Integrand size = 25, antiderivative size = 69 \[ \int \frac {e^{\coth ^{-1}(a x)} \sqrt {c-a^2 c x^2}}{x} \, dx=\frac {\sqrt {c-a^2 c x^2}}{\sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {c-a^2 c x^2} \log (x)}{a \sqrt {1-\frac {1}{a^2 x^2}} x} \]
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Time = 0.10 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {6327, 6328, 45} \[ \int \frac {e^{\coth ^{-1}(a x)} \sqrt {c-a^2 c x^2}}{x} \, dx=\frac {\sqrt {c-a^2 c x^2}}{\sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\log (x) \sqrt {c-a^2 c x^2}}{a x \sqrt {1-\frac {1}{a^2 x^2}}} \]
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Rule 45
Rule 6327
Rule 6328
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {c-a^2 c x^2} \int e^{\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}{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}\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} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.59 \[ \int \frac {e^{\coth ^{-1}(a x)} \sqrt {c-a^2 c x^2}}{x} \, dx=\frac {\sqrt {c-a^2 c x^2} \left (x+\frac {\log (x)}{a}\right )}{\sqrt {1-\frac {1}{a^2 x^2}} x} \]
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Time = 0.53 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.64
method | result | size |
default | \(\frac {\left (a x +\ln \left (x \right )\right ) \sqrt {-c \left (a^{2} x^{2}-1\right )}}{\left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}}\) | \(44\) |
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Time = 0.25 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.26 \[ \int \frac {e^{\coth ^{-1}(a x)} \sqrt {c-a^2 c x^2}}{x} \, dx=\frac {\sqrt {-a^{2} c} {\left (a x + \log \left (x\right )\right )}}{a} \]
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\[ \int \frac {e^{\coth ^{-1}(a x)} \sqrt {c-a^2 c x^2}}{x} \, dx=\int \frac {\sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )}}{x \sqrt {\frac {a x - 1}{a x + 1}}}\, dx \]
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\[ \int \frac {e^{\coth ^{-1}(a x)} \sqrt {c-a^2 c x^2}}{x} \, dx=\int { \frac {\sqrt {-a^{2} c x^{2} + c}}{x \sqrt {\frac {a x - 1}{a x + 1}}} \,d x } \]
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\[ \int \frac {e^{\coth ^{-1}(a x)} \sqrt {c-a^2 c x^2}}{x} \, dx=\int { \frac {\sqrt {-a^{2} c x^{2} + c}}{x \sqrt {\frac {a x - 1}{a x + 1}}} \,d x } \]
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Timed out. \[ \int \frac {e^{\coth ^{-1}(a x)} \sqrt {c-a^2 c x^2}}{x} \, dx=\int \frac {\sqrt {c-a^2\,c\,x^2}}{x\,\sqrt {\frac {a\,x-1}{a\,x+1}}} \,d x \]
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