Integrand size = 22, antiderivative size = 55 \[ \int e^{-2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^3 \, dx=\frac {4 c^3 (1-a x)^5}{5 a}-\frac {2 c^3 (1-a x)^6}{3 a}+\frac {c^3 (1-a x)^7}{7 a} \]
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Time = 0.07 (sec) , antiderivative size = 55, 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} \[ \int e^{-2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^3 \, dx=\frac {c^3 (1-a x)^7}{7 a}-\frac {2 c^3 (1-a x)^6}{3 a}+\frac {4 c^3 (1-a x)^5}{5 a} \]
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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 )^3 \, dx \\ & = -\left (c^3 \int (1-a x)^4 (1+a x)^2 \, dx\right ) \\ & = -\left (c^3 \int \left (4 (1-a x)^4-4 (1-a x)^5+(1-a x)^6\right ) \, dx\right ) \\ & = \frac {4 c^3 (1-a x)^5}{5 a}-\frac {2 c^3 (1-a x)^6}{3 a}+\frac {c^3 (1-a x)^7}{7 a} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.56 \[ \int e^{-2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^3 \, dx=-\frac {c^3 (-1+a x)^5 \left (29+40 a x+15 a^2 x^2\right )}{105 a} \]
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Time = 0.63 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.96
method | result | size |
gosper | \(-\frac {c^{3} x \left (15 a^{6} x^{6}-35 a^{5} x^{5}-21 a^{4} x^{4}+105 a^{3} x^{3}-35 a^{2} x^{2}-105 a x +105\right )}{105}\) | \(53\) |
default | \(c^{3} \left (-\frac {1}{7} a^{6} x^{7}+\frac {1}{3} a^{5} x^{6}+\frac {1}{5} a^{4} x^{5}-a^{3} x^{4}+\frac {1}{3} a^{2} x^{3}+a \,x^{2}-x \right )\) | \(54\) |
norman | \(a \,c^{3} x^{2}-c^{3} x +\frac {1}{3} a^{2} c^{3} x^{3}-a^{3} c^{3} x^{4}+\frac {1}{5} a^{4} c^{3} x^{5}+\frac {1}{3} a^{5} c^{3} x^{6}-\frac {1}{7} a^{6} c^{3} x^{7}\) | \(71\) |
risch | \(a \,c^{3} x^{2}-c^{3} x +\frac {1}{3} a^{2} c^{3} x^{3}-a^{3} c^{3} x^{4}+\frac {1}{5} a^{4} c^{3} x^{5}+\frac {1}{3} a^{5} c^{3} x^{6}-\frac {1}{7} a^{6} c^{3} x^{7}\) | \(71\) |
parallelrisch | \(a \,c^{3} x^{2}-c^{3} x +\frac {1}{3} a^{2} c^{3} x^{3}-a^{3} c^{3} x^{4}+\frac {1}{5} a^{4} c^{3} x^{5}+\frac {1}{3} a^{5} c^{3} x^{6}-\frac {1}{7} a^{6} c^{3} x^{7}\) | \(71\) |
meijerg | \(-\frac {c^{3} \left (\frac {a x \left (120 a^{6} x^{6}-140 a^{5} x^{5}+168 a^{4} x^{4}-210 a^{3} x^{3}+280 a^{2} x^{2}-420 a x +840\right )}{840}-\ln \left (a x +1\right )\right )}{a}+\frac {c^{3} \left (-\frac {a x \left (-70 a^{5} x^{5}+84 a^{4} x^{4}-105 a^{3} x^{3}+140 a^{2} x^{2}-210 a x +420\right )}{420}+\ln \left (a x +1\right )\right )}{a}+\frac {3 c^{3} \left (\frac {a x \left (12 a^{4} x^{4}-15 a^{3} x^{3}+20 a^{2} x^{2}-30 a x +60\right )}{60}-\ln \left (a x +1\right )\right )}{a}-\frac {3 c^{3} \left (-\frac {a x \left (-15 a^{3} x^{3}+20 a^{2} x^{2}-30 a x +60\right )}{60}+\ln \left (a x +1\right )\right )}{a}-\frac {3 c^{3} \left (\frac {a x \left (4 a^{2} x^{2}-6 a x +12\right )}{12}-\ln \left (a x +1\right )\right )}{a}+\frac {3 c^{3} \left (-\frac {a x \left (-3 a x +6\right )}{6}+\ln \left (a x +1\right )\right )}{a}+\frac {c^{3} \left (a x -\ln \left (a x +1\right )\right )}{a}-\frac {c^{3} \ln \left (a x +1\right )}{a}\) | \(310\) |
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Time = 0.24 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.27 \[ \int e^{-2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^3 \, dx=-\frac {1}{7} \, a^{6} c^{3} x^{7} + \frac {1}{3} \, a^{5} c^{3} x^{6} + \frac {1}{5} \, a^{4} c^{3} x^{5} - a^{3} c^{3} x^{4} + \frac {1}{3} \, a^{2} c^{3} x^{3} + a c^{3} x^{2} - c^{3} x \]
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Time = 0.04 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.27 \[ \int e^{-2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^3 \, dx=- \frac {a^{6} c^{3} x^{7}}{7} + \frac {a^{5} c^{3} x^{6}}{3} + \frac {a^{4} c^{3} x^{5}}{5} - a^{3} c^{3} x^{4} + \frac {a^{2} c^{3} x^{3}}{3} + a c^{3} x^{2} - c^{3} x \]
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Time = 0.19 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.27 \[ \int e^{-2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^3 \, dx=-\frac {1}{7} \, a^{6} c^{3} x^{7} + \frac {1}{3} \, a^{5} c^{3} x^{6} + \frac {1}{5} \, a^{4} c^{3} x^{5} - a^{3} c^{3} x^{4} + \frac {1}{3} \, a^{2} c^{3} x^{3} + a c^{3} x^{2} - c^{3} x \]
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Time = 0.27 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.27 \[ \int e^{-2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^3 \, dx=-\frac {1}{7} \, a^{6} c^{3} x^{7} + \frac {1}{3} \, a^{5} c^{3} x^{6} + \frac {1}{5} \, a^{4} c^{3} x^{5} - a^{3} c^{3} x^{4} + \frac {1}{3} \, a^{2} c^{3} x^{3} + a c^{3} x^{2} - c^{3} x \]
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Time = 0.06 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.27 \[ \int e^{-2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^3 \, dx=-\frac {a^6\,c^3\,x^7}{7}+\frac {a^5\,c^3\,x^6}{3}+\frac {a^4\,c^3\,x^5}{5}-a^3\,c^3\,x^4+\frac {a^2\,c^3\,x^3}{3}+a\,c^3\,x^2-c^3\,x \]
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