3.13.95 \(\int (1+x^3)^{2/3} \, dx\)

Optimal. Leaf size=94 \[ \frac {1}{3} \left (x^3+1\right )^{2/3} x-\frac {2}{9} \log \left (\sqrt [3]{x^3+1}-x\right )+\frac {2 \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^3+1}+x}\right )}{3 \sqrt {3}}+\frac {1}{9} \log \left (\sqrt [3]{x^3+1} x+\left (x^3+1\right )^{2/3}+x^2\right ) \]

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 63, normalized size of antiderivative = 0.67, number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {195, 239} \begin {gather*} \frac {1}{3} \left (x^3+1\right )^{2/3} x-\frac {1}{3} \log \left (\sqrt [3]{x^3+1}-x\right )+\frac {2 \tan ^{-1}\left (\frac {\frac {2 x}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )}{3 \sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + x^3)^(2/3),x]

[Out]

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

Rule 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

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

________________________________________________________________________________________

Mathematica [C]  time = 0.06, size = 99, normalized size = 1.05 \begin {gather*} \frac {3 (x+1) \left (x^3+1\right )^{2/3} F_1\left (\frac {5}{3};-\frac {2}{3},-\frac {2}{3};\frac {8}{3};-\frac {x+1}{-1+(-1)^{2/3}},-\frac {x+1}{-1-\sqrt [3]{-1}}\right )}{5 \left (\frac {x+1}{-1-\sqrt [3]{-1}}+1\right )^{2/3} \left (\frac {x+1}{(-1)^{2/3}-1}+1\right )^{2/3}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[(1 + x^3)^(2/3),x]

[Out]

(3*(1 + x)*(1 + x^3)^(2/3)*AppellF1[5/3, -2/3, -2/3, 8/3, -((1 + x)/(-1 + (-1)^(2/3))), -((1 + x)/(-1 - (-1)^(
1/3)))])/(5*(1 + (1 + x)/(-1 - (-1)^(1/3)))^(2/3)*(1 + (1 + x)/(-1 + (-1)^(2/3)))^(2/3))

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.18, size = 94, normalized size = 1.00 \begin {gather*} \frac {1}{3} x \left (1+x^3\right )^{2/3}+\frac {2 \tan ^{-1}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{1+x^3}}\right )}{3 \sqrt {3}}-\frac {2}{9} \log \left (-x+\sqrt [3]{1+x^3}\right )+\frac {1}{9} \log \left (x^2+x \sqrt [3]{1+x^3}+\left (1+x^3\right )^{2/3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(1 + x^3)^(2/3),x]

[Out]

(x*(1 + x^3)^(2/3))/3 + (2*ArcTan[(Sqrt[3]*x)/(x + 2*(1 + x^3)^(1/3))])/(3*Sqrt[3]) - (2*Log[-x + (1 + x^3)^(1
/3)])/9 + Log[x^2 + x*(1 + x^3)^(1/3) + (1 + x^3)^(2/3)]/9

________________________________________________________________________________________

fricas [A]  time = 0.44, size = 86, normalized size = 0.91 \begin {gather*} \frac {1}{3} \, {\left (x^{3} + 1\right )}^{\frac {2}{3}} x - \frac {2}{9} \, \sqrt {3} \arctan \left (\frac {\sqrt {3} x + 2 \, \sqrt {3} {\left (x^{3} + 1\right )}^{\frac {1}{3}}}{3 \, x}\right ) - \frac {2}{9} \, \log \left (-\frac {x - {\left (x^{3} + 1\right )}^{\frac {1}{3}}}{x}\right ) + \frac {1}{9} \, \log \left (\frac {x^{2} + {\left (x^{3} + 1\right )}^{\frac {1}{3}} x + {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{x^{2}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int {\left (x^{3} + 1\right )}^{\frac {2}{3}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((x^3 + 1)^(2/3), x)

________________________________________________________________________________________

maple [C]  time = 1.64, size = 14, normalized size = 0.15

method	result	size
meijerg	\(x \hypergeom \left (\left [-\frac {2}{3}, \frac {1}{3}\right ], \left [\frac {4}{3}\right ], -x^{3}\right )\)	\(14\)
risch	\(\frac {x \left (x^{3}+1\right )^{\frac {2}{3}}}{3}+\frac {2 x \hypergeom \left (\left [\frac {1}{3}, \frac {1}{3}\right ], \left [\frac {4}{3}\right ], -x^{3}\right )}{3}\)	\(26\)
trager	\(\frac {x \left (x^{3}+1\right )^{\frac {2}{3}}}{3}+\frac {2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \ln \left (\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{3}-3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {2}{3}} x -3 \left (x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}-4 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{3}+3 x \left (x^{3}+1\right )^{\frac {2}{3}}+3 x^{2} \left (x^{3}+1\right )^{\frac {1}{3}}+4 x^{3}-\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+2\right )}{9}-\frac {2 \ln \left (\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{3}+3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {2}{3}} x +3 \left (x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}+2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{3}+x^{3}+\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+1\right ) \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )}{9}+\frac {2 \ln \left (\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{3}+3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {2}{3}} x +3 \left (x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}+2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{3}+x^{3}+\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+1\right )}{9}\)	\(314\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3+1)^(2/3),x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.76, size = 94, normalized size = 1.00 \begin {gather*} -\frac {2}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (\frac {2 \, {\left (x^{3} + 1\right )}^{\frac {1}{3}}}{x} + 1\right )}\right ) + \frac {{\left (x^{3} + 1\right )}^{\frac {2}{3}}}{3 \, x^{2} {\left (\frac {x^{3} + 1}{x^{3}} - 1\right )}} + \frac {1}{9} \, \log \left (\frac {{\left (x^{3} + 1\right )}^{\frac {1}{3}}}{x} + \frac {{\left (x^{3} + 1\right )}^{\frac {2}{3}}}{x^{2}} + 1\right ) - \frac {2}{9} \, \log \left (\frac {{\left (x^{3} + 1\right )}^{\frac {1}{3}}}{x} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 0.77, size = 12, normalized size = 0.13 \begin {gather*} x\,{{}}_2{\mathrm {F}}_1\left (-\frac {2}{3},\frac {1}{3};\ \frac {4}{3};\ -x^3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3 + 1)^(2/3),x)

[Out]

x*hypergeom([-2/3, 1/3], 4/3, -x^3)

________________________________________________________________________________________

sympy [C]  time = 0.81, size = 29, normalized size = 0.31 \begin {gather*} \frac {x \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {x^{3} e^{i \pi }} \right )}}{3 \Gamma \left (\frac {4}{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________