Integrand size = 22, antiderivative size = 76 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^3 \, dx=-\frac {c^3}{5 a^6 x^5}-\frac {c^3}{2 a^5 x^4}+\frac {c^3}{3 a^4 x^3}+\frac {2 c^3}{a^3 x^2}+\frac {c^3}{a^2 x}+c^3 x+\frac {2 c^3 \log (x)}{a} \]
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Time = 0.11 (sec) , antiderivative size = 76, 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} \[ \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^3 \, dx=-\frac {c^3}{5 a^6 x^5}-\frac {c^3}{2 a^5 x^4}+\frac {c^3}{3 a^4 x^3}+\frac {2 c^3}{a^3 x^2}+\frac {c^3}{a^2 x}+\frac {2 c^3 \log (x)}{a}+c^3 x \]
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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 )^3 \, dx \\ & = \frac {c^3 \int \frac {e^{2 \text {arctanh}(a x)} \left (1-a^2 x^2\right )^3}{x^6} \, dx}{a^6} \\ & = \frac {c^3 \int \frac {(1-a x)^2 (1+a x)^4}{x^6} \, dx}{a^6} \\ & = \frac {c^3 \int \left (a^6+\frac {1}{x^6}+\frac {2 a}{x^5}-\frac {a^2}{x^4}-\frac {4 a^3}{x^3}-\frac {a^4}{x^2}+\frac {2 a^5}{x}\right ) \, dx}{a^6} \\ & = -\frac {c^3}{5 a^6 x^5}-\frac {c^3}{2 a^5 x^4}+\frac {c^3}{3 a^4 x^3}+\frac {2 c^3}{a^3 x^2}+\frac {c^3}{a^2 x}+c^3 x+\frac {2 c^3 \log (x)}{a} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.00 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^3 \, dx=-\frac {c^3}{5 a^6 x^5}-\frac {c^3}{2 a^5 x^4}+\frac {c^3}{3 a^4 x^3}+\frac {2 c^3}{a^3 x^2}+\frac {c^3}{a^2 x}+c^3 x+\frac {2 c^3 \log (x)}{a} \]
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Time = 0.71 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.72
method | result | size |
default | \(\frac {c^{3} \left (a^{6} x +2 a^{5} \ln \left (x \right )-\frac {a}{2 x^{4}}+\frac {a^{2}}{3 x^{3}}+\frac {2 a^{3}}{x^{2}}+\frac {a^{4}}{x}-\frac {1}{5 x^{5}}\right )}{a^{6}}\) | \(55\) |
risch | \(c^{3} x +\frac {a^{4} c^{3} x^{4}+2 a^{3} c^{3} x^{3}+\frac {1}{3} a^{2} c^{3} x^{2}-\frac {1}{2} a \,c^{3} x -\frac {1}{5} c^{3}}{a^{6} x^{5}}+\frac {2 c^{3} \ln \left (x \right )}{a}\) | \(69\) |
norman | \(\frac {a^{3} c^{3} x^{4}+a^{5} c^{3} x^{6}-\frac {c^{3}}{5 a}-\frac {c^{3} x}{2}+\frac {a \,c^{3} x^{2}}{3}+2 a^{2} c^{3} x^{3}}{a^{5} x^{5}}+\frac {2 c^{3} \ln \left (x \right )}{a}\) | \(74\) |
parallelrisch | \(\frac {30 a^{6} c^{3} x^{6}+60 c^{3} \ln \left (x \right ) a^{5} x^{5}+30 a^{4} c^{3} x^{4}+60 a^{3} c^{3} x^{3}+10 a^{2} c^{3} x^{2}-15 a \,c^{3} x -6 c^{3}}{30 a^{6} x^{5}}\) | \(79\) |
meijerg | \(-\frac {c^{3} \left (-a x -\ln \left (-a x +1\right )\right )}{a}+\frac {3 c^{3} \left (-\ln \left (-a x +1\right )+\ln \left (x \right )+\ln \left (-a \right )\right )}{a}-\frac {3 c^{3} \left (-\ln \left (-a x +1\right )+\ln \left (x \right )+\ln \left (-a \right )-\frac {1}{2 a^{2} x^{2}}-\frac {1}{a x}\right )}{a}+\frac {c^{3} \left (-\ln \left (-a x +1\right )+\ln \left (x \right )+\ln \left (-a \right )-\frac {1}{4 a^{4} x^{4}}-\frac {1}{3 x^{3} a^{3}}-\frac {1}{2 a^{2} x^{2}}-\frac {1}{a x}\right )}{a}+\frac {c^{3} \ln \left (-a x +1\right )}{a}-\frac {3 c^{3} \left (\ln \left (-a x +1\right )-\ln \left (x \right )-\ln \left (-a \right )+\frac {1}{a x}\right )}{a}+\frac {3 c^{3} \left (\ln \left (-a x +1\right )-\ln \left (x \right )-\ln \left (-a \right )+\frac {1}{3 x^{3} a^{3}}+\frac {1}{2 a^{2} x^{2}}+\frac {1}{a x}\right )}{a}-\frac {c^{3} \left (\ln \left (-a x +1\right )-\ln \left (x \right )-\ln \left (-a \right )+\frac {1}{5 x^{5} a^{5}}+\frac {1}{4 a^{4} x^{4}}+\frac {1}{3 x^{3} a^{3}}+\frac {1}{2 a^{2} x^{2}}+\frac {1}{a x}\right )}{a}\) | \(304\) |
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Time = 0.25 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.03 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^3 \, dx=\frac {30 \, a^{6} c^{3} x^{6} + 60 \, a^{5} c^{3} x^{5} \log \left (x\right ) + 30 \, a^{4} c^{3} x^{4} + 60 \, a^{3} c^{3} x^{3} + 10 \, a^{2} c^{3} x^{2} - 15 \, a c^{3} x - 6 \, c^{3}}{30 \, a^{6} x^{5}} \]
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Time = 0.17 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.00 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^3 \, dx=\frac {a^{6} c^{3} x + 2 a^{5} c^{3} \log {\left (x \right )} + \frac {30 a^{4} c^{3} x^{4} + 60 a^{3} c^{3} x^{3} + 10 a^{2} c^{3} x^{2} - 15 a c^{3} x - 6 c^{3}}{30 x^{5}}}{a^{6}} \]
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Time = 0.20 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.92 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^3 \, dx=c^{3} x + \frac {2 \, c^{3} \log \left (x\right )}{a} + \frac {30 \, a^{4} c^{3} x^{4} + 60 \, a^{3} c^{3} x^{3} + 10 \, a^{2} c^{3} x^{2} - 15 \, a c^{3} x - 6 \, c^{3}}{30 \, a^{6} x^{5}} \]
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Time = 0.28 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.93 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^3 \, dx=c^{3} x + \frac {2 \, c^{3} \log \left ({\left | x \right |}\right )}{a} + \frac {30 \, a^{4} c^{3} x^{4} + 60 \, a^{3} c^{3} x^{3} + 10 \, a^{2} c^{3} x^{2} - 15 \, a c^{3} x - 6 \, c^{3}}{30 \, a^{6} x^{5}} \]
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Time = 4.01 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.74 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^3 \, dx=\frac {c^3\,\left (\frac {a^2\,x^2}{3}-\frac {a\,x}{2}+2\,a^3\,x^3+a^4\,x^4+a^6\,x^6+2\,a^5\,x^5\,\ln \left (x\right )-\frac {1}{5}\right )}{a^6\,x^5} \]
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