Integrand size = 22, antiderivative size = 40 \[ \int e^{-2 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^2 \, dx=\frac {c^2}{a^2 x}+c^2 x+\frac {4 c^2 \log (x)}{a}-\frac {8 c^2 \log (1+a x)}{a} \]
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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, 6266, 6264, 90} \[ \int e^{-2 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^2 \, dx=\frac {c^2}{a^2 x}+\frac {4 c^2 \log (x)}{a}-\frac {8 c^2 \log (a x+1)}{a}+c^2 x \]
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Rule 90
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 )^2 \, dx \\ & = -\frac {c^2 \int \frac {e^{-2 \text {arctanh}(a x)} (1-a x)^2}{x^2} \, dx}{a^2} \\ & = -\frac {c^2 \int \frac {(1-a x)^3}{x^2 (1+a x)} \, dx}{a^2} \\ & = -\frac {c^2 \int \left (-a^2+\frac {1}{x^2}-\frac {4 a}{x}+\frac {8 a^2}{1+a x}\right ) \, dx}{a^2} \\ & = \frac {c^2}{a^2 x}+c^2 x+\frac {4 c^2 \log (x)}{a}-\frac {8 c^2 \log (1+a x)}{a} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.85 \[ \int e^{-2 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^2 \, dx=-\frac {c^2 \left (-\frac {1}{x}-a^2 x-4 a \log (x)+8 a \log (1+a x)\right )}{a^2} \]
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Time = 0.58 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.78
method | result | size |
default | \(\frac {c^{2} \left (-8 a \ln \left (a x +1\right )+a^{2} x +\frac {1}{x}+4 a \ln \left (x \right )\right )}{a^{2}}\) | \(31\) |
risch | \(\frac {c^{2}}{a^{2} x}+c^{2} x +\frac {4 c^{2} \ln \left (-x \right )}{a}-\frac {8 c^{2} \ln \left (a x +1\right )}{a}\) | \(43\) |
parallelrisch | \(\frac {a^{2} c^{2} x^{2}+4 c^{2} \ln \left (x \right ) a x -8 c^{2} \ln \left (a x +1\right ) a x +c^{2}}{a^{2} x}\) | \(44\) |
norman | \(\frac {\frac {c^{2}}{a}+a \,c^{2} x^{2}}{a x}+\frac {4 c^{2} \ln \left (x \right )}{a}-\frac {8 c^{2} \ln \left (a x +1\right )}{a}\) | \(49\) |
meijerg | \(\frac {c^{2} \left (a x -\ln \left (a x +1\right )\right )}{a}-\frac {3 c^{2} \ln \left (a x +1\right )}{a}+\frac {3 c^{2} \left (-\ln \left (a x +1\right )+\ln \left (x \right )+\ln \left (a \right )\right )}{a}-\frac {c^{2} \left (\ln \left (a x +1\right )-\ln \left (x \right )-\ln \left (a \right )-\frac {1}{a x}\right )}{a}\) | \(87\) |
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Time = 0.24 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.08 \[ \int e^{-2 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^2 \, dx=\frac {a^{2} c^{2} x^{2} - 8 \, a c^{2} x \log \left (a x + 1\right ) + 4 \, a c^{2} x \log \left (x\right ) + c^{2}}{a^{2} x} \]
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Time = 0.21 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.78 \[ \int e^{-2 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^2 \, dx=c^{2} x + \frac {4 c^{2} \left (\log {\left (x \right )} - 2 \log {\left (x + \frac {1}{a} \right )}\right )}{a} + \frac {c^{2}}{a^{2} x} \]
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Time = 0.19 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00 \[ \int e^{-2 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^2 \, dx=c^{2} x - \frac {8 \, c^{2} \log \left (a x + 1\right )}{a} + \frac {4 \, c^{2} \log \left (x\right )}{a} + \frac {c^{2}}{a^{2} x} \]
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Time = 0.28 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.05 \[ \int e^{-2 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^2 \, dx=c^{2} x - \frac {8 \, c^{2} \log \left ({\left | a x + 1 \right |}\right )}{a} + \frac {4 \, c^{2} \log \left ({\left | x \right |}\right )}{a} + \frac {c^{2}}{a^{2} x} \]
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Time = 4.25 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00 \[ \int e^{-2 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^2 \, dx=c^2\,x+\frac {c^2}{a^2\,x}+\frac {4\,c^2\,\ln \left (x\right )}{a}-\frac {8\,c^2\,\ln \left (a\,x+1\right )}{a} \]
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