{8} c^{4} x^{9}\) | \(83\) |
parallelrisch | \(-c^{4} x -a \,c^{4} x^{2}+\frac {2}{3} a^{2} c^{4} x^{3}+\frac {3}{2} a^{3} c^{4} x^{4}-a^{5} c^{4} x^{6}-\frac {2}{7} a^{6} c^{4} x^{7}+\frac {1}{4} a^{7} c^{4} x^{8}+\frac {1}{9} a^{8} c^{4} x^{9}\) | \(83\) |
meijerg | \(-\frac {c^{4} \left (-\frac {a x \left (280 a^{8} x^{8}+315 a^{7} x^{7}+360 a^{6} x^{6}+420 a^{5} x^{5}+504 a^{4} x^{4}+630 a^{3} x^{3}+840 a^{2} x^{2}+1260 a x +2520\right )}{2520}-\ln \left (-a x +1\right )\right )}{a}+\frac {4 c^{4} \left (-\frac {a x \left (120 a^{6} x^{6}+140 a^{5} x^{5}+168 a^{4} x^{4}+210 a^{3} x^{3}+280 a^{2} x^{2}+420 a x +840\right )}{840}-\ln \left (-a x +1\right )\right )}{a}-\frac {6 c^{4} \left (-\frac {a x \left (12 a^{4} x^{4}+15 a^{3} x^{3}+20 a^{2} x^{2}+30 a x +60\right )}{60}-\ln \left (-a x +1\right )\right )}{a}+\frac {4 c^{4} \left (-\frac {a x \left (4 a^{2} x^{2}+6 a x +12\right )}{12}-\ln \left (-a x +1\right )\right )}{a}-\frac {c^{4} \left (-a x -\ln \left (-a x +1\right )\right )}{a}+\frac {c^{4} \left (\frac {a x \left (315 a^{7} x^{7}+360 a^{6} x^{6}+420 a^{5} x^{5}+504 a^{4} x^{4}+630 a^{3} x^{3}+840 a^{2} x^{2}+1260 a x +2520\right )}{2520}+\ln \left (-a x +1\right )\right )}{a}-\frac {4 c^{4} \left (\frac {a x \left (70 a^{5} x^{5}+84 a^{4} x^{4}+105 a^{3} x^{3}+140 a^{2} x^{2}+210 a x +420\right )}{420}+\ln \left (-a x +1\right )\right )}{a}+\frac {6 c^{4} \left (\frac {a x \left (15 a^{3} x^{3}+20 a^{2} x^{2}+30 a x +60\right )}{60}+\ln \left (-a x +1\right )\right )}{a}-\frac {4 c^{4} \left (\frac {a x \left (3 a x +6\right )}{6}+\ln \left (-a x +1\right )\right )}{a}+\frac {c^{4} \ln \left (-a x +1\right )}{a}\) | \(477\) |
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Time = 0.23 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.19 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^4 \, dx=\frac {1}{9} \, a^{8} c^{4} x^{9} + \frac {1}{4} \, a^{7} c^{4} x^{8} - \frac {2}{7} \, a^{6} c^{4} x^{7} - a^{5} c^{4} x^{6} + \frac {3}{2} \, a^{3} c^{4} x^{4} + \frac {2}{3} \, a^{2} c^{4} x^{3} - a c^{4} x^{2} - c^{4} x \]
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Time = 0.04 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.26 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^4 \, dx=\frac {a^{8} c^{4} x^{9}}{9} + \frac {a^{7} c^{4} x^{8}}{4} - \frac {2 a^{6} c^{4} x^{7}}{7} - a^{5} c^{4} x^{6} + \frac {3 a^{3} c^{4} x^{4}}{2} + \frac {2 a^{2} c^{4} x^{3}}{3} - a c^{4} x^{2} - c^{4} x \]
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Time = 0.20 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.19 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^4 \, dx=\frac {1}{9} \, a^{8} c^{4} x^{9} + \frac {1}{4} \, a^{7} c^{4} x^{8} - \frac {2}{7} \, a^{6} c^{4} x^{7} - a^{5} c^{4} x^{6} + \frac {3}{2} \, a^{3} c^{4} x^{4} + \frac {2}{3} \, a^{2} c^{4} x^{3} - a c^{4} x^{2} - c^{4} x \]
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Time = 0.26 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.19 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^4 \, dx=\frac {1}{9} \, a^{8} c^{4} x^{9} + \frac {1}{4} \, a^{7} c^{4} x^{8} - \frac {2}{7} \, a^{6} c^{4} x^{7} - a^{5} c^{4} x^{6} + \frac {3}{2} \, a^{3} c^{4} x^{4} + \frac {2}{3} \, a^{2} c^{4} x^{3} - a c^{4} x^{2} - c^{4} x \]
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Time = 0.04 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.19 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^4 \, dx=\frac {a^8\,c^4\,x^9}{9}+\frac {a^7\,c^4\,x^8}{4}-\frac {2\,a^6\,c^4\,x^7}{7}-a^5\,c^4\,x^6+\frac {3\,a^3\,c^4\,x^4}{2}+\frac {2\,a^2\,c^4\,x^3}{3}-a\,c^4\,x^2-c^4\,x \]
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