3.176 \(\int \frac{2+2 x-x^2}{\left (2+x^2\right ) \sqrt{1-x^3}} \, dx\)

Optimal. Leaf size=20 \[ -2 \tan ^{-1}\left (\frac{1-x}{\sqrt{1-x^3}}\right ) \]

[Out]

-2*ArcTan[(1 - x)/Sqrt[1 - x^3]]

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Rubi [A]  time = 0.132165, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069 \[ -2 \tan ^{-1}\left (\frac{1-x}{\sqrt{1-x^3}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(2 + 2*x - x^2)/((2 + x^2)*Sqrt[1 - x^3]),x]

[Out]

-2*ArcTan[(1 - x)/Sqrt[1 - x^3]]

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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+2*x+2)/(x**2+2)/(-x**3+1)**(1/2),x)

[Out]

Timed out

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Mathematica [C]  time = 1.04607, size = 280, normalized size = 14. \[ \frac{2 \sqrt{\frac{1-x}{1+\sqrt [3]{-1}}} \sqrt{x^2+x+1} \left (\frac{\sqrt{3} \left (1+\sqrt [3]{-1}\right ) \left (x+\sqrt [3]{-1}\right ) F\left (\sin ^{-1}\left (\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{(-1)^{2/3} x-1}+\frac{6 \left (1+i \sqrt{2}\right ) \Pi \left (\frac{2 \sqrt{3}}{-i-2 \sqrt{2}+\sqrt{3}};\sin ^{-1}\left (\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{i+2 \sqrt{2}-\sqrt{3}}+\frac{3 \left (1-i \sqrt{2}\right ) \Pi \left (\frac{2 \sqrt{3}}{-i+2 \sqrt{2}+\sqrt{3}};\sin ^{-1}\left (\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{(-1)^{5/6}-\sqrt{2}}\right )}{3 \sqrt{1-x^3}} \]

Antiderivative was successfully verified.

[In]  Integrate[(2 + 2*x - x^2)/((2 + x^2)*Sqrt[1 - x^3]),x]

[Out]

(2*Sqrt[(1 - x)/(1 + (-1)^(1/3))]*Sqrt[1 + x + x^2]*((Sqrt[3]*(1 + (-1)^(1/3))*(
(-1)^(1/3) + x)*EllipticF[ArcSin[Sqrt[(1 - (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1
)^(1/3)])/(-1 + (-1)^(2/3)*x) + (6*(1 + I*Sqrt[2])*EllipticPi[(2*Sqrt[3])/(-I -
2*Sqrt[2] + Sqrt[3]), ArcSin[Sqrt[(1 - (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1
/3)])/(I + 2*Sqrt[2] - Sqrt[3]) + (3*(1 - I*Sqrt[2])*EllipticPi[(2*Sqrt[3])/(-I
+ 2*Sqrt[2] + Sqrt[3]), ArcSin[Sqrt[(1 - (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^
(1/3)])/((-1)^(5/6) - Sqrt[2])))/(3*Sqrt[1 - x^3])

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Maple [C]  time = 0.097, size = 732, normalized size = 36.6 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-x^2+2*x+2)/(x^2+2)/(-x^3+1)^(1/2),x)

[Out]

2/3*I*3^(1/2)*(I*(x+1/2-1/2*I*3^(1/2))*3^(1/2))^(1/2)*((-1+x)/(-3/2+1/2*I*3^(1/2
)))^(1/2)*(-I*(x+1/2+1/2*I*3^(1/2))*3^(1/2))^(1/2)/(-x^3+1)^(1/2)*EllipticF(1/3*
3^(1/2)*(I*(x+1/2-1/2*I*3^(1/2))*3^(1/2))^(1/2),(I*3^(1/2)/(-3/2+1/2*I*3^(1/2)))
^(1/2))-2/3*I*3^(1/2)*(I*3^(1/2)*x+1/2*I*3^(1/2)+3/2)^(1/2)*(1/(-3/2+1/2*I*3^(1/
2))*x-1/(-3/2+1/2*I*3^(1/2)))^(1/2)*(-I*3^(1/2)*x-1/2*I*3^(1/2)+3/2)^(1/2)/(-x^3
+1)^(1/2)/(-1/2+1/2*I*3^(1/2)-I*2^(1/2))*EllipticPi(1/3*3^(1/2)*(I*(x+1/2-1/2*I*
3^(1/2))*3^(1/2))^(1/2),I*3^(1/2)/(-1/2+1/2*I*3^(1/2)-I*2^(1/2)),(I*3^(1/2)/(-3/
2+1/2*I*3^(1/2)))^(1/2))-2/3*2^(1/2)*3^(1/2)*(I*3^(1/2)*x+1/2*I*3^(1/2)+3/2)^(1/
2)*(1/(-3/2+1/2*I*3^(1/2))*x-1/(-3/2+1/2*I*3^(1/2)))^(1/2)*(-I*3^(1/2)*x-1/2*I*3
^(1/2)+3/2)^(1/2)/(-x^3+1)^(1/2)/(-1/2+1/2*I*3^(1/2)-I*2^(1/2))*EllipticPi(1/3*3
^(1/2)*(I*(x+1/2-1/2*I*3^(1/2))*3^(1/2))^(1/2),I*3^(1/2)/(-1/2+1/2*I*3^(1/2)-I*2
^(1/2)),(I*3^(1/2)/(-3/2+1/2*I*3^(1/2)))^(1/2))+2/3*2^(1/2)*3^(1/2)*(I*3^(1/2)*x
+1/2*I*3^(1/2)+3/2)^(1/2)*(1/(-3/2+1/2*I*3^(1/2))*x-1/(-3/2+1/2*I*3^(1/2)))^(1/2
)*(-I*3^(1/2)*x-1/2*I*3^(1/2)+3/2)^(1/2)/(-x^3+1)^(1/2)/(-1/2+1/2*I*3^(1/2)+I*2^
(1/2))*EllipticPi(1/3*3^(1/2)*(I*(x+1/2-1/2*I*3^(1/2))*3^(1/2))^(1/2),I*3^(1/2)/
(-1/2+1/2*I*3^(1/2)+I*2^(1/2)),(I*3^(1/2)/(-3/2+1/2*I*3^(1/2)))^(1/2))-2/3*I*3^(
1/2)*(I*3^(1/2)*x+1/2*I*3^(1/2)+3/2)^(1/2)*(1/(-3/2+1/2*I*3^(1/2))*x-1/(-3/2+1/2
*I*3^(1/2)))^(1/2)*(-I*3^(1/2)*x-1/2*I*3^(1/2)+3/2)^(1/2)/(-x^3+1)^(1/2)/(-1/2+1
/2*I*3^(1/2)+I*2^(1/2))*EllipticPi(1/3*3^(1/2)*(I*(x+1/2-1/2*I*3^(1/2))*3^(1/2))
^(1/2),I*3^(1/2)/(-1/2+1/2*I*3^(1/2)+I*2^(1/2)),(I*3^(1/2)/(-3/2+1/2*I*3^(1/2)))
^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\int \frac{x^{2} - 2 \, x - 2}{\sqrt{-x^{3} + 1}{\left (x^{2} + 2\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(x^2 - 2*x - 2)/(sqrt(-x^3 + 1)*(x^2 + 2)),x, algorithm="maxima")

[Out]

-integrate((x^2 - 2*x - 2)/(sqrt(-x^3 + 1)*(x^2 + 2)), x)

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Fricas [A]  time = 0.306091, size = 26, normalized size = 1.3 \[ \arctan \left (\frac{x^{2} + 2 \, x}{2 \, \sqrt{-x^{3} + 1}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(x^2 - 2*x - 2)/(sqrt(-x^3 + 1)*(x^2 + 2)),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \left (- \frac{2 x}{x^{2} \sqrt{- x^{3} + 1} + 2 \sqrt{- x^{3} + 1}}\right )\, dx - \int \frac{x^{2}}{x^{2} \sqrt{- x^{3} + 1} + 2 \sqrt{- x^{3} + 1}}\, dx - \int \left (- \frac{2}{x^{2} \sqrt{- x^{3} + 1} + 2 \sqrt{- x^{3} + 1}}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-x**2+2*x+2)/(x**2+2)/(-x**3+1)**(1/2),x)

[Out]

-Integral(-2*x/(x**2*sqrt(-x**3 + 1) + 2*sqrt(-x**3 + 1)), x) - Integral(x**2/(x
**2*sqrt(-x**3 + 1) + 2*sqrt(-x**3 + 1)), x) - Integral(-2/(x**2*sqrt(-x**3 + 1)
+ 2*sqrt(-x**3 + 1)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int -\frac{x^{2} - 2 \, x - 2}{\sqrt{-x^{3} + 1}{\left (x^{2} + 2\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(x^2 - 2*x - 2)/(sqrt(-x^3 + 1)*(x^2 + 2)),x, algorithm="giac")

[Out]

integrate(-(x^2 - 2*x - 2)/(sqrt(-x^3 + 1)*(x^2 + 2)), x)