Integrand size = 27, antiderivative size = 112 \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2}}{x} \, dx=\frac {\sqrt {c-a^2 c x^2}}{\sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {c-a^2 c x^2} \log (x)}{a \sqrt {1-\frac {1}{a^2 x^2}} x}-\frac {4 \sqrt {c-a^2 c x^2} \log (1+a x)}{a \sqrt {1-\frac {1}{a^2 x^2}} x} \]
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Time = 0.12 (sec) , antiderivative size = 112, 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, 84} \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2}}{x} \, dx=\frac {\sqrt {c-a^2 c x^2}}{\sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\log (x) \sqrt {c-a^2 c x^2}}{a x \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {4 \sqrt {c-a^2 c x^2} \log (a x+1)}{a x \sqrt {1-\frac {1}{a^2 x^2}}} \]
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Rule 84
Rule 6327
Rule 6328
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {c-a^2 c x^2} \int e^{-3 \coth ^{-1}(a x)} \sqrt {1-\frac {1}{a^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 (1+a x)} \, dx}{a \sqrt {1-\frac {1}{a^2 x^2}} x} \\ & = \frac {\sqrt {c-a^2 c x^2} \int \left (a+\frac {1}{x}-\frac {4 a}{1+a x}\right ) \, dx}{a \sqrt {1-\frac {1}{a^2 x^2}} x} \\ & = \frac {\sqrt {c-a^2 c x^2}}{\sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {c-a^2 c x^2} \log (x)}{a \sqrt {1-\frac {1}{a^2 x^2}} x}-\frac {4 \sqrt {c-a^2 c x^2} \log (1+a x)}{a \sqrt {1-\frac {1}{a^2 x^2}} x} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.45 \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2}}{x} \, dx=\frac {\sqrt {c-a^2 c x^2} (a x+\log (x)-4 \log (1+a x))}{a \sqrt {1-\frac {1}{a^2 x^2}} x} \]
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Time = 0.55 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.54
method | result | size |
default | \(-\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (-a x +4 \ln \left (a x +1\right )-\ln \left (x \right )\right ) \left (a x +1\right ) \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{\left (a x -1\right )^{2}}\) | \(61\) |
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none
Time = 0.25 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.23 \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2}}{x} \, dx=\frac {\sqrt {-a^{2} c} {\left (a x - 4 \, \log \left (a x + 1\right ) + \log \left (x\right )\right )}}{a} \]
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\[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2}}{x} \, dx=\int \frac {\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} \sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )}}{x}\, dx \]
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\[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2}}{x} \, dx=\int { \frac {\sqrt {-a^{2} c x^{2} + c} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}{x} \,d x } \]
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\[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2}}{x} \, dx=\int { \frac {\sqrt {-a^{2} c x^{2} + c} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}{x} \,d x } \]
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Timed out. \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2}}{x} \, dx=\int \frac {\sqrt {c-a^2\,c\,x^2}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{x} \,d x \]
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