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

Optimal. Leaf size=38 \[ -\frac{2\ 2^{2/3} \tanh ^{-1}\left (\frac{\sqrt{3} \left (1-\sqrt [3]{2} x\right )}{\sqrt{x^3-1}}\right )}{\sqrt{3}} \]

[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 = 0.112427, antiderivative size = 38, 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} \[ -\frac{2\ 2^{2/3} \tanh ^{-1}\left (\frac{\sqrt{3} \left (1-\sqrt [3]{2} x\right )}{\sqrt{x^3-1}}\right )}{\sqrt{3}} \]

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 (\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 )\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 = 0.304679, size = 325, normalized size = 8.55 \[ -\frac{4 \sqrt [6]{2} \sqrt{-\frac{i (x-1)}{\sqrt{3}+3 i}} \left (6 i \sqrt{3} \sqrt{2 i x+\sqrt{3}+i} \sqrt{x^2+x+1} \Pi \left (\frac{2 \sqrt{3}}{i+2 i 2^{2/3}+\sqrt{3}};\sin ^{-1}\left (\frac{\sqrt{2 i x+\sqrt{3}+i}}{\sqrt{2} \sqrt [4]{3}}\right )|\frac{2 \sqrt{3}}{3 i+\sqrt{3}}\right )+\sqrt{-2 i x+\sqrt{3}-i} \left (\left (-3 i \sqrt [3]{2}+4 \sqrt{3}+\sqrt [3]{2} \sqrt{3}\right ) x-\sqrt [3]{2} \sqrt{3}+2 \sqrt{3}-3 i \sqrt [3]{2}-6 i\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{2 i x+\sqrt{3}+i}}{\sqrt{2} \sqrt [4]{3}}\right )|\frac{2 \sqrt{3}}{3 i+\sqrt{3}}\right )\right )}{\sqrt{3} \left (1+2\ 2^{2/3}-i \sqrt{3}\right ) \sqrt{2 i x+\sqrt{3}+i} \sqrt{x^3-1}} \]

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 = 0.029, size = 262, normalized size = 6.9 \begin{align*} -4\,{\frac{-3/2-i/2\sqrt{3}}{\sqrt{{x}^{3}-1}}\sqrt{{\frac{x-1}{-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{x-1}{-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{x-1}{-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{x-1}{-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))*((x-1)/(-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(((x-1)/(-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))*((x-1)/(-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(((x-1)/(-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 = 2.42924, size = 717, normalized size = 18.87 \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)