Integrand size = 22, antiderivative size = 91 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^3} \, dx=-\frac {8 e^{3 \coth ^{-1}(a x)}}{35 a c^3}-\frac {e^{3 \coth ^{-1}(a x)} (3-4 a x)}{7 a c^3 \left (1-a^2 x^2\right )^2}+\frac {12 e^{3 \coth ^{-1}(a x)} (3-2 a x)}{35 a c^3 \left (1-a^2 x^2\right )} \]
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Time = 0.09 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {6320, 6318} \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^3} \, dx=-\frac {(3-4 a x) e^{3 \coth ^{-1}(a x)}}{7 a c^3 \left (1-a^2 x^2\right )^2}+\frac {12 (3-2 a x) e^{3 \coth ^{-1}(a x)}}{35 a c^3 \left (1-a^2 x^2\right )}-\frac {8 e^{3 \coth ^{-1}(a x)}}{35 a c^3} \]
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Rule 6318
Rule 6320
Rubi steps \begin{align*} \text {integral}& = -\frac {e^{3 \coth ^{-1}(a x)} (3-4 a x)}{7 a c^3 \left (1-a^2 x^2\right )^2}+\frac {12 \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^2} \, dx}{7 c} \\ & = -\frac {e^{3 \coth ^{-1}(a x)} (3-4 a x)}{7 a c^3 \left (1-a^2 x^2\right )^2}+\frac {12 e^{3 \coth ^{-1}(a x)} (3-2 a x)}{35 a c^3 \left (1-a^2 x^2\right )}-\frac {24 \int \frac {e^{3 \coth ^{-1}(a x)}}{c-a^2 c x^2} \, dx}{35 c^2} \\ & = -\frac {8 e^{3 \coth ^{-1}(a x)}}{35 a c^3}-\frac {e^{3 \coth ^{-1}(a x)} (3-4 a x)}{7 a c^3 \left (1-a^2 x^2\right )^2}+\frac {12 e^{3 \coth ^{-1}(a x)} (3-2 a x)}{35 a c^3 \left (1-a^2 x^2\right )} \\ \end{align*}
Time = 0.41 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.73 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^3} \, dx=-\frac {\sqrt {1-\frac {1}{a^2 x^2}} x \left (-13+4 a x+20 a^2 x^2-24 a^3 x^3+8 a^4 x^4\right )}{35 c^3 (-1+a x)^4 (1+a x)} \]
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Time = 0.50 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.69
method | result | size |
trager | \(-\frac {\left (8 a^{4} x^{4}-24 a^{3} x^{3}+20 a^{2} x^{2}+4 a x -13\right ) \sqrt {-\frac {-a x +1}{a x +1}}}{35 a \,c^{3} \left (a x -1\right )^{4}}\) | \(63\) |
gosper | \(-\frac {8 a^{4} x^{4}-24 a^{3} x^{3}+20 a^{2} x^{2}+4 a x -13}{35 \left (a^{2} x^{2}-1\right )^{2} c^{3} \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} a}\) | \(65\) |
default | \(-\frac {8 a^{4} x^{4}-24 a^{3} x^{3}+20 a^{2} x^{2}+4 a x -13}{35 \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right )^{2} c^{3} a \left (a x -1\right )^{2}}\) | \(68\) |
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Time = 0.24 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.05 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^3} \, dx=-\frac {{\left (8 \, a^{4} x^{4} - 24 \, a^{3} x^{3} + 20 \, a^{2} x^{2} + 4 \, a x - 13\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{35 \, {\left (a^{5} c^{3} x^{4} - 4 \, a^{4} c^{3} x^{3} + 6 \, a^{3} c^{3} x^{2} - 4 \, a^{2} c^{3} x + a c^{3}\right )}} \]
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\[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^3} \, dx=- \frac {\int \frac {1}{\frac {a^{7} x^{7} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {a^{6} x^{6} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {3 a^{5} x^{5} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} + \frac {3 a^{4} x^{4} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} + \frac {3 a^{3} x^{3} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {3 a^{2} x^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {a x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} + \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}}\, dx}{c^{3}} \]
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Time = 0.19 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.07 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^3} \, dx=-\frac {1}{560} \, a {\left (\frac {35 \, \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} c^{3}} + \frac {\frac {28 \, {\left (a x - 1\right )}}{a x + 1} - \frac {70 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac {140 \, {\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} - 5}{a^{2} c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}}}\right )} \]
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\[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^3} \, dx=\int { -\frac {1}{{\left (a^{2} c x^{2} - c\right )}^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}} \,d x } \]
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Time = 4.31 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.66 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^3} \, dx=-\frac {8\,a^4\,x^4-24\,a^3\,x^3+20\,a^2\,x^2+4\,a\,x-13}{35\,a\,c^3\,{\left (a\,x+1\right )}^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/2}} \]
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