\(\int \frac {\sqrt [3]{-1+x^3} (1+x^3)}{x^7} \, dx\) [1373]

Optimal result

Integrand size = 18, antiderivative size = 99 \[ \int \frac {\sqrt [3]{-1+x^3} \left (1+x^3\right )}{x^7} \, dx=\frac {\left (-3-5 x^3\right ) \sqrt [3]{-1+x^3}}{18 x^6}-\frac {4 \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{-1+x^3}}{\sqrt {3}}\right )}{9 \sqrt {3}}+\frac {4}{27} \log \left (1+\sqrt [3]{-1+x^3}\right )-\frac {2}{27} \log \left (1-\sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right ) \] Output:

1/18*(-5*x^3-3)*(x^3-1)^(1/3)/x^6+4/27*arctan(-1/3*3^(1/2)+2/3*(x^3-1)^(1/ 
3)*3^(1/2))*3^(1/2)+4/27*ln(1+(x^3-1)^(1/3))-2/27*ln(1-(x^3-1)^(1/3)+(x^3- 
1)^(2/3))
 

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.93 \[ \int \frac {\sqrt [3]{-1+x^3} \left (1+x^3\right )}{x^7} \, dx=\frac {1}{54} \left (-\frac {3 \sqrt [3]{-1+x^3} \left (3+5 x^3\right )}{x^6}-8 \sqrt {3} \arctan \left (\frac {1-2 \sqrt [3]{-1+x^3}}{\sqrt {3}}\right )+8 \log \left (1+\sqrt [3]{-1+x^3}\right )-4 \log \left (1-\sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right )\right ) \] Input:

Integrate[((-1 + x^3)^(1/3)*(1 + x^3))/x^7,x]
 

Output:

((-3*(-1 + x^3)^(1/3)*(3 + 5*x^3))/x^6 - 8*Sqrt[3]*ArcTan[(1 - 2*(-1 + x^3 
)^(1/3))/Sqrt[3]] + 8*Log[1 + (-1 + x^3)^(1/3)] - 4*Log[1 - (-1 + x^3)^(1/ 
3) + (-1 + x^3)^(2/3)])/54
 

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.96, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {948, 87, 51, 70, 16, 1083, 217}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\sqrt [3]{x^3-1} \left (x^3+1\right )}{x^7} \, dx\)

\(\Big \downarrow \) 948

\(\displaystyle \frac {1}{3} \int \frac {\sqrt [3]{x^3-1} \left (x^3+1\right )}{x^9}dx^3\)

\(\Big \downarrow \) 87

\(\displaystyle \frac {1}{3} \left (\frac {4}{3} \int \frac {\sqrt [3]{x^3-1}}{x^6}dx^3+\frac {\left (x^3-1\right )^{4/3}}{2 x^6}\right )\)

\(\Big \downarrow \) 51

\(\displaystyle \frac {1}{3} \left (\frac {4}{3} \left (\frac {1}{3} \int \frac {1}{x^3 \left (x^3-1\right )^{2/3}}dx^3-\frac {\sqrt [3]{x^3-1}}{x^3}\right )+\frac {\left (x^3-1\right )^{4/3}}{2 x^6}\right )\)

\(\Big \downarrow \) 70

\(\displaystyle \frac {1}{3} \left (\frac {4}{3} \left (\frac {1}{3} \left (\frac {3}{2} \int \frac {1}{\sqrt [3]{x^3-1}+1}d\sqrt [3]{x^3-1}+\frac {3}{2} \int \frac {1}{x^6-\sqrt [3]{x^3-1}+1}d\sqrt [3]{x^3-1}-\frac {1}{2} \log \left (x^3\right )\right )-\frac {\sqrt [3]{x^3-1}}{x^3}\right )+\frac {\left (x^3-1\right )^{4/3}}{2 x^6}\right )\)

\(\Big \downarrow \) 16

\(\displaystyle \frac {1}{3} \left (\frac {4}{3} \left (\frac {1}{3} \left (\frac {3}{2} \int \frac {1}{x^6-\sqrt [3]{x^3-1}+1}d\sqrt [3]{x^3-1}-\frac {1}{2} \log \left (x^3\right )+\frac {3}{2} \log \left (\sqrt [3]{x^3-1}+1\right )\right )-\frac {\sqrt [3]{x^3-1}}{x^3}\right )+\frac {\left (x^3-1\right )^{4/3}}{2 x^6}\right )\)

\(\Big \downarrow \) 1083

\(\displaystyle \frac {1}{3} \left (\frac {4}{3} \left (\frac {1}{3} \left (-3 \int \frac {1}{-x^6-3}d\left (2 \sqrt [3]{x^3-1}-1\right )-\frac {1}{2} \log \left (x^3\right )+\frac {3}{2} \log \left (\sqrt [3]{x^3-1}+1\right )\right )-\frac {\sqrt [3]{x^3-1}}{x^3}\right )+\frac {\left (x^3-1\right )^{4/3}}{2 x^6}\right )\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {1}{3} \left (\frac {4}{3} \left (\frac {1}{3} \left (\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{x^3-1}-1}{\sqrt {3}}\right )-\frac {\log \left (x^3\right )}{2}+\frac {3}{2} \log \left (\sqrt [3]{x^3-1}+1\right )\right )-\frac {\sqrt [3]{x^3-1}}{x^3}\right )+\frac {\left (x^3-1\right )^{4/3}}{2 x^6}\right )\)

Input:

Int[((-1 + x^3)^(1/3)*(1 + x^3))/x^7,x]
 

Output:

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

Defintions of rubi rules used

rule 16

Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + 
b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
 

rule 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) 
Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n}, x 
] && ILtQ[m, -1] && FractionQ[n] && GtQ[n, 0]
 

rule 70

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> With[ 
{q = Rt[-(b*c - a*d)/b, 3]}, Simp[-Log[RemoveContent[a + b*x, x]]/(2*b*q^2) 
, x] + (Simp[3/(2*b*q)   Subst[Int[1/(q^2 - q*x + x^2), x], x, (c + d*x)^(1 
/3)], x] + Simp[3/(2*b*q^2)   Subst[Int[1/(q + x), x], x, (c + d*x)^(1/3)], 
 x])] /; FreeQ[{a, b, c, d}, x] && NegQ[(b*c - a*d)/b]
 

rule 87

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p 
+ 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p 
+ 1)))/(f*(p + 1)*(c*f - d*e))   Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] 
/; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege 
rQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ[p, n]))))
 

rule 217

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( 
-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & 
& (LtQ[a, 0] || LtQ[b, 0])
 

rule 948

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

rule 1083

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2   Subst[I 
nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, 
x]
 

Maple [C] (warning: unable to verify)

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

Time = 2.05 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.92

method	result	size
risch	\(-\frac {5 x^{6}-2 x^{3}-3}{18 x^{6} \left (x^{3}-1\right )^{\frac {2}{3}}}+\frac {4 {\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {2}{3}} \left (\frac {2 \Gamma \left (\frac {2}{3}\right ) x^{3} \operatorname {hypergeom}\left (\left [1, 1, \frac {5}{3}\right ], \left [2, 2\right ], x^{3}\right )}{3}+\left (\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+3 \ln \left (x \right )+i \pi \right ) \Gamma \left (\frac {2}{3}\right )\right )}{27 \Gamma \left (\frac {2}{3}\right ) \operatorname {signum}\left (x^{3}-1\right )^{\frac {2}{3}}}\)	\(91\)
pseudoelliptic	\(\frac {8 \sqrt {3}\, \arctan \left (\frac {\left (2 \left (x^{3}-1\right )^{\frac {1}{3}}-1\right ) \sqrt {3}}{3}\right ) x^{6}-4 \ln \left (1-\left (x^{3}-1\right )^{\frac {1}{3}}+\left (x^{3}-1\right )^{\frac {2}{3}}\right ) x^{6}+8 \ln \left (1+\left (x^{3}-1\right )^{\frac {1}{3}}\right ) x^{6}-15 x^{3} \left (x^{3}-1\right )^{\frac {1}{3}}-9 \left (x^{3}-1\right )^{\frac {1}{3}}}{54 {\left (1-\left (x^{3}-1\right )^{\frac {1}{3}}+\left (x^{3}-1\right )^{\frac {2}{3}}\right )}^{2} {\left (1+\left (x^{3}-1\right )^{\frac {1}{3}}\right )}^{2}}\)	\(120\)
meijerg	\(\frac {\operatorname {signum}\left (x^{3}-1\right )^{\frac {1}{3}} \left (-\frac {\Gamma \left (\frac {2}{3}\right ) x^{3} \operatorname {hypergeom}\left (\left [1, 1, \frac {5}{3}\right ], \left [2, 3\right ], x^{3}\right )}{3}-\left (\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}-1+3 \ln \left (x \right )+i \pi \right ) \Gamma \left (\frac {2}{3}\right )-\frac {3 \Gamma \left (\frac {2}{3}\right )}{x^{3}}\right )}{9 \Gamma \left (\frac {2}{3}\right ) {\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {1}{3}}}-\frac {\operatorname {signum}\left (x^{3}-1\right )^{\frac {1}{3}} \left (\frac {5 \Gamma \left (\frac {2}{3}\right ) x^{3} \operatorname {hypergeom}\left (\left [1, 1, \frac {8}{3}\right ], \left [2, 4\right ], x^{3}\right )}{27}+\frac {\left (\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+3 \ln \left (x \right )+i \pi \right ) \Gamma \left (\frac {2}{3}\right )}{3}+\frac {3 \Gamma \left (\frac {2}{3}\right )}{2 x^{6}}-\frac {\Gamma \left (\frac {2}{3}\right )}{x^{3}}\right )}{9 \Gamma \left (\frac {2}{3}\right ) {\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {1}{3}}}\)	\(156\)
trager	\(-\frac {\left (5 x^{3}+3\right ) \left (x^{3}-1\right )^{\frac {1}{3}}}{18 x^{6}}+\frac {4 \ln \left (-\frac {864256 \operatorname {RootOf}\left (4096 \textit {\_Z}^{2}+64 \textit {\_Z} +1\right )^{2} x^{3}-173056 \operatorname {RootOf}\left (4096 \textit {\_Z}^{2}+64 \textit {\_Z} +1\right ) x^{3}+352128 \operatorname {RootOf}\left (4096 \textit {\_Z}^{2}+64 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}}+8628 x^{3}-6914048 \operatorname {RootOf}\left (4096 \textit {\_Z}^{2}+64 \textit {\_Z} +1\right )^{2}+1263936 \left (x^{3}-1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (4096 \textit {\_Z}^{2}+64 \textit {\_Z} +1\right )+19749 \left (x^{3}-1\right )^{\frac {2}{3}}+803776 \operatorname {RootOf}\left (4096 \textit {\_Z}^{2}+64 \textit {\_Z} +1\right )+14247 \left (x^{3}-1\right )^{\frac {1}{3}}-7190}{x^{3}}\right )}{27}+\frac {256 \operatorname {RootOf}\left (4096 \textit {\_Z}^{2}+64 \textit {\_Z} +1\right ) \ln \left (\frac {6754304 \operatorname {RootOf}\left (4096 \textit {\_Z}^{2}+64 \textit {\_Z} +1\right )^{2} x^{3}+830784 \operatorname {RootOf}\left (4096 \textit {\_Z}^{2}+64 \textit {\_Z} +1\right ) x^{3}+352128 \operatorname {RootOf}\left (4096 \textit {\_Z}^{2}+64 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}}+10066 x^{3}-54034432 \operatorname {RootOf}\left (4096 \textit {\_Z}^{2}+64 \textit {\_Z} +1\right )^{2}-911808 \left (x^{3}-1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (4096 \textit {\_Z}^{2}+64 \textit {\_Z} +1\right )-14247 \left (x^{3}-1\right )^{\frac {2}{3}}-2108224 \operatorname {RootOf}\left (4096 \textit {\_Z}^{2}+64 \textit {\_Z} +1\right )-19749 \left (x^{3}-1\right )^{\frac {1}{3}}-18694}{x^{3}}\right )}{27}\)	\(301\)

Input:

int((x^3-1)^(1/3)*(x^3+1)/x^7,x,method=_RETURNVERBOSE)
 

Output:

-1/18*(5*x^6-2*x^3-3)/x^6/(x^3-1)^(2/3)+4/27/GAMMA(2/3)/signum(x^3-1)^(2/3 
)*(-signum(x^3-1))^(2/3)*(2/3*GAMMA(2/3)*x^3*hypergeom([1,1,5/3],[2,2],x^3 
)+(1/6*Pi*3^(1/2)-3/2*ln(3)+3*ln(x)+I*Pi)*GAMMA(2/3))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.89 \[ \int \frac {\sqrt [3]{-1+x^3} \left (1+x^3\right )}{x^7} \, dx=\frac {8 \, \sqrt {3} x^{6} \arctan \left (\frac {2}{3} \, \sqrt {3} {\left (x^{3} - 1\right )}^{\frac {1}{3}} - \frac {1}{3} \, \sqrt {3}\right ) - 4 \, x^{6} \log \left ({\left (x^{3} - 1\right )}^{\frac {2}{3}} - {\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1\right ) + 8 \, x^{6} \log \left ({\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1\right ) - 3 \, {\left (5 \, x^{3} + 3\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{54 \, x^{6}} \] Input:

integrate((x^3-1)^(1/3)*(x^3+1)/x^7,x, algorithm="fricas")
 

Output:

1/54*(8*sqrt(3)*x^6*arctan(2/3*sqrt(3)*(x^3 - 1)^(1/3) - 1/3*sqrt(3)) - 4* 
x^6*log((x^3 - 1)^(2/3) - (x^3 - 1)^(1/3) + 1) + 8*x^6*log((x^3 - 1)^(1/3) 
 + 1) - 3*(5*x^3 + 3)*(x^3 - 1)^(1/3))/x^6
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 86.66 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.69 \[ \int \frac {\sqrt [3]{-1+x^3} \left (1+x^3\right )}{x^7} \, dx=- \frac {\Gamma \left (\frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {2}{3} \\ \frac {5}{3} \end {matrix}\middle | {\frac {e^{2 i \pi }}{x^{3}}} \right )}}{3 x^{2} \Gamma \left (\frac {5}{3}\right )} - \frac {\Gamma \left (\frac {5}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {5}{3} \\ \frac {8}{3} \end {matrix}\middle | {\frac {e^{2 i \pi }}{x^{3}}} \right )}}{3 x^{5} \Gamma \left (\frac {8}{3}\right )} \] Input:

integrate((x**3-1)**(1/3)*(x**3+1)/x**7,x)
 

Output:

-gamma(2/3)*hyper((-1/3, 2/3), (5/3,), exp_polar(2*I*pi)/x**3)/(3*x**2*gam 
ma(5/3)) - gamma(5/3)*hyper((-1/3, 5/3), (8/3,), exp_polar(2*I*pi)/x**3)/( 
3*x**5*gamma(8/3))
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.04 \[ \int \frac {\sqrt [3]{-1+x^3} \left (1+x^3\right )}{x^7} \, dx=\frac {4}{27} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}} - 1\right )}\right ) + \frac {{\left (x^{3} - 1\right )}^{\frac {4}{3}} - 2 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{18 \, {\left (2 \, x^{3} + {\left (x^{3} - 1\right )}^{2} - 1\right )}} - \frac {{\left (x^{3} - 1\right )}^{\frac {1}{3}}}{3 \, x^{3}} - \frac {2}{27} \, \log \left ({\left (x^{3} - 1\right )}^{\frac {2}{3}} - {\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {4}{27} \, \log \left ({\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1\right ) \] Input:

integrate((x^3-1)^(1/3)*(x^3+1)/x^7,x, algorithm="maxima")
 

Output:

4/27*sqrt(3)*arctan(1/3*sqrt(3)*(2*(x^3 - 1)^(1/3) - 1)) + 1/18*((x^3 - 1) 
^(4/3) - 2*(x^3 - 1)^(1/3))/(2*x^3 + (x^3 - 1)^2 - 1) - 1/3*(x^3 - 1)^(1/3 
)/x^3 - 2/27*log((x^3 - 1)^(2/3) - (x^3 - 1)^(1/3) + 1) + 4/27*log((x^3 - 
1)^(1/3) + 1)
 

Giac [A] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.82 \[ \int \frac {\sqrt [3]{-1+x^3} \left (1+x^3\right )}{x^7} \, dx=\frac {4}{27} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}} - 1\right )}\right ) - \frac {5 \, {\left (x^{3} - 1\right )}^{\frac {4}{3}} + 8 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{18 \, x^{6}} - \frac {2}{27} \, \log \left ({\left (x^{3} - 1\right )}^{\frac {2}{3}} - {\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {4}{27} \, \log \left ({\left | {\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1 \right |}\right ) \] Input:

integrate((x^3-1)^(1/3)*(x^3+1)/x^7,x, algorithm="giac")
 

Output:

4/27*sqrt(3)*arctan(1/3*sqrt(3)*(2*(x^3 - 1)^(1/3) - 1)) - 1/18*(5*(x^3 - 
1)^(4/3) + 8*(x^3 - 1)^(1/3))/x^6 - 2/27*log((x^3 - 1)^(2/3) - (x^3 - 1)^( 
1/3) + 1) + 4/27*log(abs((x^3 - 1)^(1/3) + 1))
 

Mupad [B] (verification not implemented)

Time = 8.96 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.88 \[ \int \frac {\sqrt [3]{-1+x^3} \left (1+x^3\right )}{x^7} \, dx=\frac {\ln \left (\frac {{\left (x^3-1\right )}^{1/3}}{9}+\frac {1}{9}\right )}{9}+\frac {\ln \left (\frac {{\left (x^3-1\right )}^{1/3}}{81}+\frac {1}{81}\right )}{27}-\frac {\frac {{\left (x^3-1\right )}^{1/3}}{9}-\frac {{\left (x^3-1\right )}^{4/3}}{18}}{{\left (x^3-1\right )}^2+2\,x^3-1}+\ln \left ({\left (x^3-1\right )}^{1/3}-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-\frac {1}{18}+\frac {\sqrt {3}\,1{}\mathrm {i}}{18}\right )-\frac {{\left (x^3-1\right )}^{1/3}}{3\,x^3}-\ln \left (\frac {1}{2}-{\left (x^3-1\right )}^{1/3}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{18}+\frac {\sqrt {3}\,1{}\mathrm {i}}{18}\right )-\ln \left (\frac {1}{6}-\frac {{\left (x^3-1\right )}^{1/3}}{3}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )\,\left (\frac {1}{54}+\frac {\sqrt {3}\,1{}\mathrm {i}}{54}\right )+\ln \left (\frac {{\left (x^3-1\right )}^{1/3}}{3}-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )\,\left (-\frac {1}{54}+\frac {\sqrt {3}\,1{}\mathrm {i}}{54}\right ) \] Input:

int(((x^3 - 1)^(1/3)*(x^3 + 1))/x^7,x)
 

Output:

log((x^3 - 1)^(1/3)/9 + 1/9)/9 + log((x^3 - 1)^(1/3)/81 + 1/81)/27 - ((x^3 
 - 1)^(1/3)/9 - (x^3 - 1)^(4/3)/18)/((x^3 - 1)^2 + 2*x^3 - 1) + log((3^(1/ 
2)*1i)/2 + (x^3 - 1)^(1/3) - 1/2)*((3^(1/2)*1i)/18 - 1/18) - (x^3 - 1)^(1/ 
3)/(3*x^3) - log((3^(1/2)*1i)/2 - (x^3 - 1)^(1/3) + 1/2)*((3^(1/2)*1i)/18 
+ 1/18) - log((3^(1/2)*1i)/6 - (x^3 - 1)^(1/3)/3 + 1/6)*((3^(1/2)*1i)/54 + 
 1/54) + log((3^(1/2)*1i)/6 + (x^3 - 1)^(1/3)/3 - 1/6)*((3^(1/2)*1i)/54 - 
1/54)
 

Reduce [F]

\[ \int \frac {\sqrt [3]{-1+x^3} \left (1+x^3\right )}{x^7} \, dx=\frac {-5 \left (x^{3}-1\right )^{\frac {1}{3}} x^{3}-3 \left (x^{3}-1\right )^{\frac {1}{3}}+8 \left (\int \frac {\left (x^{3}-1\right )^{\frac {1}{3}}}{x^{4}-x}d x \right ) x^{6}}{18 x^{6}} \] Input:

int((x^3-1)^(1/3)*(x^3+1)/x^7,x)
 

Output:

( - 5*(x**3 - 1)**(1/3)*x**3 - 3*(x**3 - 1)**(1/3) + 8*int((x**3 - 1)**(1/ 
3)/(x**4 - x),x)*x**6)/(18*x**6)