Integrand size = 18, antiderivative size = 37 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{(c-a c x)^4} \, dx=-\frac {1}{2 a c^4 (1-a x)^4}+\frac {1}{3 a c^4 (1-a x)^3} \]
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Time = 0.05 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6302, 6264, 45} \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{(c-a c x)^4} \, dx=\frac {1}{3 a c^4 (1-a x)^3}-\frac {1}{2 a c^4 (1-a x)^4} \]
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Rule 45
Rule 6264
Rule 6302
Rubi steps \begin{align*} \text {integral}& = -\int \frac {e^{2 \text {arctanh}(a x)}}{(c-a c x)^4} \, dx \\ & = -\frac {\int \frac {1+a x}{(1-a x)^5} \, dx}{c^4} \\ & = -\frac {\int \left (-\frac {2}{(-1+a x)^5}-\frac {1}{(-1+a x)^4}\right ) \, dx}{c^4} \\ & = -\frac {1}{2 a c^4 (1-a x)^4}+\frac {1}{3 a c^4 (1-a x)^3} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.62 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{(c-a c x)^4} \, dx=-\frac {1+2 a x}{6 a c^4 (-1+a x)^4} \]
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Time = 0.46 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.57
method | result | size |
risch | \(\frac {-\frac {x}{3}-\frac {1}{6 a}}{c^{4} \left (a x -1\right )^{4}}\) | \(21\) |
gosper | \(-\frac {2 a x +1}{6 a \,c^{4} \left (a x -1\right )^{4}}\) | \(22\) |
default | \(\frac {-\frac {1}{2 a \left (a x -1\right )^{4}}-\frac {1}{3 a \left (a x -1\right )^{3}}}{c^{4}}\) | \(30\) |
parallelrisch | \(\frac {a^{3} x^{4}-4 a^{2} x^{3}+6 a \,x^{2}-6 x}{6 c^{4} \left (a x -1\right )^{4}}\) | \(38\) |
norman | \(\frac {\frac {a \,x^{2}}{c}-\frac {x}{c}-\frac {2 a^{2} x^{3}}{3 c}+\frac {a^{3} x^{4}}{6 c}}{\left (a x -1\right )^{4} c^{3}}\) | \(49\) |
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Time = 0.25 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.54 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{(c-a c x)^4} \, dx=-\frac {2 \, a x + 1}{6 \, {\left (a^{5} c^{4} x^{4} - 4 \, a^{4} c^{4} x^{3} + 6 \, a^{3} c^{4} x^{2} - 4 \, a^{2} c^{4} x + a c^{4}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (29) = 58\).
Time = 0.19 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.62 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{(c-a c x)^4} \, dx=\frac {- 2 a x - 1}{6 a^{5} c^{4} x^{4} - 24 a^{4} c^{4} x^{3} + 36 a^{3} c^{4} x^{2} - 24 a^{2} c^{4} x + 6 a c^{4}} \]
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Time = 0.21 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.54 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{(c-a c x)^4} \, dx=-\frac {2 \, a x + 1}{6 \, {\left (a^{5} c^{4} x^{4} - 4 \, a^{4} c^{4} x^{3} + 6 \, a^{3} c^{4} x^{2} - 4 \, a^{2} c^{4} x + a c^{4}\right )}} \]
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Time = 0.26 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.57 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{(c-a c x)^4} \, dx=-\frac {2 \, a x + 1}{6 \, {\left (a x - 1\right )}^{4} a c^{4}} \]
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Time = 0.10 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.51 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{(c-a c x)^4} \, dx=-\frac {\frac {x}{3}+\frac {1}{6\,a}}{a^4\,c^4\,x^4-4\,a^3\,c^4\,x^3+6\,a^2\,c^4\,x^2-4\,a\,c^4\,x+c^4} \]
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