3.316 \(\int \frac{-1+x^3}{\sqrt [3]{2+x^3}} \, dx\)

Optimal. Leaf size=63 \[ \frac{1}{3} \left (x^3+2\right )^{2/3} x+\frac{5}{6} \log \left (\sqrt [3]{x^3+2}-x\right )-\frac{5 \tan ^{-1}\left (\frac{\frac{2 x}{\sqrt [3]{x^3+2}}+1}{\sqrt{3}}\right )}{3 \sqrt{3}} \]

[Out]

(x*(2 + x^3)^(2/3))/3 - (5*ArcTan[(1 + (2*x)/(2 + x^3)^(1/3))/Sqrt[3]])/(3*Sqrt[3]) + (5*Log[-x + (2 + x^3)^(1
/3)])/6

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Rubi [A]  time = 0.0112845, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {388, 239} \[ \frac{1}{3} \left (x^3+2\right )^{2/3} x+\frac{5}{6} \log \left (\sqrt [3]{x^3+2}-x\right )-\frac{5 \tan ^{-1}\left (\frac{\frac{2 x}{\sqrt [3]{x^3+2}}+1}{\sqrt{3}}\right )}{3 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

Int[(-1 + x^3)/(2 + x^3)^(1/3),x]

[Out]

(x*(2 + x^3)^(2/3))/3 - (5*ArcTan[(1 + (2*x)/(2 + x^3)^(1/3))/Sqrt[3]])/(3*Sqrt[3]) + (5*Log[-x + (2 + x^3)^(1
/3)])/6

Rule 388

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(d*x*(a + b*x^n)^(p + 1))/(b*(n*
(p + 1) + 1)), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(b*(n*(p + 1) + 1)), Int[(a + b*x^n)^p, x], x] /; FreeQ[{
a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && NeQ[n*(p + 1) + 1, 0]

Rule 239

Int[((a_) + (b_.)*(x_)^3)^(-1/3), x_Symbol] :> Simp[ArcTan[(1 + (2*Rt[b, 3]*x)/(a + b*x^3)^(1/3))/Sqrt[3]]/(Sq
rt[3]*Rt[b, 3]), x] - Simp[Log[(a + b*x^3)^(1/3) - Rt[b, 3]*x]/(2*Rt[b, 3]), x] /; FreeQ[{a, b}, x]

Rubi steps

\begin{align*} \int \frac{-1+x^3}{\sqrt [3]{2+x^3}} \, dx &=\frac{1}{3} x \left (2+x^3\right )^{2/3}-\frac{5}{3} \int \frac{1}{\sqrt [3]{2+x^3}} \, dx\\ &=\frac{1}{3} x \left (2+x^3\right )^{2/3}-\frac{5 \tan ^{-1}\left (\frac{1+\frac{2 x}{\sqrt [3]{2+x^3}}}{\sqrt{3}}\right )}{3 \sqrt{3}}+\frac{5}{6} \log \left (-x+\sqrt [3]{2+x^3}\right )\\ \end{align*}

Mathematica [A]  time = 0.0338833, size = 91, normalized size = 1.44 \[ \frac{1}{18} \left (6 \left (x^3+2\right )^{2/3} x+10 \log \left (1-\frac{x}{\sqrt [3]{x^3+2}}\right )-5 \log \left (\frac{x^2}{\left (x^3+2\right )^{2/3}}+\frac{x}{\sqrt [3]{x^3+2}}+1\right )-10 \sqrt{3} \tan ^{-1}\left (\frac{\frac{2 x}{\sqrt [3]{x^3+2}}+1}{\sqrt{3}}\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(6*x*(2 + x^3)^(2/3) - 10*Sqrt[3]*ArcTan[(1 + (2*x)/(2 + x^3)^(1/3))/Sqrt[3]] + 10*Log[1 - x/(2 + x^3)^(1/3)]
- 5*Log[1 + x^2/(2 + x^3)^(2/3) + x/(2 + x^3)^(1/3)])/18

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Maple [C]  time = 0.027, size = 29, normalized size = 0.5 \begin{align*}{\frac{x}{3} \left ({x}^{3}+2 \right ) ^{{\frac{2}{3}}}}-{\frac{5\,x{2}^{2/3}}{6}{\mbox{$_2$F$_1$}({\frac{1}{3}},{\frac{1}{3}};\,{\frac{4}{3}};\,-{\frac{{x}^{3}}{2}})}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.471, size = 127, normalized size = 2.02 \begin{align*} \frac{5}{9} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (\frac{2 \,{\left (x^{3} + 2\right )}^{\frac{1}{3}}}{x} + 1\right )}\right ) + \frac{2 \,{\left (x^{3} + 2\right )}^{\frac{2}{3}}}{3 \, x^{2}{\left (\frac{x^{3} + 2}{x^{3}} - 1\right )}} - \frac{5}{18} \, \log \left (\frac{{\left (x^{3} + 2\right )}^{\frac{1}{3}}}{x} + \frac{{\left (x^{3} + 2\right )}^{\frac{2}{3}}}{x^{2}} + 1\right ) + \frac{5}{9} \, \log \left (\frac{{\left (x^{3} + 2\right )}^{\frac{1}{3}}}{x} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5/9*sqrt(3)*arctan(1/3*sqrt(3)*(2*(x^3 + 2)^(1/3)/x + 1)) + 2/3*(x^3 + 2)^(2/3)/(x^2*((x^3 + 2)/x^3 - 1)) - 5/
18*log((x^3 + 2)^(1/3)/x + (x^3 + 2)^(2/3)/x^2 + 1) + 5/9*log((x^3 + 2)^(1/3)/x - 1)

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Fricas [A]  time = 1.91578, size = 252, normalized size = 4. \begin{align*} \frac{1}{3} \,{\left (x^{3} + 2\right )}^{\frac{2}{3}} x + \frac{5}{9} \, \sqrt{3} \arctan \left (\frac{\sqrt{3} x + 2 \, \sqrt{3}{\left (x^{3} + 2\right )}^{\frac{1}{3}}}{3 \, x}\right ) + \frac{5}{9} \, \log \left (-\frac{x -{\left (x^{3} + 2\right )}^{\frac{1}{3}}}{x}\right ) - \frac{5}{18} \, \log \left (\frac{x^{2} +{\left (x^{3} + 2\right )}^{\frac{1}{3}} x +{\left (x^{3} + 2\right )}^{\frac{2}{3}}}{x^{2}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(x^3 + 2)^(2/3)*x + 5/9*sqrt(3)*arctan(1/3*(sqrt(3)*x + 2*sqrt(3)*(x^3 + 2)^(1/3))/x) + 5/9*log(-(x - (x^3
+ 2)^(1/3))/x) - 5/18*log((x^2 + (x^3 + 2)^(1/3)*x + (x^3 + 2)^(2/3))/x^2)

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Sympy [C]  time = 1.82774, size = 71, normalized size = 1.13 \begin{align*} \frac{2^{\frac{2}{3}} x^{4} \Gamma \left (\frac{4}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{3}, \frac{4}{3} \\ \frac{7}{3} \end{matrix}\middle |{\frac{x^{3} e^{i \pi }}{2}} \right )}}{6 \Gamma \left (\frac{7}{3}\right )} - \frac{2^{\frac{2}{3}} x \Gamma \left (\frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{3}, \frac{1}{3} \\ \frac{4}{3} \end{matrix}\middle |{\frac{x^{3} e^{i \pi }}{2}} \right )}}{6 \Gamma \left (\frac{4}{3}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2**(2/3)*x**4*gamma(4/3)*hyper((1/3, 4/3), (7/3,), x**3*exp_polar(I*pi)/2)/(6*gamma(7/3)) - 2**(2/3)*x*gamma(1
/3)*hyper((1/3, 1/3), (4/3,), x**3*exp_polar(I*pi)/2)/(6*gamma(4/3))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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