Integrand size = 22, antiderivative size = 89 \[ \int \frac {e^{4 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^3} \, dx=\frac {x}{c^3}-\frac {1}{a c^3 (1-a x)^4}+\frac {16}{3 a c^3 (1-a x)^3}-\frac {25}{2 a c^3 (1-a x)^2}+\frac {19}{a c^3 (1-a x)}+\frac {7 \log (1-a x)}{a c^3} \]
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Time = 0.13 (sec) , antiderivative size = 89, 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, 6266, 6264, 90} \[ \int \frac {e^{4 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^3} \, dx=\frac {19}{a c^3 (1-a x)}-\frac {25}{2 a c^3 (1-a x)^2}+\frac {16}{3 a c^3 (1-a x)^3}-\frac {1}{a c^3 (1-a x)^4}+\frac {7 \log (1-a x)}{a c^3}+\frac {x}{c^3} \]
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Rule 90
Rule 6264
Rule 6266
Rule 6302
Rubi steps \begin{align*} \text {integral}& = \int \frac {e^{4 \text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^3} \, dx \\ & = -\frac {a^3 \int \frac {e^{4 \text {arctanh}(a x)} x^3}{(1-a x)^3} \, dx}{c^3} \\ & = -\frac {a^3 \int \frac {x^3 (1+a x)^2}{(1-a x)^5} \, dx}{c^3} \\ & = -\frac {a^3 \int \left (-\frac {1}{a^3}-\frac {4}{a^3 (-1+a x)^5}-\frac {16}{a^3 (-1+a x)^4}-\frac {25}{a^3 (-1+a x)^3}-\frac {19}{a^3 (-1+a x)^2}-\frac {7}{a^3 (-1+a x)}\right ) \, dx}{c^3} \\ & = \frac {x}{c^3}-\frac {1}{a c^3 (1-a x)^4}+\frac {16}{3 a c^3 (1-a x)^3}-\frac {25}{2 a c^3 (1-a x)^2}+\frac {19}{a c^3 (1-a x)}+\frac {7 \log (1-a x)}{a c^3} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.80 \[ \int \frac {e^{4 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^3} \, dx=\frac {65-218 a x+243 a^2 x^2-78 a^3 x^3-24 a^4 x^4+6 a^5 x^5+42 (-1+a x)^4 \log (1-a x)}{6 a c^3 (-1+a x)^4} \]
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Time = 0.60 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.75
method | result | size |
risch | \(\frac {x}{c^{3}}+\frac {-19 a^{2} c^{3} x^{3}+\frac {89 a \,c^{3} x^{2}}{2}-\frac {112 c^{3} x}{3}+\frac {65 c^{3}}{6 a}}{c^{6} \left (a x -1\right )^{4}}+\frac {7 \ln \left (a x -1\right )}{a \,c^{3}}\) | \(67\) |
default | \(\frac {a^{3} \left (\frac {x}{a^{3}}-\frac {25}{2 a^{4} \left (a x -1\right )^{2}}-\frac {1}{a^{4} \left (a x -1\right )^{4}}-\frac {19}{a^{4} \left (a x -1\right )}-\frac {16}{3 a^{4} \left (a x -1\right )^{3}}+\frac {7 \ln \left (a x -1\right )}{a^{4}}\right )}{c^{3}}\) | \(73\) |
norman | \(\frac {\frac {a^{4} x^{5}}{c}+\frac {7 x}{c}-\frac {49 a \,x^{2}}{2 c}+\frac {91 a^{2} x^{3}}{3 c}-\frac {89 a^{3} x^{4}}{6 c}}{\left (a x -1\right )^{4} c^{2}}+\frac {7 \ln \left (a x -1\right )}{a \,c^{3}}\) | \(75\) |
parallelrisch | \(\frac {6 a^{5} x^{5}+42 \ln \left (a x -1\right ) x^{4} a^{4}-89 a^{4} x^{4}-168 a^{3} \ln \left (a x -1\right ) x^{3}+182 a^{3} x^{3}+252 a^{2} \ln \left (a x -1\right ) x^{2}-147 a^{2} x^{2}-168 a \ln \left (a x -1\right ) x +42 a x +42 \ln \left (a x -1\right )}{6 \left (a x -1\right )^{4} c^{3} a}\) | \(113\) |
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Time = 0.24 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.42 \[ \int \frac {e^{4 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^3} \, dx=\frac {6 \, a^{5} x^{5} - 24 \, a^{4} x^{4} - 78 \, a^{3} x^{3} + 243 \, a^{2} x^{2} - 218 \, a x + 42 \, {\left (a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x + 1\right )} \log \left (a x - 1\right ) + 65}{6 \, {\left (a^{5} c^{3} x^{4} - 4 \, a^{4} c^{3} x^{3} + 6 \, a^{3} c^{3} x^{2} - 4 \, a^{2} c^{3} x + a c^{3}\right )}} \]
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Time = 0.31 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.06 \[ \int \frac {e^{4 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^3} \, dx=\frac {- 114 a^{3} x^{3} + 267 a^{2} x^{2} - 224 a x + 65}{6 a^{5} c^{3} x^{4} - 24 a^{4} c^{3} x^{3} + 36 a^{3} c^{3} x^{2} - 24 a^{2} c^{3} x + 6 a c^{3}} + \frac {x}{c^{3}} + \frac {7 \log {\left (a x - 1 \right )}}{a c^{3}} \]
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Time = 0.20 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.04 \[ \int \frac {e^{4 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^3} \, dx=-\frac {114 \, a^{3} x^{3} - 267 \, a^{2} x^{2} + 224 \, a x - 65}{6 \, {\left (a^{5} c^{3} x^{4} - 4 \, a^{4} c^{3} x^{3} + 6 \, a^{3} c^{3} x^{2} - 4 \, a^{2} c^{3} x + a c^{3}\right )}} + \frac {x}{c^{3}} + \frac {7 \, \log \left (a x - 1\right )}{a c^{3}} \]
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Time = 0.27 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.22 \[ \int \frac {e^{4 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^3} \, dx=\frac {a x - 1}{a c^{3}} - \frac {7 \, \log \left (\frac {{\left | a x - 1 \right |}}{{\left (a x - 1\right )}^{2} {\left | a \right |}}\right )}{a c^{3}} - \frac {\frac {114 \, a^{7} c^{9}}{a x - 1} + \frac {75 \, a^{7} c^{9}}{{\left (a x - 1\right )}^{2}} + \frac {32 \, a^{7} c^{9}}{{\left (a x - 1\right )}^{3}} + \frac {6 \, a^{7} c^{9}}{{\left (a x - 1\right )}^{4}}}{6 \, a^{8} c^{12}} \]
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Time = 3.79 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.01 \[ \int \frac {e^{4 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^3} \, dx=\frac {x}{c^3}-\frac {\frac {112\,x}{3}-\frac {89\,a\,x^2}{2}-\frac {65}{6\,a}+19\,a^2\,x^3}{a^4\,c^3\,x^4-4\,a^3\,c^3\,x^3+6\,a^2\,c^3\,x^2-4\,a\,c^3\,x+c^3}+\frac {7\,\ln \left (a\,x-1\right )}{a\,c^3} \]
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