Integrand size = 22, antiderivative size = 39 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^3 \, dx=-\frac {c^3}{2 a^3 x^2}+\frac {c^3}{a^2 x}+c^3 x-\frac {c^3 \log (x)}{a} \]
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Time = 0.11 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {6302, 6266, 6264, 76} \[ \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^3 \, dx=-\frac {c^3}{2 a^3 x^2}+\frac {c^3}{a^2 x}-\frac {c^3 \log (x)}{a}+c^3 x \]
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Rule 76
Rule 6264
Rule 6266
Rule 6302
Rubi steps \begin{align*} \text {integral}& = -\int e^{2 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^3 \, dx \\ & = \frac {c^3 \int \frac {e^{2 \text {arctanh}(a x)} (1-a x)^3}{x^3} \, dx}{a^3} \\ & = \frac {c^3 \int \frac {(1-a x)^2 (1+a x)}{x^3} \, dx}{a^3} \\ & = \frac {c^3 \int \left (a^3+\frac {1}{x^3}-\frac {a}{x^2}-\frac {a^2}{x}\right ) \, dx}{a^3} \\ & = -\frac {c^3}{2 a^3 x^2}+\frac {c^3}{a^2 x}+c^3 x-\frac {c^3 \log (x)}{a} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.97 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^3 \, dx=-\frac {c^3 \left (3 a^2+\frac {1}{x^2}-\frac {2 a}{x}-2 a^3 x+2 a^2 \log (x)\right )}{2 a^3} \]
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Time = 0.59 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.79
method | result | size |
default | \(\frac {c^{3} \left (a^{3} x -a^{2} \ln \left (x \right )-\frac {1}{2 x^{2}}+\frac {a}{x}\right )}{a^{3}}\) | \(31\) |
risch | \(c^{3} x +\frac {a \,c^{3} x -\frac {1}{2} c^{3}}{a^{3} x^{2}}-\frac {c^{3} \ln \left (x \right )}{a}\) | \(36\) |
norman | \(\frac {c^{3} x +a^{2} c^{3} x^{3}-\frac {c^{3}}{2 a}}{a^{2} x^{2}}-\frac {c^{3} \ln \left (x \right )}{a}\) | \(43\) |
parallelrisch | \(-\frac {-2 a^{3} c^{3} x^{3}+2 c^{3} \ln \left (x \right ) a^{2} x^{2}-2 a \,c^{3} x +c^{3}}{2 a^{3} x^{2}}\) | \(44\) |
meijerg | \(-\frac {c^{3} \left (-a x -\ln \left (-a x +1\right )\right )}{a}-\frac {2 c^{3} \ln \left (-a x +1\right )}{a}+\frac {2 c^{3} \left (\ln \left (-a x +1\right )-\ln \left (x \right )-\ln \left (-a \right )+\frac {1}{a x}\right )}{a}+\frac {c^{3} \left (-\ln \left (-a x +1\right )+\ln \left (x \right )+\ln \left (-a \right )-\frac {1}{2 a^{2} x^{2}}-\frac {1}{a x}\right )}{a}\) | \(111\) |
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Time = 0.25 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.15 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^3 \, dx=\frac {2 \, a^{3} c^{3} x^{3} - 2 \, a^{2} c^{3} x^{2} \log \left (x\right ) + 2 \, a c^{3} x - c^{3}}{2 \, a^{3} x^{2}} \]
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Time = 0.13 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.95 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^3 \, dx=\frac {a^{3} c^{3} x - a^{2} c^{3} \log {\left (x \right )} + \frac {2 a c^{3} x - c^{3}}{2 x^{2}}}{a^{3}} \]
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Time = 0.19 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.95 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^3 \, dx=c^{3} x - \frac {c^{3} \log \left (x\right )}{a} + \frac {2 \, a c^{3} x - c^{3}}{2 \, a^{3} x^{2}} \]
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Time = 0.26 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.97 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^3 \, dx=c^{3} x - \frac {c^{3} \log \left ({\left | x \right |}\right )}{a} + \frac {2 \, a c^{3} x - c^{3}}{2 \, a^{3} x^{2}} \]
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Time = 4.27 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.90 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^3 \, dx=\frac {c^3\,\left (2\,a\,x+2\,a^3\,x^3-2\,a^2\,x^2\,\ln \left (x\right )-1\right )}{2\,a^3\,x^2} \]
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