Integrand size = 18, antiderivative size = 94 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{(c-a c x)^4} \, dx=-\frac {47 \left (a+\frac {1}{x}\right )^5}{315 a^6 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}+\frac {16 \left (a+\frac {1}{x}\right )^6}{63 a^7 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}-\frac {\left (a+\frac {1}{x}\right )^7}{9 a^8 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{9/2}} \]
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Time = 0.20 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6310, 6313, 866, 1649, 803, 665} \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{(c-a c x)^4} \, dx=-\frac {\left (a+\frac {1}{x}\right )^7}{9 a^8 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{9/2}}+\frac {16 \left (a+\frac {1}{x}\right )^6}{63 a^7 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}-\frac {47 \left (a+\frac {1}{x}\right )^5}{315 a^6 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}} \]
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Rule 665
Rule 803
Rule 866
Rule 1649
Rule 6310
Rule 6313
Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {e^{3 \coth ^{-1}(a x)}}{\left (1-\frac {1}{a x}\right )^4 x^4} \, dx}{a^4 c^4} \\ & = -\frac {\text {Subst}\left (\int \frac {x^2 \left (1-\frac {x^2}{a^2}\right )^{3/2}}{\left (1-\frac {x}{a}\right )^7} \, dx,x,\frac {1}{x}\right )}{a^4 c^4} \\ & = -\frac {\text {Subst}\left (\int \frac {x^2 \left (1+\frac {x}{a}\right )^7}{\left (1-\frac {x^2}{a^2}\right )^{11/2}} \, dx,x,\frac {1}{x}\right )}{a^4 c^4} \\ & = -\frac {\left (a+\frac {1}{x}\right )^7}{9 a^8 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{9/2}}+\frac {\text {Subst}\left (\int \frac {\left (1+\frac {x}{a}\right )^6 \left (7 a^2+9 a x\right )}{\left (1-\frac {x^2}{a^2}\right )^{9/2}} \, dx,x,\frac {1}{x}\right )}{9 a^4 c^4} \\ & = \frac {16 \left (a+\frac {1}{x}\right )^6}{63 a^7 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}-\frac {\left (a+\frac {1}{x}\right )^7}{9 a^8 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{9/2}}-\frac {47 \text {Subst}\left (\int \frac {\left (1+\frac {x}{a}\right )^5}{\left (1-\frac {x^2}{a^2}\right )^{7/2}} \, dx,x,\frac {1}{x}\right )}{63 a^2 c^4} \\ & = -\frac {47 \left (a+\frac {1}{x}\right )^5}{315 a^6 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}+\frac {16 \left (a+\frac {1}{x}\right )^6}{63 a^7 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}-\frac {\left (a+\frac {1}{x}\right )^7}{9 a^8 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{9/2}} \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.53 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{(c-a c x)^4} \, dx=-\frac {\sqrt {1-\frac {1}{a^2 x^2}} x (1+a x)^2 \left (47-14 a x+2 a^2 x^2\right )}{315 c^4 (-1+a x)^5} \]
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Time = 0.41 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.53
method | result | size |
gosper | \(-\frac {\left (2 a^{2} x^{2}-14 a x +47\right ) \left (a x +1\right )}{315 \left (a x -1\right )^{3} c^{4} \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} a}\) | \(50\) |
default | \(-\frac {\left (2 a^{2} x^{2}-14 a x +47\right ) \left (a x +1\right )}{315 \left (a x -1\right )^{3} c^{4} \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} a}\) | \(50\) |
trager | \(-\frac {\left (a x +1\right ) \left (2 a^{4} x^{4}-10 a^{3} x^{3}+21 a^{2} x^{2}+80 a x +47\right ) \sqrt {-\frac {-a x +1}{a x +1}}}{315 a \,c^{4} \left (a x -1\right )^{5}}\) | \(68\) |
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Time = 0.25 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.23 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{(c-a c x)^4} \, dx=-\frac {{\left (2 \, a^{5} x^{5} - 8 \, a^{4} x^{4} + 11 \, a^{3} x^{3} + 101 \, a^{2} x^{2} + 127 \, a x + 47\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{315 \, {\left (a^{6} c^{4} x^{5} - 5 \, a^{5} c^{4} x^{4} + 10 \, a^{4} c^{4} x^{3} - 10 \, a^{3} c^{4} x^{2} + 5 \, a^{2} c^{4} x - a c^{4}\right )}} \]
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\[ \int \frac {e^{3 \coth ^{-1}(a x)}}{(c-a c x)^4} \, dx=\frac {\int \frac {1}{\frac {a^{5} x^{5} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {5 a^{4} x^{4} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} + \frac {10 a^{3} x^{3} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {10 a^{2} x^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} + \frac {5 a x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}}\, dx}{c^{4}} \]
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Time = 0.21 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.59 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{(c-a c x)^4} \, dx=\frac {\frac {90 \, {\left (a x - 1\right )}}{a x + 1} - \frac {63 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} - 35}{1260 \, a c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {9}{2}}} \]
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Time = 0.35 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.54 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{(c-a c x)^4} \, dx=-\frac {4 \, {\left (210 \, {\left (a + \sqrt {a^{2} - \frac {1}{x^{2}}}\right )}^{6} x^{6} + 315 \, {\left (a + \sqrt {a^{2} - \frac {1}{x^{2}}}\right )}^{5} x^{5} + 441 \, {\left (a + \sqrt {a^{2} - \frac {1}{x^{2}}}\right )}^{4} x^{4} + 126 \, {\left (a + \sqrt {a^{2} - \frac {1}{x^{2}}}\right )}^{3} x^{3} + 36 \, {\left (a + \sqrt {a^{2} - \frac {1}{x^{2}}}\right )}^{2} x^{2} - 9 \, {\left (a + \sqrt {a^{2} - \frac {1}{x^{2}}}\right )} x + 1\right )}}{315 \, {\left ({\left (a + \sqrt {a^{2} - \frac {1}{x^{2}}}\right )} x - 1\right )}^{9} a c^{4}} \]
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Time = 4.18 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.60 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{(c-a c x)^4} \, dx=-\frac {\frac {{\left (a\,x-1\right )}^2}{5\,{\left (a\,x+1\right )}^2}-\frac {2\,\left (a\,x-1\right )}{7\,\left (a\,x+1\right )}+\frac {1}{9}}{4\,a\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{9/2}} \]
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