Integrand size = 22, antiderivative size = 100 \[ \int e^{4 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^4 \, dx=-\frac {c^4}{7 a^8 x^7}-\frac {2 c^4}{3 a^7 x^6}-\frac {4 c^4}{5 a^6 x^5}+\frac {c^4}{a^5 x^4}+\frac {10 c^4}{3 a^4 x^3}+\frac {2 c^4}{a^3 x^2}-\frac {4 c^4}{a^2 x}+c^4 x+\frac {4 c^4 \log (x)}{a} \]
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Time = 0.12 (sec) , antiderivative size = 100, 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} \[ \int e^{4 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^4 \, dx=-\frac {c^4}{7 a^8 x^7}-\frac {2 c^4}{3 a^7 x^6}-\frac {4 c^4}{5 a^6 x^5}+\frac {c^4}{a^5 x^4}+\frac {10 c^4}{3 a^4 x^3}+\frac {2 c^4}{a^3 x^2}-\frac {4 c^4}{a^2 x}+\frac {4 c^4 \log (x)}{a}+c^4 x \]
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Rule 90
Rule 6285
Rule 6292
Rule 6302
Rubi steps \begin{align*} \text {integral}& = \int e^{4 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^4 \, dx \\ & = \frac {c^4 \int \frac {e^{4 \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)^2 (1+a x)^6}{x^8} \, dx}{a^8} \\ & = \frac {c^4 \int \left (a^8+\frac {1}{x^8}+\frac {4 a}{x^7}+\frac {4 a^2}{x^6}-\frac {4 a^3}{x^5}-\frac {10 a^4}{x^4}-\frac {4 a^5}{x^3}+\frac {4 a^6}{x^2}+\frac {4 a^7}{x}\right ) \, dx}{a^8} \\ & = -\frac {c^4}{7 a^8 x^7}-\frac {2 c^4}{3 a^7 x^6}-\frac {4 c^4}{5 a^6 x^5}+\frac {c^4}{a^5 x^4}+\frac {10 c^4}{3 a^4 x^3}+\frac {2 c^4}{a^3 x^2}-\frac {4 c^4}{a^2 x}+c^4 x+\frac {4 c^4 \log (x)}{a} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.00 \[ \int e^{4 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^4 \, dx=-\frac {c^4}{7 a^8 x^7}-\frac {2 c^4}{3 a^7 x^6}-\frac {4 c^4}{5 a^6 x^5}+\frac {c^4}{a^5 x^4}+\frac {10 c^4}{3 a^4 x^3}+\frac {2 c^4}{a^3 x^2}-\frac {4 c^4}{a^2 x}+c^4 x+\frac {4 c^4 \log (x)}{a} \]
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Time = 0.84 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.71
method | result | size |
default | \(\frac {c^{4} \left (a^{8} x +4 a^{7} \ln \left (x \right )+\frac {a^{3}}{x^{4}}+\frac {10 a^{4}}{3 x^{3}}-\frac {2 a}{3 x^{6}}+\frac {2 a^{5}}{x^{2}}-\frac {4 a^{6}}{x}-\frac {1}{7 x^{7}}-\frac {4 a^{2}}{5 x^{5}}\right )}{a^{8}}\) | \(71\) |
risch | \(c^{4} x +\frac {-4 a^{6} c^{4} x^{6}+2 a^{5} c^{4} x^{5}+\frac {10}{3} a^{4} c^{4} x^{4}+a^{3} c^{4} x^{3}-\frac {4}{5} a^{2} c^{4} x^{2}-\frac {2}{3} a \,c^{4} x -\frac {1}{7} c^{4}}{a^{8} x^{7}}+\frac {4 c^{4} \ln \left (x \right )}{a}\) | \(91\) |
parallelrisch | \(\frac {105 a^{8} c^{4} x^{8}+420 c^{4} \ln \left (x \right ) a^{7} x^{7}-420 a^{6} c^{4} x^{6}+210 a^{5} c^{4} x^{5}+350 a^{4} c^{4} x^{4}+105 a^{3} c^{4} x^{3}-84 a^{2} c^{4} x^{2}-70 a \,c^{4} x -15 c^{4}}{105 a^{8} x^{7}}\) | \(101\) |
norman | \(\frac {-5 a^{7} c^{4} x^{8}+a^{8} c^{4} x^{9}+\frac {c^{4}}{7 a}+\frac {11 c^{4} x}{21}+\frac {2 a \,c^{4} x^{2}}{15}-\frac {9 a^{2} c^{4} x^{3}}{5}-\frac {7 a^{3} c^{4} x^{4}}{3}+\frac {4 a^{4} c^{4} x^{5}}{3}+6 a^{5} c^{4} x^{6}}{\left (a x -1\right ) a^{7} x^{7}}+\frac {4 c^{4} \ln \left (x \right )}{a}\) | \(115\) |
meijerg | \(-\frac {c^{4} \left (-\frac {a x \left (-3 a x +6\right )}{3 \left (-a x +1\right )}-2 \ln \left (-a x +1\right )\right )}{a}-\frac {3 c^{4} x}{-a x +1}-\frac {2 c^{4} \left (-\frac {3 a x}{-3 a x +3}+2 \ln \left (-a x +1\right )-1-2 \ln \left (x \right )-2 \ln \left (-a \right )+\frac {1}{a x}\right )}{a}-\frac {2 c^{4} \left (-\frac {5 a x}{-5 a x +5}+4 \ln \left (-a x +1\right )-1-4 \ln \left (x \right )-4 \ln \left (-a \right )+\frac {1}{3 x^{3} a^{3}}+\frac {1}{a^{2} x^{2}}+\frac {3}{a x}\right )}{a}+\frac {3 c^{4} \left (-\frac {7 a x}{-7 a x +7}+6 \ln \left (-a x +1\right )-1-6 \ln \left (x \right )-6 \ln \left (-a \right )+\frac {1}{5 x^{5} a^{5}}+\frac {1}{2 a^{4} x^{4}}+\frac {1}{x^{3} a^{3}}+\frac {2}{a^{2} x^{2}}+\frac {5}{a x}\right )}{a}+\frac {2 c^{4} \left (\frac {a x}{-a x +1}+\ln \left (-a x +1\right )\right )}{a}-\frac {8 c^{4} \left (\frac {2 a x}{-2 a x +2}-\ln \left (-a x +1\right )+1+\ln \left (x \right )+\ln \left (-a \right )\right )}{a}+\frac {12 c^{4} \left (\frac {4 a x}{-4 a x +4}-3 \ln \left (-a x +1\right )+1+3 \ln \left (x \right )+3 \ln \left (-a \right )-\frac {1}{2 a^{2} x^{2}}-\frac {2}{a x}\right )}{a}-\frac {8 c^{4} \left (\frac {6 a x}{-6 a x +6}-5 \ln \left (-a x +1\right )+1+5 \ln \left (x \right )+5 \ln \left (-a \right )-\frac {1}{4 a^{4} x^{4}}-\frac {2}{3 x^{3} a^{3}}-\frac {3}{2 a^{2} x^{2}}-\frac {4}{a x}\right )}{a}+\frac {2 c^{4} \left (\frac {8 a x}{-8 a x +8}-7 \ln \left (-a x +1\right )+1+7 \ln \left (x \right )+7 \ln \left (-a \right )-\frac {1}{6 a^{6} x^{6}}-\frac {2}{5 x^{5} a^{5}}-\frac {3}{4 a^{4} x^{4}}-\frac {4}{3 x^{3} a^{3}}-\frac {5}{2 a^{2} x^{2}}-\frac {6}{a x}\right )}{a}-\frac {c^{4} \left (-\frac {9 a x}{-9 a x +9}+8 \ln \left (-a x +1\right )-1-8 \ln \left (x \right )-8 \ln \left (-a \right )+\frac {1}{7 x^{7} a^{7}}+\frac {1}{3 a^{6} x^{6}}+\frac {3}{5 x^{5} a^{5}}+\frac {1}{a^{4} x^{4}}+\frac {5}{3 x^{3} a^{3}}+\frac {3}{a^{2} x^{2}}+\frac {7}{a x}\right )}{a}\) | \(623\) |
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Time = 0.23 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.00 \[ \int e^{4 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^4 \, dx=\frac {105 \, a^{8} c^{4} x^{8} + 420 \, a^{7} c^{4} x^{7} \log \left (x\right ) - 420 \, a^{6} c^{4} x^{6} + 210 \, a^{5} c^{4} x^{5} + 350 \, a^{4} c^{4} x^{4} + 105 \, a^{3} c^{4} x^{3} - 84 \, a^{2} c^{4} x^{2} - 70 \, a c^{4} x - 15 \, c^{4}}{105 \, a^{8} x^{7}} \]
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Time = 0.26 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.00 \[ \int e^{4 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^4 \, dx=\frac {a^{8} c^{4} x + 4 a^{7} c^{4} \log {\left (x \right )} + \frac {- 420 a^{6} c^{4} x^{6} + 210 a^{5} c^{4} x^{5} + 350 a^{4} c^{4} x^{4} + 105 a^{3} c^{4} x^{3} - 84 a^{2} c^{4} x^{2} - 70 a c^{4} x - 15 c^{4}}{105 x^{7}}}{a^{8}} \]
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Time = 0.20 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.92 \[ \int e^{4 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^4 \, dx=c^{4} x + \frac {4 \, c^{4} \log \left (x\right )}{a} - \frac {420 \, a^{6} c^{4} x^{6} - 210 \, a^{5} c^{4} x^{5} - 350 \, a^{4} c^{4} x^{4} - 105 \, a^{3} c^{4} x^{3} + 84 \, a^{2} c^{4} x^{2} + 70 \, a c^{4} x + 15 \, c^{4}}{105 \, a^{8} x^{7}} \]
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Time = 0.27 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.60 \[ \int e^{4 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^4 \, dx=-\frac {4 \, c^{4} \log \left (\frac {{\left | a x - 1 \right |}}{{\left (a x - 1\right )}^{2} {\left | a \right |}}\right )}{a} + \frac {4 \, c^{4} \log \left ({\left | -\frac {1}{a x - 1} - 1 \right |}\right )}{a} + \frac {{\left (105 \, c^{4} + \frac {659 \, c^{4}}{a x - 1} + \frac {1253 \, c^{4}}{{\left (a x - 1\right )}^{2}} - \frac {231 \, c^{4}}{{\left (a x - 1\right )}^{3}} - \frac {3885 \, c^{4}}{{\left (a x - 1\right )}^{4}} - \frac {5250 \, c^{4}}{{\left (a x - 1\right )}^{5}} - \frac {2730 \, c^{4}}{{\left (a x - 1\right )}^{6}} - \frac {420 \, c^{4}}{{\left (a x - 1\right )}^{7}}\right )} {\left (a x - 1\right )}}{105 \, a {\left (\frac {1}{a x - 1} + 1\right )}^{7}} \]
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Time = 4.24 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.72 \[ \int e^{4 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^4 \, dx=\frac {c^4\,\left (a^3\,x^3-\frac {4\,a^2\,x^2}{5}-\frac {2\,a\,x}{3}+\frac {10\,a^4\,x^4}{3}+2\,a^5\,x^5-4\,a^6\,x^6+a^8\,x^8+4\,a^7\,x^7\,\ln \left (x\right )-\frac {1}{7}\right )}{a^8\,x^7} \]
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