\(\int \frac {(1+x^3)^{2/3}}{x^3} \, dx\) [1282]

Optimal result

Integrand size = 13, antiderivative size = 93 \[ \int \frac {\left (1+x^3\right )^{2/3}}{x^3} \, dx=-\frac {\left (1+x^3\right )^{2/3}}{2 x^2}+\frac {\arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{1+x^3}}\right )}{\sqrt {3}}-\frac {1}{3} \log \left (-x+\sqrt [3]{1+x^3}\right )+\frac {1}{6} \log \left (x^2+x \sqrt [3]{1+x^3}+\left (1+x^3\right )^{2/3}\right ) \]

[Out]

-1/2*(x^3+1)^(2/3)/x^2+1/3*arctan(3^(1/2)*x/(x+2*(x^3+1)^(1/3)))*3^(1/2)-1/3*ln(-x+(x^3+1)^(1/3))+1/6*ln(x^2+x
*(x^3+1)^(1/3)+(x^3+1)^(2/3))

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.67, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {283, 245} \[ \int \frac {\left (1+x^3\right )^{2/3}}{x^3} \, dx=\frac {\arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{2} \log \left (\sqrt [3]{x^3+1}-x\right )-\frac {\left (x^3+1\right )^{2/3}}{2 x^2} \]

[In]

Int[(1 + x^3)^(2/3)/x^3,x]

[Out]

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

Rule 245

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]

Rule 283

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^p/(c*(m + 1
))), x] - Dist[b*n*(p/(c^n*(m + 1))), Int[(c*x)^(m + n)*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] &&
IGtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1] &&  !ILtQ[(m + n*p + n + 1)/n, 0] && IntBinomialQ[a, b, c, n, m, p, x]

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

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.97 \[ \int \frac {\left (1+x^3\right )^{2/3}}{x^3} \, dx=\frac {1}{6} \left (-\frac {3 \left (1+x^3\right )^{2/3}}{x^2}+2 \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{1+x^3}}\right )-2 \log \left (-x+\sqrt [3]{1+x^3}\right )+\log \left (x^2+x \sqrt [3]{1+x^3}+\left (1+x^3\right )^{2/3}\right )\right ) \]

[In]

Integrate[(1 + x^3)^(2/3)/x^3,x]

[Out]

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

Maple [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3.

Time = 3.17 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.18

method	result	size
meijerg	\(-\frac {\operatorname {hypergeom}\left (\left [-\frac {2}{3}, -\frac {2}{3}\right ], \left [\frac {1}{3}\right ], -x^{3}\right )}{2 x^{2}}\)	\(17\)
risch	\(-\frac {\left (x^{3}+1\right )^{\frac {2}{3}}}{2 x^{2}}+x \operatorname {hypergeom}\left (\left [\frac {1}{3}, \frac {1}{3}\right ], \left [\frac {4}{3}\right ], -x^{3}\right )\)	\(27\)
pseudoelliptic	\(\frac {-2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x +2 \left (x^{3}+1\right )^{\frac {1}{3}}\right )}{3 x}\right ) x^{2}-2 \ln \left (\frac {-x +\left (x^{3}+1\right )^{\frac {1}{3}}}{x}\right ) x^{2}+\ln \left (\frac {x^{2}+x \left (x^{3}+1\right )^{\frac {1}{3}}+\left (x^{3}+1\right )^{\frac {2}{3}}}{x^{2}}\right ) x^{2}-3 \left (x^{3}+1\right )^{\frac {2}{3}}}{6 x^{2}}\)	\(94\)
trager	\(-\frac {\left (x^{3}+1\right )^{\frac {2}{3}}}{2 x^{2}}-\frac {\ln \left (317 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{3}-555 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {2}{3}} x +2358 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {1}{3}} x^{2}-2120 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{3}-1803 x \left (x^{3}+1\right )^{\frac {2}{3}}-555 x^{2} \left (x^{3}+1\right )^{\frac {1}{3}}+2675 x^{3}-317 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2}-99 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+1070\right )}{3}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \ln \left (-13 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{3}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {2}{3}} x +57 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {1}{3}} x^{2}-41 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{3}-54 x \left (x^{3}+1\right )^{\frac {2}{3}}-3 x^{2} \left (x^{3}+1\right )^{\frac {1}{3}}+44 x^{3}+13 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2}-50 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+33\right )}{3}\)	\(275\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.42 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.13 \[ \int \frac {\left (1+x^3\right )^{2/3}}{x^3} \, dx=\frac {2 \, \sqrt {3} x^{2} \arctan \left (-\frac {25382 \, \sqrt {3} {\left (x^{3} + 1\right )}^{\frac {1}{3}} x^{2} - 13720 \, \sqrt {3} {\left (x^{3} + 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (5831 \, x^{3} + 7200\right )}}{58653 \, x^{3} + 8000}\right ) - x^{2} \log \left (3 \, {\left (x^{3} + 1\right )}^{\frac {1}{3}} x^{2} - 3 \, {\left (x^{3} + 1\right )}^{\frac {2}{3}} x + 1\right ) - 3 \, {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{6 \, x^{2}} \]

[In]

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

[Out]

1/6*(2*sqrt(3)*x^2*arctan(-(25382*sqrt(3)*(x^3 + 1)^(1/3)*x^2 - 13720*sqrt(3)*(x^3 + 1)^(2/3)*x + sqrt(3)*(583
1*x^3 + 7200))/(58653*x^3 + 8000)) - x^2*log(3*(x^3 + 1)^(1/3)*x^2 - 3*(x^3 + 1)^(2/3)*x + 1) - 3*(x^3 + 1)^(2
/3))/x^2

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.66 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.37 \[ \int \frac {\left (1+x^3\right )^{2/3}}{x^3} \, dx=\frac {\Gamma \left (- \frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, - \frac {2}{3} \\ \frac {1}{3} \end {matrix}\middle | {x^{3} e^{i \pi }} \right )}}{3 x^{2} \Gamma \left (\frac {1}{3}\right )} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.87 \[ \int \frac {\left (1+x^3\right )^{2/3}}{x^3} \, dx=-\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (\frac {2 \, {\left (x^{3} + 1\right )}^{\frac {1}{3}}}{x} + 1\right )}\right ) - \frac {{\left (x^{3} + 1\right )}^{\frac {2}{3}}}{2 \, x^{2}} + \frac {1}{6} \, \log \left (\frac {{\left (x^{3} + 1\right )}^{\frac {1}{3}}}{x} + \frac {{\left (x^{3} + 1\right )}^{\frac {2}{3}}}{x^{2}} + 1\right ) - \frac {1}{3} \, \log \left (\frac {{\left (x^{3} + 1\right )}^{\frac {1}{3}}}{x} - 1\right ) \]

[In]

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

[Out]

-1/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*(x^3 + 1)^(1/3)/x + 1)) - 1/2*(x^3 + 1)^(2/3)/x^2 + 1/6*log((x^3 + 1)^(1/3)
/x + (x^3 + 1)^(2/3)/x^2 + 1) - 1/3*log((x^3 + 1)^(1/3)/x - 1)

Giac [F]

\[ \int \frac {\left (1+x^3\right )^{2/3}}{x^3} \, dx=\int { \frac {{\left (x^{3} + 1\right )}^{\frac {2}{3}}}{x^{3}} \,d x } \]

[In]

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

[Out]

integrate((x^3 + 1)^(2/3)/x^3, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (1+x^3\right )^{2/3}}{x^3} \, dx=\int \frac {{\left (x^3+1\right )}^{2/3}}{x^3} \,d x \]

[In]

int((x^3 + 1)^(2/3)/x^3,x)

[Out]

int((x^3 + 1)^(2/3)/x^3, x)