Optimal. Leaf size=18 \[ x \cos ^{-1}(x)-\sqrt{1-x^2} \]
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Rubi [A] time = 0.0041468, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 2, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1., Rules used = {4620, 261} \[ x \cos ^{-1}(x)-\sqrt{1-x^2} \]
Antiderivative was successfully verified.
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Rule 4620
Rule 261
Rubi steps
\begin{align*} \int \cos ^{-1}(x) \, dx &=x \cos ^{-1}(x)+\int \frac{x}{\sqrt{1-x^2}} \, dx\\ &=-\sqrt{1-x^2}+x \cos ^{-1}(x)\\ \end{align*}
Mathematica [A] time = 0.0024326, size = 18, normalized size = 1. \[ x \cos ^{-1}(x)-\sqrt{1-x^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 17, normalized size = 0.9 \begin{align*} x\arccos \left ( x \right ) -\sqrt{-{x}^{2}+1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.40595, size = 22, normalized size = 1.22 \begin{align*} x \arccos \left (x\right ) - \sqrt{-x^{2} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.39014, size = 41, normalized size = 2.28 \begin{align*} x \arccos \left (x\right ) - \sqrt{-x^{2} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.117869, size = 12, normalized size = 0.67 \begin{align*} x \operatorname{acos}{\left (x \right )} - \sqrt{1 - x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.04753, size = 22, normalized size = 1.22 \begin{align*} x \arccos \left (x\right ) - \sqrt{-x^{2} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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