Optimal. Leaf size=27 \[ -\sqrt {1-x} \sqrt {x}-\frac {1}{2} \sin ^{-1}(1-2 x) \]
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Rubi [A]
time = 0.00, 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 = {52, 55, 633,
222} \begin {gather*} -\frac {1}{2} \text {ArcSin}(1-2 x)-\sqrt {1-x} \sqrt {x} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 55
Rule 222
Rule 633
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} \text {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 = 0.06, size = 35, normalized size = 1.30 \begin {gather*} -\sqrt {-((-1+x) x)}+2 \tan ^{-1}\left (\frac {\sqrt {x}}{-1+\sqrt {1-x}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 41, normalized size = 1.52
method | result | size |
meijerg | \(\frac {i \left (i \sqrt {\pi }\, \sqrt {x}\, \sqrt {1-x}-i \sqrt {\pi }\, \arcsin \left (\sqrt {x}\right )\right )}{\sqrt {\pi }}\) | \(34\) |
default | \(-\sqrt {1-x}\, \sqrt {x}+\frac {\sqrt {x \left (1-x \right )}\, \arcsin \left (2 x -1\right )}{2 \sqrt {x}\, \sqrt {1-x}}\) | \(41\) |
risch | \(\frac {\sqrt {x}\, \left (-1+x \right ) \sqrt {x \left (1-x \right )}}{\sqrt {-x \left (-1+x \right )}\, \sqrt {1-x}}+\frac {\sqrt {x \left (1-x \right )}\, \arcsin \left (2 x -1\right )}{2 \sqrt {x}\, \sqrt {1-x}}\) | \(60\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, 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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Fricas [A]
time = 1.01, 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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Sympy [C] Result contains complex when optimal does not.
time = 0.85, 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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Giac [A]
time = 2.27, 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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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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