Integrand size = 18, antiderivative size = 128 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{(c-a c x)^5} \, dx=\frac {a^3 \sqrt {1-\frac {1}{a^2 x^2}}}{7 c^5 \left (a-\frac {1}{x}\right )^4}-\frac {18 a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{35 c^5 \left (a-\frac {1}{x}\right )^3}+\frac {23 a \sqrt {1-\frac {1}{a^2 x^2}}}{35 c^5 \left (a-\frac {1}{x}\right )^2}-\frac {12 \sqrt {1-\frac {1}{a^2 x^2}}}{35 c^5 \left (a-\frac {1}{x}\right )} \]
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Time = 0.18 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6310, 6313, 1653, 807, 673, 665} \[ \int \frac {e^{-\coth ^{-1}(a x)}}{(c-a c x)^5} \, dx=-\frac {18 a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{35 c^5 \left (a-\frac {1}{x}\right )^3}+\frac {23 a \sqrt {1-\frac {1}{a^2 x^2}}}{35 c^5 \left (a-\frac {1}{x}\right )^2}-\frac {12 \sqrt {1-\frac {1}{a^2 x^2}}}{35 c^5 \left (a-\frac {1}{x}\right )}+\frac {a^3 \sqrt {1-\frac {1}{a^2 x^2}}}{7 c^5 \left (a-\frac {1}{x}\right )^4} \]
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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}{\left (1-\frac {x}{a}\right )^4 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{a^5 c^5} \\ & = \frac {a \sqrt {1-\frac {1}{a^2 x^2}}}{c^5 \left (a-\frac {1}{x}\right )^2}-\frac {\text {Subst}\left (\int \frac {\frac {2}{a^2}-\frac {3 x}{a^3}}{\left (1-\frac {x}{a}\right )^4 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{c^5} \\ & = \frac {a^3 \sqrt {1-\frac {1}{a^2 x^2}}}{7 c^5 \left (a-\frac {1}{x}\right )^4}+\frac {a \sqrt {1-\frac {1}{a^2 x^2}}}{c^5 \left (a-\frac {1}{x}\right )^2}-\frac {18 \text {Subst}\left (\int \frac {1}{\left (1-\frac {x}{a}\right )^3 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{7 a^2 c^5} \\ & = \frac {a^3 \sqrt {1-\frac {1}{a^2 x^2}}}{7 c^5 \left (a-\frac {1}{x}\right )^4}-\frac {18 a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{35 c^5 \left (a-\frac {1}{x}\right )^3}+\frac {a \sqrt {1-\frac {1}{a^2 x^2}}}{c^5 \left (a-\frac {1}{x}\right )^2}-\frac {36 \text {Subst}\left (\int \frac {1}{\left (1-\frac {x}{a}\right )^2 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{35 a^2 c^5} \\ & = \frac {a^3 \sqrt {1-\frac {1}{a^2 x^2}}}{7 c^5 \left (a-\frac {1}{x}\right )^4}-\frac {18 a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{35 c^5 \left (a-\frac {1}{x}\right )^3}+\frac {23 a \sqrt {1-\frac {1}{a^2 x^2}}}{35 c^5 \left (a-\frac {1}{x}\right )^2}-\frac {12 \text {Subst}\left (\int \frac {1}{\left (1-\frac {x}{a}\right ) \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{35 a^2 c^5} \\ & = \frac {a^3 \sqrt {1-\frac {1}{a^2 x^2}}}{7 c^5 \left (a-\frac {1}{x}\right )^4}-\frac {18 a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{35 c^5 \left (a-\frac {1}{x}\right )^3}+\frac {23 a \sqrt {1-\frac {1}{a^2 x^2}}}{35 c^5 \left (a-\frac {1}{x}\right )^2}-\frac {12 \sqrt {1-\frac {1}{a^2 x^2}}}{35 c^5 \left (a-\frac {1}{x}\right )} \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.40 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{(c-a c x)^5} \, dx=-\frac {\sqrt {1-\frac {1}{a^2 x^2}} x \left (-12+13 a x-8 a^2 x^2+2 a^3 x^3\right )}{35 c^5 (-1+a x)^4} \]
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Time = 0.42 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.45
method | result | size |
gosper | \(-\frac {\sqrt {\frac {a x -1}{a x +1}}\, \left (2 a^{3} x^{3}-8 a^{2} x^{2}+13 a x -12\right ) \left (a x +1\right )}{35 \left (a x -1\right )^{4} c^{5} a}\) | \(58\) |
default | \(-\frac {\sqrt {\frac {a x -1}{a x +1}}\, \left (2 a^{3} x^{3}-8 a^{2} x^{2}+13 a x -12\right ) \left (a x +1\right )}{35 \left (a x -1\right )^{4} c^{5} a}\) | \(58\) |
trager | \(-\frac {\left (2 a^{3} x^{3}-8 a^{2} x^{2}+13 a x -12\right ) \left (a x +1\right ) \sqrt {-\frac {-a x +1}{a x +1}}}{35 a \,c^{5} \left (a x -1\right )^{4}}\) | \(60\) |
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Time = 0.25 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.74 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{(c-a c x)^5} \, dx=-\frac {{\left (2 \, a^{4} x^{4} - 6 \, a^{3} x^{3} + 5 \, a^{2} x^{2} + a x - 12\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{35 \, {\left (a^{5} c^{5} x^{4} - 4 \, a^{4} c^{5} x^{3} + 6 \, a^{3} c^{5} x^{2} - 4 \, a^{2} c^{5} x + a c^{5}\right )}} \]
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\[ \int \frac {e^{-\coth ^{-1}(a x)}}{(c-a c x)^5} \, dx=- \frac {\int \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a^{5} x^{5} - 5 a^{4} x^{4} + 10 a^{3} x^{3} - 10 a^{2} x^{2} + 5 a x - 1}\, dx}{c^{5}} \]
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Time = 0.21 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.55 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{(c-a c x)^5} \, dx=-\frac {\frac {21 \, {\left (a x - 1\right )}}{a x + 1} - \frac {35 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac {35 \, {\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} - 5}{280 \, a c^{5} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}}} \]
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Time = 0.33 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.66 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{(c-a c x)^5} \, dx=\frac {4 \, {\left (35 \, {\left (a + \sqrt {a^{2} - \frac {1}{x^{2}}}\right )}^{3} x^{3} - 21 \, {\left (a + \sqrt {a^{2} - \frac {1}{x^{2}}}\right )}^{2} x^{2} + 7 \, {\left (a + \sqrt {a^{2} - \frac {1}{x^{2}}}\right )} x - 1\right )}}{35 \, {\left ({\left (a + \sqrt {a^{2} - \frac {1}{x^{2}}}\right )} x - 1\right )}^{7} a c^{5}} \]
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Time = 0.05 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.55 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{(c-a c x)^5} \, dx=\frac {\frac {{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}-\frac {{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}-\frac {3\,\left (a\,x-1\right )}{5\,\left (a\,x+1\right )}+\frac {1}{7}}{8\,a\,c^5\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/2}} \]
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