Optimal. Leaf size=20 \[ \sin ^{-1}\left (\frac {x}{2}\right )-\sqrt {4-x^2} \]
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Rubi [A] time = 0.00, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {641, 216} \begin {gather*} \sin ^{-1}\left (\frac {x}{2}\right )-\sqrt {4-x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 216
Rule 641
Rubi steps
\begin {align*} \int \frac {1+x}{\sqrt {4-x^2}} \, dx &=-\sqrt {4-x^2}+\int \frac {1}{\sqrt {4-x^2}} \, dx\\ &=-\sqrt {4-x^2}+\sin ^{-1}\left (\frac {x}{2}\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 20, normalized size = 1.00 \begin {gather*} \sin ^{-1}\left (\frac {x}{2}\right )-\sqrt {4-x^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.16, size = 34, normalized size = 1.70 \begin {gather*} -\sqrt {4-x^2}-2 \tan ^{-1}\left (\frac {\sqrt {4-x^2}}{x+2}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 30, normalized size = 1.50 \begin {gather*} -\sqrt {-x^{2} + 4} - 2 \, \arctan \left (\frac {\sqrt {-x^{2} + 4} - 2}{x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 16, normalized size = 0.80 \begin {gather*} -\sqrt {-x^{2} + 4} + \arcsin \left (\frac {1}{2} \, x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 17, normalized size = 0.85 \begin {gather*} \arcsin \left (\frac {x}{2}\right )-\sqrt {-x^{2}+4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.01, size = 16, normalized size = 0.80 \begin {gather*} -\sqrt {-x^{2} + 4} + \arcsin \left (\frac {1}{2} \, x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 16, normalized size = 0.80 \begin {gather*} \mathrm {asin}\left (\frac {x}{2}\right )-\sqrt {4-x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.15, size = 12, normalized size = 0.60 \begin {gather*} - \sqrt {4 - x^{2}} + \operatorname {asin}{\left (\frac {x}{2} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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