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 = 0.0107611, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1., 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 4841
Rule 12
Rule 50
Rule 53
Rule 619
Rule 216
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*}
Mathematica [A] time = 0.0150251, 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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Maple [A] time = 0.002, size = 26, normalized size = 0.7 \begin{align*} x\arccos \left ( \sqrt{x} \right ) -{\frac{1}{2}\sqrt{1-x}\sqrt{x}}+{\frac{1}{2}\arcsin \left ( \sqrt{x} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.52363, size = 34, normalized size = 0.92 \begin{align*} x \arccos \left (\sqrt{x}\right ) - \frac{1}{2} \, \sqrt{x} \sqrt{-x + 1} + \frac{1}{2} \, \arcsin \left (\sqrt{x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.61165, size = 78, normalized size = 2.11 \begin{align*} \frac{1}{2} \,{\left (2 \, x - 1\right )} \arccos \left (\sqrt{x}\right ) - \frac{1}{2} \, \sqrt{x} \sqrt{-x + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.337239, size = 29, normalized size = 0.78 \begin{align*} - \frac{\sqrt{x} \sqrt{1 - x}}{2} + x \operatorname{acos}{\left (\sqrt{x} \right )} - \frac{\operatorname{acos}{\left (\sqrt{x} \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19294, size = 34, normalized size = 0.92 \begin{align*} x \arccos \left (\sqrt{x}\right ) - \frac{1}{2} \, \sqrt{x} \sqrt{-x + 1} - \frac{1}{2} \, \arccos \left (\sqrt{x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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