Integrand size = 18, antiderivative size = 95 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{(c-a c x)^4} \, dx=-\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{5 c^4 \left (a-\frac {1}{x}\right )^3}+\frac {8 a \sqrt {1-\frac {1}{a^2 x^2}}}{15 c^4 \left (a-\frac {1}{x}\right )^2}-\frac {7 \sqrt {1-\frac {1}{a^2 x^2}}}{15 c^4 \left (a-\frac {1}{x}\right )} \]
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Time = 0.16 (sec) , antiderivative size = 95, 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, 1653, 807, 673, 665} \[ \int \frac {e^{-\coth ^{-1}(a x)}}{(c-a c x)^4} \, dx=-\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{5 c^4 \left (a-\frac {1}{x}\right )^3}+\frac {8 a \sqrt {1-\frac {1}{a^2 x^2}}}{15 c^4 \left (a-\frac {1}{x}\right )^2}-\frac {7 \sqrt {1-\frac {1}{a^2 x^2}}}{15 c^4 \left (a-\frac {1}{x}\right )} \]
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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 )^4 x^4} \, dx}{a^4 c^4} \\ & = -\frac {\text {Subst}\left (\int \frac {x^2}{\left (1-\frac {x}{a}\right )^3 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{a^4 c^4} \\ & = \frac {a \sqrt {1-\frac {1}{a^2 x^2}}}{c^4 \left (a-\frac {1}{x}\right )^2}-\frac {\text {Subst}\left (\int \frac {\frac {2}{a^2}-\frac {x}{a^3}}{\left (1-\frac {x}{a}\right )^3 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{c^4} \\ & = -\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{5 c^4 \left (a-\frac {1}{x}\right )^3}+\frac {a \sqrt {1-\frac {1}{a^2 x^2}}}{c^4 \left (a-\frac {1}{x}\right )^2}-\frac {7 \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 )}{5 a^2 c^4} \\ & = -\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{5 c^4 \left (a-\frac {1}{x}\right )^3}+\frac {8 a \sqrt {1-\frac {1}{a^2 x^2}}}{15 c^4 \left (a-\frac {1}{x}\right )^2}-\frac {7 \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 )}{15 a^2 c^4} \\ & = -\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{5 c^4 \left (a-\frac {1}{x}\right )^3}+\frac {8 a \sqrt {1-\frac {1}{a^2 x^2}}}{15 c^4 \left (a-\frac {1}{x}\right )^2}-\frac {7 \sqrt {1-\frac {1}{a^2 x^2}}}{15 c^4 \left (a-\frac {1}{x}\right )} \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.45 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{(c-a c x)^4} \, dx=-\frac {\sqrt {1-\frac {1}{a^2 x^2}} x \left (7-6 a x+2 a^2 x^2\right )}{15 c^4 (-1+a x)^3} \]
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Time = 0.42 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.53
method | result | size |
gosper | \(-\frac {\sqrt {\frac {a x -1}{a x +1}}\, \left (2 a^{2} x^{2}-6 a x +7\right ) \left (a x +1\right )}{15 \left (a x -1\right )^{3} c^{4} a}\) | \(50\) |
default | \(-\frac {\sqrt {\frac {a x -1}{a x +1}}\, \left (2 a^{2} x^{2}-6 a x +7\right ) \left (a x +1\right )}{15 \left (a x -1\right )^{3} c^{4} a}\) | \(50\) |
trager | \(-\frac {\left (2 a^{2} x^{2}-6 a x +7\right ) \left (a x +1\right ) \sqrt {-\frac {-a x +1}{a x +1}}}{15 a \,c^{4} \left (a x -1\right )^{3}}\) | \(52\) |
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Time = 0.25 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.81 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{(c-a c x)^4} \, dx=-\frac {{\left (2 \, a^{3} x^{3} - 4 \, a^{2} x^{2} + a x + 7\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{15 \, {\left (a^{4} c^{4} x^{3} - 3 \, a^{3} c^{4} x^{2} + 3 \, a^{2} c^{4} x - a c^{4}\right )}} \]
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\[ \int \frac {e^{-\coth ^{-1}(a x)}}{(c-a c x)^4} \, dx=\frac {\int \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a^{4} x^{4} - 4 a^{3} x^{3} + 6 a^{2} x^{2} - 4 a x + 1}\, dx}{c^{4}} \]
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Time = 0.21 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.58 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{(c-a c x)^4} \, dx=\frac {\frac {10 \, {\left (a x - 1\right )}}{a x + 1} - \frac {15 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} - 3}{60 \, a c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}}} \]
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Time = 0.30 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.68 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{(c-a c x)^4} \, dx=-\frac {4 \, {\left (10 \, {\left (a + \sqrt {a^{2} - \frac {1}{x^{2}}}\right )}^{2} x^{2} - 5 \, {\left (a + \sqrt {a^{2} - \frac {1}{x^{2}}}\right )} x + 1\right )}}{15 \, {\left ({\left (a + \sqrt {a^{2} - \frac {1}{x^{2}}}\right )} x - 1\right )}^{5} a c^{4}} \]
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Time = 0.04 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.58 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{(c-a c x)^4} \, dx=-\frac {\frac {{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}-\frac {2\,\left (a\,x-1\right )}{3\,\left (a\,x+1\right )}+\frac {1}{5}}{4\,a\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}} \]
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