Optimal. Leaf size=37 \[ -\frac {1}{2} \sqrt {1-x} \sqrt {x}-\frac {1}{4} \sin ^{-1}(1-2 x)+x \cos ^{-1}\left (\sqrt {x}\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {4841, 12, 50, 53, 619, 216} \[ -\frac {1}{2} \sqrt {1-x} \sqrt {x}-\frac {1}{4} \sin ^{-1}(1-2 x)+x \cos ^{-1}\left (\sqrt {x}\right ) \]
Antiderivative was successfully verified.
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Rule 12
Rule 50
Rule 53
Rule 216
Rule 619
Rule 4841
Rubi steps
\begin {align*} \int \cos ^{-1}\left (\sqrt {x}\right ) \, dx &=x \cos ^{-1}\left (\sqrt {x}\right )+\int \frac {\sqrt {x}}{2 \sqrt {1-x}} \, dx\\ &=x \cos ^{-1}\left (\sqrt {x}\right )+\frac {1}{2} \int \frac {\sqrt {x}}{\sqrt {1-x}} \, dx\\ &=-\frac {1}{2} \sqrt {1-x} \sqrt {x}+x \cos ^{-1}\left (\sqrt {x}\right )+\frac {1}{4} \int \frac {1}{\sqrt {1-x} \sqrt {x}} \, dx\\ &=-\frac {1}{2} \sqrt {1-x} \sqrt {x}+x \cos ^{-1}\left (\sqrt {x}\right )+\frac {1}{4} \int \frac {1}{\sqrt {x-x^2}} \, dx\\ &=-\frac {1}{2} \sqrt {1-x} \sqrt {x}+x \cos ^{-1}\left (\sqrt {x}\right )-\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^2}} \, dx,x,1-2 x\right )\\ &=-\frac {1}{2} \sqrt {1-x} \sqrt {x}+x \cos ^{-1}\left (\sqrt {x}\right )-\frac {1}{4} \sin ^{-1}(1-2 x)\\ \end {align*}
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Mathematica [A] time = 0.02, size = 38, normalized size = 1.03 \[ \frac {1}{2} \left (-\sqrt {-((x-1) x)}-\sin ^{-1}\left (\sqrt {1-x}\right )\right )+x \cos ^{-1}\left (\sqrt {x}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 24, normalized size = 0.65 \[ \frac {1}{2} \, {\left (2 \, x - 1\right )} \arccos \left (\sqrt {x}\right ) - \frac {1}{2} \, \sqrt {x} \sqrt {-x + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.73, size = 25, normalized size = 0.68 \[ x \arccos \left (\sqrt {x}\right ) - \frac {1}{2} \, \sqrt {x} \sqrt {-x + 1} - \frac {1}{2} \, \arccos \left (\sqrt {x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 26, normalized size = 0.70 \[ x \arccos \left (\sqrt {x}\right )-\frac {\sqrt {1-x}\, \sqrt {x}}{2}+\frac {\arcsin \left (\sqrt {x}\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.40, size = 25, normalized size = 0.68 \[ x \arccos \left (\sqrt {x}\right ) - \frac {1}{2} \, \sqrt {x} \sqrt {-x + 1} + \frac {1}{2} \, \arcsin \left (\sqrt {x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.64, size = 35, normalized size = 0.95 \[ \mathrm {atan}\left (\frac {\sqrt {x}}{\sqrt {1-x}-1}\right )+x\,\mathrm {acos}\left (\sqrt {x}\right )-\frac {\sqrt {x}\,\sqrt {1-x}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.26, size = 29, normalized size = 0.78 \[ - \frac {\sqrt {x} \sqrt {1 - x}}{2} + x \operatorname {acos}{\left (\sqrt {x} \right )} - \frac {\operatorname {acos}{\left (\sqrt {x} \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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