Integrand size = 22, antiderivative size = 110 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^3} \, dx=\frac {x}{c^3}+\frac {1}{12 a c^3 (1-a x)^3}-\frac {5}{8 a c^3 (1-a x)^2}+\frac {39}{16 a c^3 (1-a x)}-\frac {1}{16 a c^3 (1+a x)}+\frac {9 \log (1-a x)}{4 a c^3}-\frac {\log (1+a x)}{4 a c^3} \]
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Time = 0.15 (sec) , antiderivative size = 110, 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} \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^3} \, dx=\frac {39}{16 a c^3 (1-a x)}-\frac {1}{16 a c^3 (a x+1)}-\frac {5}{8 a c^3 (1-a x)^2}+\frac {1}{12 a c^3 (1-a x)^3}+\frac {9 \log (1-a x)}{4 a c^3}-\frac {\log (a x+1)}{4 a c^3}+\frac {x}{c^3} \]
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Rule 90
Rule 6285
Rule 6292
Rule 6302
Rubi steps \begin{align*} \text {integral}& = -\int \frac {e^{2 \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^3} \, dx \\ & = \frac {a^6 \int \frac {e^{2 \text {arctanh}(a x)} x^6}{\left (1-a^2 x^2\right )^3} \, dx}{c^3} \\ & = \frac {a^6 \int \frac {x^6}{(1-a x)^4 (1+a x)^2} \, dx}{c^3} \\ & = \frac {a^6 \int \left (\frac {1}{a^6}+\frac {1}{4 a^6 (-1+a x)^4}+\frac {5}{4 a^6 (-1+a x)^3}+\frac {39}{16 a^6 (-1+a x)^2}+\frac {9}{4 a^6 (-1+a x)}+\frac {1}{16 a^6 (1+a x)^2}-\frac {1}{4 a^6 (1+a x)}\right ) \, dx}{c^3} \\ & = \frac {x}{c^3}+\frac {1}{12 a c^3 (1-a x)^3}-\frac {5}{8 a c^3 (1-a x)^2}+\frac {39}{16 a c^3 (1-a x)}-\frac {1}{16 a c^3 (1+a x)}+\frac {9 \log (1-a x)}{4 a c^3}-\frac {\log (1+a x)}{4 a c^3} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.75 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^3} \, dx=\frac {\frac {2 \left (-11+7 a x+24 a^2 x^2-15 a^3 x^3-12 a^4 x^4+6 a^5 x^5\right )}{(-1+a x)^3 (1+a x)}+27 \log (1-a x)-3 \log (1+a x)}{12 a c^3} \]
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Time = 0.55 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.76
method | result | size |
default | \(\frac {a^{6} \left (-\frac {\ln \left (a x +1\right )}{4 a^{7}}-\frac {1}{16 a^{7} \left (a x +1\right )}+\frac {x}{a^{6}}-\frac {39}{16 a^{7} \left (a x -1\right )}+\frac {9 \ln \left (a x -1\right )}{4 a^{7}}-\frac {1}{12 a^{7} \left (a x -1\right )^{3}}-\frac {5}{8 a^{7} \left (a x -1\right )^{2}}\right )}{c^{3}}\) | \(84\) |
risch | \(\frac {x}{c^{3}}+\frac {-\frac {5 a^{2} c^{3} x^{3}}{2}+2 a \,c^{3} x^{2}+\frac {13 c^{3} x}{6}-\frac {11 c^{3}}{6 a}}{c^{6} \left (a x -1\right )^{2} \left (a^{2} x^{2}-1\right )}+\frac {9 \ln \left (-a x +1\right )}{4 a \,c^{3}}-\frac {\ln \left (a x +1\right )}{4 a \,c^{3}}\) | \(93\) |
norman | \(\frac {\frac {a^{5} x^{6}}{c}-\frac {5 x}{2 c}+\frac {3 a \,x^{2}}{2 c}+\frac {31 a^{2} x^{3}}{6 c}-\frac {8 a^{3} x^{4}}{3 c}-\frac {17 a^{4} x^{5}}{6 c}}{\left (a x -1\right )^{3} \left (a x +1\right )^{2} c^{2}}+\frac {9 \ln \left (a x -1\right )}{4 a \,c^{3}}-\frac {\ln \left (a x +1\right )}{4 a \,c^{3}}\) | \(107\) |
parallelrisch | \(\frac {12 a^{5} x^{5}+27 \ln \left (a x -1\right ) x^{4} a^{4}-3 \ln \left (a x +1\right ) x^{4} a^{4}-46 a^{4} x^{4}-54 a^{3} \ln \left (a x -1\right ) x^{3}+6 a^{3} \ln \left (a x +1\right ) x^{3}+14 a^{3} x^{3}+48 a^{2} x^{2}+54 a \ln \left (a x -1\right ) x -6 a \ln \left (a x +1\right ) x -30 a x -27 \ln \left (a x -1\right )+3 \ln \left (a x +1\right )}{12 c^{3} \left (a x -1\right )^{2} \left (a^{2} x^{2}-1\right ) a}\) | \(156\) |
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Time = 0.25 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.25 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^3} \, dx=\frac {12 \, a^{5} x^{5} - 24 \, a^{4} x^{4} - 30 \, a^{3} x^{3} + 48 \, a^{2} x^{2} + 14 \, a x - 3 \, {\left (a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a x - 1\right )} \log \left (a x + 1\right ) + 27 \, {\left (a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a x - 1\right )} \log \left (a x - 1\right ) - 22}{12 \, {\left (a^{5} c^{3} x^{4} - 2 \, a^{4} c^{3} x^{3} + 2 \, a^{2} c^{3} x - a c^{3}\right )}} \]
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Time = 0.33 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.93 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^3} \, dx=a^{6} \left (\frac {- 15 a^{3} x^{3} + 12 a^{2} x^{2} + 13 a x - 11}{6 a^{11} c^{3} x^{4} - 12 a^{10} c^{3} x^{3} + 12 a^{8} c^{3} x - 6 a^{7} c^{3}} + \frac {x}{a^{6} c^{3}} + \frac {\frac {9 \log {\left (x - \frac {1}{a} \right )}}{4} - \frac {\log {\left (x + \frac {1}{a} \right )}}{4}}{a^{7} c^{3}}\right ) \]
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Time = 0.21 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.88 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^3} \, dx=-\frac {15 \, a^{3} x^{3} - 12 \, a^{2} x^{2} - 13 \, a x + 11}{6 \, {\left (a^{5} c^{3} x^{4} - 2 \, a^{4} c^{3} x^{3} + 2 \, a^{2} c^{3} x - a c^{3}\right )}} + \frac {x}{c^{3}} - \frac {\log \left (a x + 1\right )}{4 \, a c^{3}} + \frac {9 \, \log \left (a x - 1\right )}{4 \, a c^{3}} \]
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Time = 0.28 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.73 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^3} \, dx=\frac {x}{c^{3}} - \frac {\log \left ({\left | a x + 1 \right |}\right )}{4 \, a c^{3}} + \frac {9 \, \log \left ({\left | a x - 1 \right |}\right )}{4 \, a c^{3}} - \frac {15 \, a^{3} x^{3} - 12 \, a^{2} x^{2} - 13 \, a x + 11}{6 \, {\left (a x + 1\right )} {\left (a x - 1\right )}^{3} a c^{3}} \]
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Time = 4.19 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.85 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^3} \, dx=\frac {x}{c^3}-\frac {\frac {13\,x}{6}+2\,a\,x^2-\frac {11}{6\,a}-\frac {5\,a^2\,x^3}{2}}{-a^4\,c^3\,x^4+2\,a^3\,c^3\,x^3-2\,a\,c^3\,x+c^3}+\frac {9\,\ln \left (a\,x-1\right )}{4\,a\,c^3}-\frac {\ln \left (a\,x+1\right )}{4\,a\,c^3} \]
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