Integrand size = 24, antiderivative size = 69 \[ \int e^{-\coth ^{-1}(a x)} \sqrt {c-a^2 c x^2} \, dx=-\frac {\sqrt {c-a^2 c x^2}}{a \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {x \sqrt {c-a^2 c x^2}}{2 \sqrt {1-\frac {1}{a^2 x^2}}} \]
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Time = 0.09 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {6327, 6328} \[ \int e^{-\coth ^{-1}(a x)} \sqrt {c-a^2 c x^2} \, dx=\frac {x \sqrt {c-a^2 c x^2}}{2 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {\sqrt {c-a^2 c x^2}}{a \sqrt {1-\frac {1}{a^2 x^2}}} \]
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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}} x \, dx}{\sqrt {1-\frac {1}{a^2 x^2}} x} \\ & = \frac {\sqrt {c-a^2 c x^2} \int (-1+a x) \, 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}}}+\frac {x \sqrt {c-a^2 c x^2}}{2 \sqrt {1-\frac {1}{a^2 x^2}}} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.59 \[ \int e^{-\coth ^{-1}(a x)} \sqrt {c-a^2 c x^2} \, dx=\frac {(-2+a x) \sqrt {c-a^2 c x^2}}{2 a \sqrt {1-\frac {1}{a^2 x^2}}} \]
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Time = 0.52 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.64
method | result | size |
gosper | \(\frac {x \left (a x -2\right ) \sqrt {-a^{2} c \,x^{2}+c}\, \sqrt {\frac {a x -1}{a x +1}}}{2 a x -2}\) | \(44\) |
default | \(\frac {\left (a x -2\right ) x \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \sqrt {\frac {a x -1}{a x +1}}}{2 a x -2}\) | \(45\) |
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Time = 0.25 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.32 \[ \int e^{-\coth ^{-1}(a x)} \sqrt {c-a^2 c x^2} \, dx=\frac {\sqrt {-a^{2} c} {\left (a x^{2} - 2 \, x\right )}}{2 \, a} \]
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\[ \int e^{-\coth ^{-1}(a x)} \sqrt {c-a^2 c x^2} \, dx=\int \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )}\, dx \]
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\[ \int e^{-\coth ^{-1}(a x)} \sqrt {c-a^2 c x^2} \, dx=\int { \sqrt {-a^{2} c x^{2} + c} \sqrt {\frac {a x - 1}{a x + 1}} \,d x } \]
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\[ \int e^{-\coth ^{-1}(a x)} \sqrt {c-a^2 c x^2} \, dx=\int { \sqrt {-a^{2} c x^{2} + c} \sqrt {\frac {a x - 1}{a x + 1}} \,d x } \]
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Timed out. \[ \int e^{-\coth ^{-1}(a x)} \sqrt {c-a^2 c x^2} \, dx=\int \sqrt {c-a^2\,c\,x^2}\,\sqrt {\frac {a\,x-1}{a\,x+1}} \,d x \]
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