3.180 \(\int \frac{2+2 x-x^2}{\left (2-d+d x+x^2\right ) \sqrt{1-x^3}} \, dx\)
Optimal. Leaf size=38 \[ -\frac{2 \tan ^{-1}\left (\frac{\sqrt{1-d} (1-x)}{\sqrt{1-x^3}}\right )}{\sqrt{1-d}} \]
[Out]
(-2*ArcTan[(Sqrt[1 - d]*(1 - x))/Sqrt[1 - x^3]])/Sqrt[1 - d]
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Rubi [A] time = 0.182616, antiderivative size = 38, normalized size of antiderivative = 1.,
number of steps used = 2, number of rules used = 2, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.057
\[ -\frac{2 \tan ^{-1}\left (\frac{\sqrt{1-d} (1-x)}{\sqrt{1-x^3}}\right )}{\sqrt{1-d}} \]
Antiderivative was successfully verified.
[In] Int[(2 + 2*x - x^2)/((2 - d + d*x + x^2)*Sqrt[1 - x^3]),x]
[Out]
(-2*ArcTan[(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 = 2.0837, size = 427, normalized size = 11.24 \[ \frac{\sqrt{\frac{1-x}{1+\sqrt [3]{-1}}} \sqrt{x^2+x+1} \left (\frac{2 \sqrt{3} \left (1+\sqrt [3]{-1}\right ) \left (x+\sqrt [3]{-1}\right ) F\left (\sin ^{-1}\left (\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{(-1)^{2/3} x-1}+\frac{3 i \left (\left (-\left (1+\sqrt [3]{-1}\right ) d^2+\left (1+\sqrt [3]{-1}\right ) \left (\sqrt{d^2+4 d-8}-4\right ) d+2 \sqrt [3]{-1} \sqrt{d^2+4 d-8}-4 \sqrt{d^2+4 d-8}+8 \sqrt [3]{-1}+8\right ) \Pi \left (\frac{2 i \sqrt{3}}{-d+\sqrt{d^2+4 d-8}+2 \sqrt [3]{-1}};\sin ^{-1}\left (\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )+\left (\left (1+\sqrt [3]{-1}\right ) d^2+\left (1+\sqrt [3]{-1}\right ) \left (\sqrt{d^2+4 d-8}+4\right ) d+2 \sqrt [3]{-1} \sqrt{d^2+4 d-8}-4 \sqrt{d^2+4 d-8}-8 \sqrt [3]{-1}-8\right ) \Pi \left (-\frac{2 i \sqrt{3}}{d+\sqrt{d^2+4 d-8}-2 \sqrt [3]{-1}};\sin ^{-1}\left (\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )\right )}{\left (\sqrt [3]{-1} d+d-(-1)^{2/3}-2\right ) \sqrt{d^2+4 d-8}}\right )}{3 \sqrt{1-x^3}} \]
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 + S
qrt[-8 + 4*d + d^2]), ArcSin[Sqrt[(1 - (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1
/3)] + (-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]))*E
llipticPi[((-2*I)*Sqrt[3])/(-2*(-1)^(1/3) + d + Sqrt[-8 + 4*d + d^2]), ArcSin[Sq
rt[(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 = 0.069, size = 1908, normalized size = 50.2 \[ \text{result too large to display} \]
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))*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)+1/2*d-1/2*(d^2+4*d-8)^(1/2))*Ell
ipticPi(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))*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)+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))*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)+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))*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)+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))*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)+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))*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)+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))*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)+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))*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)+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))*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)+
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))*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)+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 = 0.289378, 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. \[ - \int \left (- \frac{2 x}{d x \sqrt{- x^{3} + 1} - d \sqrt{- x^{3} + 1} + x^{2} \sqrt{- x^{3} + 1} + 2 \sqrt{- x^{3} + 1}}\right )\, dx - \int \frac{x^{2}}{d x \sqrt{- x^{3} + 1} - d \sqrt{- x^{3} + 1} + x^{2} \sqrt{- x^{3} + 1} + 2 \sqrt{- x^{3} + 1}}\, dx - \int \left (- \frac{2}{d x \sqrt{- x^{3} + 1} - d \sqrt{- x^{3} + 1} + x^{2} \sqrt{- x^{3} + 1} + 2 \sqrt{- x^{3} + 1}}\right )\, dx \]
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)