Optimal. Leaf size=22 \[ -2 x-x \sqrt {1-x^2}-\sin ^{-1}(x) \]
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Rubi [A]
time = 0.04, 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 = {6820, 6874,
201, 222} \begin {gather*} -\text {ArcSin}(x)-\sqrt {1-x^2} x-2 x \end {gather*}
Antiderivative was successfully verified.
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Rule 201
Rule 222
Rule 6820
Rule 6874
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 = 38, normalized size = 1.73 \begin {gather*} -2-x \left (2+\sqrt {1-x^2}\right )-2 \tan ^{-1}\left (\frac {\sqrt {1+x}}{\sqrt {1-x}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(58\) vs.
\(2(20)=40\).
time = 0.25, size = 59, normalized size = 2.68
method | result | size |
default | \(-2 x -\sqrt {1-x}\, \left (1+x \right )^{\frac {3}{2}}+\sqrt {1-x}\, \sqrt {1+x}-\frac {\sqrt {\left (1-x \right ) \left (1+x \right )}\, \arcsin \left (x \right )}{\sqrt {1+x}\, \sqrt {1-x}}\) | \(59\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 20, normalized size = 0.91 \begin {gather*} -\sqrt {-x^{2} + 1} x - 2 \, x - \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. 41 vs.
\(2 (20) = 40\).
time = 0.37, 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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Sympy [A]
time = 21.12, size = 63, normalized size = 2.86 \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}\: \sqrt {x + 1} > - \sqrt {2} \wedge \sqrt {x + 1} < \sqrt {2} \end {cases}\right ) - 2 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 49 vs.
\(2 (20) = 40\).
time = 3.38, 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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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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