Integrand size = 22, antiderivative size = 105 \[ \int \frac {e^{4 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^4} \, dx=\frac {x}{c^4}+\frac {4}{5 a c^4 (1-a x)^5}-\frac {5}{a c^4 (1-a x)^4}+\frac {41}{3 a c^4 (1-a x)^3}-\frac {22}{a c^4 (1-a x)^2}+\frac {26}{a c^4 (1-a x)}+\frac {8 \log (1-a x)}{a c^4} \]
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Time = 0.13 (sec) , antiderivative size = 105, 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} \[ \int \frac {e^{4 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^4} \, dx=\frac {26}{a c^4 (1-a x)}-\frac {22}{a c^4 (1-a x)^2}+\frac {41}{3 a c^4 (1-a x)^3}-\frac {5}{a c^4 (1-a x)^4}+\frac {4}{5 a c^4 (1-a x)^5}+\frac {8 \log (1-a x)}{a c^4}+\frac {x}{c^4} \]
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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 )^4} \, dx \\ & = \frac {a^4 \int \frac {e^{4 \text {arctanh}(a x)} x^4}{(1-a x)^4} \, dx}{c^4} \\ & = \frac {a^4 \int \frac {x^4 (1+a x)^2}{(1-a x)^6} \, dx}{c^4} \\ & = \frac {a^4 \int \left (\frac {1}{a^4}+\frac {4}{a^4 (-1+a x)^6}+\frac {20}{a^4 (-1+a x)^5}+\frac {41}{a^4 (-1+a x)^4}+\frac {44}{a^4 (-1+a x)^3}+\frac {26}{a^4 (-1+a x)^2}+\frac {8}{a^4 (-1+a x)}\right ) \, dx}{c^4} \\ & = \frac {x}{c^4}+\frac {4}{5 a c^4 (1-a x)^5}-\frac {5}{a c^4 (1-a x)^4}+\frac {41}{3 a c^4 (1-a x)^3}-\frac {22}{a c^4 (1-a x)^2}+\frac {26}{a c^4 (1-a x)}+\frac {8 \log (1-a x)}{a c^4} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.75 \[ \int \frac {e^{4 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^4} \, dx=\frac {-202+890 a x-1480 a^2 x^2+1080 a^3 x^3-240 a^4 x^4-75 a^5 x^5+15 a^6 x^6+120 (-1+a x)^5 \log (1-a x)}{15 a c^4 (-1+a x)^5} \]
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Time = 0.59 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.74
method | result | size |
risch | \(\frac {x}{c^{4}}+\frac {-26 a^{3} c^{4} x^{4}+82 a^{2} c^{4} x^{3}-\frac {311 a \,c^{4} x^{2}}{3}+\frac {181 c^{4} x}{3}-\frac {202 c^{4}}{15 a}}{c^{8} \left (a x -1\right )^{5}}+\frac {8 \ln \left (a x -1\right )}{a \,c^{4}}\) | \(78\) |
default | \(\frac {a^{4} \left (\frac {x}{a^{4}}-\frac {4}{5 a^{5} \left (a x -1\right )^{5}}-\frac {22}{a^{5} \left (a x -1\right )^{2}}-\frac {5}{a^{5} \left (a x -1\right )^{4}}-\frac {26}{a^{5} \left (a x -1\right )}-\frac {41}{3 a^{5} \left (a x -1\right )^{3}}+\frac {8 \ln \left (a x -1\right )}{a^{5}}\right )}{c^{4}}\) | \(85\) |
norman | \(\frac {\frac {a^{5} x^{6}}{c}-\frac {8 x}{c}+\frac {36 a \,x^{2}}{c}-\frac {188 a^{2} x^{3}}{3 c}+\frac {154 a^{3} x^{4}}{3 c}-\frac {277 a^{4} x^{5}}{15 c}}{\left (a x -1\right )^{5} c^{3}}+\frac {8 \ln \left (a x -1\right )}{a \,c^{4}}\) | \(86\) |
parallelrisch | \(\frac {15 a^{6} x^{6}+120 \ln \left (a x -1\right ) x^{5} a^{5}-277 a^{5} x^{5}-600 \ln \left (a x -1\right ) x^{4} a^{4}+770 a^{4} x^{4}+1200 a^{3} \ln \left (a x -1\right ) x^{3}-940 a^{3} x^{3}-1200 a^{2} \ln \left (a x -1\right ) x^{2}+540 a^{2} x^{2}+600 a \ln \left (a x -1\right ) x -120 a x -120 \ln \left (a x -1\right )}{15 \left (a x -1\right )^{5} c^{4} a}\) | \(135\) |
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Time = 0.24 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.47 \[ \int \frac {e^{4 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^4} \, dx=\frac {15 \, a^{6} x^{6} - 75 \, a^{5} x^{5} - 240 \, a^{4} x^{4} + 1080 \, a^{3} x^{3} - 1480 \, a^{2} x^{2} + 890 \, a x + 120 \, {\left (a^{5} x^{5} - 5 \, a^{4} x^{4} + 10 \, a^{3} x^{3} - 10 \, a^{2} x^{2} + 5 \, a x - 1\right )} \log \left (a x - 1\right ) - 202}{15 \, {\left (a^{6} c^{4} x^{5} - 5 \, a^{5} c^{4} x^{4} + 10 \, a^{4} c^{4} x^{3} - 10 \, a^{3} c^{4} x^{2} + 5 \, a^{2} c^{4} x - a c^{4}\right )}} \]
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Time = 0.43 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.09 \[ \int \frac {e^{4 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^4} \, dx=\frac {- 390 a^{4} x^{4} + 1230 a^{3} x^{3} - 1555 a^{2} x^{2} + 905 a x - 202}{15 a^{6} c^{4} x^{5} - 75 a^{5} c^{4} x^{4} + 150 a^{4} c^{4} x^{3} - 150 a^{3} c^{4} x^{2} + 75 a^{2} c^{4} x - 15 a c^{4}} + \frac {x}{c^{4}} + \frac {8 \log {\left (a x - 1 \right )}}{a c^{4}} \]
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Time = 0.19 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.08 \[ \int \frac {e^{4 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^4} \, dx=-\frac {390 \, a^{4} x^{4} - 1230 \, a^{3} x^{3} + 1555 \, a^{2} x^{2} - 905 \, a x + 202}{15 \, {\left (a^{6} c^{4} x^{5} - 5 \, a^{5} c^{4} x^{4} + 10 \, a^{4} c^{4} x^{3} - 10 \, a^{3} c^{4} x^{2} + 5 \, a^{2} c^{4} x - a c^{4}\right )}} + \frac {x}{c^{4}} + \frac {8 \, \log \left (a x - 1\right )}{a c^{4}} \]
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Time = 0.27 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.18 \[ \int \frac {e^{4 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^4} \, dx=\frac {a x - 1}{a c^{4}} - \frac {8 \, \log \left (\frac {{\left | a x - 1 \right |}}{{\left (a x - 1\right )}^{2} {\left | a \right |}}\right )}{a c^{4}} - \frac {\frac {390 \, a^{9} c^{16}}{a x - 1} + \frac {330 \, a^{9} c^{16}}{{\left (a x - 1\right )}^{2}} + \frac {205 \, a^{9} c^{16}}{{\left (a x - 1\right )}^{3}} + \frac {75 \, a^{9} c^{16}}{{\left (a x - 1\right )}^{4}} + \frac {12 \, a^{9} c^{16}}{{\left (a x - 1\right )}^{5}}}{15 \, a^{10} c^{20}} \]
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Time = 3.86 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.04 \[ \int \frac {e^{4 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^4} \, dx=\frac {x}{c^4}+\frac {\frac {311\,a\,x^2}{3}-\frac {181\,x}{3}+\frac {202}{15\,a}-82\,a^2\,x^3+26\,a^3\,x^4}{-a^5\,c^4\,x^5+5\,a^4\,c^4\,x^4-10\,a^3\,c^4\,x^3+10\,a^2\,c^4\,x^2-5\,a\,c^4\,x+c^4}+\frac {8\,\ln \left (a\,x-1\right )}{a\,c^4} \]
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