Optimal. Leaf size=12 \[ -2 E\left (\left .\sin ^{-1}\left (\sqrt {-x}\right )\right |-1\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {714, 110} \[ -2 E\left (\left .\sin ^{-1}\left (\sqrt {-x}\right )\right |-1\right ) \]
Antiderivative was successfully verified.
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Rule 110
Rule 714
Rubi steps
\begin {align*} \int \frac {\sqrt {1-x}}{\sqrt {-x-x^2}} \, dx &=\int \frac {\sqrt {1-x}}{\sqrt {-x} \sqrt {1+x}} \, dx\\ &=-2 E\left (\left .\sin ^{-1}\left (\sqrt {-x}\right )\right |-1\right )\\ \end {align*}
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Mathematica [C] time = 0.01, size = 66, normalized size = 5.50 \[ -\frac {2 x \sqrt {1-x^2} \left (x \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};x^2\right )-3 \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};x^2\right )\right )}{3 \sqrt {1-x} \sqrt {-x (x+1)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.96, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-x^{2} - x} \sqrt {-x + 1}}{x^{2} + x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {-x + 1}}{\sqrt {-x^{2} - x}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 50, normalized size = 4.17 \[ \frac {2 \left (-x +1\right ) \sqrt {-\left (x +1\right ) x}\, \sqrt {x +1}\, \sqrt {2}\, \sqrt {-x}\, \EllipticE \left (\sqrt {x +1}, \frac {\sqrt {2}}{2}\right )}{\left (x^{2}-1\right ) x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {-x + 1}}{\sqrt {-x^{2} - x}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.08 \[ \int \frac {\sqrt {1-x}}{\sqrt {-x^2-x}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {1 - x}}{\sqrt {- x \left (x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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