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

Optimal. Leaf size=44 \[ -\frac{2 \tan ^{-1}\left (\frac{\sqrt{2 \sqrt{3}-3} (x+1)}{\sqrt{-x^3-1}}\right )}{\sqrt{2 \sqrt{3}-3}} \]

[Out]

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

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Rubi [A]  time = 0.161418, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062 \[ -\frac{2 \tan ^{-1}\left (\frac{\sqrt{2 \sqrt{3}-3} (x+1)}{\sqrt{-x^3-1}}\right )}{\sqrt{2 \sqrt{3}-3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 16.9534, size = 80, normalized size = 1.82 \[ \frac{2 \tilde{\infty } \sqrt{\frac{x^{2} - x + 1}{\left (x - \sqrt{3} + 1\right )^{2}}} \left (x + 1\right ) F\left (\operatorname{asin}{\left (\frac{x + 1 + \sqrt{3}}{x - \sqrt{3} + 1} \right )}\middle | -7 + 4 \sqrt{3}\right )}{\sqrt{\frac{- x - 1}{\left (x - \sqrt{3} + 1\right )^{2}}} \sqrt{- x^{3} - 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*zoo*sqrt((x**2 - x + 1)/(x - sqrt(3) + 1)**2)*(x + 1)*elliptic_f(asin((x + 1 +
sqrt(3))/(x - sqrt(3) + 1)), -7 + 4*sqrt(3))/(sqrt((-x - 1)/(x - sqrt(3) + 1)**
2)*sqrt(-x**3 - 1))

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Mathematica [C]  time = 0.500513, size = 269, normalized size = 6.11 \[ -\frac{2 \sqrt{6} \sqrt{\frac{i (x+1)}{\sqrt{3}+3 i}} \left (4 i \sqrt{-2 i x+\sqrt{3}+i} \sqrt{x^2-x+1} \Pi \left (\frac{2 i \sqrt{3}}{-3+(2+i) \sqrt{3}};\sin ^{-1}\left (\frac{\sqrt{-2 i x+\sqrt{3}+i}}{\sqrt{2} \sqrt [4]{3}}\right )|\frac{2 \sqrt{3}}{3 i+\sqrt{3}}\right )+\sqrt{2 i x+\sqrt{3}-i} \left (\left (\sqrt{3}+(-2-i)\right ) x-i \sqrt{3}+(1+2 i)\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{-2 i x+\sqrt{3}+i}}{\sqrt{2} \sqrt [4]{3}}\right )|\frac{2 \sqrt{3}}{3 i+\sqrt{3}}\right )\right )}{\left (-3+(2+i) \sqrt{3}\right ) \sqrt{-2 i x+\sqrt{3}+i} \sqrt{-x^3-1}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(-2*Sqrt[6]*Sqrt[(I*(1 + x))/(3*I + Sqrt[3])]*(Sqrt[-I + Sqrt[3] + (2*I)*x]*((1
+ 2*I) - I*Sqrt[3] + ((-2 - I) + Sqrt[3])*x)*EllipticF[ArcSin[Sqrt[I + Sqrt[3] -
(2*I)*x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(3*I + Sqrt[3])] + (4*I)*Sqrt[I + Sqrt
[3] - (2*I)*x]*Sqrt[1 - x + x^2]*EllipticPi[((2*I)*Sqrt[3])/(-3 + (2 + I)*Sqrt[3
]), ArcSin[Sqrt[I + Sqrt[3] - (2*I)*x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(3*I + Sq
rt[3])]))/((-3 + (2 + I)*Sqrt[3])*Sqrt[I + Sqrt[3] - (2*I)*x]*Sqrt[-1 - x^3])

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Maple [C]  time = 0.09, size = 247, normalized size = 5.6 \[{-{\frac{2\,i}{3}}\sqrt{3}\sqrt{i \left ( x-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}\sqrt{{\frac{1+x}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}}\sqrt{-i \left ( x-{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}{\it EllipticF} \left ({\frac{\sqrt{3}}{3}\sqrt{i \left ( x-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}},\sqrt{{\frac{i\sqrt{3}}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}} \right ){\frac{1}{\sqrt{-{x}^{3}-1}}}}-{\frac{4\,i}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}-\sqrt{3}}\sqrt{i \left ( x-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}\sqrt{{\frac{1+x}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}}\sqrt{-i \left ( x-{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}{\it EllipticPi} \left ({\frac{\sqrt{3}}{3}\sqrt{i \left ( x-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}},{\frac{i\sqrt{3}}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}-\sqrt{3}}},\sqrt{{\frac{i\sqrt{3}}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}} \right ){\frac{1}{\sqrt{-{x}^{3}-1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1+x+3^(1/2))/(1+x-3^(1/2))/(-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))-4*I*(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)/(3/2+1/2*I*3^(1/2)-3
^(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)
/(3/2+1/2*I*3^(1/2)-3^(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. \[ \int \frac{x + \sqrt{3} + 1}{\sqrt{-x^{3} - 1}{\left (x - \sqrt{3} + 1\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.349794, size = 143, normalized size = 3.25 \[ -\frac{1}{3} \, \sqrt{3} \sqrt{2 \, \sqrt{3} + 3} \arctan \left (\frac{2340 \, x^{4} - 4680 \, x^{3} + 6516 \, x^{2} + \sqrt{3}{\left (1351 \, x^{4} - 2702 \, x^{3} + 3762 \, x^{2} - 3284 \, x + 1060\right )} - 5688 \, x + 1836}{2 \, \sqrt{-x^{3} - 1}{\left (627 \, x^{2} + 2 \, \sqrt{3}{\left (181 \, x^{2} - 265 \, x + 97\right )} - 918 \, x + 336\right )} \sqrt{2 \, \sqrt{3} + 3}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*sqrt(3)*sqrt(2*sqrt(3) + 3)*arctan(1/2*(2340*x^4 - 4680*x^3 + 6516*x^2 + sq
rt(3)*(1351*x^4 - 2702*x^3 + 3762*x^2 - 3284*x + 1060) - 5688*x + 1836)/(sqrt(-x
^3 - 1)*(627*x^2 + 2*sqrt(3)*(181*x^2 - 265*x + 97) - 918*x + 336)*sqrt(2*sqrt(3
) + 3)))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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