Integrand size = 18, antiderivative size = 21 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{(c-a c x)^3} \, dx=\frac {1}{a c^3 \sqrt {1-\frac {1}{a^2 x^2}}} \]
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Time = 0.07 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6310, 6313, 267} \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{(c-a c x)^3} \, dx=\frac {1}{a c^3 \sqrt {1-\frac {1}{a^2 x^2}}} \]
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Rule 267
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 )^3 x^3} \, dx}{a^3 c^3} \\ & = \frac {\text {Subst}\left (\int \frac {x}{\left (1-\frac {x^2}{a^2}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{a^3 c^3} \\ & = \frac {1}{a c^3 \sqrt {1-\frac {1}{a^2 x^2}}} \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.57 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{(c-a c x)^3} \, dx=\frac {a \sqrt {1-\frac {1}{a^2 x^2}} x^2}{c^3 \left (-1+a^2 x^2\right )} \]
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Time = 0.42 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.43
method | result | size |
trager | \(\frac {x \sqrt {-\frac {-a x +1}{a x +1}}}{c^{3} \left (a x -1\right )}\) | \(30\) |
gosper | \(\frac {\left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right ) x}{\left (a x -1\right )^{2} c^{3}}\) | \(33\) |
default | \(\frac {\left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right ) x}{\left (a x -1\right )^{2} c^{3}}\) | \(33\) |
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Time = 0.25 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.48 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{(c-a c x)^3} \, dx=\frac {x \sqrt {\frac {a x - 1}{a x + 1}}}{a c^{3} x - c^{3}} \]
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\[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{(c-a c x)^3} \, dx=- \frac {\int \left (- \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a^{4} x^{4} - 2 a^{3} x^{3} + 2 a x - 1}\right )\, dx + \int \frac {a x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a^{4} x^{4} - 2 a^{3} x^{3} + 2 a x - 1}\, dx}{c^{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (19) = 38\).
Time = 0.20 (sec) , antiderivative size = 48, normalized size of antiderivative = 2.29 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{(c-a c x)^3} \, dx=\frac {1}{2} \, a {\left (\frac {\sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} c^{3}} + \frac {1}{a^{2} c^{3} \sqrt {\frac {a x - 1}{a x + 1}}}\right )} \]
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Time = 0.28 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.05 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{(c-a c x)^3} \, dx=\frac {x \mathrm {sgn}\left (a x + 1\right )}{\sqrt {a^{2} x^{2} - 1} c^{3}} \]
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Time = 4.20 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.81 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{(c-a c x)^3} \, dx=\frac {\frac {a\,x-1}{a\,x+1}+1}{2\,a\,c^3\,\sqrt {\frac {a\,x-1}{a\,x+1}}} \]
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