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

Optimal. Leaf size=18 \[ 2 \tanh ^{-1}\left (\frac{x+1}{\sqrt{-x^3-1}}\right ) \]

[Out]

2*ArcTanh[(1 + x)/Sqrt[-1 - x^3]]

_______________________________________________________________________________________

Rubi [A]  time = 0.123203, antiderivative size = 18, 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 \[ 2 \tanh ^{-1}\left (\frac{x+1}{\sqrt{-x^3-1}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

2*ArcTanh[(1 + x)/Sqrt[-1 - x^3]]

_______________________________________________________________________________________

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

_______________________________________________________________________________________

Mathematica [C]  time = 1.18729, size = 298, normalized size = 16.56 \[ \frac{2 \sqrt{\frac{x+1}{1+\sqrt [3]{-1}}} \sqrt{x^2-x+1} \left (\frac{\sqrt{3} \left (1+\sqrt [3]{-1}\right ) \left (\sqrt [3]{-1}-x\right ) F\left (\sin ^{-1}\left (\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{(-1)^{2/3} x+1}-\frac{3 i \left (\sqrt{2}-i\right ) \Pi \left (\frac{2 \sqrt{3}}{-i-2 \sqrt{2}+\sqrt{3}};\sin ^{-1}\left (\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{(-1)^{5/6}+\sqrt{2}}+\frac{3 \left (5+i \sqrt{2}+i \sqrt{3}+\sqrt{6}\right ) \Pi \left (\frac{2 \sqrt{3}}{-i+2 \sqrt{2}+\sqrt{3}};\sin ^{-1}\left (\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{5 i+2 \sqrt{2}+\sqrt{3}+2 i \sqrt{6}}\right )}{3 \sqrt{-x^3-1}} \]

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) - ((3*I)*(-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*(5 + I*Sqrt[2] + I*Sqrt[3] + Sqrt[6])*Ellipt
icPi[(2*Sqrt[3])/(-I + 2*Sqrt[2] + Sqrt[3]), ArcSin[Sqrt[(1 + (-1)^(2/3)*x)/(1 +
(-1)^(1/3))]], (-1)^(1/3)])/(5*I + 2*Sqrt[2] + Sqrt[3] + (2*I)*Sqrt[6])))/(3*Sq
rt[-1 - x^3])

_______________________________________________________________________________________

Maple [C]  time = 0.092, size = 724, normalized size = 40.2 \[ \text{result too large to display} \]

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*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))+1/(3/2+1/2*I*3^(1/2))*x)^(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))+1/(3/2+1/2*I*3^(1/2))*x)^(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))+1/(3/2+1/2*I*3^(1/2))*x)^(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))*EllipticP
i(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))+1/(3/2+1/2*I*3^(1/2))*x)^(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))

_______________________________________________________________________________________

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)

_______________________________________________________________________________________

Fricas [A]  time = 0.269407, size = 38, normalized size = 2.11 \[ \log \left (-\frac{x^{2} - 2 \, x - 2 \, \sqrt{-x^{3} - 1}}{x^{2} + 2}\right ) \]

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]

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

_______________________________________________________________________________________

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

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)

_______________________________________________________________________________________

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)