\(\int \frac {\sqrt [3]{-1+x^6} (1+x^6)}{x^3} \, dx\) [1594]

Optimal result

Integrand size = 18, antiderivative size = 109 \[ \int \frac {\sqrt [3]{-1+x^6} \left (1+x^6\right )}{x^3} \, dx=\frac {\left (-3+x^6\right ) \sqrt [3]{-1+x^6}}{6 x^2}-\frac {\arctan \left (\frac {\sqrt {3} x^2}{x^2+2 \sqrt [3]{-1+x^6}}\right )}{3 \sqrt {3}}-\frac {1}{9} \log \left (-x^2+\sqrt [3]{-1+x^6}\right )+\frac {1}{18} \log \left (x^4+x^2 \sqrt [3]{-1+x^6}+\left (-1+x^6\right )^{2/3}\right ) \]

[Out]

1/6*(x^6-3)*(x^6-1)^(1/3)/x^2-1/9*arctan(3^(1/2)*x^2/(x^2+2*(x^6-1)^(1/3)))*3^(1/2)-1/9*ln(-x^2+(x^6-1)^(1/3))
+1/18*ln(x^4+x^2*(x^6-1)^(1/3)+(x^6-1)^(2/3))

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.78, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {464, 281, 285, 337} \[ \int \frac {\sqrt [3]{-1+x^6} \left (1+x^6\right )}{x^3} \, dx=-\frac {\arctan \left (\frac {\frac {2 x^2}{\sqrt [3]{x^6-1}}+1}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {1}{3} \sqrt [3]{x^6-1} x^4+\frac {\left (x^6-1\right )^{4/3}}{2 x^2}-\frac {1}{6} \log \left (x^2-\sqrt [3]{x^6-1}\right ) \]

[In]

Int[((-1 + x^6)^(1/3)*(1 + x^6))/x^3,x]

[Out]

-1/3*(x^4*(-1 + x^6)^(1/3)) + (-1 + x^6)^(4/3)/(2*x^2) - ArcTan[(1 + (2*x^2)/(-1 + x^6)^(1/3))/Sqrt[3]]/(3*Sqr
t[3]) - Log[x^2 - (-1 + x^6)^(1/3)]/6

Rule 281

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

Rule 285

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

Rule 337

Int[(x_)/((a_) + (b_.)*(x_)^3)^(2/3), x_Symbol] :> With[{q = Rt[b, 3]}, Simp[-ArcTan[(1 + 2*q*(x/(a + b*x^3)^(
1/3)))/Sqrt[3]]/(Sqrt[3]*q^2), x] - Simp[Log[q*x - (a + b*x^3)^(1/3)]/(2*q^2), 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^6\right )^{4/3}}{2 x^2}-2 \int x^3 \sqrt [3]{-1+x^6} \, dx \\ & = \frac {\left (-1+x^6\right )^{4/3}}{2 x^2}-\text {Subst}\left (\int x \sqrt [3]{-1+x^3} \, dx,x,x^2\right ) \\ & = -\frac {1}{3} x^4 \sqrt [3]{-1+x^6}+\frac {\left (-1+x^6\right )^{4/3}}{2 x^2}+\frac {1}{3} \text {Subst}\left (\int \frac {x}{\left (-1+x^3\right )^{2/3}} \, dx,x,x^2\right ) \\ & = -\frac {1}{3} x^4 \sqrt [3]{-1+x^6}+\frac {\left (-1+x^6\right )^{4/3}}{2 x^2}-\frac {\arctan \left (\frac {1+\frac {2 x^2}{\sqrt [3]{-1+x^6}}}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {1}{6} \log \left (x^2-\sqrt [3]{-1+x^6}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.40 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.94 \[ \int \frac {\sqrt [3]{-1+x^6} \left (1+x^6\right )}{x^3} \, dx=\frac {1}{18} \left (\frac {3 \left (-3+x^6\right ) \sqrt [3]{-1+x^6}}{x^2}-2 \sqrt {3} \arctan \left (\frac {\sqrt {3} x^2}{x^2+2 \sqrt [3]{-1+x^6}}\right )-2 \log \left (-x^2+\sqrt [3]{-1+x^6}\right )+\log \left (x^4+x^2 \sqrt [3]{-1+x^6}+\left (-1+x^6\right )^{2/3}\right )\right ) \]

[In]

Integrate[((-1 + x^6)^(1/3)*(1 + x^6))/x^3,x]

[Out]

((3*(-3 + x^6)*(-1 + x^6)^(1/3))/x^2 - 2*Sqrt[3]*ArcTan[(Sqrt[3]*x^2)/(x^2 + 2*(-1 + x^6)^(1/3))] - 2*Log[-x^2
+ (-1 + x^6)^(1/3)] + Log[x^4 + x^2*(-1 + x^6)^(1/3) + (-1 + x^6)^(2/3)])/18

Maple [C] (warning: unable to verify)

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

Time = 4.10 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.51

method	result	size
risch	\(\frac {x^{12}-4 x^{6}+3}{6 x^{2} \left (x^{6}-1\right )^{\frac {2}{3}}}+\frac {{\left (-\operatorname {signum}\left (x^{6}-1\right )\right )}^{\frac {2}{3}} x^{4} \operatorname {hypergeom}\left (\left [\frac {2}{3}, \frac {2}{3}\right ], \left [\frac {5}{3}\right ], x^{6}\right )}{6 \operatorname {signum}\left (x^{6}-1\right )^{\frac {2}{3}}}\)	\(56\)
meijerg	\(\frac {\operatorname {signum}\left (x^{6}-1\right )^{\frac {1}{3}} x^{4} \operatorname {hypergeom}\left (\left [-\frac {1}{3}, \frac {2}{3}\right ], \left [\frac {5}{3}\right ], x^{6}\right )}{4 {\left (-\operatorname {signum}\left (x^{6}-1\right )\right )}^{\frac {1}{3}}}-\frac {\operatorname {signum}\left (x^{6}-1\right )^{\frac {1}{3}} \operatorname {hypergeom}\left (\left [-\frac {1}{3}, -\frac {1}{3}\right ], \left [\frac {2}{3}\right ], x^{6}\right )}{2 {\left (-\operatorname {signum}\left (x^{6}-1\right )\right )}^{\frac {1}{3}} x^{2}}\)	\(66\)
pseudoelliptic	\(\frac {3 \left (x^{6}-1\right )^{\frac {1}{3}} x^{6}+2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{2}+2 \left (x^{6}-1\right )^{\frac {1}{3}}\right )}{3 x^{2}}\right ) x^{2}+\ln \left (\frac {x^{4}+x^{2} \left (x^{6}-1\right )^{\frac {1}{3}}+\left (x^{6}-1\right )^{\frac {2}{3}}}{x^{4}}\right ) x^{2}-2 \ln \left (\frac {-x^{2}+\left (x^{6}-1\right )^{\frac {1}{3}}}{x^{2}}\right ) x^{2}-9 \left (x^{6}-1\right )^{\frac {1}{3}}}{18 \left (x^{2}-\left (x^{6}-1\right )^{\frac {1}{3}}\right ) x^{2} \left (\left (x^{6}-1\right )^{\frac {2}{3}}+x^{2} \left (x^{2}+\left (x^{6}-1\right )^{\frac {1}{3}}\right )\right )}\)	\(152\)
trager	\(\frac {\left (x^{6}-3\right ) \left (x^{6}-1\right )^{\frac {1}{3}}}{6 x^{2}}-\frac {\ln \left (-23398198288 \operatorname {RootOf}\left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right )^{2} x^{6}-1099291994096 \operatorname {RootOf}\left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right ) x^{6}[phone] \operatorname {RootOf}\left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {1}{3}} x^{4}[phone] x^{6}-1203968676864 \operatorname {RootOf}\left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {2}{3}} x^{2}-300992169216 x^{4} \left (x^{6}-1\right )^{\frac {1}{3}}-276285385917 x^{2} \left (x^{6}-1\right )^{\frac {2}{3}}[phone] \operatorname {RootOf}\left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right )^{2}-407313550340 \operatorname {RootOf}\left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right )-190928187198\right )}{9}+\frac {\ln \left (48489698336 \operatorname {RootOf}\left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right )^{2} x^{6}-2321232645116 \operatorname {RootOf}\left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right ) x^{6}[phone] \operatorname {RootOf}\left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {1}{3}} x^{4}[phone] x^{6}[phone] \operatorname {RootOf}\left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {2}{3}} x^{2}[phone] x^{4} \left (x^{6}-1\right )^{\frac {1}{3}}-577277555133 x^{2} \left (x^{6}-1\right )^{\frac {2}{3}}-3103340693504 \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 )-185723198911\right )}{9}-\frac {4 \ln \left (48489698336 \operatorname {RootOf}\left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right )^{2} x^{6}-2321232645116 \operatorname {RootOf}\left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right ) x^{6}[phone] \operatorname {RootOf}\left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {1}{3}} x^{4}[phone] x^{6}[phone] \operatorname {RootOf}\left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {2}{3}} x^{2}[phone] x^{4} \left (x^{6}-1\right )^{\frac {1}{3}}-577277555133 x^{2} \left (x^{6}-1\right )^{\frac {2}{3}}-3103340693504 \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 )-185723198911\right ) \operatorname {RootOf}\left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right )}{9}\)	\(456\)

[In]

int((x^6-1)^(1/3)*(x^6+1)/x^3,x,method=_RETURNVERBOSE)

[Out]

1/6*(x^12-4*x^6+3)/x^2/(x^6-1)^(2/3)+1/6/signum(x^6-1)^(2/3)*(-signum(x^6-1))^(2/3)*x^4*hypergeom([2/3,2/3],[5
/3],x^6)

Fricas [A] (verification not implemented)

none

Time = 0.44 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.04 \[ \int \frac {\sqrt [3]{-1+x^6} \left (1+x^6\right )}{x^3} \, dx=-\frac {2 \, \sqrt {3} x^{2} \arctan \left (-\frac {25382 \, \sqrt {3} {\left (x^{6} - 1\right )}^{\frac {1}{3}} x^{4} - 13720 \, \sqrt {3} {\left (x^{6} - 1\right )}^{\frac {2}{3}} x^{2} + \sqrt {3} {\left (5831 \, x^{6} - 7200\right )}}{58653 \, x^{6} - 8000}\right ) + x^{2} \log \left (-3 \, {\left (x^{6} - 1\right )}^{\frac {1}{3}} x^{4} + 3 \, {\left (x^{6} - 1\right )}^{\frac {2}{3}} x^{2} + 1\right ) - 3 \, {\left (x^{6} - 1\right )}^{\frac {1}{3}} {\left (x^{6} - 3\right )}}{18 \, x^{2}} \]

[In]

integrate((x^6-1)^(1/3)*(x^6+1)/x^3,x, algorithm="fricas")

[Out]

-1/18*(2*sqrt(3)*x^2*arctan(-(25382*sqrt(3)*(x^6 - 1)^(1/3)*x^4 - 13720*sqrt(3)*(x^6 - 1)^(2/3)*x^2 + sqrt(3)*
(5831*x^6 - 7200))/(58653*x^6 - 8000)) + x^2*log(-3*(x^6 - 1)^(1/3)*x^4 + 3*(x^6 - 1)^(2/3)*x^2 + 1) - 3*(x^6
- 1)^(1/3)*(x^6 - 3))/x^2

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.87 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.65 \[ \int \frac {\sqrt [3]{-1+x^6} \left (1+x^6\right )}{x^3} \, dx=- \frac {x^{4} e^{- \frac {2 i \pi }{3}} \Gamma \left (\frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {2}{3} \\ \frac {5}{3} \end {matrix}\middle | {x^{6}} \right )}}{6 \Gamma \left (\frac {5}{3}\right )} + \frac {e^{\frac {i \pi }{3}} \Gamma \left (- \frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, - \frac {1}{3} \\ \frac {2}{3} \end {matrix}\middle | {x^{6}} \right )}}{6 x^{2} \Gamma \left (\frac {2}{3}\right )} \]

[In]

integrate((x**6-1)**(1/3)*(x**6+1)/x**3,x)

[Out]

-x**4*exp(-2*I*pi/3)*gamma(2/3)*hyper((-1/3, 2/3), (5/3,), x**6)/(6*gamma(5/3)) + exp(I*pi/3)*gamma(-1/3)*hype
r((-1/3, -1/3), (2/3,), x**6)/(6*x**2*gamma(2/3))

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.97 \[ \int \frac {\sqrt [3]{-1+x^6} \left (1+x^6\right )}{x^3} \, dx=\frac {1}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (\frac {2 \, {\left (x^{6} - 1\right )}^{\frac {1}{3}}}{x^{2}} + 1\right )}\right ) - \frac {{\left (x^{6} - 1\right )}^{\frac {1}{3}}}{2 \, x^{2}} - \frac {{\left (x^{6} - 1\right )}^{\frac {1}{3}}}{6 \, x^{2} {\left (\frac {x^{6} - 1}{x^{6}} - 1\right )}} + \frac {1}{18} \, \log \left (\frac {{\left (x^{6} - 1\right )}^{\frac {1}{3}}}{x^{2}} + \frac {{\left (x^{6} - 1\right )}^{\frac {2}{3}}}{x^{4}} + 1\right ) - \frac {1}{9} \, \log \left (\frac {{\left (x^{6} - 1\right )}^{\frac {1}{3}}}{x^{2}} - 1\right ) \]

[In]

integrate((x^6-1)^(1/3)*(x^6+1)/x^3,x, algorithm="maxima")

[Out]

1/9*sqrt(3)*arctan(1/3*sqrt(3)*(2*(x^6 - 1)^(1/3)/x^2 + 1)) - 1/2*(x^6 - 1)^(1/3)/x^2 - 1/6*(x^6 - 1)^(1/3)/(x
^2*((x^6 - 1)/x^6 - 1)) + 1/18*log((x^6 - 1)^(1/3)/x^2 + (x^6 - 1)^(2/3)/x^4 + 1) - 1/9*log((x^6 - 1)^(1/3)/x^
2 - 1)

Giac [F]

\[ \int \frac {\sqrt [3]{-1+x^6} \left (1+x^6\right )}{x^3} \, dx=\int { \frac {{\left (x^{6} + 1\right )} {\left (x^{6} - 1\right )}^{\frac {1}{3}}}{x^{3}} \,d x } \]

[In]

integrate((x^6-1)^(1/3)*(x^6+1)/x^3,x, algorithm="giac")

[Out]

integrate((x^6 + 1)*(x^6 - 1)^(1/3)/x^3, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt [3]{-1+x^6} \left (1+x^6\right )}{x^3} \, dx=\int \frac {{\left (x^6-1\right )}^{1/3}\,\left (x^6+1\right )}{x^3} \,d x \]

[In]

int(((x^6 - 1)^(1/3)*(x^6 + 1))/x^3,x)

[Out]

int(((x^6 - 1)^(1/3)*(x^6 + 1))/x^3, x)