Integrand size = 12, antiderivative size = 33 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{x^3} \, dx=\frac {1}{2 x^2}+\frac {2 a}{x}-2 a^2 \log (x)+2 a^2 \log (1-a x) \]
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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} \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{x^3} \, dx=-2 a^2 \log (x)+2 a^2 \log (1-a x)+\frac {2 a}{x}+\frac {1}{2 x^2} \]
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Rule 78
Rule 6261
Rule 6302
Rubi steps \begin{align*} \text {integral}& = -\int \frac {e^{2 \text {arctanh}(a x)}}{x^3} \, dx \\ & = -\int \frac {1+a x}{x^3 (1-a x)} \, dx \\ & = -\int \left (\frac {1}{x^3}+\frac {2 a}{x^2}+\frac {2 a^2}{x}-\frac {2 a^3}{-1+a x}\right ) \, dx \\ & = \frac {1}{2 x^2}+\frac {2 a}{x}-2 a^2 \log (x)+2 a^2 \log (1-a x) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{x^3} \, dx=\frac {1}{2 x^2}+\frac {2 a}{x}-2 a^2 \log (x)+2 a^2 \log (1-a x) \]
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Time = 0.43 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.91
method | result | size |
norman | \(\frac {\frac {1}{2}+2 a x}{x^{2}}-2 a^{2} \ln \left (x \right )+2 a^{2} \ln \left (a x -1\right )\) | \(30\) |
default | \(\frac {1}{2 x^{2}}+\frac {2 a}{x}-2 a^{2} \ln \left (x \right )+2 a^{2} \ln \left (a x -1\right )\) | \(31\) |
risch | \(\frac {\frac {1}{2}+2 a x}{x^{2}}+2 a^{2} \ln \left (-a x +1\right )-2 a^{2} \ln \left (x \right )\) | \(31\) |
parallelrisch | \(-\frac {4 a^{2} \ln \left (x \right ) x^{2}-4 a^{2} \ln \left (a x -1\right ) x^{2}-1-4 a x}{2 x^{2}}\) | \(36\) |
meijerg | \(a^{2} \left (\ln \left (-a x +1\right )-\ln \left (x \right )-\ln \left (-a \right )+\frac {1}{a x}\right )-a^{2} \left (-\ln \left (-a x +1\right )+\ln \left (x \right )+\ln \left (-a \right )-\frac {1}{2 a^{2} x^{2}}-\frac {1}{a x}\right )\) | \(68\) |
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Time = 0.25 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.06 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{x^3} \, dx=\frac {4 \, a^{2} x^{2} \log \left (a x - 1\right ) - 4 \, a^{2} x^{2} \log \left (x\right ) + 4 \, a x + 1}{2 \, x^{2}} \]
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Time = 0.12 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.79 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{x^3} \, dx=2 a^{2} \left (- \log {\left (x \right )} + \log {\left (x - \frac {1}{a} \right )}\right ) + \frac {4 a x + 1}{2 x^{2}} \]
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Time = 0.19 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.91 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{x^3} \, dx=2 \, a^{2} \log \left (a x - 1\right ) - 2 \, a^{2} \log \left (x\right ) + \frac {4 \, a x + 1}{2 \, x^{2}} \]
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Time = 0.26 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.97 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{x^3} \, dx=2 \, a^{2} \log \left ({\left | a x - 1 \right |}\right ) - 2 \, a^{2} \log \left ({\left | x \right |}\right ) + \frac {4 \, a x + 1}{2 \, x^{2}} \]
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Time = 0.05 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.70 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{x^3} \, dx=\frac {2\,a\,x+\frac {1}{2}}{x^2}-4\,a^2\,\mathrm {atanh}\left (2\,a\,x-1\right ) \]
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