Integrand size = 22, antiderivative size = 111 \[ \int \frac {e^{4 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^3} \, dx=\frac {x}{c^3}-\frac {1}{8 a c^3 (1-a x)^4}+\frac {11}{12 a c^3 (1-a x)^3}-\frac {49}{16 a c^3 (1-a x)^2}+\frac {111}{16 a c^3 (1-a x)}+\frac {129 \log (1-a x)}{32 a c^3}-\frac {\log (1+a x)}{32 a c^3} \]
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Time = 0.14 (sec) , antiderivative size = 111, 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} \[ \int \frac {e^{4 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^3} \, dx=\frac {111}{16 a c^3 (1-a x)}-\frac {49}{16 a c^3 (1-a x)^2}+\frac {11}{12 a c^3 (1-a x)^3}-\frac {1}{8 a c^3 (1-a x)^4}+\frac {129 \log (1-a x)}{32 a c^3}-\frac {\log (a x+1)}{32 a c^3}+\frac {x}{c^3} \]
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Rule 90
Rule 6285
Rule 6292
Rule 6302
Rubi steps \begin{align*} \text {integral}& = \int \frac {e^{4 \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^3} \, dx \\ & = -\frac {a^6 \int \frac {e^{4 \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)^5 (1+a x)} \, dx}{c^3} \\ & = -\frac {a^6 \int \left (-\frac {1}{a^6}-\frac {1}{2 a^6 (-1+a x)^5}-\frac {11}{4 a^6 (-1+a x)^4}-\frac {49}{8 a^6 (-1+a x)^3}-\frac {111}{16 a^6 (-1+a x)^2}-\frac {129}{32 a^6 (-1+a x)}+\frac {1}{32 a^6 (1+a x)}\right ) \, dx}{c^3} \\ & = \frac {x}{c^3}-\frac {1}{8 a c^3 (1-a x)^4}+\frac {11}{12 a c^3 (1-a x)^3}-\frac {49}{16 a c^3 (1-a x)^2}+\frac {111}{16 a c^3 (1-a x)}+\frac {129 \log (1-a x)}{32 a c^3}-\frac {\log (1+a x)}{32 a c^3} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.80 \[ \int \frac {e^{4 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^3} \, dx=\frac {2 \left (224-701 a x+660 a^2 x^2-45 a^3 x^3-192 a^4 x^4+48 a^5 x^5\right )+387 (-1+a x)^4 \log (1-a x)-3 (-1+a x)^4 \log (1+a x)}{96 a c^3 (-1+a x)^4} \]
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Time = 0.59 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.74
method | result | size |
risch | \(\frac {x}{c^{3}}+\frac {-\frac {111 a^{2} c^{3} x^{3}}{16}+\frac {71 a \,c^{3} x^{2}}{4}-\frac {749 c^{3} x}{48}+\frac {14 c^{3}}{3 a}}{c^{6} \left (a x -1\right )^{4}}+\frac {129 \ln \left (-a x +1\right )}{32 a \,c^{3}}-\frac {\ln \left (a x +1\right )}{32 a \,c^{3}}\) | \(82\) |
default | \(\frac {a^{6} \left (-\frac {\ln \left (a x +1\right )}{32 a^{7}}+\frac {x}{a^{6}}-\frac {1}{8 a^{7} \left (a x -1\right )^{4}}-\frac {11}{12 a^{7} \left (a x -1\right )^{3}}-\frac {49}{16 a^{7} \left (a x -1\right )^{2}}-\frac {111}{16 a^{7} \left (a x -1\right )}+\frac {129 \ln \left (a x -1\right )}{32 a^{7}}\right )}{c^{3}}\) | \(84\) |
norman | \(\frac {\frac {a^{6} x^{7}}{c}+\frac {65 x}{16 c}-\frac {49 a \,x^{2}}{8 c}-\frac {161 a^{2} x^{3}}{24 c}+\frac {301 a^{3} x^{4}}{24 c}+\frac {67 a^{4} x^{5}}{48 c}-\frac {20 a^{5} x^{6}}{3 c}}{\left (a x +1\right )^{2} \left (a x -1\right )^{4} c^{2}}+\frac {129 \ln \left (a x -1\right )}{32 a \,c^{3}}-\frac {\ln \left (a x +1\right )}{32 a \,c^{3}}\) | \(118\) |
parallelrisch | \(\frac {12 a \ln \left (a x +1\right ) x -18 a^{2} \ln \left (a x +1\right ) x^{2}+96 a^{5} x^{5}+1702 a^{3} x^{3}-3 \ln \left (a x +1\right ) x^{4} a^{4}+387 \ln \left (a x -1\right ) x^{4} a^{4}+12 a^{3} \ln \left (a x +1\right ) x^{3}+390 a x -1548 a^{3} \ln \left (a x -1\right ) x^{3}+2322 a^{2} \ln \left (a x -1\right ) x^{2}-1548 a \ln \left (a x -1\right ) x -832 a^{4} x^{4}+387 \ln \left (a x -1\right )-3 \ln \left (a x +1\right )-1368 a^{2} x^{2}}{96 \left (a x -1\right )^{4} c^{3} a}\) | \(173\) |
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Time = 0.24 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.47 \[ \int \frac {e^{4 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^3} \, dx=\frac {96 \, a^{5} x^{5} - 384 \, a^{4} x^{4} - 90 \, a^{3} x^{3} + 1320 \, a^{2} x^{2} - 1402 \, a x - 3 \, {\left (a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x + 1\right )} \log \left (a x + 1\right ) + 387 \, {\left (a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x + 1\right )} \log \left (a x - 1\right ) + 448}{96 \, {\left (a^{5} c^{3} x^{4} - 4 \, a^{4} c^{3} x^{3} + 6 \, a^{3} c^{3} x^{2} - 4 \, a^{2} c^{3} x + a c^{3}\right )}} \]
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Time = 0.34 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.03 \[ \int \frac {e^{4 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^3} \, dx=a^{6} \left (\frac {- 333 a^{3} x^{3} + 852 a^{2} x^{2} - 749 a x + 224}{48 a^{11} c^{3} x^{4} - 192 a^{10} c^{3} x^{3} + 288 a^{9} c^{3} x^{2} - 192 a^{8} c^{3} x + 48 a^{7} c^{3}} + \frac {x}{a^{6} c^{3}} + \frac {\frac {129 \log {\left (x - \frac {1}{a} \right )}}{32} - \frac {\log {\left (x + \frac {1}{a} \right )}}{32}}{a^{7} c^{3}}\right ) \]
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Time = 0.19 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.96 \[ \int \frac {e^{4 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^3} \, dx=-\frac {333 \, a^{3} x^{3} - 852 \, a^{2} x^{2} + 749 \, a x - 224}{48 \, {\left (a^{5} c^{3} x^{4} - 4 \, a^{4} c^{3} x^{3} + 6 \, a^{3} c^{3} x^{2} - 4 \, a^{2} c^{3} x + a c^{3}\right )}} + \frac {x}{c^{3}} - \frac {\log \left (a x + 1\right )}{32 \, a c^{3}} + \frac {129 \, \log \left (a x - 1\right )}{32 \, a c^{3}} \]
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Time = 0.27 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.17 \[ \int \frac {e^{4 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^3} \, dx=\frac {a x - 1}{a c^{3}} - \frac {4 \, \log \left (\frac {{\left | a x - 1 \right |}}{{\left (a x - 1\right )}^{2} {\left | a \right |}}\right )}{a c^{3}} - \frac {\log \left ({\left | -\frac {2}{a x - 1} - 1 \right |}\right )}{32 \, a c^{3}} - \frac {\frac {333 \, a^{11} c^{9}}{a x - 1} + \frac {147 \, a^{11} c^{9}}{{\left (a x - 1\right )}^{2}} + \frac {44 \, a^{11} c^{9}}{{\left (a x - 1\right )}^{3}} + \frac {6 \, a^{11} c^{9}}{{\left (a x - 1\right )}^{4}}}{48 \, a^{12} c^{12}} \]
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Time = 0.12 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.94 \[ \int \frac {e^{4 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^3} \, dx=\frac {x}{c^3}-\frac {\frac {749\,x}{48}-\frac {71\,a\,x^2}{4}-\frac {14}{3\,a}+\frac {111\,a^2\,x^3}{16}}{a^4\,c^3\,x^4-4\,a^3\,c^3\,x^3+6\,a^2\,c^3\,x^2-4\,a\,c^3\,x+c^3}+\frac {129\,\ln \left (a\,x-1\right )}{32\,a\,c^3}-\frac {\ln \left (a\,x+1\right )}{32\,a\,c^3} \]
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