Integrand size = 22, antiderivative size = 55 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^2} \, dx=-\frac {2 e^{3 \coth ^{-1}(a x)}}{15 a c^2}+\frac {e^{3 \coth ^{-1}(a x)} (3-2 a x)}{5 a c^2 \left (1-a^2 x^2\right )} \]
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Time = 0.06 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {6320, 6318} \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^2} \, dx=\frac {(3-2 a x) e^{3 \coth ^{-1}(a x)}}{5 a c^2 \left (1-a^2 x^2\right )}-\frac {2 e^{3 \coth ^{-1}(a x)}}{15 a c^2} \]
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Rule 6318
Rule 6320
Rubi steps \begin{align*} \text {integral}& = \frac {e^{3 \coth ^{-1}(a x)} (3-2 a x)}{5 a c^2 \left (1-a^2 x^2\right )}-\frac {2 \int \frac {e^{3 \coth ^{-1}(a x)}}{c-a^2 c x^2} \, dx}{5 c} \\ & = -\frac {2 e^{3 \coth ^{-1}(a x)}}{15 a c^2}+\frac {e^{3 \coth ^{-1}(a x)} (3-2 a x)}{5 a c^2 \left (1-a^2 x^2\right )} \\ \end{align*}
Time = 0.34 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.78 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^2} \, dx=-\frac {\sqrt {1-\frac {1}{a^2 x^2}} x \left (7-6 a x+2 a^2 x^2\right )}{15 c^2 (-1+a x)^3} \]
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Time = 0.50 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.89
method | result | size |
gosper | \(-\frac {2 a^{2} x^{2}-6 a x +7}{15 \left (a^{2} x^{2}-1\right ) c^{2} \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} a}\) | \(49\) |
default | \(-\frac {2 a^{2} x^{2}-6 a x +7}{15 \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right ) c^{2} \left (a x -1\right ) a}\) | \(52\) |
trager | \(-\frac {\left (a x +1\right ) \left (2 a^{2} x^{2}-6 a x +7\right ) \sqrt {-\frac {-a x +1}{a x +1}}}{15 a \,c^{2} \left (a x -1\right )^{3}}\) | \(52\) |
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Time = 0.24 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.40 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^2} \, dx=-\frac {{\left (2 \, a^{3} x^{3} - 4 \, a^{2} x^{2} + a x + 7\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{15 \, {\left (a^{4} c^{2} x^{3} - 3 \, a^{3} c^{2} x^{2} + 3 \, a^{2} c^{2} x - a c^{2}\right )}} \]
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\[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^2} \, dx=\frac {\int \frac {1}{\frac {a^{5} x^{5} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {a^{4} x^{4} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {2 a^{3} x^{3} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} + \frac {2 a^{2} x^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} + \frac {a x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}}\, dx}{c^{2}} \]
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Time = 0.19 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^2} \, dx=\frac {\frac {10 \, {\left (a x - 1\right )}}{a x + 1} - \frac {15 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} - 3}{60 \, a c^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}}} \]
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Time = 0.31 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.18 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^2} \, dx=-\frac {4 \, {\left (10 \, {\left (a + \sqrt {a^{2} - \frac {1}{x^{2}}}\right )}^{2} x^{2} - 5 \, {\left (a + \sqrt {a^{2} - \frac {1}{x^{2}}}\right )} x + 1\right )}}{15 \, {\left ({\left (a + \sqrt {a^{2} - \frac {1}{x^{2}}}\right )} x - 1\right )}^{5} a c^{2}} \]
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Time = 4.37 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^2} \, dx=-\frac {\frac {{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}-\frac {2\,\left (a\,x-1\right )}{3\,\left (a\,x+1\right )}+\frac {1}{5}}{4\,a\,c^2\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}} \]
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