Integrand size = 18, antiderivative size = 35 \[ \int e^{4 \coth ^{-1}(a x)} (c-a c x)^3 \, dx=\frac {2 c^3 (1+a x)^3}{3 a}-\frac {c^3 (1+a x)^4}{4 a} \]
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Time = 0.04 (sec) , antiderivative size = 35, 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} \[ \int e^{4 \coth ^{-1}(a x)} (c-a c x)^3 \, dx=\frac {2 c^3 (a x+1)^3}{3 a}-\frac {c^3 (a x+1)^4}{4 a} \]
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Rule 45
Rule 6264
Rule 6302
Rubi steps \begin{align*} \text {integral}& = \int e^{4 \text {arctanh}(a x)} (c-a c x)^3 \, dx \\ & = c^3 \int (1-a x) (1+a x)^2 \, dx \\ & = c^3 \int \left (2 (1+a x)^2-(1+a x)^3\right ) \, dx \\ & = \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 = 0.86 \[ \int e^{4 \coth ^{-1}(a x)} (c-a c x)^3 \, dx=-\frac {1}{12} c^3 x \left (-12-6 a x+4 a^2 x^2+3 a^3 x^3\right ) \]
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Time = 0.56 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.83
method | result | size |
gosper | \(-\frac {x \left (3 a^{3} x^{3}+4 a^{2} x^{2}-6 a x -12\right ) c^{3}}{12}\) | \(29\) |
default | \(c^{3} \left (-\frac {1}{4} a^{3} x^{4}-\frac {1}{3} a^{2} x^{3}+\frac {1}{2} a \,x^{2}+x \right )\) | \(29\) |
risch | \(-\frac {1}{4} a^{3} c^{3} x^{4}-\frac {1}{3} a^{2} c^{3} x^{3}+\frac {1}{2} a \,c^{3} x^{2}+c^{3} x\) | \(38\) |
parallelrisch | \(-\frac {1}{4} a^{3} c^{3} x^{4}-\frac {1}{3} a^{2} c^{3} x^{3}+\frac {1}{2} a \,c^{3} x^{2}+c^{3} x\) | \(38\) |
norman | \(\frac {-c^{3} x +\frac {1}{2} a \,c^{3} x^{2}+\frac {5}{6} a^{2} c^{3} x^{3}-\frac {1}{12} a^{3} c^{3} x^{4}-\frac {1}{4} a^{4} c^{3} x^{5}}{a x -1}\) | \(58\) |
meijerg | \(-\frac {c^{3} \left (\frac {a x \left (-3 a^{4} x^{4}-5 a^{3} x^{3}-10 a^{2} x^{2}-30 a x +60\right )}{-12 a x +12}+5 \ln \left (-a x +1\right )\right )}{a}-\frac {c^{3} \left (-\frac {a x \left (-5 a^{3} x^{3}-10 a^{2} x^{2}-30 a x +60\right )}{15 \left (-a x +1\right )}-4 \ln \left (-a x +1\right )\right )}{a}+\frac {2 c^{3} \left (\frac {a x \left (-2 a^{2} x^{2}-6 a x +12\right )}{-4 a x +4}+3 \ln \left (-a x +1\right )\right )}{a}+\frac {2 c^{3} \left (-\frac {a x \left (-3 a x +6\right )}{3 \left (-a x +1\right )}-2 \ln \left (-a x +1\right )\right )}{a}-\frac {c^{3} \left (\frac {a x}{-a x +1}+\ln \left (-a x +1\right )\right )}{a}+\frac {c^{3} x}{-a x +1}\) | \(234\) |
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Time = 0.23 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.06 \[ \int e^{4 \coth ^{-1}(a x)} (c-a c x)^3 \, dx=-\frac {1}{4} \, a^{3} c^{3} x^{4} - \frac {1}{3} \, a^{2} c^{3} x^{3} + \frac {1}{2} \, a c^{3} x^{2} + c^{3} x \]
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Time = 0.05 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.06 \[ \int e^{4 \coth ^{-1}(a x)} (c-a c x)^3 \, dx=- \frac {a^{3} c^{3} x^{4}}{4} - \frac {a^{2} c^{3} x^{3}}{3} + \frac {a c^{3} x^{2}}{2} + c^{3} x \]
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Time = 0.21 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.06 \[ \int e^{4 \coth ^{-1}(a x)} (c-a c x)^3 \, dx=-\frac {1}{4} \, a^{3} c^{3} x^{4} - \frac {1}{3} \, a^{2} c^{3} x^{3} + \frac {1}{2} \, a c^{3} x^{2} + c^{3} x \]
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Time = 0.26 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.20 \[ \int e^{4 \coth ^{-1}(a x)} (c-a c x)^3 \, dx=-\frac {{\left (3 \, c^{3} + \frac {16 \, c^{3}}{a x - 1} + \frac {24 \, c^{3}}{{\left (a x - 1\right )}^{2}}\right )} {\left (a x - 1\right )}^{4}}{12 \, a} \]
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Time = 0.05 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.06 \[ \int e^{4 \coth ^{-1}(a x)} (c-a c x)^3 \, dx=-\frac {a^3\,c^3\,x^4}{4}-\frac {a^2\,c^3\,x^3}{3}+\frac {a\,c^3\,x^2}{2}+c^3\,x \]
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