Optimal. Leaf size=23 \[ \frac{x^2}{2}-\frac{1}{3} \left (1-x^2\right )^{3/2} \]
[Out]
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Rubi [A] time = 0.0159755, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{x^2}{2}-\frac{1}{3} \left (1-x^2\right )^{3/2} \]
Antiderivative was successfully verified.
[In] Int[x*(1 + Sqrt[1 - x^2]),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \frac{\left (- x^{2} + 1\right )^{\frac{3}{2}}}{3} + \int x\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x*(1+(-x**2+1)**(1/2)),x)
[Out]
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Mathematica [A] time = 0.0188422, size = 23, normalized size = 1. \[ \frac{x^2}{2}-\frac{1}{3} \left (1-x^2\right )^{3/2} \]
Antiderivative was successfully verified.
[In] Integrate[x*(1 + Sqrt[1 - x^2]),x]
[Out]
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Maple [A] time = 0.001, size = 18, normalized size = 0.8 \[{\frac{{x}^{2}}{2}}-{\frac{1}{3} \left ( -{x}^{2}+1 \right ) ^{{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x*(1+(-x^2+1)^(1/2)),x)
[Out]
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Maxima [A] time = 0.718612, size = 23, normalized size = 1. \[ \frac{1}{2} \, x^{2} - \frac{1}{3} \,{\left (-x^{2} + 1\right )}^{\frac{3}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*(sqrt(-x^2 + 1) + 1),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.264869, size = 70, normalized size = 3.04 \[ \frac{2 \, x^{6} + 3 \, \sqrt{-x^{2} + 1} x^{4} - 3 \, x^{4}}{6 \,{\left (3 \, x^{2} -{\left (x^{2} - 4\right )} \sqrt{-x^{2} + 1} - 4\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*(sqrt(-x^2 + 1) + 1),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.459458, size = 27, normalized size = 1.17 \[ \frac{x^{2} \sqrt{- x^{2} + 1}}{3} + \frac{x^{2}}{2} - \frac{\sqrt{- x^{2} + 1}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*(1+(-x**2+1)**(1/2)),x)
[Out]
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GIAC/XCAS [A] time = 0.261545, size = 24, normalized size = 1.04 \[ \frac{1}{2} \, x^{2} - \frac{1}{3} \,{\left (-x^{2} + 1\right )}^{\frac{3}{2}} - \frac{1}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*(sqrt(-x^2 + 1) + 1),x, algorithm="giac")
[Out]