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

Optimal. Leaf size=39 \[ \frac{2\ 2^{2/3} \tanh ^{-1}\left (\frac{\sqrt{3} \left (\sqrt [3]{2} x+1\right )}{\sqrt{-x^3-1}}\right )}{\sqrt{3}} \]

[Out]

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

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Rubi [A]  time = 0.113543, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {2137, 206} \[ \frac{2\ 2^{2/3} \tanh ^{-1}\left (\frac{\sqrt{3} \left (\sqrt [3]{2} x+1\right )}{\sqrt{-x^3-1}}\right )}{\sqrt{3}} \]

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)*ArcTanh[(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 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[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} \tanh ^{-1}\left (\frac{\sqrt{3} \left (1+\sqrt [3]{2} x\right )}{\sqrt{-1-x^3}}\right )}{\sqrt{3}}\\ \end{align*}

Mathematica [C]  time = 0.283514, size = 328, normalized size = 8.41 \[ -\frac{4 \sqrt [6]{2} \sqrt{\frac{i (x+1)}{\sqrt{3}+3 i}} \left (\sqrt{2 i x+\sqrt{3}-i} \left (\left (-3 i \sqrt [3]{2}+4 \sqrt{3}+\sqrt [3]{2} \sqrt{3}\right ) x+\sqrt [3]{2} \sqrt{3}-2 \sqrt{3}+3 i \sqrt [3]{2}+6 i\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{-2 i x+\sqrt{3}+i}}{\sqrt{2} \sqrt [4]{3}}\right )|\frac{2 \sqrt{3}}{3 i+\sqrt{3}}\right )-6 i \sqrt{3} \sqrt{-2 i x+\sqrt{3}+i} \sqrt{x^2-x+1} \Pi \left (\frac{2 \sqrt{3}}{i+2 i 2^{2/3}+\sqrt{3}};\sin ^{-1}\left (\frac{\sqrt{-2 i x+\sqrt{3}+i}}{\sqrt{2} \sqrt [4]{3}}\right )|\frac{2 \sqrt{3}}{3 i+\sqrt{3}}\right )\right )}{\sqrt{3} \left (1+2\ 2^{2/3}-i \sqrt{3}\right ) \sqrt{-2 i x+\sqrt{3}+i} \sqrt{-x^3-1}} \]

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 = 0.032, size = 249, normalized size = 6.4 \begin{align*}{{\frac{4\,i}{3}}\sqrt{3}\sqrt{i \left ( x-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}\sqrt{{\frac{1+x}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}}\sqrt{-i \left ( x-{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}{\it EllipticF} \left ({\frac{\sqrt{3}}{3}\sqrt{i \left ( x-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}},\sqrt{{\frac{i\sqrt{3}}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}} \right ){\frac{1}{\sqrt{-{x}^{3}-1}}}}-{\frac{2\,i{2}^{{\frac{2}{3}}}\sqrt{3}}{{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3}+{2}^{{\frac{2}{3}}}}\sqrt{i \left ( x-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}\sqrt{{\frac{1+x}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}}\sqrt{-i \left ( x-{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}{\it EllipticPi} \left ({\frac{\sqrt{3}}{3}\sqrt{i \left ( x-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}},{\frac{i\sqrt{3}}{{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3}+{2}^{{\frac{2}{3}}}}},\sqrt{{\frac{i\sqrt{3}}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}} \right ){\frac{1}{\sqrt{-{x}^{3}-1}}}} \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*I*3^(1/2)*(I*(x-1/2-1/2*I*3^(1/2))*3^(1/2))^(1/2)*((1+x)/(3/2+1/2*I*3^(1/2)))^(1/2)*(-I*(x-1/2+1/2*I*3^(1/
2))*3^(1/2))^(1/2)/(-x^3-1)^(1/2)*EllipticF(1/3*3^(1/2)*(I*(x-1/2-1/2*I*3^(1/2))*3^(1/2))^(1/2),(I*3^(1/2)/(3/
2+1/2*I*3^(1/2)))^(1/2))-2*I*2^(2/3)*3^(1/2)*(I*(x-1/2-1/2*I*3^(1/2))*3^(1/2))^(1/2)*((1+x)/(3/2+1/2*I*3^(1/2)
))^(1/2)*(-I*(x-1/2+1/2*I*3^(1/2))*3^(1/2))^(1/2)/(-x^3-1)^(1/2)/(1/2+1/2*I*3^(1/2)+2^(2/3))*EllipticPi(1/3*3^
(1/2)*(I*(x-1/2-1/2*I*3^(1/2))*3^(1/2))^(1/2),I*3^(1/2)/(1/2+1/2*I*3^(1/2)+2^(2/3)),(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 = 2.22002, size = 718, normalized size = 18.41 \begin{align*} \frac{1}{6} \, \sqrt{6} 2^{\frac{1}{6}} \log \left (\frac{x^{18} - 1440 \, x^{15} + 17400 \, x^{12} + 21056 \, x^{9} - 10368 \, x^{6} - 15360 \, x^{3} - 2 \, \sqrt{6} 2^{\frac{1}{6}}{\left (126 \, x^{14} - 2664 \, x^{11} + 4608 \, x^{5} + 2304 \, x^{2} + 2^{\frac{2}{3}}{\left (x^{16} - 310 \, x^{13} + 2332 \, x^{10} + 2656 \, x^{7} - 256 \, x^{4} - 512 \, x\right )} - 2^{\frac{1}{3}}{\left (17 \, x^{15} - 1058 \, x^{12} + 2528 \, x^{9} + 5408 \, x^{6} + 2560 \, x^{3} + 512\right )}\right )} \sqrt{-x^{3} - 1} - 24 \cdot 2^{\frac{2}{3}}{\left (x^{17} - 121 \, x^{14} + 478 \, x^{11} + 1144 \, x^{8} + 608 \, x^{5} + 64 \, x^{2}\right )} + 48 \cdot 2^{\frac{1}{3}}{\left (5 \, x^{16} - 176 \, x^{13} + 83 \, x^{10} + 680 \, x^{7} + 544 \, x^{4} + 128 \, x\right )} - 2048}{x^{18} + 24 \, x^{15} + 240 \, x^{12} + 1280 \, x^{9} + 3840 \, x^{6} + 6144 \, x^{3} + 4096}\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]

1/6*sqrt(6)*2^(1/6)*log((x^18 - 1440*x^15 + 17400*x^12 + 21056*x^9 - 10368*x^6 - 15360*x^3 - 2*sqrt(6)*2^(1/6)
*(126*x^14 - 2664*x^11 + 4608*x^5 + 2304*x^2 + 2^(2/3)*(x^16 - 310*x^13 + 2332*x^10 + 2656*x^7 - 256*x^4 - 512
*x) - 2^(1/3)*(17*x^15 - 1058*x^12 + 2528*x^9 + 5408*x^6 + 2560*x^3 + 512))*sqrt(-x^3 - 1) - 24*2^(2/3)*(x^17
- 121*x^14 + 478*x^11 + 1144*x^8 + 608*x^5 + 64*x^2) + 48*2^(1/3)*(5*x^16 - 176*x^13 + 83*x^10 + 680*x^7 + 544
*x^4 + 128*x) - 2048)/(x^18 + 24*x^15 + 240*x^12 + 1280*x^9 + 3840*x^6 + 6144*x^3 + 4096))

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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)