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

Optimal. Leaf size=210 \[ -\frac{\left (2+\sqrt{3}\right ) \tan ^{-1}\left (\frac{\sqrt [4]{3} \left (-2 x-\sqrt{3}+1\right )}{\sqrt{2} \sqrt{x^3+1}}\right )}{3 \sqrt{2} \sqrt [4]{3}}-\frac{\left (2+\sqrt{3}\right ) \tan ^{-1}\left (\frac{\sqrt [4]{3} \left (1+\sqrt{3}\right ) (x+1)}{\sqrt{2} \sqrt{x^3+1}}\right )}{6 \sqrt{2} \sqrt [4]{3}}+\frac{\left (2+\sqrt{3}\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{3} \left (1-\sqrt{3}\right ) (x+1)}{\sqrt{2} \sqrt{x^3+1}}\right )}{2 \sqrt{2} 3^{3/4}}+\frac{\left (2+\sqrt{3}\right ) \tanh ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt{x^3+1}}{\sqrt{2} 3^{3/4}}\right )}{3 \sqrt{2} 3^{3/4}} \]

[Out]

-((2 + Sqrt[3])*ArcTan[(3^(1/4)*(1 - Sqrt[3] - 2*x))/(Sqrt[2]*Sqrt[1 + x^3])])/(
3*Sqrt[2]*3^(1/4)) - ((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])*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 = 0.104841, antiderivative size = 210, 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 \[ -\frac{\left (2+\sqrt{3}\right ) \tan ^{-1}\left (\frac{\sqrt [4]{3} \left (-2 x-\sqrt{3}+1\right )}{\sqrt{2} \sqrt{x^3+1}}\right )}{3 \sqrt{2} \sqrt [4]{3}}-\frac{\left (2+\sqrt{3}\right ) \tan ^{-1}\left (\frac{\sqrt [4]{3} \left (1+\sqrt{3}\right ) (x+1)}{\sqrt{2} \sqrt{x^3+1}}\right )}{6 \sqrt{2} \sqrt [4]{3}}+\frac{\left (2+\sqrt{3}\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{3} \left (1-\sqrt{3}\right ) (x+1)}{\sqrt{2} \sqrt{x^3+1}}\right )}{2 \sqrt{2} 3^{3/4}}+\frac{\left (2+\sqrt{3}\right ) \tanh ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt{x^3+1}}{\sqrt{2} 3^{3/4}}\right )}{3 \sqrt{2} 3^{3/4}} \]

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] - 2*x))/(Sqrt[2]*Sqrt[1 + x^3])])/(
3*Sqrt[2]*3^(1/4)) - ((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])*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 = 7.69253, size = 34, normalized size = 0.16 \[ \frac{x^{2} \operatorname{appellf_{1}}{\left (\frac{2}{3},\frac{1}{2},1,\frac{5}{3},- x^{3},- \frac{x^{3}}{- 6 \sqrt{3} + 10} \right )}}{2 \left (- 6 \sqrt{3} + 10\right )} \]

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

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Mathematica [C]  time = 0.656094, size = 207, normalized size = 0.99 \[ \frac{10 \left (26-15 \sqrt{3}\right ) x^2 F_1\left (\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-x^3,\frac{1}{4} \left (5+3 \sqrt{3}\right ) x^3\right )}{\left (3 \sqrt{3}-5\right ) \left (-x^3+6 \sqrt{3}-10\right ) \sqrt{x^3+1} \left (\left (50-30 \sqrt{3}\right ) F_1\left (\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-x^3,\frac{1}{4} \left (5+3 \sqrt{3}\right ) x^3\right )-3 x^3 \left (F_1\left (\frac{5}{3};\frac{1}{2},2;\frac{8}{3};-x^3,\frac{1}{4} \left (5+3 \sqrt{3}\right ) x^3\right )+\left (5-3 \sqrt{3}\right ) F_1\left (\frac{5}{3};\frac{3}{2},1;\frac{8}{3};-x^3,\frac{1}{4} \left (5+3 \sqrt{3}\right ) x^3\right )\right )\right )} \]

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, ((5 + 3*Sqrt[3])*x^3)
/4])/((-5 + 3*Sqrt[3])*(-10 + 6*Sqrt[3] - x^3)*Sqrt[1 + x^3]*((50 - 30*Sqrt[3])*
AppellF1[2/3, 1/2, 1, 5/3, -x^3, ((5 + 3*Sqrt[3])*x^3)/4] - 3*x^3*(AppellF1[5/3,
1/2, 2, 8/3, -x^3, ((5 + 3*Sqrt[3])*x^3)/4] + (5 - 3*Sqrt[3])*AppellF1[5/3, 3/2
, 1, 8/3, -x^3, ((5 + 3*Sqrt[3])*x^3)/4])))

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Maple [C]  time = 0.266, size = 350, normalized size = 1.7 \[{\frac{ \left ( \sqrt{3}-1 \right ) \left ({\frac{3}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}{-18+9\,\sqrt{3}}\sqrt{{\frac{1+x}{{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}}}\sqrt{{\frac{1}{-{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}} \left ( x-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) }}\sqrt{{\frac{1}{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}} \left ( x-{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) }}{\it EllipticPi} \left ( \sqrt{{\frac{1+x}{{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}}},-{\frac{ \left ( -{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}{3}},\sqrt{{\frac{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}{-{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}}} \right ){\frac{1}{\sqrt{{x}^{3}+1}}}}-{\frac{\sqrt{2}}{18}\sum _{{\it \_alpha}={\it RootOf} \left ({{\it \_Z}}^{2}+ \left ( \sqrt{3}-1 \right ){\it \_Z}-2\,\sqrt{3}+4 \right ) }{\frac{ \left ( -{\it \_alpha}\,\sqrt{3}-{\it \_alpha}+2 \right ) \left ( -i\sqrt{3}+3 \right ) \left ( -1+2\,{\it \_alpha}+{\it \_alpha}\,\sqrt{3} \right ) }{-\sqrt{3}-2\,{\it \_alpha}+1}\sqrt{{\frac{1+x}{-i\sqrt{3}+3}}}\sqrt{{\frac{2\,x-1-i\sqrt{3}}{-i\sqrt{3}-3}}}\sqrt{{\frac{2\,x-1+i\sqrt{3}}{i\sqrt{3}-3}}}{\it EllipticPi} \left ( \sqrt{{\frac{1+x}{{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}}},{\frac{i}{2}}{\it \_alpha}+{\frac{i}{3}}{\it \_alpha}\,\sqrt{3}-{\frac{{\it \_alpha}\,\sqrt{3}}{2}}-{\it \_alpha}-{\frac{i}{6}}\sqrt{3}+{\frac{1}{2}},\sqrt{{\frac{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}{-{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}}} \right ){\frac{1}{\sqrt{{x}^{3}+1}}}}} \]

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*(3^(1/2)-1)/(-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*_alph
a*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+(3^(1/2)-1)*_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)