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

Optimal. Leaf size=32 \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{d+1} (x+1)}{\sqrt{-x^3-1}}\right )}{\sqrt{d+1}} \]

[Out]

(2*ArcTanh[(Sqrt[1 + d]*(1 + x))/Sqrt[-1 - x^3]])/Sqrt[1 + d]

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Rubi [A]  time = 0.149565, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.061 \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{d+1} (x+1)}{\sqrt{-x^3-1}}\right )}{\sqrt{d+1}} \]

Antiderivative was successfully verified.

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

[Out]

(2*ArcTanh[(Sqrt[1 + d]*(1 + x))/Sqrt[-1 - x^3]])/Sqrt[1 + d]

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

[Out]

Timed out

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Mathematica [C]  time = 2.06181, size = 426, normalized size = 13.31 \[ \frac{\sqrt{\frac{x+1}{1+\sqrt [3]{-1}}} \sqrt{x^2-x+1} \left (\frac{2 \sqrt{3} \left (1+\sqrt [3]{-1}\right ) \left (\sqrt [3]{-1}-x\right ) F\left (\sin ^{-1}\left (\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{(-1)^{2/3} x+1}-\frac{3 i \left (\left (-\left (1+\sqrt [3]{-1}\right ) d^2+\left (1+\sqrt [3]{-1}\right ) \left (\sqrt{d^2-4 d-8}+4\right ) d-2 \sqrt [3]{-1} \sqrt{d^2-4 d-8}+4 \sqrt{d^2-4 d-8}+8 \sqrt [3]{-1}+8\right ) \Pi \left (\frac{2 i \sqrt{3}}{d-\sqrt{d^2-4 d-8}+2 \sqrt [3]{-1}};\sin ^{-1}\left (\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )+\left (\left (1+\sqrt [3]{-1}\right ) d^2+\left (1+\sqrt [3]{-1}\right ) \left (\sqrt{d^2-4 d-8}-4\right ) d-2 \left (\sqrt [3]{-1} \sqrt{d^2-4 d-8}-2 \sqrt{d^2-4 d-8}+4 \sqrt [3]{-1}+4\right )\right ) \Pi \left (\frac{2 i \sqrt{3}}{d+\sqrt{d^2-4 d-8}+2 \sqrt [3]{-1}};\sin ^{-1}\left (\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )\right )}{\left (\sqrt [3]{-1} d+d+(-1)^{2/3}+2\right ) \sqrt{d^2-4 d-8}}\right )}{3 \sqrt{-x^3-1}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(Sqrt[(1 + x)/(1 + (-1)^(1/3))]*Sqrt[1 - x + x^2]*((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)*((8 + 8*(-1)^(1/3) - (1 + (-1)^(1/3))*d^2
+ 4*Sqrt[-8 - 4*d + d^2] - 2*(-1)^(1/3)*Sqrt[-8 - 4*d + d^2] + (1 + (-1)^(1/3))*
d*(4 + Sqrt[-8 - 4*d + d^2]))*EllipticPi[((2*I)*Sqrt[3])/(2*(-1)^(1/3) + d - Sqr
t[-8 - 4*d + d^2]), ArcSin[Sqrt[(1 + (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3
)] + ((1 + (-1)^(1/3))*d^2 + (1 + (-1)^(1/3))*d*(-4 + Sqrt[-8 - 4*d + d^2]) - 2*
(4 + 4*(-1)^(1/3) - 2*Sqrt[-8 - 4*d + d^2] + (-1)^(1/3)*Sqrt[-8 - 4*d + d^2]))*E
llipticPi[((2*I)*Sqrt[3])/(2*(-1)^(1/3) + d + Sqrt[-8 - 4*d + d^2]), ArcSin[Sqrt
[(1 + (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)]))/((2 + (-1)^(2/3) + d + (-1
)^(1/3)*d)*Sqrt[-8 - 4*d + d^2])))/(3*Sqrt[-1 - x^3])

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Maple [C]  time = 0.053, size = 1888, normalized size = 59. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-x^2-2*x+2)/(d*x+x^2+d+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))+1/3*I/(d^2-4*d-8)^(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)+1/2*d-1/2*(d^2-4*d-8)^(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)+1/2*d-1/2*(d^2-4*d-8)^(1/2)),(I*3^(1/2)/(3/2+1/2*I*3^(1/2)))^(1/2))*d^2-1/3*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)+1/2*d-1/2*(d^2-4*d-8)^(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)+1/2*d-1/2*(d^2-4*d-8)^(1/
2)),(I*3^(1/2)/(3/2+1/2*I*3^(1/2)))^(1/2))*d-4/3*I/(d^2-4*d-8)^(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
)+1/2*d-1/2*(d^2-4*d-8)^(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)+1/2*d-1/2*(d^2-4*d-8)^(1/2)),(I*3^(1/
2)/(3/2+1/2*I*3^(1/2)))^(1/2))*d+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)+1/2*d-1/2*(d^2-4*d-8)^(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)+1/2*d-1/2*(d^2-4*d-8)^(1/2)),(I*3^(1/2)/(3/2+1/2*I*3^(1/2)))^(1/2))
-8/3*I/(d^2-4*d-8)^(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)+1/2*d-1/2*(d^2-4*d-8)^(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)+1
/2*d-1/2*(d^2-4*d-8)^(1/2)),(I*3^(1/2)/(3/2+1/2*I*3^(1/2)))^(1/2))-1/3*I/(d^2-4*
d-8)^(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)+1/2*d+1/2*(d^2-4*d-8)^(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)+1/2*d+1/2*(d^2-
4*d-8)^(1/2)),(I*3^(1/2)/(3/2+1/2*I*3^(1/2)))^(1/2))*d^2-1/3*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)+1/2*
d+1/2*(d^2-4*d-8)^(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)+1/2*d+1/2*(d^2-4*d-8)^(1/2)),(I*3^(1/2)/(3/
2+1/2*I*3^(1/2)))^(1/2))*d+4/3*I/(d^2-4*d-8)^(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)+1/2*d+1/2*(d^2-4
*d-8)^(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)+1/2*d+1/2*(d^2-4*d-8)^(1/2)),(I*3^(1/2)/(3/2+1/2*I*3^(1
/2)))^(1/2))*d+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)+1/2*d+1/2*(d^2-4*d-8)^(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)+1/2*d
+1/2*(d^2-4*d-8)^(1/2)),(I*3^(1/2)/(3/2+1/2*I*3^(1/2)))^(1/2))+8/3*I/(d^2-4*d-8)
^(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)+1/2*d+1/2*(d^2-4*d-8)^(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)+1/2*d+1/2*(d^2-4*d-
8)^(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. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.295897, size = 1, normalized size = 0.03 \[ \left [\frac{\log \left (-\frac{4 \,{\left ({\left (d + 1\right )} x^{2} - d^{2} -{\left (d^{2} + 3 \, d + 2\right )} x - d\right )} \sqrt{-x^{3} - 1} +{\left (2 \,{\left (3 \, d + 4\right )} x^{3} - x^{4} -{\left (d^{2} + 2 \, d + 4\right )} x^{2} - d^{2} - 2 \,{\left (d^{2} + 2 \, d\right )} x + 4 \, d + 4\right )} \sqrt{d + 1}}{2 \, d x^{3} + x^{4} +{\left (d^{2} + 2 \, d + 4\right )} x^{2} + d^{2} + 2 \,{\left (d^{2} + 2 \, d\right )} x + 4 \, d + 4}\right )}{2 \, \sqrt{d + 1}}, -\frac{\arctan \left (-\frac{{\left ({\left (d + 2\right )} x - x^{2} + d\right )} \sqrt{-d - 1}}{2 \, \sqrt{-x^{3} - 1}{\left (d + 1\right )}}\right )}{\sqrt{-d - 1}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/2*log(-(4*((d + 1)*x^2 - d^2 - (d^2 + 3*d + 2)*x - d)*sqrt(-x^3 - 1) + (2*(3*
d + 4)*x^3 - x^4 - (d^2 + 2*d + 4)*x^2 - d^2 - 2*(d^2 + 2*d)*x + 4*d + 4)*sqrt(d
+ 1))/(2*d*x^3 + x^4 + (d^2 + 2*d + 4)*x^2 + d^2 + 2*(d^2 + 2*d)*x + 4*d + 4))/
sqrt(d + 1), -arctan(-1/2*((d + 2)*x - x^2 + d)*sqrt(-d - 1)/(sqrt(-x^3 - 1)*(d
+ 1)))/sqrt(-d - 1)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-Integral(2*x/(d*x*sqrt(-x**3 - 1) + d*sqrt(-x**3 - 1) + x**2*sqrt(-x**3 - 1) +
2*sqrt(-x**3 - 1)), x) - Integral(x**2/(d*x*sqrt(-x**3 - 1) + d*sqrt(-x**3 - 1)
+ x**2*sqrt(-x**3 - 1) + 2*sqrt(-x**3 - 1)), x) - Integral(-2/(d*x*sqrt(-x**3 -
1) + d*sqrt(-x**3 - 1) + 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. \[ \int -\frac{x^{2} + 2 \, x - 2}{\sqrt{-x^{3} - 1}{\left (d x + x^{2} + d + 2\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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