Optimal. Leaf size=30 \[ \frac {1}{4} (1+2 x) \sqrt {3-4 x-4 x^2}+\sin ^{-1}\left (\frac {1}{2}+x\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {626, 633, 222}
\begin {gather*} \text {ArcSin}\left (x+\frac {1}{2}\right )+\frac {1}{4} \sqrt {-4 x^2-4 x+3} (2 x+1) \end {gather*}
Antiderivative was successfully verified.
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Rule 222
Rule 626
Rule 633
Rubi steps
\begin {align*} \int \sqrt {3-4 x-4 x^2} \, dx &=\frac {1}{4} (1+2 x) \sqrt {3-4 x-4 x^2}+2 \int \frac {1}{\sqrt {3-4 x-4 x^2}} \, dx\\ &=\frac {1}{4} (1+2 x) \sqrt {3-4 x-4 x^2}-\frac {1}{8} \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{64}}} \, dx,x,-4-8 x\right )\\ &=\frac {1}{4} (1+2 x) \sqrt {3-4 x-4 x^2}+\sin ^{-1}\left (\frac {1}{2}+x\right )\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 49, normalized size = 1.63 \begin {gather*} \frac {1}{4} (1+2 x) \sqrt {3-4 x-4 x^2}-2 \tan ^{-1}\left (\frac {\sqrt {3-4 x-4 x^2}}{3+2 x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.57, size = 25, normalized size = 0.83
method | result | size |
default | \(-\frac {\left (-8 x -4\right ) \sqrt {-4 x^{2}-4 x +3}}{16}+\arcsin \left (x +\frac {1}{2}\right )\) | \(25\) |
risch | \(-\frac {\left (4 x^{2}+4 x -3\right ) \left (2 x +1\right )}{4 \sqrt {-4 x^{2}-4 x +3}}+\arcsin \left (x +\frac {1}{2}\right )\) | \(35\) |
trager | \(\left (\frac {x}{2}+\frac {1}{4}\right ) \sqrt {-4 x^{2}-4 x +3}+\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-2 x \RootOf \left (\textit {\_Z}^{2}+1\right )+\sqrt {-4 x^{2}-4 x +3}-\RootOf \left (\textit {\_Z}^{2}+1\right )\right )\) | \(58\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 38, normalized size = 1.27 \begin {gather*} \frac {1}{2} \, \sqrt {-4 \, x^{2} - 4 \, x + 3} x + \frac {1}{4} \, \sqrt {-4 \, x^{2} - 4 \, x + 3} - \arcsin \left (-x - \frac {1}{2}\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. 53 vs.
\(2 (24) = 48\).
time = 1.68, size = 53, normalized size = 1.77 \begin {gather*} \frac {1}{4} \, \sqrt {-4 \, x^{2} - 4 \, x + 3} {\left (2 \, x + 1\right )} - \arctan \left (\frac {\sqrt {-4 \, x^{2} - 4 \, x + 3} {\left (2 \, x + 1\right )}}{4 \, x^{2} + 4 \, x - 3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {- 4 x^{2} - 4 x + 3}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.39, size = 24, normalized size = 0.80 \begin {gather*} \frac {1}{4} \, \sqrt {-4 \, x^{2} - 4 \, x + 3} {\left (2 \, x + 1\right )} + \arcsin \left (x + \frac {1}{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 23, normalized size = 0.77 \begin {gather*} \mathrm {asin}\left (x+\frac {1}{2}\right )+\left (\frac {x}{2}+\frac {1}{4}\right )\,\sqrt {-4\,x^2-4\,x+3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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