Integrand size = 27, antiderivative size = 114 \[ \int \frac {e^{3 \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2}}{x^2} \, dx=\frac {\sqrt {c-a^2 c x^2}}{a \sqrt {1-\frac {1}{a^2 x^2}} x^2}-\frac {3 \sqrt {c-a^2 c x^2} \log (x)}{\sqrt {1-\frac {1}{a^2 x^2}} x}+\frac {4 \sqrt {c-a^2 c x^2} \log (1-a x)}{\sqrt {1-\frac {1}{a^2 x^2}} x} \]
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Time = 0.16 (sec) , antiderivative size = 114, 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} \[ \int \frac {e^{3 \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2}}{x^2} \, dx=\frac {\sqrt {c-a^2 c x^2}}{a x^2 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {3 \log (x) \sqrt {c-a^2 c x^2}}{x \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {4 \sqrt {c-a^2 c x^2} \log (1-a x)}{x \sqrt {1-\frac {1}{a^2 x^2}}} \]
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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} \, 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^2 (-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^2}-\frac {3 a}{x}+\frac {4 a^2}{-1+a x}\right ) \, dx}{a \sqrt {1-\frac {1}{a^2 x^2}} x} \\ & = \frac {\sqrt {c-a^2 c x^2}}{a \sqrt {1-\frac {1}{a^2 x^2}} x^2}-\frac {3 \sqrt {c-a^2 c x^2} \log (x)}{\sqrt {1-\frac {1}{a^2 x^2}} x}+\frac {4 \sqrt {c-a^2 c x^2} \log (1-a x)}{\sqrt {1-\frac {1}{a^2 x^2}} x} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.47 \[ \int \frac {e^{3 \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2}}{x^2} \, dx=\frac {\sqrt {c-a^2 c x^2} \left (\frac {1}{a x}-3 \log (x)+4 \log (1-a x)\right )}{\sqrt {1-\frac {1}{a^2 x^2}} x} \]
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Time = 0.52 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.57
method | result | size |
default | \(-\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (3 a \ln \left (x \right ) x -4 a \ln \left (a x -1\right ) x -1\right ) \left (a x -1\right )}{\left (a x +1\right )^{2} x \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}\) | \(65\) |
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Time = 0.26 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.29 \[ \int \frac {e^{3 \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2}}{x^2} \, dx=\frac {\sqrt {-a^{2} c} {\left (4 \, a x \log \left (a x - 1\right ) - 3 \, a x \log \left (x\right ) + 1\right )}}{a x} \]
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Timed out. \[ \int \frac {e^{3 \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2}}{x^2} \, dx=\text {Timed out} \]
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\[ \int \frac {e^{3 \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2}}{x^2} \, dx=\int { \frac {\sqrt {-a^{2} c x^{2} + c}}{x^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {e^{3 \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2}}{x^2} \, dx=\int { \frac {\sqrt {-a^{2} c x^{2} + c}}{x^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {e^{3 \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2}}{x^2} \, dx=\int \frac {\sqrt {c-a^2\,c\,x^2}}{x^2\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}} \,d x \]
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