Optimal. Leaf size=16 \[ \sin ^{-1}(x)-\sqrt {1-x^2} \]
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Rubi [A] time = 0.00, antiderivative size = 16, 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}(x)-\sqrt {1-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 {1-x^2}} \, dx &=-\sqrt {1-x^2}+\int \frac {1}{\sqrt {1-x^2}} \, dx\\ &=-\sqrt {1-x^2}+\sin ^{-1}(x)\\ \end {align*}
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Mathematica [A] time = 0.01, size = 16, normalized size = 1.00 \begin {gather*} \sin ^{-1}(x)-\sqrt {1-x^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [B] time = 0.13, size = 34, normalized size = 2.12 \begin {gather*} -\sqrt {1-x^2}-2 \tan ^{-1}\left (\frac {\sqrt {1-x^2}}{x+1}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.40, size = 30, normalized size = 1.88 \begin {gather*} -\sqrt {-x^{2} + 1} - 2 \, \arctan \left (\frac {\sqrt {-x^{2} + 1} - 1}{x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 14, normalized size = 0.88 \begin {gather*} -\sqrt {-x^{2} + 1} + \arcsin \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 15, normalized size = 0.94 \begin {gather*} \arcsin \relax (x )-\sqrt {-x^{2}+1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.96, size = 14, normalized size = 0.88 \begin {gather*} -\sqrt {-x^{2} + 1} + \arcsin \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 14, normalized size = 0.88 \begin {gather*} \mathrm {asin}\relax (x)-\sqrt {1-x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.15, size = 10, normalized size = 0.62 \begin {gather*} - \sqrt {1 - x^{2}} + \operatorname {asin}{\relax (x )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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