Integrand size = 27, antiderivative size = 108 \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x} \, dx=-\frac {\sqrt {c-\frac {c}{a^2 x^2}}}{a \sqrt {1-\frac {1}{a^2 x^2}} x}-\frac {3 \sqrt {c-\frac {c}{a^2 x^2}} \log (x)}{\sqrt {1-\frac {1}{a^2 x^2}}}+\frac {4 \sqrt {c-\frac {c}{a^2 x^2}} \log (1+a x)}{\sqrt {1-\frac {1}{a^2 x^2}}} \]
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Time = 0.20 (sec) , antiderivative size = 108, 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 = {6332, 6328, 90} \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x} \, dx=-\frac {\sqrt {c-\frac {c}{a^2 x^2}}}{a x \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {3 \log (x) \sqrt {c-\frac {c}{a^2 x^2}}}{\sqrt {1-\frac {1}{a^2 x^2}}}+\frac {4 \sqrt {c-\frac {c}{a^2 x^2}} \log (a x+1)}{\sqrt {1-\frac {1}{a^2 x^2}}} \]
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Rule 90
Rule 6328
Rule 6332
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {c-\frac {c}{a^2 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}}} \\ & = \frac {\sqrt {c-\frac {c}{a^2 x^2}} \int \frac {(-1+a x)^2}{x^2 (1+a x)} \, dx}{a \sqrt {1-\frac {1}{a^2 x^2}}} \\ & = \frac {\sqrt {c-\frac {c}{a^2 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}}} \\ & = -\frac {\sqrt {c-\frac {c}{a^2 x^2}}}{a \sqrt {1-\frac {1}{a^2 x^2}} x}-\frac {3 \sqrt {c-\frac {c}{a^2 x^2}} \log (x)}{\sqrt {1-\frac {1}{a^2 x^2}}}+\frac {4 \sqrt {c-\frac {c}{a^2 x^2}} \log (1+a x)}{\sqrt {1-\frac {1}{a^2 x^2}}} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.47 \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x} \, dx=\frac {\sqrt {c-\frac {c}{a^2 x^2}} \left (-\frac {1}{a x}-3 \log (x)+4 \log (1+a x)\right )}{\sqrt {1-\frac {1}{a^2 x^2}}} \]
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Time = 0.06 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.61
method | result | size |
default | \(\frac {\left (4 a \ln \left (a x +1\right ) x -3 a \ln \left (x \right ) x -1\right ) \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}\, \left (a x +1\right ) \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{\left (a x -1\right )^{2}}\) | \(66\) |
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Time = 0.26 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.30 \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x} \, dx=\frac {\sqrt {a^{2} c} {\left (4 \, a x \log \left (a x + 1\right ) - 3 \, a x \log \left (x\right ) - 1\right )}}{a^{2} x} \]
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Timed out. \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x} \, dx=\text {Timed out} \]
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\[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x} \, dx=\int { \frac {\sqrt {c - \frac {c}{a^{2} x^{2}}} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}{x} \,d x } \]
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\[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x} \, dx=\int { \frac {\sqrt {c - \frac {c}{a^{2} x^{2}}} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}{x} \,d x } \]
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Timed out. \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x} \, dx=\int \frac {\sqrt {c-\frac {c}{a^2\,x^2}}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{x} \,d x \]
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