Integrand size = 19, antiderivative size = 10 \[ \int \frac {\sqrt {1+x}}{\sqrt {x-x^2}} \, dx=2 E\left (\left .\arcsin \left (\sqrt {x}\right )\right |-1\right ) \]
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Time = 0.01 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {728, 111} \[ \int \frac {\sqrt {1+x}}{\sqrt {x-x^2}} \, dx=2 E\left (\left .\arcsin \left (\sqrt {x}\right )\right |-1\right ) \]
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Rule 111
Rule 728
Rubi steps \begin{align*} \text {integral}& = \int \frac {\sqrt {1+x}}{\sqrt {1-x} \sqrt {x}} \, dx \\ & = 2 E\left (\left .\sin ^{-1}\left (\sqrt {x}\right )\right |-1\right ) \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.00 (sec) , antiderivative size = 64, normalized size of antiderivative = 6.40 \[ \int \frac {\sqrt {1+x}}{\sqrt {x-x^2}} \, dx=\frac {2 x \sqrt {1-x^2} \left (3 \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},x^2\right )+x \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},x^2\right )\right )}{3 \sqrt {-((-1+x) x)} \sqrt {1+x}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(55\) vs. \(2(8)=16\).
Time = 1.46 (sec) , antiderivative size = 56, normalized size of antiderivative = 5.60
method | result | size |
default | \(-\frac {2 \left (F\left (\sqrt {1+x}, \frac {\sqrt {2}}{2}\right )-E\left (\sqrt {1+x}, \frac {\sqrt {2}}{2}\right )\right ) \sqrt {-x}\, \sqrt {2-2 x}\, \sqrt {-\left (-1+x \right ) x}}{\left (-1+x \right ) x}\) | \(56\) |
elliptic | \(\frac {\sqrt {-x \left (x^{2}-1\right )}\, \left (\frac {\sqrt {1+x}\, \sqrt {2-2 x}\, \sqrt {-x}\, F\left (\sqrt {1+x}, \frac {\sqrt {2}}{2}\right )}{\sqrt {-x^{3}+x}}+\frac {\sqrt {1+x}\, \sqrt {2-2 x}\, \sqrt {-x}\, \left (-2 E\left (\sqrt {1+x}, \frac {\sqrt {2}}{2}\right )+F\left (\sqrt {1+x}, \frac {\sqrt {2}}{2}\right )\right )}{\sqrt {-x^{3}+x}}\right )}{\sqrt {1+x}\, \sqrt {-\left (-1+x \right ) x}}\) | \(116\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.07 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.60 \[ \int \frac {\sqrt {1+x}}{\sqrt {x-x^2}} \, dx=-2 i \, {\rm weierstrassPInverse}\left (4, 0, x\right ) + 2 i \, {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, x\right )\right ) \]
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\[ \int \frac {\sqrt {1+x}}{\sqrt {x-x^2}} \, dx=\int \frac {\sqrt {x + 1}}{\sqrt {- x \left (x - 1\right )}}\, dx \]
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\[ \int \frac {\sqrt {1+x}}{\sqrt {x-x^2}} \, dx=\int { \frac {\sqrt {x + 1}}{\sqrt {-x^{2} + x}} \,d x } \]
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\[ \int \frac {\sqrt {1+x}}{\sqrt {x-x^2}} \, dx=\int { \frac {\sqrt {x + 1}}{\sqrt {-x^{2} + x}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {1+x}}{\sqrt {x-x^2}} \, dx=\int \frac {\sqrt {x+1}}{\sqrt {x-x^2}} \,d x \]
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