Integrand size = 18, antiderivative size = 37 \[ \int e^{2 \coth ^{-1}(a x)} (c-a c x)^3 \, dx=\frac {2 c^3 (1-a x)^3}{3 a}-\frac {c^3 (1-a x)^4}{4 a} \]
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Time = 0.05 (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} \[ \int e^{2 \coth ^{-1}(a x)} (c-a c x)^3 \, dx=\frac {2 c^3 (1-a x)^3}{3 a}-\frac {c^3 (1-a x)^4}{4 a} \]
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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)^3 \, dx \\ & = -\left (c^3 \int (1-a x)^2 (1+a x) \, dx\right ) \\ & = -\left (c^3 \int \left (2 (1-a x)^2-(1-a x)^3\right ) \, dx\right ) \\ & = \frac {2 c^3 (1-a x)^3}{3 a}-\frac {c^3 (1-a x)^4}{4 a} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.81 \[ \int e^{2 \coth ^{-1}(a x)} (c-a c x)^3 \, dx=-\frac {1}{12} c^3 x \left (12-6 a x-4 a^2 x^2+3 a^3 x^3\right ) \]
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Time = 0.48 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.78
method | result | size |
gosper | \(-\frac {\left (3 a^{3} x^{3}-4 a^{2} x^{2}-6 a x +12\right ) c^{3} x}{12}\) | \(29\) |
default | \(c^{3} \left (-\frac {1}{4} a^{3} x^{4}+\frac {1}{3} a^{2} x^{3}+\frac {1}{2} a \,x^{2}-x \right )\) | \(31\) |
norman | \(-c^{3} x +\frac {1}{2} a \,c^{3} x^{2}+\frac {1}{3} a^{2} c^{3} x^{3}-\frac {1}{4} a^{3} c^{3} x^{4}\) | \(39\) |
risch | \(-c^{3} x +\frac {1}{2} a \,c^{3} x^{2}+\frac {1}{3} a^{2} c^{3} x^{3}-\frac {1}{4} a^{3} c^{3} x^{4}\) | \(39\) |
parallelrisch | \(-c^{3} x +\frac {1}{2} a \,c^{3} x^{2}+\frac {1}{3} a^{2} c^{3} x^{3}-\frac {1}{4} a^{3} c^{3} x^{4}\) | \(39\) |
meijerg | \(-\frac {c^{3} \left (\frac {a x \left (15 a^{3} x^{3}+20 a^{2} x^{2}+30 a x +60\right )}{60}+\ln \left (-a x +1\right )\right )}{a}-\frac {2 c^{3} \left (-\frac {a x \left (4 a^{2} x^{2}+6 a x +12\right )}{12}-\ln \left (-a x +1\right )\right )}{a}+\frac {2 c^{3} \left (-a x -\ln \left (-a x +1\right )\right )}{a}+\frac {c^{3} \ln \left (-a x +1\right )}{a}\) | \(116\) |
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Time = 0.24 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.03 \[ \int e^{2 \coth ^{-1}(a x)} (c-a c x)^3 \, dx=-\frac {1}{4} \, a^{3} c^{3} x^{4} + \frac {1}{3} \, a^{2} c^{3} x^{3} + \frac {1}{2} \, a c^{3} x^{2} - c^{3} x \]
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Time = 0.05 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00 \[ \int e^{2 \coth ^{-1}(a x)} (c-a c x)^3 \, dx=- \frac {a^{3} c^{3} x^{4}}{4} + \frac {a^{2} c^{3} x^{3}}{3} + \frac {a c^{3} x^{2}}{2} - c^{3} x \]
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Time = 0.19 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.03 \[ \int e^{2 \coth ^{-1}(a x)} (c-a c x)^3 \, dx=-\frac {1}{4} \, a^{3} c^{3} x^{4} + \frac {1}{3} \, a^{2} c^{3} x^{3} + \frac {1}{2} \, a c^{3} x^{2} - c^{3} x \]
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Time = 0.26 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.03 \[ \int e^{2 \coth ^{-1}(a x)} (c-a c x)^3 \, dx=-\frac {1}{4} \, a^{3} c^{3} x^{4} + \frac {1}{3} \, a^{2} c^{3} x^{3} + \frac {1}{2} \, a c^{3} x^{2} - c^{3} x \]
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Time = 0.05 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.03 \[ \int e^{2 \coth ^{-1}(a x)} (c-a c x)^3 \, dx=-\frac {a^3\,c^3\,x^4}{4}+\frac {a^2\,c^3\,x^3}{3}+\frac {a\,c^3\,x^2}{2}-c^3\,x \]
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