Optimal. Leaf size=27 \[ -\sqrt {1-x} \sqrt {x}-\frac {1}{2} \sin ^{-1}(1-2 x) \]
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Rubi [A] time = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {50, 53, 619, 216} \begin {gather*} -\sqrt {1-x} \sqrt {x}-\frac {1}{2} \sin ^{-1}(1-2 x) \end {gather*}
Antiderivative was successfully verified.
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Rule 50
Rule 53
Rule 216
Rule 619
Rubi steps
\begin {align*} \int \frac {\sqrt {x}}{\sqrt {1-x}} \, dx &=-\sqrt {1-x} \sqrt {x}+\frac {1}{2} \int \frac {1}{\sqrt {1-x} \sqrt {x}} \, dx\\ &=-\sqrt {1-x} \sqrt {x}+\frac {1}{2} \int \frac {1}{\sqrt {x-x^2}} \, dx\\ &=-\sqrt {1-x} \sqrt {x}-\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^2}} \, dx,x,1-2 x\right )\\ &=-\sqrt {1-x} \sqrt {x}-\frac {1}{2} \sin ^{-1}(1-2 x)\\ \end {align*}
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Mathematica [A] time = 0.01, size = 25, normalized size = 0.93 \begin {gather*} -\sqrt {-((x-1) x)}-\sin ^{-1}\left (\sqrt {1-x}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.08, size = 39, normalized size = 1.44 \begin {gather*} 2 \tan ^{-1}\left (\frac {\sqrt {x}}{\sqrt {1-x}-1}\right )-\sqrt {1-x} \sqrt {x} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.52, size = 27, normalized size = 1.00 \begin {gather*} -\sqrt {x} \sqrt {-x + 1} - \arctan \left (\frac {\sqrt {-x + 1}}{\sqrt {x}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.17, size = 17, normalized size = 0.63 \begin {gather*} -\sqrt {x} \sqrt {-x + 1} + \arcsin \left (\sqrt {x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 41, normalized size = 1.52 \begin {gather*} -\sqrt {-x +1}\, \sqrt {x}+\frac {\sqrt {\left (-x +1\right ) x}\, \arcsin \left (2 x -1\right )}{2 \sqrt {-x +1}\, \sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.00, size = 37, normalized size = 1.37 \begin {gather*} \frac {\sqrt {-x + 1}}{\sqrt {x} {\left (\frac {x - 1}{x} - 1\right )}} - \arctan \left (\frac {\sqrt {-x + 1}}{\sqrt {x}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.57, size = 31, normalized size = 1.15 \begin {gather*} 2\,\mathrm {atan}\left (\frac {\sqrt {x}}{\sqrt {1-x}-1}\right )-\sqrt {x}\,\sqrt {1-x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.65, size = 54, normalized size = 2.00 \begin {gather*} \begin {cases} - i \sqrt {x} \sqrt {x - 1} - i \operatorname {acosh}{\left (\sqrt {x} \right )} & \text {for}\: \left |{x}\right | > 1 \\\frac {x^{\frac {3}{2}}}{\sqrt {1 - x}} - \frac {\sqrt {x}}{\sqrt {1 - x}} + \operatorname {asin}{\left (\sqrt {x} \right )} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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