Integrand size = 22, antiderivative size = 71 \[ \int \frac {e^{4 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^2} \, dx=\frac {x}{c^2}+\frac {4}{3 a c^2 (1-a x)^3}-\frac {6}{a c^2 (1-a x)^2}+\frac {13}{a c^2 (1-a x)}+\frac {6 \log (1-a x)}{a c^2} \]
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Time = 0.12 (sec) , antiderivative size = 71, 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} \[ \int \frac {e^{4 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^2} \, dx=\frac {13}{a c^2 (1-a x)}-\frac {6}{a c^2 (1-a x)^2}+\frac {4}{3 a c^2 (1-a x)^3}+\frac {6 \log (1-a x)}{a c^2}+\frac {x}{c^2} \]
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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 )^2} \, dx \\ & = \frac {a^2 \int \frac {e^{4 \text {arctanh}(a x)} x^2}{(1-a x)^2} \, dx}{c^2} \\ & = \frac {a^2 \int \frac {x^2 (1+a x)^2}{(1-a x)^4} \, dx}{c^2} \\ & = \frac {a^2 \int \left (\frac {1}{a^2}+\frac {4}{a^2 (-1+a x)^4}+\frac {12}{a^2 (-1+a x)^3}+\frac {13}{a^2 (-1+a x)^2}+\frac {6}{a^2 (-1+a x)}\right ) \, dx}{c^2} \\ & = \frac {x}{c^2}+\frac {4}{3 a c^2 (1-a x)^3}-\frac {6}{a c^2 (1-a x)^2}+\frac {13}{a c^2 (1-a x)}+\frac {6 \log (1-a x)}{a c^2} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.89 \[ \int \frac {e^{4 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^2} \, dx=\frac {-25+57 a x-30 a^2 x^2-9 a^3 x^3+3 a^4 x^4+18 (-1+a x)^3 \log (1-a x)}{3 a c^2 (-1+a x)^3} \]
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Time = 0.59 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.79
method | result | size |
risch | \(\frac {x}{c^{2}}+\frac {-13 a \,c^{2} x^{2}+20 c^{2} x -\frac {25 c^{2}}{3 a}}{c^{4} \left (a x -1\right )^{3}}+\frac {6 \ln \left (a x -1\right )}{a \,c^{2}}\) | \(56\) |
default | \(\frac {a^{2} \left (\frac {x}{a^{2}}-\frac {6}{a^{3} \left (a x -1\right )^{2}}-\frac {13}{a^{3} \left (a x -1\right )}-\frac {4}{3 a^{3} \left (a x -1\right )^{3}}+\frac {6 \ln \left (a x -1\right )}{a^{3}}\right )}{c^{2}}\) | \(61\) |
norman | \(\frac {\frac {a^{3} x^{4}}{c}-\frac {6 x}{c}+\frac {15 a \,x^{2}}{c}-\frac {34 a^{2} x^{3}}{3 c}}{\left (a x -1\right )^{3} c}+\frac {6 \ln \left (a x -1\right )}{a \,c^{2}}\) | \(64\) |
parallelrisch | \(\frac {3 a^{4} x^{4}+18 a^{3} \ln \left (a x -1\right ) x^{3}-34 a^{3} x^{3}-54 a^{2} \ln \left (a x -1\right ) x^{2}+45 a^{2} x^{2}+54 a \ln \left (a x -1\right ) x -18 a x -18 \ln \left (a x -1\right )}{3 \left (a x -1\right )^{3} c^{2} a}\) | \(91\) |
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Time = 0.25 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.41 \[ \int \frac {e^{4 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^2} \, dx=\frac {3 \, a^{4} x^{4} - 9 \, a^{3} x^{3} - 30 \, a^{2} x^{2} + 57 \, a x + 18 \, {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \log \left (a x - 1\right ) - 25}{3 \, {\left (a^{4} c^{2} x^{3} - 3 \, a^{3} c^{2} x^{2} + 3 \, a^{2} c^{2} x - a c^{2}\right )}} \]
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Time = 0.19 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.03 \[ \int \frac {e^{4 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^2} \, dx=\frac {- 39 a^{2} x^{2} + 60 a x - 25}{3 a^{4} c^{2} x^{3} - 9 a^{3} c^{2} x^{2} + 9 a^{2} c^{2} x - 3 a c^{2}} + \frac {x}{c^{2}} + \frac {6 \log {\left (a x - 1 \right )}}{a c^{2}} \]
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Time = 0.18 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.06 \[ \int \frac {e^{4 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^2} \, dx=-\frac {39 \, a^{2} x^{2} - 60 \, a x + 25}{3 \, {\left (a^{4} c^{2} x^{3} - 3 \, a^{3} c^{2} x^{2} + 3 \, a^{2} c^{2} x - a c^{2}\right )}} + \frac {x}{c^{2}} + \frac {6 \, \log \left (a x - 1\right )}{a c^{2}} \]
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Time = 0.27 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.32 \[ \int \frac {e^{4 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^2} \, dx=\frac {a x - 1}{a c^{2}} - \frac {6 \, \log \left (\frac {{\left | a x - 1 \right |}}{{\left (a x - 1\right )}^{2} {\left | a \right |}}\right )}{a c^{2}} - \frac {\frac {39 \, a^{5} c^{4}}{a x - 1} + \frac {18 \, a^{5} c^{4}}{{\left (a x - 1\right )}^{2}} + \frac {4 \, a^{5} c^{4}}{{\left (a x - 1\right )}^{3}}}{3 \, a^{6} c^{6}} \]
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Time = 3.86 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.00 \[ \int \frac {e^{4 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^2} \, dx=\frac {13\,a\,x^2-20\,x+\frac {25}{3\,a}}{-a^3\,c^2\,x^3+3\,a^2\,c^2\,x^2-3\,a\,c^2\,x+c^2}+\frac {x}{c^2}+\frac {6\,\ln \left (a\,x-1\right )}{a\,c^2} \]
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