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

Optimal. Leaf size=103 \[ \frac{2 \sqrt{2+\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{\sqrt [4]{3} \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{x^3+1}} \]

[Out]

(2*Sqrt[2 + Sqrt[3]]*(1 + x)*Sqrt[(1 - x + x^2)/(1 + Sqrt[3] + x)^2]*EllipticF[A
rcSin[(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])

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Rubi [A]  time = 0.0284474, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \frac{2 \sqrt{2+\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{\sqrt [4]{3} \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{x^3+1}} \]

Antiderivative was successfully verified.

[In]  Int[1/Sqrt[1 + x^3],x]

[Out]

(2*Sqrt[2 + Sqrt[3]]*(1 + x)*Sqrt[(1 - x + x^2)/(1 + Sqrt[3] + x)^2]*EllipticF[A
rcSin[(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])

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Rubi in Sympy [A]  time = 1.94914, size = 95, normalized size = 0.92 \[ \frac{2 \cdot 3^{\frac{3}{4}} \sqrt{\frac{x^{2} - x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (x + 1\right ) F\left (\operatorname{asin}{\left (\frac{x - \sqrt{3} + 1}{x + 1 + \sqrt{3}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{3 \sqrt{\frac{x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \sqrt{x^{3} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*3**(3/4)*sqrt((x**2 - x + 1)/(x + 1 + sqrt(3))**2)*sqrt(sqrt(3) + 2)*(x + 1)*e
lliptic_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))

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Mathematica [A]  time = 0.0723485, size = 88, normalized size = 0.85 \[ \frac{2 \sqrt [6]{-1} \sqrt{-\sqrt [6]{-1} \left (x+(-1)^{2/3}\right )} \sqrt{(-1)^{2/3} x^2+\sqrt [3]{-1} x+1} F\left (\sin ^{-1}\left (\frac{\sqrt{-(-1)^{5/6} (x+1)}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )}{\sqrt [4]{3} \sqrt{x^3+1}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[1/Sqrt[1 + x^3],x]

[Out]

(2*(-1)^(1/6)*Sqrt[-((-1)^(1/6)*((-1)^(2/3) + x))]*Sqrt[1 + (-1)^(1/3)*x + (-1)^
(2/3)*x^2]*EllipticF[ArcSin[Sqrt[-((-1)^(5/6)*(1 + x))]/3^(1/4)], (-1)^(1/3)])/(
3^(1/4)*Sqrt[1 + x^3])

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Maple [A]  time = 0.02, size = 116, normalized size = 1.1 \[ 2\,{\frac{3/2-i/2\sqrt{3}}{\sqrt{{x}^{3}+1}}\sqrt{{\frac{1+x}{3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x-1/2-i/2\sqrt{3}}{-3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x-1/2+i/2\sqrt{3}}{-3/2+i/2\sqrt{3}}}}{\it EllipticF} \left ( \sqrt{{\frac{1+x}{3/2-i/2\sqrt{3}}}},\sqrt{{\frac{-3/2+i/2\sqrt{3}}{-3/2-i/2\sqrt{3}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(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))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/sqrt(x^3 + 1), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(1/sqrt(x^3 + 1), x)

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Sympy [A]  time = 1.67638, size = 27, normalized size = 0.26 \[ \frac{x \Gamma \left (\frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{3}, \frac{1}{2} \\ \frac{4}{3} \end{matrix}\middle |{x^{3} e^{i \pi }} \right )}}{3 \Gamma \left (\frac{4}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(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))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/sqrt(x^3 + 1), x)