Optimal. Leaf size=22 \[ -\sqrt {1-x^2} x-2 x-\sin ^{-1}(x) \]
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Rubi [A] time = 0.05, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {6688, 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 6688
Rule 6742
Rubi steps
\begin {align*} \int \left (-\sqrt {1-x}-\sqrt {1+x}\right ) \left (\sqrt {1-x}+\sqrt {1+x}\right ) \, dx &=-\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 = 21, 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 = 2.68 \begin {gather*} -2 (x+1)+\sqrt {1-x} \left (\sqrt {x+1}-(x+1)^{3/2}\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 = 0.39, size = 41, normalized size = 1.86 \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 = 0.37, size = 49, normalized size = 2.23 \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 = 0.00, size = 59, normalized size = 2.68 \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 = 1.39, size = 20, normalized size = 0.91 \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 = 3.71, size = 205, normalized size = 9.32 \begin {gather*} 4\,\mathrm {atan}\left (\frac {\sqrt {1-x}-1}{\sqrt {x+1}-1}\right )-2\,x+\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 = 37.87, size = 46, normalized size = 2.09 \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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