\(\int \frac {(-1+x^3)^{2/3} (1+x^3)}{x^3} \, dx\) [1461]

Optimal result

Integrand size = 18, antiderivative size = 103 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (1+x^3\right )}{x^3} \, dx=\frac {\left (-1+x^3\right )^{2/3} \left (-3+2 x^3\right )}{6 x^2}+\frac {\arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-1+x^3}}\right )}{3 \sqrt {3}}-\frac {1}{9} \log \left (-x+\sqrt [3]{-1+x^3}\right )+\frac {1}{18} \log \left (x^2+x \sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right ) \]

[Out]

1/6*(x^3-1)^(2/3)*(2*x^3-3)/x^2+1/9*arctan(3^(1/2)*x/(x+2*(x^3-1)^(1/3)))*3^(1/2)-1/9*ln(-x+(x^3-1)^(1/3))+1/1
8*ln(x^2+x*(x^3-1)^(1/3)+(x^3-1)^(2/3))

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.77, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {464, 201, 245} \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (1+x^3\right )}{x^3} \, dx=\frac {\arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {1}{6} x \left (x^3-1\right )^{2/3}-\frac {1}{6} \log \left (\sqrt [3]{x^3-1}-x\right )+\frac {\left (x^3-1\right )^{5/3}}{2 x^2} \]

[In]

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

[Out]

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

Rule 201

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 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 464

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[c*(e*x)^(m +
1)*((a + b*x^n)^(p + 1)/(a*e*(m + 1))), x] + Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*e^n*(m + 1)), In
t[(e*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && (IntegerQ[n] ||
GtQ[e, 0]) && ((GtQ[n, 0] && LtQ[m, -1]) || (LtQ[n, 0] && GtQ[m + n, -1])) &&  !ILtQ[p, -1]

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

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.94 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (1+x^3\right )}{x^3} \, dx=\frac {1}{18} \left (\frac {3 \left (-1+x^3\right )^{2/3} \left (-3+2 x^3\right )}{x^2}+2 \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-1+x^3}}\right )-2 \log \left (-x+\sqrt [3]{-1+x^3}\right )+\log \left (x^2+x \sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right )\right ) \]

[In]

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

[Out]

((3*(-1 + x^3)^(2/3)*(-3 + 2*x^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)])/18

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 1.54 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.54

method	result	size
risch	\(\frac {2 x^{6}-5 x^{3}+3}{6 x^{2} \left (x^{3}-1\right )^{\frac {1}{3}}}+\frac {{\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {1}{3}} x \operatorname {hypergeom}\left (\left [\frac {1}{3}, \frac {1}{3}\right ], \left [\frac {4}{3}\right ], x^{3}\right )}{3 \operatorname {signum}\left (x^{3}-1\right )^{\frac {1}{3}}}\)	\(56\)
meijerg	\(\frac {\operatorname {signum}\left (x^{3}-1\right )^{\frac {2}{3}} x \operatorname {hypergeom}\left (\left [-\frac {2}{3}, \frac {1}{3}\right ], \left [\frac {4}{3}\right ], x^{3}\right )}{{\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {2}{3}}}-\frac {\operatorname {signum}\left (x^{3}-1\right )^{\frac {2}{3}} \operatorname {hypergeom}\left (\left [-\frac {2}{3}, -\frac {2}{3}\right ], \left [\frac {1}{3}\right ], x^{3}\right )}{2 {\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {2}{3}} x^{2}}\)	\(63\)
pseudoelliptic	\(\frac {6 x^{3} \left (x^{3}-1\right )^{\frac {2}{3}}-2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x +2 \left (x^{3}-1\right )^{\frac {1}{3}}\right )}{3 x}\right ) x^{2}-2 \ln \left (\frac {-x +\left (x^{3}-1\right )^{\frac {1}{3}}}{x}\right ) x^{2}+\ln \left (\frac {x^{2}+x \left (x^{3}-1\right )^{\frac {1}{3}}+\left (x^{3}-1\right )^{\frac {2}{3}}}{x^{2}}\right ) x^{2}-9 \left (x^{3}-1\right )^{\frac {2}{3}}}{18 \left (\left (x^{3}-1\right )^{\frac {2}{3}}+x \left (x +\left (x^{3}-1\right )^{\frac {1}{3}}\right )\right ) x^{2} \left (x -\left (x^{3}-1\right )^{\frac {1}{3}}\right )}\)	\(140\)
trager	\(\frac {\left (x^{3}-1\right )^{\frac {2}{3}} \left (2 x^{3}-3\right )}{6 x^{2}}+\frac {4 \operatorname {RootOf}\left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right ) \ln \left (182271728 \operatorname {RootOf}\left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right )^{2} x^{3}[phone] \operatorname {RootOf}\left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}} x [phone] \operatorname {RootOf}\left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}} x^{2}[phone] \operatorname {RootOf}\left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right ) x^{3}-1508552373 x \left (x^{3}-1\right )^{\frac {2}{3}}-1508552373 x^{2} \left (x^{3}-1\right )^{\frac {1}{3}}-1497160390 x^{3}-1458173824 \operatorname {RootOf}\left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right )^{2}[phone] \operatorname {RootOf}\left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right )[phone]\right )}{9}+\frac {\ln \left (182271728 \operatorname {RootOf}\left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right )^{2} x^{3}-775851456 \operatorname {RootOf}\left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}} x -775851456 \operatorname {RootOf}\left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}} x^{2}-821419388 \operatorname {RootOf}\left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right ) x^{3}-1314589509 x \left (x^{3}-1\right )^{\frac {2}{3}}-1314589509 x^{2} \left (x^{3}-1\right )^{\frac {1}{3}}-1303197526 x^{3}-1458173824 \operatorname {RootOf}\left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right )^{2}-1129625404 \operatorname {RootOf}\left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right )[phone]\right )}{9}-\frac {4 \ln \left (182271728 \operatorname {RootOf}\left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right )^{2} x^{3}-775851456 \operatorname {RootOf}\left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}} x -775851456 \operatorname {RootOf}\left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}} x^{2}-821419388 \operatorname {RootOf}\left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right ) x^{3}-1314589509 x \left (x^{3}-1\right )^{\frac {2}{3}}-1314589509 x^{2} \left (x^{3}-1\right )^{\frac {1}{3}}-1303197526 x^{3}-1458173824 \operatorname {RootOf}\left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right )^{2}-1129625404 \operatorname {RootOf}\left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right )[phone]\right ) \operatorname {RootOf}\left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right )}{9}\)	\(457\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.51 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.09 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (1+x^3\right )}{x^3} \, dx=\frac {2 \, \sqrt {3} x^{2} \arctan \left (-\frac {25382 \, \sqrt {3} {\left (x^{3} - 1\right )}^{\frac {1}{3}} x^{2} - 13720 \, \sqrt {3} {\left (x^{3} - 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (5831 \, x^{3} - 7200\right )}}{58653 \, x^{3} - 8000}\right ) - x^{2} \log \left (-3 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}} x^{2} + 3 \, {\left (x^{3} - 1\right )}^{\frac {2}{3}} x + 1\right ) + 3 \, {\left (2 \, x^{3} - 3\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{18 \, x^{2}} \]

[In]

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

[Out]

1/18*(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)*(58
31*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*(2*x^3 - 3
)*(x^3 - 1)^(2/3))/x^2

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.52 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.68 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (1+x^3\right )}{x^3} \, dx=- \frac {x e^{- \frac {i \pi }{3}} \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {x^{3}} \right )}}{3 \Gamma \left (\frac {4}{3}\right )} + \frac {e^{\frac {2 i \pi }{3}} \Gamma \left (- \frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, - \frac {2}{3} \\ \frac {1}{3} \end {matrix}\middle | {x^{3}} \right )}}{3 x^{2} \Gamma \left (\frac {1}{3}\right )} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.03 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (1+x^3\right )}{x^3} \, dx=-\frac {1}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (\frac {2 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x} + 1\right )}\right ) - \frac {{\left (x^{3} - 1\right )}^{\frac {2}{3}}}{2 \, x^{2}} - \frac {{\left (x^{3} - 1\right )}^{\frac {2}{3}}}{3 \, x^{2} {\left (\frac {x^{3} - 1}{x^{3}} - 1\right )}} + \frac {1}{18} \, \log \left (\frac {{\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x} + \frac {{\left (x^{3} - 1\right )}^{\frac {2}{3}}}{x^{2}} + 1\right ) - \frac {1}{9} \, \log \left (\frac {{\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x} - 1\right ) \]

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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