3.86 \(\int \frac{x}{\sqrt{-1+x^3} \left (-10-6 \sqrt{3}+x^3\right )} \, dx\)

Optimal. Leaf size=222 \[ \frac{\left (2-\sqrt{3}\right ) \tan ^{-1}\left (\frac{\sqrt [4]{3} \left (1-\sqrt{3}\right ) (1-x)}{\sqrt{2} \sqrt{x^3-1}}\right )}{6 \sqrt{2} \sqrt [4]{3}}+\frac{\left (2-\sqrt{3}\right ) \tan ^{-1}\left (\frac{\sqrt [4]{3} \left (2 x+\sqrt{3}+1\right )}{\sqrt{2} \sqrt{x^3-1}}\right )}{3 \sqrt{2} \sqrt [4]{3}}+\frac{\left (2-\sqrt{3}\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{3} \left (1+\sqrt{3}\right ) (1-x)}{\sqrt{2} \sqrt{x^3-1}}\right )}{2 \sqrt{2} 3^{3/4}}-\frac{\left (2-\sqrt{3}\right ) \tanh ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt{x^3-1}}{\sqrt{2} 3^{3/4}}\right )}{3 \sqrt{2} 3^{3/4}} \]

[Out]

((2 - Sqrt[3])*ArcTan[(3^(1/4)*(1 - Sqrt[3])*(1 - x))/(Sqrt[2]*Sqrt[-1 + x^3])])
/(6*Sqrt[2]*3^(1/4)) + ((2 - Sqrt[3])*ArcTan[(3^(1/4)*(1 + Sqrt[3] + 2*x))/(Sqrt
[2]*Sqrt[-1 + x^3])])/(3*Sqrt[2]*3^(1/4)) + ((2 - Sqrt[3])*ArcTanh[(3^(1/4)*(1 +
Sqrt[3])*(1 - x))/(Sqrt[2]*Sqrt[-1 + x^3])])/(2*Sqrt[2]*3^(3/4)) - ((2 - Sqrt[3
])*ArcTanh[((1 - Sqrt[3])*Sqrt[-1 + x^3])/(Sqrt[2]*3^(3/4))])/(3*Sqrt[2]*3^(3/4)
)

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Rubi [A]  time = 0.100172, antiderivative size = 222, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04 \[ \frac{\left (2-\sqrt{3}\right ) \tan ^{-1}\left (\frac{\sqrt [4]{3} \left (1-\sqrt{3}\right ) (1-x)}{\sqrt{2} \sqrt{x^3-1}}\right )}{6 \sqrt{2} \sqrt [4]{3}}+\frac{\left (2-\sqrt{3}\right ) \tan ^{-1}\left (\frac{\sqrt [4]{3} \left (2 x+\sqrt{3}+1\right )}{\sqrt{2} \sqrt{x^3-1}}\right )}{3 \sqrt{2} \sqrt [4]{3}}+\frac{\left (2-\sqrt{3}\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{3} \left (1+\sqrt{3}\right ) (1-x)}{\sqrt{2} \sqrt{x^3-1}}\right )}{2 \sqrt{2} 3^{3/4}}-\frac{\left (2-\sqrt{3}\right ) \tanh ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt{x^3-1}}{\sqrt{2} 3^{3/4}}\right )}{3 \sqrt{2} 3^{3/4}} \]

Antiderivative was successfully verified.

[In]  Int[x/(Sqrt[-1 + x^3]*(-10 - 6*Sqrt[3] + x^3)),x]

[Out]

((2 - Sqrt[3])*ArcTan[(3^(1/4)*(1 - Sqrt[3])*(1 - x))/(Sqrt[2]*Sqrt[-1 + x^3])])
/(6*Sqrt[2]*3^(1/4)) + ((2 - Sqrt[3])*ArcTan[(3^(1/4)*(1 + Sqrt[3] + 2*x))/(Sqrt
[2]*Sqrt[-1 + x^3])])/(3*Sqrt[2]*3^(1/4)) + ((2 - Sqrt[3])*ArcTanh[(3^(1/4)*(1 +
Sqrt[3])*(1 - x))/(Sqrt[2]*Sqrt[-1 + x^3])])/(2*Sqrt[2]*3^(3/4)) - ((2 - Sqrt[3
])*ArcTanh[((1 - Sqrt[3])*Sqrt[-1 + x^3])/(Sqrt[2]*3^(3/4))])/(3*Sqrt[2]*3^(3/4)
)

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Rubi in Sympy [A]  time = 11.817, size = 53, normalized size = 0.24 \[ \frac{x^{2} \sqrt{x^{3} - 1} \operatorname{appellf_{1}}{\left (\frac{2}{3},\frac{1}{2},1,\frac{5}{3},x^{3},- \frac{x^{3}}{- 6 \sqrt{3} - 10} \right )}}{2 \left (10 + 6 \sqrt{3}\right ) \sqrt{- x^{3} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x/(-10+x**3-6*3**(1/2))/(x**3-1)**(1/2),x)

[Out]

x**2*sqrt(x**3 - 1)*appellf1(2/3, 1/2, 1, 5/3, x**3, -x**3/(-6*sqrt(3) - 10))/(2
*(10 + 6*sqrt(3))*sqrt(-x**3 + 1))

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Mathematica [C]  time = 0.623126, size = 196, normalized size = 0.88 \[ -\frac{10 \left (26+15 \sqrt{3}\right ) x^2 F_1\left (\frac{2}{3};\frac{1}{2},1;\frac{5}{3};x^3,\frac{x^3}{10+6 \sqrt{3}}\right )}{\left (5+3 \sqrt{3}\right ) \left (-x^3+6 \sqrt{3}+10\right ) \sqrt{x^3-1} \left (3 x^3 \left (F_1\left (\frac{5}{3};\frac{1}{2},2;\frac{8}{3};x^3,\frac{x^3}{10+6 \sqrt{3}}\right )+\left (5+3 \sqrt{3}\right ) F_1\left (\frac{5}{3};\frac{3}{2},1;\frac{8}{3};x^3,\frac{x^3}{10+6 \sqrt{3}}\right )\right )+10 \left (5+3 \sqrt{3}\right ) F_1\left (\frac{2}{3};\frac{1}{2},1;\frac{5}{3};x^3,\frac{x^3}{10+6 \sqrt{3}}\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[x/(Sqrt[-1 + x^3]*(-10 - 6*Sqrt[3] + x^3)),x]

[Out]

(-10*(26 + 15*Sqrt[3])*x^2*AppellF1[2/3, 1/2, 1, 5/3, x^3, x^3/(10 + 6*Sqrt[3])]
)/((5 + 3*Sqrt[3])*(10 + 6*Sqrt[3] - x^3)*Sqrt[-1 + x^3]*(10*(5 + 3*Sqrt[3])*App
ellF1[2/3, 1/2, 1, 5/3, x^3, x^3/(10 + 6*Sqrt[3])] + 3*x^3*(AppellF1[5/3, 1/2, 2
, 8/3, x^3, x^3/(10 + 6*Sqrt[3])] + (5 + 3*Sqrt[3])*AppellF1[5/3, 3/2, 1, 8/3, x
^3, x^3/(10 + 6*Sqrt[3])])))

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Maple [C]  time = 0.257, size = 349, normalized size = 1.6 \[{\frac{ \left ( -1-\sqrt{3} \right ) \left ( -{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}{18+9\,\sqrt{3}}\sqrt{{\frac{-1+x}{-{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}}}\sqrt{{\frac{1}{{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}} \left ( x+{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) }}\sqrt{{\frac{1}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}} \left ( x+{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) }}{\it EllipticPi} \left ( \sqrt{{\frac{-1+x}{-{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}}},-{\frac{ \left ({\frac{3}{2}}+{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}{3}},\sqrt{{\frac{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}{{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}}} \right ){\frac{1}{\sqrt{{x}^{3}-1}}}}-{\frac{\sqrt{2}}{18}\sum _{{\it \_alpha}={\it RootOf} \left ({{\it \_Z}}^{2}+ \left ( 1+\sqrt{3} \right ){\it \_Z}+2\,\sqrt{3}+4 \right ) }{\frac{ \left ( -{\it \_alpha}\,\sqrt{3}+{\it \_alpha}+2 \right ) \left ( -i\sqrt{3}-3 \right ) \left ( 1+2\,{\it \_alpha}-{\it \_alpha}\,\sqrt{3} \right ) }{-\sqrt{3}-2\,{\it \_alpha}-1}\sqrt{{\frac{-1+x}{-i\sqrt{3}-3}}}\sqrt{{\frac{2\,x+1-i\sqrt{3}}{-i\sqrt{3}+3}}}\sqrt{{\frac{2\,x+1+i\sqrt{3}}{i\sqrt{3}+3}}}{\it EllipticPi} \left ( \sqrt{{\frac{-1+x}{-{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}}},-{\frac{i}{2}}{\it \_alpha}+{\frac{i}{3}}{\it \_alpha}\,\sqrt{3}-{\frac{{\it \_alpha}\,\sqrt{3}}{2}}+{\it \_alpha}+{\frac{i}{6}}\sqrt{3}+{\frac{1}{2}},\sqrt{{\frac{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}{{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}}} \right ){\frac{1}{\sqrt{{x}^{3}-1}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(-10+x^3-6*3^(1/2))/(x^3-1)^(1/2),x)

[Out]

1/9*(-1-3^(1/2))/(2+3^(1/2))*(-3/2-1/2*I*3^(1/2))*((-1+x)/(-3/2-1/2*I*3^(1/2)))^
(1/2)*((x+1/2-1/2*I*3^(1/2))/(3/2-1/2*I*3^(1/2)))^(1/2)*((x+1/2+1/2*I*3^(1/2))/(
3/2+1/2*I*3^(1/2)))^(1/2)/(x^3-1)^(1/2)*3^(1/2)*EllipticPi(((-1+x)/(-3/2-1/2*I*3
^(1/2)))^(1/2),-1/3*(3/2+1/2*I*3^(1/2))*3^(1/2),((3/2+1/2*I*3^(1/2))/(3/2-1/2*I*
3^(1/2)))^(1/2))-1/18*2^(1/2)*sum((-_alpha*3^(1/2)+_alpha+2)/(-3^(1/2)-2*_alpha-
1)*(-I*3^(1/2)-3)*((-1+x)/(-I*3^(1/2)-3))^(1/2)*((2*x+1-I*3^(1/2))/(-I*3^(1/2)+3
))^(1/2)*((2*x+1+I*3^(1/2))/(I*3^(1/2)+3))^(1/2)/(x^3-1)^(1/2)*(1+2*_alpha-_alph
a*3^(1/2))*EllipticPi(((-1+x)/(-3/2-1/2*I*3^(1/2)))^(1/2),-1/2*I*_alpha+1/3*I*_a
lpha*3^(1/2)-1/2*_alpha*3^(1/2)+_alpha+1/6*I*3^(1/2)+1/2,((3/2+1/2*I*3^(1/2))/(3
/2-1/2*I*3^(1/2)))^(1/2)),_alpha=RootOf(_Z^2+(1+3^(1/2))*_Z+2*3^(1/2)+4))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{{\left (x^{3} - 6 \, \sqrt{3} - 10\right )} \sqrt{x^{3} - 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/((x^3 - 6*sqrt(3) - 10)*sqrt(x^3 - 1)),x, algorithm="maxima")

[Out]

integrate(x/((x^3 - 6*sqrt(3) - 10)*sqrt(x^3 - 1)), x)

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/((x^3 - 6*sqrt(3) - 10)*sqrt(x^3 - 1)),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{\sqrt{\left (x - 1\right ) \left (x^{2} + x + 1\right )} \left (x^{3} - 6 \sqrt{3} - 10\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(-10+x**3-6*3**(1/2))/(x**3-1)**(1/2),x)

[Out]

Integral(x/(sqrt((x - 1)*(x**2 + x + 1))*(x**3 - 6*sqrt(3) - 10)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{{\left (x^{3} - 6 \, \sqrt{3} - 10\right )} \sqrt{x^{3} - 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/((x^3 - 6*sqrt(3) - 10)*sqrt(x^3 - 1)),x, algorithm="giac")

[Out]

integrate(x/((x^3 - 6*sqrt(3) - 10)*sqrt(x^3 - 1)), x)