Integrand size = 27, antiderivative size = 153 \[ \int \frac {e^{3 \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2}}{x^3} \, dx=\frac {\sqrt {c-a^2 c x^2}}{2 a \sqrt {1-\frac {1}{a^2 x^2}} x^3}+\frac {3 \sqrt {c-a^2 c x^2}}{\sqrt {1-\frac {1}{a^2 x^2}} x^2}-\frac {4 a \sqrt {c-a^2 c x^2} \log (x)}{\sqrt {1-\frac {1}{a^2 x^2}} x}+\frac {4 a \sqrt {c-a^2 c x^2} \log (1-a x)}{\sqrt {1-\frac {1}{a^2 x^2}} x} \]
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Time = 0.17 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {6327, 6328, 90} \[ \int \frac {e^{3 \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2}}{x^3} \, dx=\frac {3 \sqrt {c-a^2 c x^2}}{x^2 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {4 a \log (x) \sqrt {c-a^2 c x^2}}{x \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {4 a \sqrt {c-a^2 c x^2} \log (1-a x)}{x \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {c-a^2 c x^2}}{2 a x^3 \sqrt {1-\frac {1}{a^2 x^2}}} \]
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Rule 90
Rule 6327
Rule 6328
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {c-a^2 c x^2} \int \frac {e^{3 \coth ^{-1}(a x)} \sqrt {1-\frac {1}{a^2 x^2}}}{x^2} \, dx}{\sqrt {1-\frac {1}{a^2 x^2}} x} \\ & = \frac {\sqrt {c-a^2 c x^2} \int \frac {(1+a x)^2}{x^3 (-1+a x)} \, dx}{a \sqrt {1-\frac {1}{a^2 x^2}} x} \\ & = \frac {\sqrt {c-a^2 c x^2} \int \left (-\frac {1}{x^3}-\frac {3 a}{x^2}-\frac {4 a^2}{x}+\frac {4 a^3}{-1+a x}\right ) \, dx}{a \sqrt {1-\frac {1}{a^2 x^2}} x} \\ & = \frac {\sqrt {c-a^2 c x^2}}{2 a \sqrt {1-\frac {1}{a^2 x^2}} x^3}+\frac {3 \sqrt {c-a^2 c x^2}}{\sqrt {1-\frac {1}{a^2 x^2}} x^2}-\frac {4 a \sqrt {c-a^2 c x^2} \log (x)}{\sqrt {1-\frac {1}{a^2 x^2}} x}+\frac {4 a \sqrt {c-a^2 c x^2} \log (1-a x)}{\sqrt {1-\frac {1}{a^2 x^2}} x} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.42 \[ \int \frac {e^{3 \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2}}{x^3} \, dx=\frac {\sqrt {c-a^2 c x^2} \left (\frac {1}{2 a x^2}+\frac {3}{x}-4 a \log (x)+4 a \log (1-a x)\right )}{\sqrt {1-\frac {1}{a^2 x^2}} x} \]
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Time = 0.54 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.50
method | result | size |
default | \(-\frac {\left (8 a^{2} \ln \left (x \right ) x^{2}-8 a^{2} \ln \left (a x -1\right ) x^{2}-6 a x -1\right ) \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (a x -1\right )}{2 \left (a x +1\right )^{2} x^{2} \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}\) | \(77\) |
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Time = 0.26 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.59 \[ \int \frac {e^{3 \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2}}{x^3} \, dx=\frac {8 \, a^{3} \sqrt {-c} x^{2} \log \left (\frac {2 \, a^{3} c x^{2} - 2 \, a^{2} c x + \sqrt {-a^{2} c} {\left (2 \, a x - 1\right )} \sqrt {-c} + a c}{a x^{2} - x}\right ) + \sqrt {-a^{2} c} {\left (6 \, a x + 1\right )}}{2 \, a x^{2}} \]
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Timed out. \[ \int \frac {e^{3 \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2}}{x^3} \, dx=\text {Timed out} \]
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\[ \int \frac {e^{3 \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2}}{x^3} \, dx=\int { \frac {\sqrt {-a^{2} c x^{2} + c}}{x^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {e^{3 \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2}}{x^3} \, dx=\int { \frac {\sqrt {-a^{2} c x^{2} + c}}{x^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {e^{3 \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2}}{x^3} \, dx=\int \frac {\sqrt {c-a^2\,c\,x^2}}{x^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}} \,d x \]
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