Integrand size = 12, antiderivative size = 33 \[ \int e^{2 \coth ^{-1}(a x)} x^2 \, dx=\frac {2 x}{a^2}+\frac {x^2}{a}+\frac {x^3}{3}+\frac {2 \log (1-a x)}{a^3} \]
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Time = 0.04 (sec) , antiderivative size = 33, 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} \[ \int e^{2 \coth ^{-1}(a x)} x^2 \, dx=\frac {2 \log (1-a x)}{a^3}+\frac {2 x}{a^2}+\frac {x^2}{a}+\frac {x^3}{3} \]
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Rule 78
Rule 6261
Rule 6302
Rubi steps \begin{align*} \text {integral}& = -\int e^{2 \text {arctanh}(a x)} x^2 \, dx \\ & = -\int \frac {x^2 (1+a x)}{1-a x} \, dx \\ & = -\int \left (-\frac {2}{a^2}-\frac {2 x}{a}-x^2-\frac {2}{a^2 (-1+a x)}\right ) \, dx \\ & = \frac {2 x}{a^2}+\frac {x^2}{a}+\frac {x^3}{3}+\frac {2 \log (1-a x)}{a^3} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00 \[ \int e^{2 \coth ^{-1}(a x)} x^2 \, dx=\frac {2 x}{a^2}+\frac {x^2}{a}+\frac {x^3}{3}+\frac {2 \log (1-a x)}{a^3} \]
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Time = 0.40 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.94
method | result | size |
norman | \(\frac {x^{2}}{a}+\frac {x^{3}}{3}+\frac {2 x}{a^{2}}+\frac {2 \ln \left (a x -1\right )}{a^{3}}\) | \(31\) |
risch | \(\frac {x^{2}}{a}+\frac {x^{3}}{3}+\frac {2 x}{a^{2}}+\frac {2 \ln \left (a x -1\right )}{a^{3}}\) | \(31\) |
default | \(\frac {\frac {1}{3} a^{2} x^{3}+a \,x^{2}+2 x}{a^{2}}+\frac {2 \ln \left (a x -1\right )}{a^{3}}\) | \(34\) |
parallelrisch | \(\frac {a^{3} x^{3}+3 a^{2} x^{2}+6 a x +6 \ln \left (a x -1\right )}{3 a^{3}}\) | \(34\) |
meijerg | \(-\frac {-\frac {a x \left (4 a^{2} x^{2}+6 a x +12\right )}{12}-\ln \left (-a x +1\right )}{a^{3}}+\frac {\frac {a x \left (3 a x +6\right )}{6}+\ln \left (-a x +1\right )}{a^{3}}\) | \(57\) |
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Time = 0.23 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00 \[ \int e^{2 \coth ^{-1}(a x)} x^2 \, dx=\frac {a^{3} x^{3} + 3 \, a^{2} x^{2} + 6 \, a x + 6 \, \log \left (a x - 1\right )}{3 \, a^{3}} \]
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Time = 0.06 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.82 \[ \int e^{2 \coth ^{-1}(a x)} x^2 \, dx=\frac {x^{3}}{3} + \frac {x^{2}}{a} + \frac {2 x}{a^{2}} + \frac {2 \log {\left (a x - 1 \right )}}{a^{3}} \]
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Time = 0.21 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.03 \[ \int e^{2 \coth ^{-1}(a x)} x^2 \, dx=\frac {a^{2} x^{3} + 3 \, a x^{2} + 6 \, x}{3 \, a^{2}} + \frac {2 \, \log \left (a x - 1\right )}{a^{3}} \]
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Time = 0.25 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.15 \[ \int e^{2 \coth ^{-1}(a x)} x^2 \, dx=\frac {a^{3} x^{3} + 3 \, a^{2} x^{2} + 6 \, a x}{3 \, a^{3}} + \frac {2 \, \log \left ({\left | a x - 1 \right |}\right )}{a^{3}} \]
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Time = 0.04 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.91 \[ \int e^{2 \coth ^{-1}(a x)} x^2 \, dx=\frac {2\,\ln \left (a\,x-1\right )}{a^3}+\frac {2\,x}{a^2}+\frac {x^3}{3}+\frac {x^2}{a} \]
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