Optimal. Leaf size=14 \[ \sqrt {1-x^2}+\sin ^{-1}(x) \]
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Rubi [A]
time = 0.00, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {679, 222}
\begin {gather*} \text {ArcSin}(x)+\sqrt {1-x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 222
Rule 679
Rubi steps
\begin {align*} \int \frac {\sqrt {1-x^2}}{1+x} \, 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 [B] Leaf count is larger than twice the leaf count of optimal. \(32\) vs. \(2(14)=28\).
time = 0.07, size = 32, normalized size = 2.29 \begin {gather*} \sqrt {1-x^2}-2 \tan ^{-1}\left (\frac {\sqrt {1-x^2}}{1+x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.47, size = 18, normalized size = 1.29
method | result | size |
default | \(\sqrt {-\left (x +1\right )^{2}+2 x +2}+\arcsin \left (x \right )\) | \(18\) |
risch | \(-\frac {x^{2}-1}{\sqrt {-x^{2}+1}}+\arcsin \left (x \right )\) | \(20\) |
trager | \(\sqrt {-x^{2}+1}+\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {-x^{2}+1}+x \right )\) | \(37\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 12, normalized size = 0.86 \begin {gather*} \sqrt {-x^{2} + 1} + \arcsin \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 28 vs.
\(2 (12) = 24\).
time = 2.43, size = 28, normalized size = 2.00 \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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Sympy [A]
time = 1.05, size = 15, normalized size = 1.07 \begin {gather*} \begin {cases} \sqrt {1 - x^{2}} + \operatorname {asin}{\left (x \right )} & \text {for}\: x > -1 \wedge x < 1 \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.46, size = 12, normalized size = 0.86 \begin {gather*} \sqrt {-x^{2} + 1} + \arcsin \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.48, size = 12, normalized size = 0.86 \begin {gather*} \mathrm {asin}\left (x\right )+\sqrt {1-x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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