Optimal. Leaf size=19 \[ \sqrt {1-x^2} x+2 x+\sin ^{-1}(x) \]
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Rubi [A] time = 0.02, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {6742, 195, 216} \begin {gather*} \sqrt {1-x^2} x+2 x+\sin ^{-1}(x) \end {gather*}
Antiderivative was successfully verified.
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Rule 195
Rule 216
Rule 6742
Rubi steps
\begin {align*} \int \left (\sqrt {1-x}+\sqrt {1+x}\right )^2 \, dx &=\int \left (2+2 \sqrt {1-x^2}\right ) \, dx\\ &=2 x+2 \int \sqrt {1-x^2} \, dx\\ &=2 x+x \sqrt {1-x^2}+\int \frac {1}{\sqrt {1-x^2}} \, dx\\ &=2 x+x \sqrt {1-x^2}+\sin ^{-1}(x)\\ \end {align*}
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Mathematica [A] time = 0.01, size = 18, normalized size = 0.95 \begin {gather*} x \left (\sqrt {1-x^2}+2\right )+\sin ^{-1}(x) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [C] time = 0.14, size = 59, normalized size = 3.11 \begin {gather*} 2 (x+1)+\sqrt {1-x} \left ((x+1)^{3/2}-\sqrt {x+1}\right )+2 i \log \left (\sqrt {1-x}-i \sqrt {x+1}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.59, size = 40, normalized size = 2.11 \begin {gather*} \sqrt {x + 1} x \sqrt {-x + 1} + 2 \, x - 2 \, \arctan \left (\frac {\sqrt {x + 1} \sqrt {-x + 1} - 1}{x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.20, size = 48, normalized size = 2.53 \begin {gather*} \sqrt {x + 1} {\left (x - 2\right )} \sqrt {-x + 1} + 2 \, x + 2 \, \sqrt {x + 1} \sqrt {-x + 1} + 2 \, \arcsin \left (\frac {1}{2} \, \sqrt {2} \sqrt {x + 1}\right ) + 2 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 58, normalized size = 3.05 \begin {gather*} 2 x +\frac {\sqrt {\left (x +1\right ) \left (-x +1\right )}\, \arcsin \relax (x )}{\sqrt {-x +1}\, \sqrt {x +1}}-\sqrt {x +1}\, \left (-x +1\right )^{\frac {3}{2}}+\sqrt {-x +1}\, \sqrt {x +1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.42, size = 17, normalized size = 0.89 \begin {gather*} \sqrt {-x^{2} + 1} x + 2 \, x + \arcsin \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 7.72, size = 206, normalized size = 10.84 \begin {gather*} 2\,x-4\,\mathrm {atan}\left (\frac {\sqrt {1-x}-1}{\sqrt {x+1}-1}\right )-\frac {\frac {4\,\left (\sqrt {1-x}-1\right )}{\sqrt {x+1}-1}-\frac {28\,{\left (\sqrt {1-x}-1\right )}^3}{{\left (\sqrt {x+1}-1\right )}^3}+\frac {28\,{\left (\sqrt {1-x}-1\right )}^5}{{\left (\sqrt {x+1}-1\right )}^5}-\frac {4\,{\left (\sqrt {1-x}-1\right )}^7}{{\left (\sqrt {x+1}-1\right )}^7}}{\frac {4\,{\left (\sqrt {1-x}-1\right )}^2}{{\left (\sqrt {x+1}-1\right )}^2}+\frac {6\,{\left (\sqrt {1-x}-1\right )}^4}{{\left (\sqrt {x+1}-1\right )}^4}+\frac {4\,{\left (\sqrt {1-x}-1\right )}^6}{{\left (\sqrt {x+1}-1\right )}^6}+\frac {{\left (\sqrt {1-x}-1\right )}^8}{{\left (\sqrt {x+1}-1\right )}^8}+1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 31.43, size = 44, normalized size = 2.32 \begin {gather*} 2 x + 4 \left (\begin {cases} \frac {x \sqrt {1 - x} \sqrt {x + 1}}{4} + \frac {\operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )}}{2} & \text {for}\: x \geq -1 \wedge x < 1 \end {cases}\right ) + 2 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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