Optimal. Leaf size=25 \[ 2 \sqrt{x} \cos ^{-1}\left (\sqrt{x}\right )-2 \sqrt{1-x} \]
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Rubi [A] time = 0.0187569, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {6715, 4620, 261} \[ 2 \sqrt{x} \cos ^{-1}\left (\sqrt{x}\right )-2 \sqrt{1-x} \]
Antiderivative was successfully verified.
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Rule 6715
Rule 4620
Rule 261
Rubi steps
\begin{align*} \int \frac{\cos ^{-1}\left (\sqrt{x}\right )}{\sqrt{x}} \, dx &=2 \operatorname{Subst}\left (\int \cos ^{-1}(x) \, dx,x,\sqrt{x}\right )\\ &=2 \sqrt{x} \cos ^{-1}\left (\sqrt{x}\right )+2 \operatorname{Subst}\left (\int \frac{x}{\sqrt{1-x^2}} \, dx,x,\sqrt{x}\right )\\ &=-2 \sqrt{1-x}+2 \sqrt{x} \cos ^{-1}\left (\sqrt{x}\right )\\ \end{align*}
Mathematica [A] time = 0.0078178, size = 25, normalized size = 1. \[ 2 \sqrt{x} \cos ^{-1}\left (\sqrt{x}\right )-2 \sqrt{1-x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 20, normalized size = 0.8 \begin{align*} -2\,\sqrt{1-x}+2\,\arccos \left ( \sqrt{x} \right ) \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.48198, size = 26, normalized size = 1.04 \begin{align*} 2 \, \sqrt{x} \arccos \left (\sqrt{x}\right ) - 2 \, \sqrt{-x + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.53294, size = 59, normalized size = 2.36 \begin{align*} 2 \, \sqrt{x} \arccos \left (\sqrt{x}\right ) - 2 \, \sqrt{-x + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.365094, size = 20, normalized size = 0.8 \begin{align*} 2 \sqrt{x} \operatorname{acos}{\left (\sqrt{x} \right )} - 2 \sqrt{1 - x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.2737, size = 26, normalized size = 1.04 \begin{align*} 2 \, \sqrt{x} \arccos \left (\sqrt{x}\right ) - 2 \, \sqrt{-x + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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