Integrand size = 12, antiderivative size = 43 \[ \int e^{2 \coth ^{-1}(a x)} x^3 \, dx=\frac {2 x}{a^3}+\frac {x^2}{a^2}+\frac {2 x^3}{3 a}+\frac {x^4}{4}+\frac {2 \log (1-a x)}{a^4} \]
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Time = 0.05 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6302, 6261, 78} \[ \int e^{2 \coth ^{-1}(a x)} x^3 \, dx=\frac {2 \log (1-a x)}{a^4}+\frac {2 x}{a^3}+\frac {x^2}{a^2}+\frac {2 x^3}{3 a}+\frac {x^4}{4} \]
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Rule 78
Rule 6261
Rule 6302
Rubi steps \begin{align*} \text {integral}& = -\int e^{2 \text {arctanh}(a x)} x^3 \, dx \\ & = -\int \frac {x^3 (1+a x)}{1-a x} \, dx \\ & = -\int \left (-\frac {2}{a^3}-\frac {2 x}{a^2}-\frac {2 x^2}{a}-x^3-\frac {2}{a^3 (-1+a x)}\right ) \, dx \\ & = \frac {2 x}{a^3}+\frac {x^2}{a^2}+\frac {2 x^3}{3 a}+\frac {x^4}{4}+\frac {2 \log (1-a x)}{a^4} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00 \[ \int e^{2 \coth ^{-1}(a x)} x^3 \, dx=\frac {2 x}{a^3}+\frac {x^2}{a^2}+\frac {2 x^3}{3 a}+\frac {x^4}{4}+\frac {2 \log (1-a x)}{a^4} \]
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Time = 0.39 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.91
method | result | size |
norman | \(\frac {x^{2}}{a^{2}}+\frac {x^{4}}{4}+\frac {2 x}{a^{3}}+\frac {2 x^{3}}{3 a}+\frac {2 \ln \left (a x -1\right )}{a^{4}}\) | \(39\) |
risch | \(\frac {x^{2}}{a^{2}}+\frac {x^{4}}{4}+\frac {2 x}{a^{3}}+\frac {2 x^{3}}{3 a}+\frac {2 \ln \left (a x -1\right )}{a^{4}}\) | \(39\) |
default | \(\frac {\frac {1}{4} a^{3} x^{4}+\frac {2}{3} a^{2} x^{3}+a \,x^{2}+2 x}{a^{3}}+\frac {2 \ln \left (a x -1\right )}{a^{4}}\) | \(42\) |
parallelrisch | \(\frac {3 a^{4} x^{4}+8 a^{3} x^{3}+12 a^{2} x^{2}+24 a x +24 \ln \left (a x -1\right )}{12 a^{4}}\) | \(43\) |
meijerg | \(\frac {\frac {a x \left (15 a^{3} x^{3}+20 a^{2} x^{2}+30 a x +60\right )}{60}+\ln \left (-a x +1\right )}{a^{4}}-\frac {-\frac {a x \left (4 a^{2} x^{2}+6 a x +12\right )}{12}-\ln \left (-a x +1\right )}{a^{4}}\) | \(73\) |
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none
Time = 0.24 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.98 \[ \int e^{2 \coth ^{-1}(a x)} x^3 \, dx=\frac {3 \, a^{4} x^{4} + 8 \, a^{3} x^{3} + 12 \, a^{2} x^{2} + 24 \, a x + 24 \, \log \left (a x - 1\right )}{12 \, a^{4}} \]
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Time = 0.06 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.86 \[ \int e^{2 \coth ^{-1}(a x)} x^3 \, dx=\frac {x^{4}}{4} + \frac {2 x^{3}}{3 a} + \frac {x^{2}}{a^{2}} + \frac {2 x}{a^{3}} + \frac {2 \log {\left (a x - 1 \right )}}{a^{4}} \]
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none
Time = 0.20 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00 \[ \int e^{2 \coth ^{-1}(a x)} x^3 \, dx=\frac {3 \, a^{3} x^{4} + 8 \, a^{2} x^{3} + 12 \, a x^{2} + 24 \, x}{12 \, a^{3}} + \frac {2 \, \log \left (a x - 1\right )}{a^{4}} \]
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none
Time = 0.26 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.09 \[ \int e^{2 \coth ^{-1}(a x)} x^3 \, dx=\frac {3 \, a^{4} x^{4} + 8 \, a^{3} x^{3} + 12 \, a^{2} x^{2} + 24 \, a x}{12 \, a^{4}} + \frac {2 \, \log \left ({\left | a x - 1 \right |}\right )}{a^{4}} \]
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Time = 0.04 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.88 \[ \int e^{2 \coth ^{-1}(a x)} x^3 \, dx=\frac {2\,\ln \left (a\,x-1\right )}{a^4}+\frac {2\,x}{a^3}+\frac {x^4}{4}+\frac {2\,x^3}{3\,a}+\frac {x^2}{a^2} \]
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