Optimal. Leaf size=70 \[ -\frac{2}{3} \left (-x-\sqrt{x}+1\right )^{3/2}-\frac{1}{4} \left (2 \sqrt{x}+1\right ) \sqrt{-x-\sqrt{x}+1}-\frac{5}{8} \sin ^{-1}\left (\frac{2 \sqrt{x}+1}{\sqrt{5}}\right ) \]
[Out]
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Rubi [A] time = 0.0702571, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312 \[ -\frac{2}{3} \left (-x-\sqrt{x}+1\right )^{3/2}-\frac{1}{4} \left (2 \sqrt{x}+1\right ) \sqrt{-x-\sqrt{x}+1}-\frac{5}{8} \sin ^{-1}\left (\frac{2 \sqrt{x}+1}{\sqrt{5}}\right ) \]
Antiderivative was successfully verified.
[In] Int[Sqrt[1 - Sqrt[x] - x],x]
[Out]
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Rubi in Sympy [A] time = 2.67464, size = 68, normalized size = 0.97 \[ - \frac{\left (2 \sqrt{x} + 1\right ) \sqrt{- \sqrt{x} - x + 1}}{4} - \frac{2 \left (- \sqrt{x} - x + 1\right )^{\frac{3}{2}}}{3} - \frac{5 \operatorname{atan}{\left (- \frac{- 2 \sqrt{x} - 1}{2 \sqrt{- \sqrt{x} - x + 1}} \right )}}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((1-x-x**(1/2))**(1/2),x)
[Out]
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Mathematica [A] time = 0.0414724, size = 53, normalized size = 0.76 \[ \frac{1}{12} \sqrt{-x-\sqrt{x}+1} \left (8 x+2 \sqrt{x}-11\right )+\frac{5}{8} \sin ^{-1}\left (\frac{-2 \sqrt{x}-1}{\sqrt{5}}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[1 - Sqrt[x] - x],x]
[Out]
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Maple [A] time = 0.009, size = 50, normalized size = 0.7 \[ -{\frac{2}{3} \left ( 1-x-\sqrt{x} \right ) ^{{\frac{3}{2}}}}+{\frac{1}{4} \left ( -2\,\sqrt{x}-1 \right ) \sqrt{1-x-\sqrt{x}}}-{\frac{5}{8}\arcsin \left ({\frac{2\,\sqrt{5}}{5} \left ( \sqrt{x}+{\frac{1}{2}} \right ) } \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((1-x-x^(1/2))^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{-x - \sqrt{x} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-x - sqrt(x) + 1),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.977254, size = 82, normalized size = 1.17 \[ \frac{1}{12} \,{\left (8 \, x + 2 \, \sqrt{x} - 11\right )} \sqrt{-x - \sqrt{x} + 1} - \frac{5}{16} \, \arctan \left (\frac{8 \, x + 8 \, \sqrt{x} - 3}{4 \, \sqrt{-x - \sqrt{x} + 1}{\left (2 \, \sqrt{x} + 1\right )}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-x - sqrt(x) + 1),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{- \sqrt{x} - x + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((1-x-x**(1/2))**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.274965, size = 59, normalized size = 0.84 \[ \frac{1}{12} \,{\left (2 \, \sqrt{x}{\left (4 \, \sqrt{x} + 1\right )} - 11\right )} \sqrt{-x - \sqrt{x} + 1} - \frac{5}{8} \, \arcsin \left (\frac{1}{5} \, \sqrt{5}{\left (2 \, \sqrt{x} + 1\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-x - sqrt(x) + 1),x, algorithm="giac")
[Out]