Integrand size = 21, antiderivative size = 29 \[ \int \frac {\sqrt {1+\frac {1}{x}}}{\sqrt {1-x^2}} \, dx=-\frac {\sqrt {1+\frac {1}{x}} \sqrt {x} \arcsin (1-2 x)}{\sqrt {1+x}} \]
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Time = 0.01 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {1462, 26, 55, 633, 222} \[ \int \frac {\sqrt {1+\frac {1}{x}}}{\sqrt {1-x^2}} \, dx=-\frac {\sqrt {\frac {1}{x}+1} \sqrt {x} \arcsin (1-2 x)}{\sqrt {x+1}} \]
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Rule 26
Rule 55
Rule 222
Rule 633
Rule 1462
Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {1+\frac {1}{x}} \sqrt {x}\right ) \int \frac {\sqrt {1+x}}{\sqrt {x} \sqrt {1-x^2}} \, dx}{\sqrt {1+x}} \\ & = \frac {\left (\sqrt {1+\frac {1}{x}} \sqrt {x}\right ) \int \frac {1}{\sqrt {1-x} \sqrt {x}} \, dx}{\sqrt {1+x}} \\ & = \frac {\left (\sqrt {1+\frac {1}{x}} \sqrt {x}\right ) \int \frac {1}{\sqrt {x-x^2}} \, dx}{\sqrt {1+x}} \\ & = -\frac {\left (\sqrt {1+\frac {1}{x}} \sqrt {x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2}} \, dx,x,1-2 x\right )}{\sqrt {1+x}} \\ & = -\frac {\sqrt {1+\frac {1}{x}} \sqrt {x} \sin ^{-1}(1-2 x)}{\sqrt {1+x}} \\ \end{align*}
Time = 1.76 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.41 \[ \int \frac {\sqrt {1+\frac {1}{x}}}{\sqrt {1-x^2}} \, dx=-\arctan \left (\frac {\sqrt {\frac {1+x}{x}} (-1+2 x) \sqrt {1-x^2}}{2 \left (-1+x^2\right )}\right ) \]
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Time = 1.10 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.38
method | result | size |
default | \(\frac {\sqrt {\frac {x +1}{x}}\, x \sqrt {-x^{2}+1}\, \arcsin \left (2 x -1\right )}{\left (x +1\right ) \sqrt {-\left (x -1\right ) x}}\) | \(40\) |
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none
Time = 0.29 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.17 \[ \int \frac {\sqrt {1+\frac {1}{x}}}{\sqrt {1-x^2}} \, dx=-\arctan \left (\frac {2 \, \sqrt {-x^{2} + 1} x \sqrt {\frac {x + 1}{x}}}{2 \, x^{2} + x - 1}\right ) \]
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\[ \int \frac {\sqrt {1+\frac {1}{x}}}{\sqrt {1-x^2}} \, dx=\int \frac {\sqrt {1 + \frac {1}{x}}}{\sqrt {- \left (x - 1\right ) \left (x + 1\right )}}\, dx \]
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\[ \int \frac {\sqrt {1+\frac {1}{x}}}{\sqrt {1-x^2}} \, dx=\int { \frac {\sqrt {\frac {1}{x} + 1}}{\sqrt {-x^{2} + 1}} \,d x } \]
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\[ \int \frac {\sqrt {1+\frac {1}{x}}}{\sqrt {1-x^2}} \, dx=\int { \frac {\sqrt {\frac {1}{x} + 1}}{\sqrt {-x^{2} + 1}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {1+\frac {1}{x}}}{\sqrt {1-x^2}} \, dx=\int \frac {\sqrt {\frac {1}{x}+1}}{\sqrt {1-x^2}} \,d x \]
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