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

Optimal. Leaf size=122 \[ -\frac{\sqrt{x^3+1}}{2 x^2}-\frac{\sqrt{2+\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{2 \sqrt [4]{3} \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{x^3+1}} \]

[Out]

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

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Rubi [A]  time = 0.0563906, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ -\frac{\sqrt{x^3+1}}{2 x^2}-\frac{\sqrt{2+\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{2 \sqrt [4]{3} \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{x^3+1}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^3*Sqrt[1 + x^3]),x]

[Out]

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

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Rubi in Sympy [A]  time = 4.084, size = 109, normalized size = 0.89 \[ - \frac{3^{\frac{3}{4}} \sqrt{\frac{x^{2} - x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (x + 1\right ) F\left (\operatorname{asin}{\left (\frac{x - \sqrt{3} + 1}{x + 1 + \sqrt{3}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{6 \sqrt{\frac{x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \sqrt{x^{3} + 1}} - \frac{\sqrt{x^{3} + 1}}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3**(3/4)*sqrt((x**2 - x + 1)/(x + 1 + sqrt(3))**2)*sqrt(sqrt(3) + 2)*(x + 1)*el
liptic_f(asin((x - sqrt(3) + 1)/(x + 1 + sqrt(3))), -7 - 4*sqrt(3))/(6*sqrt((x +
1)/(x + 1 + sqrt(3))**2)*sqrt(x**3 + 1)) - sqrt(x**3 + 1)/(2*x**2)

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Mathematica [A]  time = 0.183252, size = 104, normalized size = 0.85 \[ -\frac{3 x^3+\sqrt [6]{-1} 3^{3/4} \sqrt{-\sqrt [6]{-1} \left (x+(-1)^{2/3}\right )} \sqrt{(-1)^{2/3} x^2+\sqrt [3]{-1} x+1} x^2 F\left (\sin ^{-1}\left (\frac{\sqrt{-(-1)^{5/6} (x+1)}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )+3}{6 x^2 \sqrt{x^3+1}} \]

Warning: Unable to verify antiderivative.

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

[Out]

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

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Maple [A]  time = 0.028, size = 129, normalized size = 1.1 \[ -{\frac{1}{2\,{x}^{2}}\sqrt{{x}^{3}+1}}-{\frac{{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}{2}\sqrt{{\frac{1+x}{{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}}}\sqrt{{\frac{1}{-{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}} \left ( x-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) }}\sqrt{{\frac{1}{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}} \left ( x-{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) }}{\it EllipticF} \left ( \sqrt{{\frac{1+x}{{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}}},\sqrt{{\frac{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}{-{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}}} \right ){\frac{1}{\sqrt{{x}^{3}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^3/(x^3+1)^(1/2),x)

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/(sqrt(x^3 + 1)*x^3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(1/(sqrt(x^3 + 1)*x^3), x)

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Sympy [A]  time = 2.18446, size = 32, normalized size = 0.26 \[ \frac{\Gamma \left (- \frac{2}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{2}{3}, \frac{1}{2} \\ \frac{1}{3} \end{matrix}\middle |{x^{3} e^{i \pi }} \right )}}{3 x^{2} \Gamma \left (\frac{1}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/(sqrt(x^3 + 1)*x^3), x)