\(\int \frac {1}{x (1-x^3)^{4/3} (1+x^3)} \, dx\) [857]

Optimal result

Integrand size = 22, antiderivative size = 154 \[ \int \frac {1}{x \left (1-x^3\right )^{4/3} \left (1+x^3\right )} \, dx=\frac {1}{2 \sqrt [3]{1-x^3}}+\frac {\arctan \left (\frac {1+2 \sqrt [3]{1-x^3}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\arctan \left (\frac {1+2^{2/3} \sqrt [3]{1-x^3}}{\sqrt {3}}\right )}{2 \sqrt [3]{2} \sqrt {3}}-\frac {\log (x)}{2}+\frac {\log \left (1+x^3\right )}{12 \sqrt [3]{2}}+\frac {1}{2} \log \left (1-\sqrt [3]{1-x^3}\right )-\frac {\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{4 \sqrt [3]{2}} \] Output:

1/2/(-x^3+1)^(1/3)+1/3*arctan(1/3*(1+2*(-x^3+1)^(1/3))*3^(1/2))*3^(1/2)-1/ 
12*arctan(1/3*(1+2^(2/3)*(-x^3+1)^(1/3))*3^(1/2))*2^(2/3)*3^(1/2)-1/2*ln(x 
)+1/24*ln(x^3+1)*2^(2/3)+1/2*ln(1-(-x^3+1)^(1/3))-1/8*ln(2^(1/3)-(-x^3+1)^ 
(1/3))*2^(2/3)
 

Mathematica [A] (verified)

Time = 0.28 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.29 \[ \int \frac {1}{x \left (1-x^3\right )^{4/3} \left (1+x^3\right )} \, dx=\frac {1}{24} \left (\frac {12}{\sqrt [3]{1-x^3}}+8 \sqrt {3} \arctan \left (\frac {1+2 \sqrt [3]{1-x^3}}{\sqrt {3}}\right )-2\ 2^{2/3} \sqrt {3} \arctan \left (\frac {1+2^{2/3} \sqrt [3]{1-x^3}}{\sqrt {3}}\right )+8 \log \left (-1+\sqrt [3]{1-x^3}\right )-2\ 2^{2/3} \log \left (-2+2^{2/3} \sqrt [3]{1-x^3}\right )-4 \log \left (1+\sqrt [3]{1-x^3}+\left (1-x^3\right )^{2/3}\right )+2^{2/3} \log \left (2+2^{2/3} \sqrt [3]{1-x^3}+\sqrt [3]{2} \left (1-x^3\right )^{2/3}\right )\right ) \] Input:

Integrate[1/(x*(1 - x^3)^(4/3)*(1 + x^3)),x]
 

Output:

(12/(1 - x^3)^(1/3) + 8*Sqrt[3]*ArcTan[(1 + 2*(1 - x^3)^(1/3))/Sqrt[3]] - 
2*2^(2/3)*Sqrt[3]*ArcTan[(1 + 2^(2/3)*(1 - x^3)^(1/3))/Sqrt[3]] + 8*Log[-1 
 + (1 - x^3)^(1/3)] - 2*2^(2/3)*Log[-2 + 2^(2/3)*(1 - x^3)^(1/3)] - 4*Log[ 
1 + (1 - x^3)^(1/3) + (1 - x^3)^(2/3)] + 2^(2/3)*Log[2 + 2^(2/3)*(1 - x^3) 
^(1/3) + 2^(1/3)*(1 - x^3)^(2/3)])/24
 

Rubi [A] (verified)

Time = 0.49 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.08, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {948, 96, 174, 67, 16, 1082, 217, 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.

\(\displaystyle \int \frac {1}{x \left (1-x^3\right )^{4/3} \left (x^3+1\right )} \, dx\)

\(\Big \downarrow \) 948

\(\displaystyle \frac {1}{3} \int \frac {1}{x^3 \left (1-x^3\right )^{4/3} \left (x^3+1\right )}dx^3\)

\(\Big \downarrow \) 96

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

\(\Big \downarrow \) 174

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

\(\Big \downarrow \) 67

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

\(\Big \downarrow \) 16

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

\(\Big \downarrow \) 1082

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

\(\Big \downarrow \) 217

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

\(\Big \downarrow \) 1083

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

\(\Big \downarrow \) 217

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

Input:

Int[1/(x*(1 - x^3)^(4/3)*(1 + x^3)),x]
 

Output:

(3/(2*(1 - x^3)^(1/3)) + (-((Sqrt[3]*ArcTan[(1 + 2^(2/3)*(1 - x^3)^(1/3))/ 
Sqrt[3]])/2^(1/3)) + Log[1 + x^3]/(2*2^(1/3)) + 2*(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*Log[2^(1/3) - (1 - x^3)^(1/3)])/(2*2^(1/3)))/2)/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 67

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

rule 96

Int[((e_.) + (f_.)*(x_))^(p_)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), 
x_] :> Simp[f*((e + f*x)^(p + 1)/((p + 1)*(b*e - a*f)*(d*e - c*f))), x] + S 
imp[1/((b*e - a*f)*(d*e - c*f))   Int[(b*d*e - b*c*f - a*d*f - b*d*f*x)*((e 
 + f*x)^(p + 1)/((a + b*x)*(c + d*x))), x], x] /; FreeQ[{a, b, c, d, e, f}, 
 x] && LtQ[p, -1]
 

rule 174

Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))* 
((c_.) + (d_.)*(x_))), x_] :> Simp[(b*g - a*h)/(b*c - a*d)   Int[(e + f*x)^ 
p/(a + b*x), x], x] - Simp[(d*g - c*h)/(b*c - a*d)   Int[(e + f*x)^p/(c + d 
*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]
 

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 1082

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S 
implify[a*(c/b^2)]}, Simp[-2/b   Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b 
)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /; Fre 
eQ[{a, b, c}, x]
 

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 [A] (verified)

Time = 5.30 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.36

method	result	size
pseudoelliptic	\(\frac {-2 \arctan \left (\frac {\left (1+2^{\frac {2}{3}} \left (-x^{3}+1\right )^{\frac {1}{3}}\right ) \sqrt {3}}{3}\right ) 2^{\frac {2}{3}} \sqrt {3}\, \left (-x^{3}+1\right )^{\frac {1}{3}}-2 \,2^{\frac {2}{3}} \ln \left (\left (-x^{3}+1\right )^{\frac {1}{3}}-2^{\frac {1}{3}}\right ) \left (-x^{3}+1\right )^{\frac {1}{3}}+2^{\frac {2}{3}} \ln \left (\left (-x^{3}+1\right )^{\frac {2}{3}}+2^{\frac {1}{3}} \left (-x^{3}+1\right )^{\frac {1}{3}}+2^{\frac {2}{3}}\right ) \left (-x^{3}+1\right )^{\frac {1}{3}}+8 \arctan \left (\frac {\left (1+2 \left (-x^{3}+1\right )^{\frac {1}{3}}\right ) \sqrt {3}}{3}\right ) \sqrt {3}\, \left (-x^{3}+1\right )^{\frac {1}{3}}+8 \ln \left (-1+\left (-x^{3}+1\right )^{\frac {1}{3}}\right ) \left (-x^{3}+1\right )^{\frac {1}{3}}-4 \ln \left (\left (-x^{3}+1\right )^{\frac {2}{3}}+\left (-x^{3}+1\right )^{\frac {1}{3}}+1\right ) \left (-x^{3}+1\right )^{\frac {1}{3}}+12}{24 \left (-x^{3}+1\right )^{\frac {1}{3}}}\)	\(210\)

Input:

int(1/x/(-x^3+1)^(4/3)/(x^3+1),x,method=_RETURNVERBOSE)
 

Output:

1/24*(-2*arctan(1/3*(1+2^(2/3)*(-x^3+1)^(1/3))*3^(1/2))*2^(2/3)*3^(1/2)*(- 
x^3+1)^(1/3)-2*2^(2/3)*ln((-x^3+1)^(1/3)-2^(1/3))*(-x^3+1)^(1/3)+2^(2/3)*l 
n((-x^3+1)^(2/3)+2^(1/3)*(-x^3+1)^(1/3)+2^(2/3))*(-x^3+1)^(1/3)+8*arctan(1 
/3*(1+2*(-x^3+1)^(1/3))*3^(1/2))*3^(1/2)*(-x^3+1)^(1/3)+8*ln(-1+(-x^3+1)^( 
1/3))*(-x^3+1)^(1/3)-4*ln((-x^3+1)^(2/3)+(-x^3+1)^(1/3)+1)*(-x^3+1)^(1/3)+ 
12)/(-x^3+1)^(1/3)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.29 \[ \int \frac {1}{x \left (1-x^3\right )^{4/3} \left (1+x^3\right )} \, dx=-\frac {12 \cdot 2^{\frac {1}{6}} \sqrt {\frac {1}{6}} {\left (x^{3} - 1\right )} \arctan \left (2^{\frac {1}{6}} \sqrt {\frac {1}{6}} {\left (2^{\frac {1}{3}} + 2 \, {\left (-x^{3} + 1\right )}^{\frac {1}{3}}\right )}\right ) - 2^{\frac {2}{3}} {\left (x^{3} - 1\right )} \log \left (2^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + {\left (-x^{3} + 1\right )}^{\frac {2}{3}}\right ) + 2 \cdot 2^{\frac {2}{3}} {\left (x^{3} - 1\right )} \log \left (-2^{\frac {1}{3}} + {\left (-x^{3} + 1\right )}^{\frac {1}{3}}\right ) - 8 \, \sqrt {3} {\left (x^{3} - 1\right )} \arctan \left (\frac {2}{3} \, \sqrt {3} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + \frac {1}{3} \, \sqrt {3}\right ) + 4 \, {\left (x^{3} - 1\right )} \log \left ({\left (-x^{3} + 1\right )}^{\frac {2}{3}} + {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + 1\right ) - 8 \, {\left (x^{3} - 1\right )} \log \left ({\left (-x^{3} + 1\right )}^{\frac {1}{3}} - 1\right ) + 12 \, {\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{24 \, {\left (x^{3} - 1\right )}} \] Input:

integrate(1/x/(-x^3+1)^(4/3)/(x^3+1),x, algorithm="fricas")
 

Output:

-1/24*(12*2^(1/6)*sqrt(1/6)*(x^3 - 1)*arctan(2^(1/6)*sqrt(1/6)*(2^(1/3) + 
2*(-x^3 + 1)^(1/3))) - 2^(2/3)*(x^3 - 1)*log(2^(2/3) + 2^(1/3)*(-x^3 + 1)^ 
(1/3) + (-x^3 + 1)^(2/3)) + 2*2^(2/3)*(x^3 - 1)*log(-2^(1/3) + (-x^3 + 1)^ 
(1/3)) - 8*sqrt(3)*(x^3 - 1)*arctan(2/3*sqrt(3)*(-x^3 + 1)^(1/3) + 1/3*sqr 
t(3)) + 4*(x^3 - 1)*log((-x^3 + 1)^(2/3) + (-x^3 + 1)^(1/3) + 1) - 8*(x^3 
- 1)*log((-x^3 + 1)^(1/3) - 1) + 12*(-x^3 + 1)^(2/3))/(x^3 - 1)
 

Sympy [F]

\[ \int \frac {1}{x \left (1-x^3\right )^{4/3} \left (1+x^3\right )} \, dx=\int \frac {1}{x \left (- \left (x - 1\right ) \left (x^{2} + x + 1\right )\right )^{\frac {4}{3}} \left (x + 1\right ) \left (x^{2} - x + 1\right )}\, dx \] Input:

integrate(1/x/(-x**3+1)**(4/3)/(x**3+1),x)
 

Output:

Integral(1/(x*(-(x - 1)*(x**2 + x + 1))**(4/3)*(x + 1)*(x**2 - x + 1)), x)
 

Maxima [F]

\[ \int \frac {1}{x \left (1-x^3\right )^{4/3} \left (1+x^3\right )} \, dx=\int { \frac {1}{{\left (x^{3} + 1\right )} {\left (-x^{3} + 1\right )}^{\frac {4}{3}} x} \,d x } \] Input:

integrate(1/x/(-x^3+1)^(4/3)/(x^3+1),x, algorithm="maxima")
 

Output:

integrate(1/((x^3 + 1)*(-x^3 + 1)^(4/3)*x), x)
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.04 \[ \int \frac {1}{x \left (1-x^3\right )^{4/3} \left (1+x^3\right )} \, dx=-\frac {1}{12} \, \sqrt {3} 2^{\frac {2}{3}} \arctan \left (\frac {1}{6} \, \sqrt {3} 2^{\frac {2}{3}} {\left (2^{\frac {1}{3}} + 2 \, {\left (-x^{3} + 1\right )}^{\frac {1}{3}}\right )}\right ) + \frac {1}{24} \cdot 2^{\frac {2}{3}} \log \left (2^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + {\left (-x^{3} + 1\right )}^{\frac {2}{3}}\right ) - \frac {1}{12} \cdot 2^{\frac {2}{3}} \log \left ({\left | -2^{\frac {1}{3}} + {\left (-x^{3} + 1\right )}^{\frac {1}{3}} \right |}\right ) + \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) + \frac {1}{2 \, {\left (-x^{3} + 1\right )}^{\frac {1}{3}}} - \frac {1}{6} \, \log \left ({\left (-x^{3} + 1\right )}^{\frac {2}{3}} + {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {1}{3} \, \log \left ({\left | {\left (-x^{3} + 1\right )}^{\frac {1}{3}} - 1 \right |}\right ) \] Input:

integrate(1/x/(-x^3+1)^(4/3)/(x^3+1),x, algorithm="giac")
 

Output:

-1/12*sqrt(3)*2^(2/3)*arctan(1/6*sqrt(3)*2^(2/3)*(2^(1/3) + 2*(-x^3 + 1)^( 
1/3))) + 1/24*2^(2/3)*log(2^(2/3) + 2^(1/3)*(-x^3 + 1)^(1/3) + (-x^3 + 1)^ 
(2/3)) - 1/12*2^(2/3)*log(abs(-2^(1/3) + (-x^3 + 1)^(1/3))) + 1/3*sqrt(3)* 
arctan(1/3*sqrt(3)*(2*(-x^3 + 1)^(1/3) + 1)) + 1/2/(-x^3 + 1)^(1/3) - 1/6* 
log((-x^3 + 1)^(2/3) + (-x^3 + 1)^(1/3) + 1) + 1/3*log(abs((-x^3 + 1)^(1/3 
) - 1))
 

Mupad [B] (verification not implemented)

Time = 3.47 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.64 \[ \int \frac {1}{x \left (1-x^3\right )^{4/3} \left (1+x^3\right )} \, dx=\frac {\ln \left (\frac {17}{4}-\frac {17\,{\left (1-x^3\right )}^{1/3}}{4}\right )}{3}+\ln \left (\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )\,\left (1458\,{\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )}^2-\frac {459\,{\left (1-x^3\right )}^{1/3}}{4}\right )-\frac {63}{4}\right )\,\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )-\ln \left (\left (\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )\,\left (1458\,{\left (\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )}^2-\frac {459\,{\left (1-x^3\right )}^{1/3}}{4}\right )+\frac {63}{4}\right )\,\left (\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )-\frac {2^{2/3}\,\ln \left (\frac {2^{2/3}\,\left (\frac {81\,2^{1/3}}{4}-\frac {459\,{\left (1-x^3\right )}^{1/3}}{4}\right )}{12}+\frac {63}{4}\right )}{12}+\frac {1}{2\,{\left (1-x^3\right )}^{1/3}}+\frac {{\left (-1\right )}^{1/3}\,2^{2/3}\,\ln \left (\frac {{\left (-1\right )}^{1/3}\,2^{2/3}\,\left (\frac {81\,{\left (-1\right )}^{2/3}\,2^{1/3}}{4}-\frac {459\,{\left (1-x^3\right )}^{1/3}}{4}\right )}{12}-\frac {63}{4}\right )}{12}-\frac {{\left (-1\right )}^{1/3}\,2^{2/3}\,\ln \left (\frac {{\left (-1\right )}^{1/3}\,2^{2/3}\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,\left (\frac {459\,{\left (1-x^3\right )}^{1/3}}{4}-\frac {81\,{\left (-1\right )}^{2/3}\,2^{1/3}\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^2}{16}\right )}{24}-\frac {63}{4}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{24} \] Input:

int(1/(x*(1 - x^3)^(4/3)*(x^3 + 1)),x)
 

Output:

log(17/4 - (17*(1 - x^3)^(1/3))/4)/3 + log(((3^(1/2)*1i)/6 - 1/6)*(1458*(( 
3^(1/2)*1i)/6 - 1/6)^2 - (459*(1 - x^3)^(1/3))/4) - 63/4)*((3^(1/2)*1i)/6 
- 1/6) - log(((3^(1/2)*1i)/6 + 1/6)*(1458*((3^(1/2)*1i)/6 + 1/6)^2 - (459* 
(1 - x^3)^(1/3))/4) + 63/4)*((3^(1/2)*1i)/6 + 1/6) - (2^(2/3)*log((2^(2/3) 
*((81*2^(1/3))/4 - (459*(1 - x^3)^(1/3))/4))/12 + 63/4))/12 + 1/(2*(1 - x^ 
3)^(1/3)) + ((-1)^(1/3)*2^(2/3)*log(((-1)^(1/3)*2^(2/3)*((81*(-1)^(2/3)*2^ 
(1/3))/4 - (459*(1 - x^3)^(1/3))/4))/12 - 63/4))/12 - ((-1)^(1/3)*2^(2/3)* 
log(((-1)^(1/3)*2^(2/3)*(3^(1/2)*1i + 1)*((459*(1 - x^3)^(1/3))/4 - (81*(- 
1)^(2/3)*2^(1/3)*(3^(1/2)*1i + 1)^2)/16))/24 - 63/4)*(3^(1/2)*1i + 1))/24
 

Reduce [F]

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

int(1/x/(-x^3+1)^(4/3)/(x^3+1),x)
 

Output:

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