3.129 \(\int \frac{1+\sqrt{3}+x}{x \sqrt{1+x^3}} \, dx\)

Optimal. Leaf size=125 \[ \frac{2 \sqrt{2+\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{\sqrt [4]{3} \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{x^3+1}}-\frac{2}{3} \left (1+\sqrt{3}\right ) \tanh ^{-1}\left (\sqrt{x^3+1}\right ) \]

[Out]

(-2*(1 + Sqrt[3])*ArcTanh[Sqrt[1 + x^3]])/3 + (2*Sqrt[2 + Sqrt[3]]*(1 + x)*Sqrt[
(1 - x + x^2)/(1 + Sqrt[3] + x)^2]*EllipticF[ArcSin[(1 - Sqrt[3] + x)/(1 + Sqrt[
3] + x)], -7 - 4*Sqrt[3]])/(3^(1/4)*Sqrt[(1 + x)/(1 + Sqrt[3] + x)^2]*Sqrt[1 + x
^3])

_______________________________________________________________________________________

Rubi [A]  time = 0.102632, antiderivative size = 125, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238 \[ \frac{2 \sqrt{2+\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{\sqrt [4]{3} \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{x^3+1}}-\frac{2}{3} \left (1+\sqrt{3}\right ) \tanh ^{-1}\left (\sqrt{x^3+1}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-2*(1 + Sqrt[3])*ArcTanh[Sqrt[1 + x^3]])/3 + (2*Sqrt[2 + Sqrt[3]]*(1 + x)*Sqrt[
(1 - x + x^2)/(1 + Sqrt[3] + x)^2]*EllipticF[ArcSin[(1 - Sqrt[3] + x)/(1 + Sqrt[
3] + x)], -7 - 4*Sqrt[3]])/(3^(1/4)*Sqrt[(1 + x)/(1 + Sqrt[3] + x)^2]*Sqrt[1 + x
^3])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 10.7831, size = 117, normalized size = 0.94 \[ - \left (\frac{2}{3} + \frac{2 \sqrt{3}}{3}\right ) \operatorname{atanh}{\left (\sqrt{x^{3} + 1} \right )} + \frac{2 \cdot 3^{\frac{3}{4}} \sqrt{\frac{x^{2} - x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (x + 1\right ) F\left (\operatorname{asin}{\left (\frac{x - \sqrt{3} + 1}{x + 1 + \sqrt{3}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{3 \sqrt{\frac{x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \sqrt{x^{3} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.787178, size = 149, normalized size = 1.19 \[ -\frac{2 \tanh ^{-1}\left (\sqrt{x^3+1}\right )}{\sqrt{3}}-\frac{2}{3} \tanh ^{-1}\left (\sqrt{x^3+1}\right )-\frac{2 \left (\sqrt [3]{-1}-x\right ) \sqrt{\frac{x+1}{1+\sqrt [3]{-1}}} \sqrt{-\frac{(-1)^{2/3} \left (x+(-1)^{2/3}\right )}{1+\sqrt [3]{-1}}} F\left (\sin ^{-1}\left (\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt [3]{-1}}} \sqrt{x^3+1}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*ArcTanh[Sqrt[1 + x^3]])/3 - (2*ArcTanh[Sqrt[1 + x^3]])/Sqrt[3] - (2*((-1)^(1
/3) - x)*Sqrt[(1 + x)/(1 + (-1)^(1/3))]*Sqrt[-(((-1)^(2/3)*((-1)^(2/3) + x))/(1
+ (-1)^(1/3)))]*EllipticF[ArcSin[Sqrt[(1 + (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1
)^(1/3)])/(Sqrt[(1 + (-1)^(2/3)*x)/(1 + (-1)^(1/3))]*Sqrt[1 + x^3])

_______________________________________________________________________________________

Maple [A]  time = 0.041, size = 132, normalized size = 1.1 \[ 2\,{\frac{3/2-i/2\sqrt{3}}{\sqrt{{x}^{3}+1}}\sqrt{{\frac{1+x}{3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x-1/2-i/2\sqrt{3}}{-3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x-1/2+i/2\sqrt{3}}{-3/2+i/2\sqrt{3}}}}{\it EllipticF} \left ( \sqrt{{\frac{1+x}{3/2-i/2\sqrt{3}}}},\sqrt{{\frac{-3/2+i/2\sqrt{3}}{-3/2-i/2\sqrt{3}}}} \right ) }-{\frac{2+2\,\sqrt{3}}{3}{\it Artanh} \left ( \sqrt{{x}^{3}+1} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1+x+3^(1/2))/x/(x^3+1)^(1/2),x)

[Out]

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)*EllipticF(((1+x)/(3/2-1/2*I*3^(1/2)))^(1/2),((-3/2+1/2*I*3^(1/2))/(-
3/2-1/2*I*3^(1/2)))^(1/2))-2/3*arctanh((x^3+1)^(1/2))*(1+3^(1/2))

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x + \sqrt{3} + 1}{\sqrt{x^{3} + 1} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x + \sqrt{3} + 1}{\sqrt{x^{3} + 1} x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((x + sqrt(3) + 1)/(sqrt(x^3 + 1)*x), x)

_______________________________________________________________________________________

Sympy [A]  time = 6.84871, size = 56, normalized size = 0.45 \[ \frac{x \Gamma \left (\frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{3}, \frac{1}{2} \\ \frac{4}{3} \end{matrix}\middle |{x^{3} e^{i \pi }} \right )}}{3 \Gamma \left (\frac{4}{3}\right )} - \frac{2 \sqrt{3} \operatorname{asinh}{\left (\frac{1}{x^{\frac{3}{2}}} \right )}}{3} - \frac{2 \operatorname{asinh}{\left (\frac{1}{x^{\frac{3}{2}}} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*gamma(1/3)*hyper((1/3, 1/2), (4/3,), x**3*exp_polar(I*pi))/(3*gamma(4/3)) - 2*
sqrt(3)*asinh(x**(-3/2))/3 - 2*asinh(x**(-3/2))/3

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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