3.284 \(\int x^2 \left (-\sqrt{1-x}-\sqrt{1+x}\right ) \left (\sqrt{1-x}+\sqrt{1+x}\right ) \, dx\)

Optimal. Leaf size=48 \[ -\frac{2 x^3}{3}+\frac{1}{4} \sqrt{1-x^2} x-\frac{1}{2} \sqrt{1-x^2} x^3-\frac{1}{4} \sin ^{-1}(x) \]

[Out]

(-2*x^3)/3 + (x*Sqrt[1 - x^2])/4 - (x^3*Sqrt[1 - x^2])/2 - ArcSin[x]/4

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Rubi [A]  time = 0.566827, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 42, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.119 \[ -\frac{2 x^3}{3}+\frac{1}{4} \sqrt{1-x^2} x-\frac{1}{2} \sqrt{1-x^2} x^3-\frac{1}{4} \sin ^{-1}(x) \]

Antiderivative was successfully verified.

[In]  Int[x^2*(-Sqrt[1 - x] - Sqrt[1 + x])*(Sqrt[1 - x] + Sqrt[1 + x]),x]

[Out]

(-2*x^3)/3 + (x*Sqrt[1 - x^2])/4 - (x^3*Sqrt[1 - x^2])/2 - ArcSin[x]/4

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2*(-(1-x)**(1/2)-(1+x)**(1/2))*((1-x)**(1/2)+(1+x)**(1/2)),x)

[Out]

Timed out

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Mathematica [A]  time = 0.0590548, size = 56, normalized size = 1.17 \[ \frac{1}{12} \left (3 \sqrt{1-x^2} x-\left (6 \sqrt{1-x^2}+8\right ) x^3-6 \sin ^{-1}\left (\frac{\sqrt{x+1}}{\sqrt{2}}\right )-8\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^2*(-Sqrt[1 - x] - Sqrt[1 + x])*(Sqrt[1 - x] + Sqrt[1 + x]),x]

[Out]

(-8 + 3*x*Sqrt[1 - x^2] - x^3*(8 + 6*Sqrt[1 - x^2]) - 6*ArcSin[Sqrt[1 + x]/Sqrt[
2]])/12

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Maple [A]  time = 0.002, size = 59, normalized size = 1.2 \[ -{\frac{2\,{x}^{3}}{3}}-{\frac{1}{4}\sqrt{1-x}\sqrt{1+x} \left ( 2\,{x}^{3}\sqrt{-{x}^{2}+1}-x\sqrt{-{x}^{2}+1}+\arcsin \left ( x \right ) \right ){\frac{1}{\sqrt{-{x}^{2}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(-(1-x)^(1/2)-(1+x)^(1/2))*((1-x)^(1/2)+(1+x)^(1/2)),x)

[Out]

-2/3*x^3-1/4*(1-x)^(1/2)*(1+x)^(1/2)*(2*x^3*(-x^2+1)^(1/2)-x*(-x^2+1)^(1/2)+arcs
in(x))/(-x^2+1)^(1/2)

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Maxima [A]  time = 0.763824, size = 46, normalized size = 0.96 \[ -\frac{2}{3} \, x^{3} + \frac{1}{2} \,{\left (-x^{2} + 1\right )}^{\frac{3}{2}} x - \frac{1}{4} \, \sqrt{-x^{2} + 1} x - \frac{1}{4} \, \arcsin \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-x^2*(sqrt(x + 1) + sqrt(-x + 1))^2,x, algorithm="maxima")

[Out]

-2/3*x^3 + 1/2*(-x^2 + 1)^(3/2)*x - 1/4*sqrt(-x^2 + 1)*x - 1/4*arcsin(x)

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Fricas [A]  time = 0.273695, size = 184, normalized size = 3.83 \[ \frac{16 \, x^{7} - 20 \, x^{5} + 20 \, x^{3} -{\left (6 \, x^{7} - 19 \, x^{5} + 8 \, x^{3} - 24 \, x\right )} \sqrt{x + 1} \sqrt{-x + 1} + 6 \,{\left (x^{4} - 8 \, x^{2} + 4 \,{\left (x^{2} - 2\right )} \sqrt{x + 1} \sqrt{-x + 1} + 8\right )} \arctan \left (\frac{\sqrt{x + 1} \sqrt{-x + 1} - 1}{x}\right ) - 24 \, x}{12 \,{\left (x^{4} - 8 \, x^{2} + 4 \,{\left (x^{2} - 2\right )} \sqrt{x + 1} \sqrt{-x + 1} + 8\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-x^2*(sqrt(x + 1) + sqrt(-x + 1))^2,x, algorithm="fricas")

[Out]

1/12*(16*x^7 - 20*x^5 + 20*x^3 - (6*x^7 - 19*x^5 + 8*x^3 - 24*x)*sqrt(x + 1)*sqr
t(-x + 1) + 6*(x^4 - 8*x^2 + 4*(x^2 - 2)*sqrt(x + 1)*sqrt(-x + 1) + 8)*arctan((s
qrt(x + 1)*sqrt(-x + 1) - 1)/x) - 24*x)/(x^4 - 8*x^2 + 4*(x^2 - 2)*sqrt(x + 1)*s
qrt(-x + 1) + 8)

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Sympy [A]  time = 147.376, size = 219, normalized size = 4.56 \[ \frac{x^{4}}{4} - \frac{x^{3}}{3} - \frac{\left (x + 1\right )^{4}}{4} + \frac{2 \left (x + 1\right )^{3}}{3} - \frac{\left (x + 1\right )^{2}}{2} - 4 \left (\begin{cases} \frac{x \sqrt{- x + 1} \sqrt{x + 1}}{4} + \frac{\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{x + 1}}{2} \right )}}{2} & \text{for}\: x \geq -1 \wedge x < 1 \end{cases}\right ) + 8 \left (\begin{cases} \frac{x \sqrt{- x + 1} \sqrt{x + 1}}{4} - \frac{\left (- x + 1\right )^{\frac{3}{2}} \left (x + 1\right )^{\frac{3}{2}}}{6} + \frac{\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{x + 1}}{2} \right )}}{2} & \text{for}\: x \geq -1 \wedge x < 1 \end{cases}\right ) - 4 \left (\begin{cases} \frac{x \sqrt{- x + 1} \sqrt{x + 1}}{4} - \frac{\left (- x + 1\right )^{\frac{3}{2}} \left (x + 1\right )^{\frac{3}{2}}}{3} - \frac{\sqrt{- x + 1} \sqrt{x + 1} \left (- 5 x - 2 \left (x + 1\right )^{3} + 6 \left (x + 1\right )^{2} - 4\right )}{16} + \frac{5 \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{x + 1}}{2} \right )}}{8} & \text{for}\: x \geq -1 \wedge x < 1 \end{cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2*(-(1-x)**(1/2)-(1+x)**(1/2))*((1-x)**(1/2)+(1+x)**(1/2)),x)

[Out]

x**4/4 - x**3/3 - (x + 1)**4/4 + 2*(x + 1)**3/3 - (x + 1)**2/2 - 4*Piecewise((x*
sqrt(-x + 1)*sqrt(x + 1)/4 + asin(sqrt(2)*sqrt(x + 1)/2)/2, (x >= -1) & (x < 1))
) + 8*Piecewise((x*sqrt(-x + 1)*sqrt(x + 1)/4 - (-x + 1)**(3/2)*(x + 1)**(3/2)/6
+ asin(sqrt(2)*sqrt(x + 1)/2)/2, (x >= -1) & (x < 1))) - 4*Piecewise((x*sqrt(-x
+ 1)*sqrt(x + 1)/4 - (-x + 1)**(3/2)*(x + 1)**(3/2)/3 - sqrt(-x + 1)*sqrt(x + 1
)*(-5*x - 2*(x + 1)**3 + 6*(x + 1)**2 - 4)/16 + 5*asin(sqrt(2)*sqrt(x + 1)/2)/8,
(x >= -1) & (x < 1)))

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GIAC/XCAS [A]  time = 0.291801, size = 84, normalized size = 1.75 \[ -\frac{2}{3} \,{\left (x + 1\right )}^{3} + 2 \,{\left (x + 1\right )}^{2} - \frac{1}{4} \,{\left ({\left (2 \,{\left (x + 1\right )}{\left (x - 2\right )} + 5\right )}{\left (x + 1\right )} - 1\right )} \sqrt{x + 1} \sqrt{-x + 1} - 2 \, x - \frac{1}{2} \, \arcsin \left (\frac{1}{2} \, \sqrt{2} \sqrt{x + 1}\right ) - 2 \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-x^2*(sqrt(x + 1) + sqrt(-x + 1))^2,x, algorithm="giac")

[Out]

-2/3*(x + 1)^3 + 2*(x + 1)^2 - 1/4*((2*(x + 1)*(x - 2) + 5)*(x + 1) - 1)*sqrt(x
+ 1)*sqrt(-x + 1) - 2*x - 1/2*arcsin(1/2*sqrt(2)*sqrt(x + 1)) - 2