Integrand size = 18, antiderivative size = 62 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{(c-a c x)^3} \, dx=\frac {a \sqrt {1-\frac {1}{a^2 x^2}}}{3 c^3 \left (a-\frac {1}{x}\right )^2}-\frac {2 \sqrt {1-\frac {1}{a^2 x^2}}}{3 c^3 \left (a-\frac {1}{x}\right )} \]
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Time = 0.10 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6310, 6313, 807, 665} \[ \int \frac {e^{-\coth ^{-1}(a x)}}{(c-a c x)^3} \, dx=\frac {a \sqrt {1-\frac {1}{a^2 x^2}}}{3 c^3 \left (a-\frac {1}{x}\right )^2}-\frac {2 \sqrt {1-\frac {1}{a^2 x^2}}}{3 c^3 \left (a-\frac {1}{x}\right )} \]
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Rule 665
Rule 807
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 )^3 x^3} \, dx}{a^3 c^3} \\ & = \frac {\text {Subst}\left (\int \frac {x}{\left (1-\frac {x}{a}\right )^2 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{a^3 c^3} \\ & = \frac {a \sqrt {1-\frac {1}{a^2 x^2}}}{3 c^3 \left (a-\frac {1}{x}\right )^2}-\frac {2 \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 )}{3 a^2 c^3} \\ & = \frac {a \sqrt {1-\frac {1}{a^2 x^2}}}{3 c^3 \left (a-\frac {1}{x}\right )^2}-\frac {2 \sqrt {1-\frac {1}{a^2 x^2}}}{3 c^3 \left (a-\frac {1}{x}\right )} \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.55 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{(c-a c x)^3} \, dx=-\frac {\sqrt {1-\frac {1}{a^2 x^2}} x (-2+a x)}{3 c^3 (-1+a x)^2} \]
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Time = 0.44 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.66
method | result | size |
gosper | \(-\frac {\sqrt {\frac {a x -1}{a x +1}}\, \left (a x -2\right ) \left (a x +1\right )}{3 \left (a x -1\right )^{2} c^{3} a}\) | \(41\) |
default | \(-\frac {\sqrt {\frac {a x -1}{a x +1}}\, \left (a x -2\right ) \left (a x +1\right )}{3 \left (a x -1\right )^{2} c^{3} a}\) | \(41\) |
trager | \(-\frac {\left (a x -2\right ) \left (a x +1\right ) \sqrt {-\frac {-a x +1}{a x +1}}}{3 a \,c^{3} \left (a x -1\right )^{2}}\) | \(43\) |
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Time = 0.24 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.92 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{(c-a c x)^3} \, dx=-\frac {{\left (a^{2} x^{2} - a x - 2\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{3 \, {\left (a^{3} c^{3} x^{2} - 2 \, a^{2} c^{3} x + a c^{3}\right )}} \]
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\[ \int \frac {e^{-\coth ^{-1}(a x)}}{(c-a c x)^3} \, dx=- \frac {\int \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a^{3} x^{3} - 3 a^{2} x^{2} + 3 a x - 1}\, dx}{c^{3}} \]
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Time = 0.20 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.63 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{(c-a c x)^3} \, dx=-\frac {\frac {3 \, {\left (a x - 1\right )}}{a x + 1} - 1}{6 \, a c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}} \]
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Time = 0.29 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.73 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{(c-a c x)^3} \, dx=\frac {2 \, {\left (3 \, {\left (a + \sqrt {a^{2} - \frac {1}{x^{2}}}\right )} x - 1\right )}}{3 \, {\left ({\left (a + \sqrt {a^{2} - \frac {1}{x^{2}}}\right )} x - 1\right )}^{3} a c^{3}} \]
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Time = 0.04 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.61 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{(c-a c x)^3} \, dx=-\frac {\frac {a\,x-1}{a\,x+1}-\frac {1}{3}}{2\,a\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}} \]
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