Optimal. Leaf size=18 \[ x \cos ^{-1}(x)-\sqrt {1-x^2} \]
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Rubi [A] time = 0.00, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {4620, 261} \[ x \cos ^{-1}(x)-\sqrt {1-x^2} \]
Antiderivative was successfully verified.
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Rule 261
Rule 4620
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*}
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Mathematica [A] time = 0.00, size = 18, normalized size = 1.00 \[ x \cos ^{-1}(x)-\sqrt {1-x^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 16, normalized size = 0.89 \[ x \arccos \relax (x) - \sqrt {-x^{2} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.82, size = 16, normalized size = 0.89 \[ x \arccos \relax (x) - \sqrt {-x^{2} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 17, normalized size = 0.94 \[ x \arccos \relax (x )-\sqrt {-x^{2}+1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.98, size = 16, normalized size = 0.89 \[ x \arccos \relax (x) - \sqrt {-x^{2} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.17, size = 16, normalized size = 0.89 \[ x\,\mathrm {acos}\relax (x)-\sqrt {1-x^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.13, size = 12, normalized size = 0.67 \[ x \operatorname {acos}{\relax (x )} - \sqrt {1 - x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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