Integrand size = 18, antiderivative size = 61 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{(c-a c x)^4} \, dx=\frac {2}{3 a c^4 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {1}{3 a^2 c^4 \sqrt {1-\frac {1}{a^2 x^2}} \left (a-\frac {1}{x}\right ) x^2} \]
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Time = 0.10 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {6310, 6313, 869, 12, 267} \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{(c-a c x)^4} \, dx=\frac {2}{3 a c^4 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {1}{3 a^2 c^4 x^2 \sqrt {1-\frac {1}{a^2 x^2}} \left (a-\frac {1}{x}\right )} \]
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Rule 12
Rule 267
Rule 869
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 )^4 x^4} \, dx}{a^4 c^4} \\ & = -\frac {\text {Subst}\left (\int \frac {x^2}{\left (1-\frac {x}{a}\right ) \left (1-\frac {x^2}{a^2}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{a^4 c^4} \\ & = -\frac {1}{3 a^2 c^4 \sqrt {1-\frac {1}{a^2 x^2}} \left (a-\frac {1}{x}\right ) x^2}+\frac {\text {Subst}\left (\int \frac {2 x}{\left (1-\frac {x^2}{a^2}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{3 a^3 c^4} \\ & = -\frac {1}{3 a^2 c^4 \sqrt {1-\frac {1}{a^2 x^2}} \left (a-\frac {1}{x}\right ) x^2}+\frac {2 \text {Subst}\left (\int \frac {x}{\left (1-\frac {x^2}{a^2}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{3 a^3 c^4} \\ & = \frac {2}{3 a c^4 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {1}{3 a^2 c^4 \sqrt {1-\frac {1}{a^2 x^2}} \left (a-\frac {1}{x}\right ) x^2} \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.82 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{(c-a c x)^4} \, dx=\frac {\sqrt {1-\frac {1}{a^2 x^2}} x \left (-1-2 a x+2 a^2 x^2\right )}{3 c^4 (-1+a x)^2 (1+a x)} \]
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Time = 0.41 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.74
method | result | size |
default | \(\frac {\left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (2 a^{3} x^{3}-3 a x -1\right )}{3 \left (a x -1\right )^{3} c^{4} a}\) | \(45\) |
trager | \(\frac {\left (2 a^{2} x^{2}-2 a x -1\right ) \sqrt {-\frac {-a x +1}{a x +1}}}{3 a \,c^{4} \left (a x -1\right )^{2}}\) | \(47\) |
gosper | \(\frac {\left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (2 a^{2} x^{2}-2 a x -1\right ) \left (a x +1\right )}{3 \left (a x -1\right )^{3} c^{4} a}\) | \(50\) |
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Time = 0.25 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.95 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{(c-a c x)^4} \, dx=\frac {{\left (2 \, a^{2} x^{2} - 2 \, a x - 1\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{3 \, {\left (a^{3} c^{4} x^{2} - 2 \, a^{2} c^{4} x + a c^{4}\right )}} \]
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\[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{(c-a c x)^4} \, dx=\frac {\int \left (- \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a^{5} x^{5} - 3 a^{4} x^{4} + 2 a^{3} x^{3} + 2 a^{2} x^{2} - 3 a x + 1}\right )\, dx + \int \frac {a x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a^{5} x^{5} - 3 a^{4} x^{4} + 2 a^{3} x^{3} + 2 a^{2} x^{2} - 3 a x + 1}\, dx}{c^{4}} \]
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Time = 0.21 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.07 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{(c-a c x)^4} \, dx=\frac {1}{12} \, a {\left (\frac {3 \, \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} c^{4}} + \frac {\frac {6 \, {\left (a x - 1\right )}}{a x + 1} - 1}{a^{2} c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}\right )} \]
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\[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{(c-a c x)^4} \, dx=\int { \frac {\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}{{\left (a c x - c\right )}^{4}} \,d x } \]
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Time = 4.55 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.82 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{(c-a c x)^4} \, dx=\frac {-2\,a^2\,x^2+2\,a\,x+1}{\left (3\,a\,c^4-3\,a^3\,c^4\,x^2\right )\,\sqrt {\frac {a\,x-1}{a\,x+1}}} \]
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