Integrand size = 16, antiderivative size = 133 \[ \int \frac {e^{\coth ^{-1}(a x)}}{(c-a c x)^5} \, dx=\frac {a^5 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{9 c^5 \left (a-\frac {1}{x}\right )^6}-\frac {8 a^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{21 c^5 \left (a-\frac {1}{x}\right )^5}+\frac {47 a^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{105 c^5 \left (a-\frac {1}{x}\right )^4}-\frac {58 a^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{315 c^5 \left (a-\frac {1}{x}\right )^3} \]
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Time = 0.25 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6310, 6313, 1653, 807, 673, 665} \[ \int \frac {e^{\coth ^{-1}(a x)}}{(c-a c x)^5} \, dx=-\frac {58 a^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{315 c^5 \left (a-\frac {1}{x}\right )^3}+\frac {a^5 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{9 c^5 \left (a-\frac {1}{x}\right )^6}-\frac {8 a^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{21 c^5 \left (a-\frac {1}{x}\right )^5}+\frac {47 a^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{105 c^5 \left (a-\frac {1}{x}\right )^4} \]
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Rule 665
Rule 673
Rule 807
Rule 1653
Rule 6310
Rule 6313
Rubi steps \begin{align*} \text {integral}& = -\frac {\int \frac {e^{\coth ^{-1}(a x)}}{\left (1-\frac {1}{a x}\right )^5 x^5} \, dx}{a^5 c^5} \\ & = \frac {\text {Subst}\left (\int \frac {x^3 \sqrt {1-\frac {x^2}{a^2}}}{\left (1-\frac {x}{a}\right )^6} \, dx,x,\frac {1}{x}\right )}{a^5 c^5} \\ & = \frac {a^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{c^5 \left (a-\frac {1}{x}\right )^4}-\frac {\text {Subst}\left (\int \frac {\left (\frac {4}{a^2}-\frac {7 x}{a^3}+\frac {2 x^2}{a^4}\right ) \sqrt {1-\frac {x^2}{a^2}}}{\left (1-\frac {x}{a}\right )^6} \, dx,x,\frac {1}{x}\right )}{c^5} \\ & = \frac {a^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{c^5 \left (a-\frac {1}{x}\right )^5}+\frac {a^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{c^5 \left (a-\frac {1}{x}\right )^4}-\frac {a^4 \text {Subst}\left (\int \frac {\left (\frac {18}{a^6}-\frac {20 x}{a^7}\right ) \sqrt {1-\frac {x^2}{a^2}}}{\left (1-\frac {x}{a}\right )^6} \, dx,x,\frac {1}{x}\right )}{2 c^5} \\ & = \frac {a^5 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{9 c^5 \left (a-\frac {1}{x}\right )^6}+\frac {a^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{c^5 \left (a-\frac {1}{x}\right )^5}+\frac {a^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{c^5 \left (a-\frac {1}{x}\right )^4}-\frac {29 \text {Subst}\left (\int \frac {\sqrt {1-\frac {x^2}{a^2}}}{\left (1-\frac {x}{a}\right )^5} \, dx,x,\frac {1}{x}\right )}{3 a^2 c^5} \\ & = \frac {a^5 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{9 c^5 \left (a-\frac {1}{x}\right )^6}-\frac {8 a^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{21 c^5 \left (a-\frac {1}{x}\right )^5}+\frac {a^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{c^5 \left (a-\frac {1}{x}\right )^4}-\frac {58 \text {Subst}\left (\int \frac {\sqrt {1-\frac {x^2}{a^2}}}{\left (1-\frac {x}{a}\right )^4} \, dx,x,\frac {1}{x}\right )}{21 a^2 c^5} \\ & = \frac {a^5 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{9 c^5 \left (a-\frac {1}{x}\right )^6}-\frac {8 a^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{21 c^5 \left (a-\frac {1}{x}\right )^5}+\frac {47 a^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{105 c^5 \left (a-\frac {1}{x}\right )^4}-\frac {58 \text {Subst}\left (\int \frac {\sqrt {1-\frac {x^2}{a^2}}}{\left (1-\frac {x}{a}\right )^3} \, dx,x,\frac {1}{x}\right )}{105 a^2 c^5} \\ & = \frac {a^5 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{9 c^5 \left (a-\frac {1}{x}\right )^6}-\frac {8 a^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{21 c^5 \left (a-\frac {1}{x}\right )^5}+\frac {47 a^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{105 c^5 \left (a-\frac {1}{x}\right )^4}-\frac {58 a^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{315 c^5 \left (a-\frac {1}{x}\right )^3} \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.44 \[ \int \frac {e^{\coth ^{-1}(a x)}}{(c-a c x)^5} \, dx=-\frac {\sqrt {1-\frac {1}{a^2 x^2}} x \left (-58-25 a x+21 a^2 x^2-10 a^3 x^3+2 a^4 x^4\right )}{315 c^5 (-1+a x)^5} \]
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Time = 0.40 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.44
method | result | size |
gosper | \(-\frac {\left (2 a^{3} x^{3}-12 a^{2} x^{2}+33 a x -58\right ) \left (a x +1\right )}{315 \left (a x -1\right )^{4} c^{5} \sqrt {\frac {a x -1}{a x +1}}\, a}\) | \(58\) |
default | \(-\frac {\left (2 a^{3} x^{3}-12 a^{2} x^{2}+33 a x -58\right ) \left (a x +1\right )}{315 \left (a x -1\right )^{4} c^{5} \sqrt {\frac {a x -1}{a x +1}}\, a}\) | \(58\) |
trager | \(-\frac {\left (a x +1\right ) \left (2 a^{4} x^{4}-10 a^{3} x^{3}+21 a^{2} x^{2}-25 a x -58\right ) \sqrt {-\frac {-a x +1}{a x +1}}}{315 a \,c^{5} \left (a x -1\right )^{5}}\) | \(68\) |
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Time = 0.24 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.87 \[ \int \frac {e^{\coth ^{-1}(a x)}}{(c-a c x)^5} \, dx=-\frac {{\left (2 \, a^{5} x^{5} - 8 \, a^{4} x^{4} + 11 \, a^{3} x^{3} - 4 \, a^{2} x^{2} - 83 \, a x - 58\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{315 \, {\left (a^{6} c^{5} x^{5} - 5 \, a^{5} c^{5} x^{4} + 10 \, a^{4} c^{5} x^{3} - 10 \, a^{3} c^{5} x^{2} + 5 \, a^{2} c^{5} x - a c^{5}\right )}} \]
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\[ \int \frac {e^{\coth ^{-1}(a x)}}{(c-a c x)^5} \, dx=- \frac {\int \frac {1}{a^{5} x^{5} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}} - 5 a^{4} x^{4} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}} + 10 a^{3} x^{3} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}} - 10 a^{2} x^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}} + 5 a x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}} - \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}\, dx}{c^{5}} \]
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Time = 0.21 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.53 \[ \int \frac {e^{\coth ^{-1}(a x)}}{(c-a c x)^5} \, dx=-\frac {\frac {135 \, {\left (a x - 1\right )}}{a x + 1} - \frac {189 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac {105 \, {\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} - 35}{2520 \, a c^{5} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {9}{2}}} \]
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Time = 0.34 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.94 \[ \int \frac {e^{\coth ^{-1}(a x)}}{(c-a c x)^5} \, dx=\frac {4 \, {\left (315 \, {\left (a + \sqrt {a^{2} - \frac {1}{x^{2}}}\right )}^{5} x^{5} + 189 \, {\left (a + \sqrt {a^{2} - \frac {1}{x^{2}}}\right )}^{4} x^{4} + 84 \, {\left (a + \sqrt {a^{2} - \frac {1}{x^{2}}}\right )}^{3} x^{3} - 36 \, {\left (a + \sqrt {a^{2} - \frac {1}{x^{2}}}\right )}^{2} x^{2} + 9 \, {\left (a + \sqrt {a^{2} - \frac {1}{x^{2}}}\right )} x - 1\right )}}{315 \, {\left ({\left (a + \sqrt {a^{2} - \frac {1}{x^{2}}}\right )} x - 1\right )}^{9} a c^{5}} \]
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Time = 0.05 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.54 \[ \int \frac {e^{\coth ^{-1}(a x)}}{(c-a c x)^5} \, dx=\frac {\frac {3\,{\left (a\,x-1\right )}^2}{5\,{\left (a\,x+1\right )}^2}-\frac {{\left (a\,x-1\right )}^3}{3\,{\left (a\,x+1\right )}^3}-\frac {3\,\left (a\,x-1\right )}{7\,\left (a\,x+1\right )}+\frac {1}{9}}{8\,a\,c^5\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{9/2}} \]
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