Optimal. Leaf size=25 \[ \sin ^{-1}\left (\sqrt {x}\right )-\left (\sqrt {x}+2\right ) \sqrt {1-x} \]
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Rubi [A] time = 0.03, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {1398, 785, 780, 216} \begin {gather*} \sin ^{-1}\left (\sqrt {x}\right )-\left (\sqrt {x}+2\right ) \sqrt {1-x} \end {gather*}
Antiderivative was successfully verified.
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Rule 216
Rule 780
Rule 785
Rule 1398
Rubi steps
\begin {align*} \int \frac {\sqrt {1-x}}{1-\sqrt {x}} \, dx &=2 \operatorname {Subst}\left (\int \frac {x \sqrt {1-x^2}}{1-x} \, dx,x,\sqrt {x}\right )\\ &=-\left (2 \operatorname {Subst}\left (\int \frac {(-1-x) x}{\sqrt {1-x^2}} \, dx,x,\sqrt {x}\right )\right )\\ &=-\left (\left (2+\sqrt {x}\right ) \sqrt {1-x}\right )+\operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^2}} \, dx,x,\sqrt {x}\right )\\ &=-\left (\left (2+\sqrt {x}\right ) \sqrt {1-x}\right )+\sin ^{-1}\left (\sqrt {x}\right )\\ \end {align*}
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Mathematica [A] time = 0.03, size = 26, normalized size = 1.04 \begin {gather*} \sqrt {1-x} \left (-\sqrt {x}-2\right )+\sin ^{-1}\left (\sqrt {x}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.18, size = 42, normalized size = 1.68 \begin {gather*} \sqrt {1-x} \left (-\sqrt {x}-2\right )+2 \tan ^{-1}\left (\frac {\sqrt {x}}{\sqrt {1-x}-1}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 36, normalized size = 1.44 \begin {gather*} -\sqrt {x} \sqrt {-x + 1} - 2 \, \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 = 0.40, size = 32, normalized size = 1.28 \begin {gather*} -\sqrt {x} \sqrt {-x + 1} - 2 \, \sqrt {-x + 1} - \arcsin \left (\sqrt {-x + 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 48, normalized size = 1.92 \begin {gather*} \frac {\sqrt {-x +1}\, \left (\arcsin \left (2 x -1\right )-2 \sqrt {-\left (x -1\right ) x}\right ) \sqrt {x}}{2 \sqrt {-\left (x -1\right ) x}}-2 \sqrt {-x +1} \end {gather*}
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 \begin {gather*} -\int \frac {\sqrt {-x + 1}}{\sqrt {x} - 1}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.65, size = 40, normalized size = 1.60 \begin {gather*} 2\,\mathrm {atan}\left (\frac {\sqrt {x}}{\sqrt {1-x}-1}\right )-2\,\sqrt {1-x}-\sqrt {x}\,\sqrt {1-x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.77, size = 87, normalized size = 3.48 \begin {gather*} 2 \left (\begin {cases} - \sqrt {1 - x} + \frac {i \operatorname {acosh}{\left (\sqrt {1 - x} \right )}}{2} - \frac {i \left (1 - x\right )^{\frac {3}{2}}}{2 \sqrt {- x}} + \frac {i \sqrt {1 - x}}{2 \sqrt {- x}} & \text {for}\: \left |{x - 1}\right | > 1 \\\frac {\sqrt {x} \sqrt {1 - x}}{2} - \sqrt {1 - x} + \frac {\operatorname {asin}{\left (\sqrt {1 - x} \right )}}{2} & \text {otherwise} \end {cases}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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