Integrand size = 22, antiderivative size = 90 \[ \int e^{-2 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^4 \, dx=\frac {c^4}{7 a^8 x^7}-\frac {c^4}{3 a^7 x^6}-\frac {2 c^4}{5 a^6 x^5}+\frac {3 c^4}{2 a^5 x^4}-\frac {3 c^4}{a^3 x^2}+\frac {2 c^4}{a^2 x}+c^4 x-\frac {2 c^4 \log (x)}{a} \]
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Time = 0.13 (sec) , antiderivative size = 90, 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, 90} \[ \int e^{-2 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^4 \, dx=\frac {c^4}{7 a^8 x^7}-\frac {c^4}{3 a^7 x^6}-\frac {2 c^4}{5 a^6 x^5}+\frac {3 c^4}{2 a^5 x^4}-\frac {3 c^4}{a^3 x^2}+\frac {2 c^4}{a^2 x}-\frac {2 c^4 \log (x)}{a}+c^4 x \]
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Rule 90
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 )^4 \, dx \\ & = -\frac {c^4 \int \frac {e^{-2 \text {arctanh}(a x)} \left (1-a^2 x^2\right )^4}{x^8} \, dx}{a^8} \\ & = -\frac {c^4 \int \frac {(1-a x)^5 (1+a x)^3}{x^8} \, dx}{a^8} \\ & = -\frac {c^4 \int \left (-a^8+\frac {1}{x^8}-\frac {2 a}{x^7}-\frac {2 a^2}{x^6}+\frac {6 a^3}{x^5}-\frac {6 a^5}{x^3}+\frac {2 a^6}{x^2}+\frac {2 a^7}{x}\right ) \, dx}{a^8} \\ & = \frac {c^4}{7 a^8 x^7}-\frac {c^4}{3 a^7 x^6}-\frac {2 c^4}{5 a^6 x^5}+\frac {3 c^4}{2 a^5 x^4}-\frac {3 c^4}{a^3 x^2}+\frac {2 c^4}{a^2 x}+c^4 x-\frac {2 c^4 \log (x)}{a} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.00 \[ \int e^{-2 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^4 \, dx=\frac {c^4}{7 a^8 x^7}-\frac {c^4}{3 a^7 x^6}-\frac {2 c^4}{5 a^6 x^5}+\frac {3 c^4}{2 a^5 x^4}-\frac {3 c^4}{a^3 x^2}+\frac {2 c^4}{a^2 x}+c^4 x-\frac {2 c^4 \log (x)}{a} \]
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Time = 0.78 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.71
method | result | size |
default | \(\frac {c^{4} \left (a^{8} x -2 a^{7} \ln \left (x \right )+\frac {3 a^{3}}{2 x^{4}}-\frac {a}{3 x^{6}}-\frac {3 a^{5}}{x^{2}}+\frac {2 a^{6}}{x}+\frac {1}{7 x^{7}}-\frac {2 a^{2}}{5 x^{5}}\right )}{a^{8}}\) | \(64\) |
risch | \(c^{4} x +\frac {2 a^{6} c^{4} x^{6}-3 a^{5} c^{4} x^{5}+\frac {3}{2} a^{3} c^{4} x^{3}-\frac {2}{5} a^{2} c^{4} x^{2}-\frac {1}{3} a \,c^{4} x +\frac {1}{7} c^{4}}{a^{8} x^{7}}-\frac {2 c^{4} \ln \left (x \right )}{a}\) | \(81\) |
norman | \(\frac {a^{7} c^{4} x^{8}+\frac {c^{4}}{7 a}-\frac {c^{4} x}{3}-\frac {2 a \,c^{4} x^{2}}{5}+\frac {3 a^{2} c^{4} x^{3}}{2}-3 a^{4} c^{4} x^{5}+2 a^{5} c^{4} x^{6}}{a^{7} x^{7}}-\frac {2 c^{4} \ln \left (x \right )}{a}\) | \(86\) |
parallelrisch | \(-\frac {-210 a^{8} c^{4} x^{8}+420 c^{4} \ln \left (x \right ) a^{7} x^{7}-420 a^{6} c^{4} x^{6}+630 a^{5} c^{4} x^{5}-315 a^{3} c^{4} x^{3}+84 a^{2} c^{4} x^{2}+70 a \,c^{4} x -30 c^{4}}{210 a^{8} x^{7}}\) | \(90\) |
meijerg | \(\frac {c^{4} \left (a x -\ln \left (a x +1\right )\right )}{a}-\frac {c^{4} \ln \left (a x +1\right )}{a}-\frac {4 c^{4} \left (-\ln \left (a x +1\right )+\ln \left (x \right )+\ln \left (a \right )\right )}{a}+\frac {4 c^{4} \left (\ln \left (a x +1\right )-\ln \left (x \right )-\ln \left (a \right )-\frac {1}{a x}\right )}{a}+\frac {6 c^{4} \left (-\ln \left (a x +1\right )+\ln \left (x \right )+\ln \left (a \right )-\frac {1}{2 a^{2} x^{2}}+\frac {1}{a x}\right )}{a}-\frac {6 c^{4} \left (\ln \left (a x +1\right )-\ln \left (x \right )-\ln \left (a \right )-\frac {1}{3 x^{3} a^{3}}+\frac {1}{2 a^{2} x^{2}}-\frac {1}{a x}\right )}{a}-\frac {4 c^{4} \left (-\ln \left (a x +1\right )+\ln \left (x \right )+\ln \left (a \right )-\frac {1}{4 a^{4} x^{4}}+\frac {1}{3 x^{3} a^{3}}-\frac {1}{2 a^{2} x^{2}}+\frac {1}{a x}\right )}{a}+\frac {4 c^{4} \left (\ln \left (a x +1\right )-\ln \left (x \right )-\ln \left (a \right )-\frac {1}{5 x^{5} a^{5}}+\frac {1}{4 a^{4} x^{4}}-\frac {1}{3 x^{3} a^{3}}+\frac {1}{2 a^{2} x^{2}}-\frac {1}{a x}\right )}{a}+\frac {c^{4} \left (-\ln \left (a x +1\right )+\ln \left (x \right )+\ln \left (a \right )-\frac {1}{6 a^{6} x^{6}}+\frac {1}{5 x^{5} a^{5}}-\frac {1}{4 a^{4} x^{4}}+\frac {1}{3 x^{3} a^{3}}-\frac {1}{2 a^{2} x^{2}}+\frac {1}{a x}\right )}{a}-\frac {c^{4} \left (\ln \left (a x +1\right )-\ln \left (x \right )-\ln \left (a \right )-\frac {1}{7 x^{7} a^{7}}+\frac {1}{6 a^{6} x^{6}}-\frac {1}{5 x^{5} a^{5}}+\frac {1}{4 a^{4} x^{4}}-\frac {1}{3 x^{3} a^{3}}+\frac {1}{2 a^{2} x^{2}}-\frac {1}{a x}\right )}{a}\) | \(431\) |
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Time = 0.25 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.99 \[ \int e^{-2 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^4 \, dx=\frac {210 \, a^{8} c^{4} x^{8} - 420 \, a^{7} c^{4} x^{7} \log \left (x\right ) + 420 \, a^{6} c^{4} x^{6} - 630 \, a^{5} c^{4} x^{5} + 315 \, a^{3} c^{4} x^{3} - 84 \, a^{2} c^{4} x^{2} - 70 \, a c^{4} x + 30 \, c^{4}}{210 \, a^{8} x^{7}} \]
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Time = 0.25 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.98 \[ \int e^{-2 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^4 \, dx=\frac {a^{8} c^{4} x - 2 a^{7} c^{4} \log {\left (x \right )} + \frac {420 a^{6} c^{4} x^{6} - 630 a^{5} c^{4} x^{5} + 315 a^{3} c^{4} x^{3} - 84 a^{2} c^{4} x^{2} - 70 a c^{4} x + 30 c^{4}}{210 x^{7}}}{a^{8}} \]
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Time = 0.19 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.90 \[ \int e^{-2 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^4 \, dx=c^{4} x - \frac {2 \, c^{4} \log \left (x\right )}{a} + \frac {420 \, a^{6} c^{4} x^{6} - 630 \, a^{5} c^{4} x^{5} + 315 \, a^{3} c^{4} x^{3} - 84 \, a^{2} c^{4} x^{2} - 70 \, a c^{4} x + 30 \, c^{4}}{210 \, a^{8} x^{7}} \]
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Time = 0.28 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.91 \[ \int e^{-2 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^4 \, dx=c^{4} x - \frac {2 \, c^{4} \log \left ({\left | x \right |}\right )}{a} + \frac {420 \, a^{6} c^{4} x^{6} - 630 \, a^{5} c^{4} x^{5} + 315 \, a^{3} c^{4} x^{3} - 84 \, a^{2} c^{4} x^{2} - 70 \, a c^{4} x + 30 \, c^{4}}{210 \, a^{8} x^{7}} \]
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Time = 4.03 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.74 \[ \int e^{-2 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^4 \, dx=-\frac {c^4\,\left (\frac {a\,x}{3}+\frac {2\,a^2\,x^2}{5}-\frac {3\,a^3\,x^3}{2}+3\,a^5\,x^5-2\,a^6\,x^6-a^8\,x^8+2\,a^7\,x^7\,\ln \left (x\right )-\frac {1}{7}\right )}{a^8\,x^7} \]
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