Integrand size = 22, antiderivative size = 67 \[ \int e^{\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} \, dx=\frac {\sqrt {c-\frac {c}{a^2 x^2}} x}{\sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {c-\frac {c}{a^2 x^2}} \log (x)}{a \sqrt {1-\frac {1}{a^2 x^2}}} \]
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Time = 0.08 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {6332, 6328, 45} \[ \int e^{\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} \, dx=\frac {x \sqrt {c-\frac {c}{a^2 x^2}}}{\sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\log (x) \sqrt {c-\frac {c}{a^2 x^2}}}{a \sqrt {1-\frac {1}{a^2 x^2}}} \]
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Rule 45
Rule 6328
Rule 6332
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {c-\frac {c}{a^2 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}}} \\ & = \frac {\sqrt {c-\frac {c}{a^2 x^2}} \int \frac {1+a x}{x} \, dx}{a \sqrt {1-\frac {1}{a^2 x^2}}} \\ & = \frac {\sqrt {c-\frac {c}{a^2 x^2}} \int \left (a+\frac {1}{x}\right ) \, dx}{a \sqrt {1-\frac {1}{a^2 x^2}}} \\ & = \frac {\sqrt {c-\frac {c}{a^2 x^2}} x}{\sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {c-\frac {c}{a^2 x^2}} \log (x)}{a \sqrt {1-\frac {1}{a^2 x^2}}} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.57 \[ \int e^{\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} \, dx=\frac {\sqrt {c-\frac {c}{a^2 x^2}} \left (x+\frac {\log (x)}{a}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}} \]
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Time = 0.04 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.75
method | result | size |
default | \(\frac {\left (a x +\ln \left (x \right )\right ) x \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}}{\left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}}\) | \(50\) |
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Time = 0.25 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.25 \[ \int e^{\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} \, dx=\frac {\sqrt {a^{2} c} {\left (a x + \log \left (x\right )\right )}}{a^{2}} \]
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\[ \int e^{\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} \, dx=\int \frac {\sqrt {- c \left (-1 + \frac {1}{a x}\right ) \left (1 + \frac {1}{a x}\right )}}{\sqrt {\frac {a x - 1}{a x + 1}}}\, dx \]
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\[ \int e^{\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} \, dx=\int { \frac {\sqrt {c - \frac {c}{a^{2} x^{2}}}}{\sqrt {\frac {a x - 1}{a x + 1}}} \,d x } \]
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\[ \int e^{\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} \, dx=\int { \frac {\sqrt {c - \frac {c}{a^{2} x^{2}}}}{\sqrt {\frac {a x - 1}{a x + 1}}} \,d x } \]
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Timed out. \[ \int e^{\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} \, dx=\int \frac {\sqrt {c-\frac {c}{a^2\,x^2}}}{\sqrt {\frac {a\,x-1}{a\,x+1}}} \,d x \]
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