Optimal. Leaf size=31 \[ -\frac {x^2}{2}+\frac {1}{2} \sqrt {1-x^2} x+x+\frac {1}{2} \sin ^{-1}(x) \]
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Rubi [A] time = 0.05, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {6688, 195, 216} \[ -\frac {x^2}{2}+\frac {1}{2} \sqrt {1-x^2} x+x+\frac {1}{2} \sin ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 195
Rule 216
Rule 6688
Rubi steps
\begin {align*} \int \sqrt {1-x} \left (\sqrt {1-x}+\sqrt {1+x}\right ) \, dx &=\int \left (1-x+\sqrt {1-x^2}\right ) \, dx\\ &=x-\frac {x^2}{2}+\int \sqrt {1-x^2} \, dx\\ &=x-\frac {x^2}{2}+\frac {1}{2} x \sqrt {1-x^2}+\frac {1}{2} \int \frac {1}{\sqrt {1-x^2}} \, dx\\ &=x-\frac {x^2}{2}+\frac {1}{2} x \sqrt {1-x^2}+\frac {1}{2} \sin ^{-1}(x)\\ \end {align*}
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Mathematica [A] time = 0.01, size = 31, normalized size = 1.00 \[ -\frac {x^2}{2}+\frac {1}{2} \sqrt {1-x^2} x+x+\frac {1}{2} \sin ^{-1}(x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 44, normalized size = 1.42 \[ -\frac {1}{2} \, x^{2} + \frac {1}{2} \, \sqrt {x + 1} x \sqrt {-x + 1} + x - \arctan \left (\frac {\sqrt {x + 1} \sqrt {-x + 1} - 1}{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.61, size = 54, normalized size = 1.74 \[ -\frac {1}{2} \, {\left (x - 1\right )}^{2} + \frac {1}{2} \, {\left (x + 2\right )} \sqrt {x + 1} \sqrt {-x + 1} - \sqrt {x + 1} \sqrt {-x + 1} - \arcsin \left (\frac {1}{2} \, \sqrt {2} \sqrt {-x + 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.00, size = 63, normalized size = 2.03 \[ -\frac {x^{2}}{2}+x +\frac {\sqrt {\left (x +1\right ) \left (-x +1\right )}\, \arcsin \relax (x )}{2 \sqrt {-x +1}\, \sqrt {x +1}}-\frac {\sqrt {x +1}\, \left (-x +1\right )^{\frac {3}{2}}}{2}+\frac {\sqrt {-x +1}\, \sqrt {x +1}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.49, size = 23, normalized size = 0.74 \[ -\frac {1}{2} \, x^{2} + \frac {1}{2} \, \sqrt {-x^{2} + 1} x + x + \frac {1}{2} \, \arcsin \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.12, size = 209, normalized size = 6.74 \[ x-2\,\mathrm {atan}\left (\frac {\sqrt {1-x}-1}{\sqrt {x+1}-1}\right )-\frac {\frac {2\,\left (\sqrt {1-x}-1\right )}{\sqrt {x+1}-1}-\frac {14\,{\left (\sqrt {1-x}-1\right )}^3}{{\left (\sqrt {x+1}-1\right )}^3}+\frac {14\,{\left (\sqrt {1-x}-1\right )}^5}{{\left (\sqrt {x+1}-1\right )}^5}-\frac {2\,{\left (\sqrt {1-x}-1\right )}^7}{{\left (\sqrt {x+1}-1\right )}^7}}{\frac {4\,{\left (\sqrt {1-x}-1\right )}^2}{{\left (\sqrt {x+1}-1\right )}^2}+\frac {6\,{\left (\sqrt {1-x}-1\right )}^4}{{\left (\sqrt {x+1}-1\right )}^4}+\frac {4\,{\left (\sqrt {1-x}-1\right )}^6}{{\left (\sqrt {x+1}-1\right )}^6}+\frac {{\left (\sqrt {1-x}-1\right )}^8}{{\left (\sqrt {x+1}-1\right )}^8}+1}-\frac {x^2}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.07, size = 48, normalized size = 1.55 \[ - \frac {\left (1 - x\right )^{2}}{2} - 2 \left (\begin {cases} - \frac {x \sqrt {1 - x} \sqrt {x + 1}}{4} + \frac {\operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {1 - x}}{2} \right )}}{2} & \text {for}\: x \leq 1 \wedge x > -1 \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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