Integrand size = 22, antiderivative size = 40 \[ \int e^{-2 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^2 \, dx=\frac {c^2}{3 a^4 x^3}-\frac {c^2}{a^3 x^2}+c^2 x-\frac {2 c^2 \log (x)}{a} \]
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Time = 0.13 (sec) , antiderivative size = 40, 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, 6292, 6285, 76} \[ \int e^{-2 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^2 \, dx=\frac {c^2}{3 a^4 x^3}-\frac {c^2}{a^3 x^2}-\frac {2 c^2 \log (x)}{a}+c^2 x \]
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Rule 76
Rule 6285
Rule 6292
Rule 6302
Rubi steps \begin{align*} \text {integral}& = -\int e^{-2 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^2 \, dx \\ & = -\frac {c^2 \int \frac {e^{-2 \text {arctanh}(a x)} \left (1-a^2 x^2\right )^2}{x^4} \, dx}{a^4} \\ & = -\frac {c^2 \int \frac {(1-a x)^3 (1+a x)}{x^4} \, dx}{a^4} \\ & = -\frac {c^2 \int \left (-a^4+\frac {1}{x^4}-\frac {2 a}{x^3}+\frac {2 a^3}{x}\right ) \, dx}{a^4} \\ & = \frac {c^2}{3 a^4 x^3}-\frac {c^2}{a^3 x^2}+c^2 x-\frac {2 c^2 \log (x)}{a} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00 \[ \int e^{-2 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^2 \, dx=\frac {c^2}{3 a^4 x^3}-\frac {c^2}{a^3 x^2}+c^2 x-\frac {2 c^2 \log (x)}{a} \]
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Time = 0.66 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.80
method | result | size |
default | \(\frac {c^{2} \left (a^{4} x -2 a^{3} \ln \left (x \right )+\frac {1}{3 x^{3}}-\frac {a}{x^{2}}\right )}{a^{4}}\) | \(32\) |
risch | \(c^{2} x +\frac {-a \,c^{2} x +\frac {1}{3} c^{2}}{a^{4} x^{3}}-\frac {2 c^{2} \ln \left (x \right )}{a}\) | \(37\) |
norman | \(\frac {a^{3} c^{2} x^{4}+\frac {c^{2}}{3 a}-c^{2} x}{a^{3} x^{3}}-\frac {2 c^{2} \ln \left (x \right )}{a}\) | \(44\) |
parallelrisch | \(-\frac {-3 a^{4} c^{2} x^{4}+6 c^{2} \ln \left (x \right ) a^{3} x^{3}+3 a \,c^{2} x -c^{2}}{3 a^{4} x^{3}}\) | \(46\) |
meijerg | \(\frac {c^{2} \left (a x -\ln \left (a x +1\right )\right )}{a}-\frac {c^{2} \ln \left (a x +1\right )}{a}-\frac {2 c^{2} \left (-\ln \left (a x +1\right )+\ln \left (x \right )+\ln \left (a \right )\right )}{a}+\frac {2 c^{2} \left (\ln \left (a x +1\right )-\ln \left (x \right )-\ln \left (a \right )-\frac {1}{a x}\right )}{a}+\frac {c^{2} \left (-\ln \left (a x +1\right )+\ln \left (x \right )+\ln \left (a \right )-\frac {1}{2 a^{2} x^{2}}+\frac {1}{a x}\right )}{a}-\frac {c^{2} \left (\ln \left (a x +1\right )-\ln \left (x \right )-\ln \left (a \right )-\frac {1}{3 x^{3} a^{3}}+\frac {1}{2 a^{2} x^{2}}-\frac {1}{a x}\right )}{a}\) | \(169\) |
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Time = 0.23 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.08 \[ \int e^{-2 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^2 \, dx=\frac {3 \, a^{4} c^{2} x^{4} - 6 \, a^{3} c^{2} x^{3} \log \left (x\right ) - 3 \, a c^{2} x + c^{2}}{3 \, a^{4} x^{3}} \]
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Time = 0.09 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.98 \[ \int e^{-2 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^2 \, dx=\frac {a^{4} c^{2} x - 2 a^{3} c^{2} \log {\left (x \right )} + \frac {- 3 a c^{2} x + c^{2}}{3 x^{3}}}{a^{4}} \]
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Time = 0.19 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.92 \[ \int e^{-2 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^2 \, dx=c^{2} x - \frac {2 \, c^{2} \log \left (x\right )}{a} - \frac {3 \, a c^{2} x - c^{2}}{3 \, a^{4} x^{3}} \]
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Time = 0.27 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.95 \[ \int e^{-2 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^2 \, dx=c^{2} x - \frac {2 \, c^{2} \log \left ({\left | x \right |}\right )}{a} - \frac {3 \, a c^{2} x - c^{2}}{3 \, a^{4} x^{3}} \]
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Time = 0.08 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.88 \[ \int e^{-2 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^2 \, dx=-\frac {c^2\,\left (3\,a\,x-3\,a^4\,x^4+6\,a^3\,x^3\,\ln \left (x\right )-1\right )}{3\,a^4\,x^3} \]
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