Integrand size = 22, antiderivative size = 127 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^5 \, dx=-\frac {c^5}{9 a^{10} x^9}-\frac {c^5}{4 a^9 x^8}+\frac {3 c^5}{7 a^8 x^7}+\frac {4 c^5}{3 a^7 x^6}-\frac {2 c^5}{5 a^6 x^5}-\frac {3 c^5}{a^5 x^4}-\frac {2 c^5}{3 a^4 x^3}+\frac {4 c^5}{a^3 x^2}+\frac {3 c^5}{a^2 x}+c^5 x+\frac {2 c^5 \log (x)}{a} \]
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Time = 0.13 (sec) , antiderivative size = 127, 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} \[ \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^5 \, dx=-\frac {c^5}{9 a^{10} x^9}-\frac {c^5}{4 a^9 x^8}+\frac {3 c^5}{7 a^8 x^7}+\frac {4 c^5}{3 a^7 x^6}-\frac {2 c^5}{5 a^6 x^5}-\frac {3 c^5}{a^5 x^4}-\frac {2 c^5}{3 a^4 x^3}+\frac {4 c^5}{a^3 x^2}+\frac {3 c^5}{a^2 x}+\frac {2 c^5 \log (x)}{a}+c^5 x \]
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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 )^5 \, dx \\ & = \frac {c^5 \int \frac {e^{2 \text {arctanh}(a x)} \left (1-a^2 x^2\right )^5}{x^{10}} \, dx}{a^{10}} \\ & = \frac {c^5 \int \frac {(1-a x)^4 (1+a x)^6}{x^{10}} \, dx}{a^{10}} \\ & = \frac {c^5 \int \left (a^{10}+\frac {1}{x^{10}}+\frac {2 a}{x^9}-\frac {3 a^2}{x^8}-\frac {8 a^3}{x^7}+\frac {2 a^4}{x^6}+\frac {12 a^5}{x^5}+\frac {2 a^6}{x^4}-\frac {8 a^7}{x^3}-\frac {3 a^8}{x^2}+\frac {2 a^9}{x}\right ) \, dx}{a^{10}} \\ & = -\frac {c^5}{9 a^{10} x^9}-\frac {c^5}{4 a^9 x^8}+\frac {3 c^5}{7 a^8 x^7}+\frac {4 c^5}{3 a^7 x^6}-\frac {2 c^5}{5 a^6 x^5}-\frac {3 c^5}{a^5 x^4}-\frac {2 c^5}{3 a^4 x^3}+\frac {4 c^5}{a^3 x^2}+\frac {3 c^5}{a^2 x}+c^5 x+\frac {2 c^5 \log (x)}{a} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.00 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^5 \, dx=-\frac {c^5}{9 a^{10} x^9}-\frac {c^5}{4 a^9 x^8}+\frac {3 c^5}{7 a^8 x^7}+\frac {4 c^5}{3 a^7 x^6}-\frac {2 c^5}{5 a^6 x^5}-\frac {3 c^5}{a^5 x^4}-\frac {2 c^5}{3 a^4 x^3}+\frac {4 c^5}{a^3 x^2}+\frac {3 c^5}{a^2 x}+c^5 x+\frac {2 c^5 \log (x)}{a} \]
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Time = 0.85 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.69
method | result | size |
default | \(\frac {c^{5} \left (a^{10} x -\frac {1}{9 x^{9}}+2 a^{9} \ln \left (x \right )-\frac {3 a^{5}}{x^{4}}-\frac {2 a^{6}}{3 x^{3}}+\frac {4 a^{3}}{3 x^{6}}+\frac {4 a^{7}}{x^{2}}+\frac {3 a^{8}}{x}+\frac {3 a^{2}}{7 x^{7}}-\frac {a}{4 x^{8}}-\frac {2 a^{4}}{5 x^{5}}\right )}{a^{10}}\) | \(88\) |
risch | \(c^{5} x +\frac {3 a^{8} c^{5} x^{8}+4 a^{7} c^{5} x^{7}-\frac {2}{3} a^{6} c^{5} x^{6}-3 a^{5} c^{5} x^{5}-\frac {2}{5} a^{4} c^{5} x^{4}+\frac {4}{3} a^{3} c^{5} x^{3}+\frac {3}{7} a^{2} c^{5} x^{2}-\frac {1}{4} a \,c^{5} x -\frac {1}{9} c^{5}}{a^{10} x^{9}}+\frac {2 c^{5} \ln \left (x \right )}{a}\) | \(114\) |
norman | \(\frac {a^{9} c^{5} x^{10}-\frac {c^{5}}{9 a}-\frac {c^{5} x}{4}+\frac {3 a \,c^{5} x^{2}}{7}-\frac {2 a^{3} c^{5} x^{4}}{5}-3 a^{4} c^{5} x^{5}-\frac {2 a^{5} c^{5} x^{6}}{3}+4 a^{6} c^{5} x^{7}+3 a^{7} c^{5} x^{8}+\frac {4 c^{5} a^{2} x^{3}}{3}}{a^{9} x^{9}}+\frac {2 c^{5} \ln \left (x \right )}{a}\) | \(119\) |
parallelrisch | \(\frac {1260 a^{10} c^{5} x^{10}+2520 c^{5} \ln \left (x \right ) a^{9} x^{9}+3780 a^{8} c^{5} x^{8}+5040 a^{7} c^{5} x^{7}-840 a^{6} c^{5} x^{6}-3780 a^{5} c^{5} x^{5}-504 a^{4} c^{5} x^{4}+1680 a^{3} c^{5} x^{3}+540 a^{2} c^{5} x^{2}-315 a \,c^{5} x -140 c^{5}}{1260 a^{10} x^{9}}\) | \(123\) |
meijerg | \(-\frac {c^{5} \left (-a x -\ln \left (-a x +1\right )\right )}{a}+\frac {5 c^{5} \left (-\ln \left (-a x +1\right )+\ln \left (x \right )+\ln \left (-a \right )\right )}{a}-\frac {10 c^{5} \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 {10 c^{5} \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 {5 c^{5} \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^{5} \left (-\ln \left (-a x +1\right )+\ln \left (x \right )+\ln \left (-a \right )-\frac {1}{8 a^{8} x^{8}}-\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}+\frac {c^{5} \ln \left (-a x +1\right )}{a}-\frac {5 c^{5} \left (\ln \left (-a x +1\right )-\ln \left (x \right )-\ln \left (-a \right )+\frac {1}{a x}\right )}{a}+\frac {10 c^{5} \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 {10 c^{5} \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 {5 c^{5} \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}-\frac {c^{5} \left (\ln \left (-a x +1\right )-\ln \left (x \right )-\ln \left (-a \right )+\frac {1}{9 x^{9} a^{9}}+\frac {1}{8 a^{8} x^{8}}+\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}\) | \(642\) |
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Time = 0.24 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.96 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^5 \, dx=\frac {1260 \, a^{10} c^{5} x^{10} + 2520 \, a^{9} c^{5} x^{9} \log \left (x\right ) + 3780 \, a^{8} c^{5} x^{8} + 5040 \, a^{7} c^{5} x^{7} - 840 \, a^{6} c^{5} x^{6} - 3780 \, a^{5} c^{5} x^{5} - 504 \, a^{4} c^{5} x^{4} + 1680 \, a^{3} c^{5} x^{3} + 540 \, a^{2} c^{5} x^{2} - 315 \, a c^{5} x - 140 \, c^{5}}{1260 \, a^{10} x^{9}} \]
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Time = 0.37 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.98 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^5 \, dx=\frac {a^{10} c^{5} x + 2 a^{9} c^{5} \log {\left (x \right )} + \frac {3780 a^{8} c^{5} x^{8} + 5040 a^{7} c^{5} x^{7} - 840 a^{6} c^{5} x^{6} - 3780 a^{5} c^{5} x^{5} - 504 a^{4} c^{5} x^{4} + 1680 a^{3} c^{5} x^{3} + 540 a^{2} c^{5} x^{2} - 315 a c^{5} x - 140 c^{5}}{1260 x^{9}}}{a^{10}} \]
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Time = 0.20 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.90 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^5 \, dx=c^{5} x + \frac {2 \, c^{5} \log \left (x\right )}{a} + \frac {3780 \, a^{8} c^{5} x^{8} + 5040 \, a^{7} c^{5} x^{7} - 840 \, a^{6} c^{5} x^{6} - 3780 \, a^{5} c^{5} x^{5} - 504 \, a^{4} c^{5} x^{4} + 1680 \, a^{3} c^{5} x^{3} + 540 \, a^{2} c^{5} x^{2} - 315 \, a c^{5} x - 140 \, c^{5}}{1260 \, a^{10} x^{9}} \]
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Time = 0.26 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.91 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^5 \, dx=c^{5} x + \frac {2 \, c^{5} \log \left ({\left | x \right |}\right )}{a} + \frac {3780 \, a^{8} c^{5} x^{8} + 5040 \, a^{7} c^{5} x^{7} - 840 \, a^{6} c^{5} x^{6} - 3780 \, a^{5} c^{5} x^{5} - 504 \, a^{4} c^{5} x^{4} + 1680 \, a^{3} c^{5} x^{3} + 540 \, a^{2} c^{5} x^{2} - 315 \, a c^{5} x - 140 \, c^{5}}{1260 \, a^{10} x^{9}} \]
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Time = 0.11 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.70 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^5 \, dx=\frac {c^5\,\left (\frac {3\,a^2\,x^2}{7}-\frac {a\,x}{4}+\frac {4\,a^3\,x^3}{3}-\frac {2\,a^4\,x^4}{5}-3\,a^5\,x^5-\frac {2\,a^6\,x^6}{3}+4\,a^7\,x^7+3\,a^8\,x^8+a^{10}\,x^{10}+2\,a^9\,x^9\,\ln \left (x\right )-\frac {1}{9}\right )}{a^{10}\,x^9} \]
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