3.43 \(\int \frac{2^{2/3}-2 x}{(2^{2/3}+x) \sqrt{1+x^3}} \, dx\)

Optimal. Leaf size=37 \[ \frac{2\ 2^{2/3} \tan ^{-1}\left (\frac{\sqrt{3} \left (\sqrt [3]{2} x+1\right )}{\sqrt{x^3+1}}\right )}{\sqrt{3}} \]

[Out]

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

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Rubi [A]  time = 0.104899, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {2137, 203} \[ \frac{2\ 2^{2/3} \tan ^{-1}\left (\frac{\sqrt{3} \left (\sqrt [3]{2} x+1\right )}{\sqrt{x^3+1}}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2137

Int[((e_) + (f_.)*(x_))/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^3]), x_Symbol] :> Dist[(2*e)/d, Subst[Int[
1/(1 + 3*a*x^2), x], x, (1 + (2*d*x)/c)/Sqrt[a + b*x^3]], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[d*e - c*f,
0] && EqQ[b*c^3 - 4*a*d^3, 0] && EqQ[2*d*e + c*f, 0]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{2^{2/3}-2 x}{\left (2^{2/3}+x\right ) \sqrt{1+x^3}} \, dx &=\left (2\ 2^{2/3}\right ) \operatorname{Subst}\left (\int \frac{1}{1+3 x^2} \, dx,x,\frac{1+\sqrt [3]{2} x}{\sqrt{1+x^3}}\right )\\ &=\frac{2\ 2^{2/3} \tan ^{-1}\left (\frac{\sqrt{3} \left (1+\sqrt [3]{2} x\right )}{\sqrt{1+x^3}}\right )}{\sqrt{3}}\\ \end{align*}

Mathematica [C]  time = 0.430227, size = 326, normalized size = 8.81 \[ -\frac{4 \sqrt [6]{2} \sqrt{\frac{i (x+1)}{\sqrt{3}+3 i}} \left (\sqrt{2 i x+\sqrt{3}-i} \left (\left (-3 i \sqrt [3]{2}+4 \sqrt{3}+\sqrt [3]{2} \sqrt{3}\right ) x+\sqrt [3]{2} \sqrt{3}-2 \sqrt{3}+3 i \sqrt [3]{2}+6 i\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{-2 i x+\sqrt{3}+i}}{\sqrt{2} \sqrt [4]{3}}\right )|\frac{2 \sqrt{3}}{3 i+\sqrt{3}}\right )-6 i \sqrt{3} \sqrt{-2 i x+\sqrt{3}+i} \sqrt{x^2-x+1} \Pi \left (\frac{2 \sqrt{3}}{i+2 i 2^{2/3}+\sqrt{3}};\sin ^{-1}\left (\frac{\sqrt{-2 i x+\sqrt{3}+i}}{\sqrt{2} \sqrt [4]{3}}\right )|\frac{2 \sqrt{3}}{3 i+\sqrt{3}}\right )\right )}{\sqrt{3} \left (1+2\ 2^{2/3}-i \sqrt{3}\right ) \sqrt{-2 i x+\sqrt{3}+i} \sqrt{x^3+1}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [C]  time = 0.034, size = 258, normalized size = 7. \begin{align*} -4\,{\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 ) }+6\,{\frac{{2}^{2/3} \left ( 3/2-i/2\sqrt{3} \right ) }{\sqrt{{x}^{3}+1} \left ({2}^{2/3}-1 \right ) }\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 EllipticPi} \left ( \sqrt{{\frac{1+x}{3/2-i/2\sqrt{3}}}},{\frac{-3/2+i/2\sqrt{3}}{{2}^{2/3}-1}},\sqrt{{\frac{-3/2+i/2\sqrt{3}}{-3/2-i/2\sqrt{3}}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2^(2/3)-2*x)/(2^(2/3)+x)/(x^3+1)^(1/2),x)

[Out]

-4*(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))+6*2^(2/3)*(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)/(2^(2/3)-1)*EllipticPi(((1+x)/(3/2-1/2*I*3^(1/2)))^(1/2),(-3/2+1/2*I*3^(1/2))/(2^(2/3)-1),((-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. \begin{align*} -\int \frac{2 \, x - 2^{\frac{2}{3}}}{\sqrt{x^{3} + 1}{\left (x + 2^{\frac{2}{3}}\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2^(2/3)-2*x)/(2^(2/3)+x)/(x^3+1)^(1/2),x, algorithm="maxima")

[Out]

-integrate((2*x - 2^(2/3))/(sqrt(x^3 + 1)*(x + 2^(2/3))), x)

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Fricas [B]  time = 3.099, size = 198, normalized size = 5.35 \begin{align*} \frac{1}{3} \, \sqrt{6} 2^{\frac{1}{6}} \arctan \left (-\frac{\sqrt{6} 2^{\frac{1}{6}}{\left (2 \, x^{5} + 2 \, x^{2} - 2^{\frac{2}{3}}{\left (7 \, x^{4} + 4 \, x\right )} - 2^{\frac{1}{3}}{\left (5 \, x^{3} + 2\right )}\right )} \sqrt{x^{3} + 1}}{12 \,{\left (2 \, x^{6} + 3 \, x^{3} + 1\right )}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2^(2/3)-2*x)/(2^(2/3)+x)/(x^3+1)^(1/2),x, algorithm="fricas")

[Out]

1/3*sqrt(6)*2^(1/6)*arctan(-1/12*sqrt(6)*2^(1/6)*(2*x^5 + 2*x^2 - 2^(2/3)*(7*x^4 + 4*x) - 2^(1/3)*(5*x^3 + 2))
*sqrt(x^3 + 1)/(2*x^6 + 3*x^3 + 1))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int - \frac{2^{\frac{2}{3}}}{x \sqrt{x^{3} + 1} + 2^{\frac{2}{3}} \sqrt{x^{3} + 1}}\, dx - \int \frac{2 x}{x \sqrt{x^{3} + 1} + 2^{\frac{2}{3}} \sqrt{x^{3} + 1}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2**(2/3)-2*x)/(2**(2/3)+x)/(x**3+1)**(1/2),x)

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{2 \, x - 2^{\frac{2}{3}}}{\sqrt{x^{3} + 1}{\left (x + 2^{\frac{2}{3}}\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2^(2/3)-2*x)/(2^(2/3)+x)/(x^3+1)^(1/2),x, algorithm="giac")

[Out]

integrate(-(2*x - 2^(2/3))/(sqrt(x^3 + 1)*(x + 2^(2/3))), x)