Integrand size = 18, antiderivative size = 52 \[ \int \frac {e^{4 \coth ^{-1}(a x)}}{(c-a c x)^3} \, dx=\frac {1}{a c^3 (1-a x)^4}-\frac {4}{3 a c^3 (1-a x)^3}+\frac {1}{2 a c^3 (1-a x)^2} \]
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Time = 0.05 (sec) , antiderivative size = 52, 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} \[ \int \frac {e^{4 \coth ^{-1}(a x)}}{(c-a c x)^3} \, dx=\frac {1}{2 a c^3 (1-a x)^2}-\frac {4}{3 a c^3 (1-a x)^3}+\frac {1}{a c^3 (1-a x)^4} \]
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Rule 45
Rule 6264
Rule 6302
Rubi steps \begin{align*} \text {integral}& = \int \frac {e^{4 \text {arctanh}(a x)}}{(c-a c x)^3} \, dx \\ & = \frac {\int \frac {(1+a x)^2}{(1-a x)^5} \, dx}{c^3} \\ & = \frac {\int \left (-\frac {4}{(-1+a x)^5}-\frac {4}{(-1+a x)^4}-\frac {1}{(-1+a x)^3}\right ) \, dx}{c^3} \\ & = \frac {1}{a c^3 (1-a x)^4}-\frac {4}{3 a c^3 (1-a x)^3}+\frac {1}{2 a c^3 (1-a x)^2} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.60 \[ \int \frac {e^{4 \coth ^{-1}(a x)}}{(c-a c x)^3} \, dx=\frac {1+2 a x+3 a^2 x^2}{6 a c^3 (-1+a x)^4} \]
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Time = 0.54 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.52
method | result | size |
risch | \(\frac {\frac {a \,x^{2}}{2}+\frac {x}{3}+\frac {1}{6 a}}{\left (a x -1\right )^{4} c^{3}}\) | \(27\) |
gosper | \(\frac {3 a^{2} x^{2}+2 a x +1}{6 \left (a x -1\right )^{4} a \,c^{3}}\) | \(30\) |
parallelrisch | \(\frac {-a^{3} x^{4}+4 a^{2} x^{3}-3 a \,x^{2}+6 x}{6 \left (a x -1\right )^{4} c^{3}}\) | \(39\) |
default | \(\frac {\frac {1}{a \left (a x -1\right )^{4}}+\frac {1}{2 \left (a x -1\right )^{2} a}+\frac {4}{3 a \left (a x -1\right )^{3}}}{c^{3}}\) | \(41\) |
norman | \(\frac {\frac {x}{c}-\frac {a \,x^{2}}{2 c}+\frac {2 a^{2} x^{3}}{3 c}-\frac {a^{3} x^{4}}{6 c}}{\left (a x -1\right )^{4} c^{2}}\) | \(49\) |
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Time = 0.23 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.25 \[ \int \frac {e^{4 \coth ^{-1}(a x)}}{(c-a c x)^3} \, dx=\frac {3 \, a^{2} x^{2} + 2 \, a x + 1}{6 \, {\left (a^{5} c^{3} x^{4} - 4 \, a^{4} c^{3} x^{3} + 6 \, a^{3} c^{3} x^{2} - 4 \, a^{2} c^{3} x + a c^{3}\right )}} \]
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Time = 0.20 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.35 \[ \int \frac {e^{4 \coth ^{-1}(a x)}}{(c-a c x)^3} \, dx=- \frac {- 3 a^{2} x^{2} - 2 a x - 1}{6 a^{5} c^{3} x^{4} - 24 a^{4} c^{3} x^{3} + 36 a^{3} c^{3} x^{2} - 24 a^{2} c^{3} x + 6 a c^{3}} \]
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Time = 0.20 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.25 \[ \int \frac {e^{4 \coth ^{-1}(a x)}}{(c-a c x)^3} \, dx=\frac {3 \, a^{2} x^{2} + 2 \, a x + 1}{6 \, {\left (a^{5} c^{3} x^{4} - 4 \, a^{4} c^{3} x^{3} + 6 \, a^{3} c^{3} x^{2} - 4 \, a^{2} c^{3} x + a c^{3}\right )}} \]
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Time = 0.26 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.81 \[ \int \frac {e^{4 \coth ^{-1}(a x)}}{(c-a c x)^3} \, dx=\frac {\frac {3}{{\left (a x - 1\right )}^{2} a} + \frac {8}{{\left (a x - 1\right )}^{3} a} + \frac {6}{{\left (a x - 1\right )}^{4} a}}{6 \, c^{3}} \]
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Time = 4.44 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.56 \[ \int \frac {e^{4 \coth ^{-1}(a x)}}{(c-a c x)^3} \, dx=\frac {3\,a^2\,x^2+2\,a\,x+1}{6\,a\,c^3\,{\left (a\,x-1\right )}^4} \]
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