\(\int \frac {\sqrt [3]{1-x^3}}{1+x^3} \, dx\) [116]

Optimal result

Integrand size = 19, antiderivative size = 272 \[ \int \frac {\sqrt [3]{1-x^3}}{1+x^3} \, dx=\frac {\sqrt [3]{2} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {\arctan \left (\frac {1+\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3}}+\frac {\log \left (2^{2/3}-\frac {1-x}{\sqrt [3]{1-x^3}}\right )}{3\ 2^{2/3}}-\frac {\log \left (1+\frac {2^{2/3} (1-x)^2}{\left (1-x^3\right )^{2/3}}-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}\right )}{3\ 2^{2/3}}+\frac {1}{3} \sqrt [3]{2} \log \left (1+\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}\right )-\frac {\log \left (2 \sqrt [3]{2}+\frac {(1-x)^2}{\left (1-x^3\right )^{2/3}}+\frac {2^{2/3} (1-x)}{\sqrt [3]{1-x^3}}\right )}{6\ 2^{2/3}} \] Output:

1/6*ln(2^(2/3)+(-1+x)/(-x^3+1)^(1/3))*2^(1/3)-1/6*ln(1+2^(2/3)*(1-x)^2/(-x 
^3+1)^(2/3)-2^(1/3)*(1-x)/(-x^3+1)^(1/3))*2^(1/3)+1/3*2^(1/3)*ln(1+2^(1/3) 
*(1-x)/(-x^3+1)^(1/3))-1/12*ln(2*2^(1/3)+(1-x)^2/(-x^3+1)^(2/3)+2^(2/3)*(1 
-x)/(-x^3+1)^(1/3))*2^(1/3)+1/3*2^(1/3)*arctan(1/3*(1-2*2^(1/3)*(1-x)/(-x^ 
3+1)^(1/3))*3^(1/2))*3^(1/2)+1/6*arctan(1/3*(1+2^(1/3)*(1-x)/(-x^3+1)^(1/3 
))*3^(1/2))*2^(1/3)*3^(1/2)
 

Mathematica [A] (verified)

Time = 1.88 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.04 \[ \int \frac {\sqrt [3]{1-x^3}}{1+x^3} \, dx=-\frac {2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{1-x^3}}{\sqrt [3]{2}-\sqrt [3]{2} x+\sqrt [3]{1-x^3}}\right )+4 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{1-x^3}}{-2 \sqrt [3]{2}+2 \sqrt [3]{2} x+\sqrt [3]{1-x^3}}\right )-4 \log \left (-\sqrt [3]{2}+\sqrt [3]{2} x-\sqrt [3]{1-x^3}\right )-2 \log \left (-\sqrt [3]{2}+\sqrt [3]{2} x+2 \sqrt [3]{1-x^3}\right )+2 \log \left (2^{2/3}-2\ 2^{2/3} x+2^{2/3} x^2+(-1+x) \sqrt [3]{2-2 x^3}+\left (1-x^3\right )^{2/3}\right )+\log \left (2^{2/3}-2\ 2^{2/3} x+2^{2/3} x^2-2 (-1+x) \sqrt [3]{2-2 x^3}+4 \left (1-x^3\right )^{2/3}\right )}{6\ 2^{2/3}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.08, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.526, Rules used = {927, 982, 821, 16, 1142, 25, 27, 1082, 217, 1103}

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.

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

\(\Big \downarrow \) 927

\(\displaystyle -9 \int \frac {1-x}{\sqrt [3]{1-x^3} \left (4-\frac {(1-x)^3}{1-x^3}\right ) \left (\frac {2 (1-x)^3}{1-x^3}+1\right )}d\frac {1-x}{\sqrt [3]{1-x^3}}\)

\(\Big \downarrow \) 982

\(\displaystyle -9 \left (\frac {1}{9} \int \frac {1-x}{\sqrt [3]{1-x^3} \left (4-\frac {(1-x)^3}{1-x^3}\right )}d\frac {1-x}{\sqrt [3]{1-x^3}}+\frac {2}{9} \int \frac {1-x}{\sqrt [3]{1-x^3} \left (\frac {2 (1-x)^3}{1-x^3}+1\right )}d\frac {1-x}{\sqrt [3]{1-x^3}}\right )\)

\(\Big \downarrow \) 821

\(\displaystyle -9 \left (\frac {2}{9} \left (\frac {\int \frac {\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}{\frac {2^{2/3} (1-x)^2}{\left (1-x^3\right )^{2/3}}-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}d\frac {1-x}{\sqrt [3]{1-x^3}}}{3 \sqrt [3]{2}}-\frac {\int \frac {1}{\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}d\frac {1-x}{\sqrt [3]{1-x^3}}}{3 \sqrt [3]{2}}\right )+\frac {1}{9} \left (\frac {\int \frac {1}{2^{2/3}-\frac {1-x}{\sqrt [3]{1-x^3}}}d\frac {1-x}{\sqrt [3]{1-x^3}}}{3\ 2^{2/3}}-\frac {\int \frac {2^{2/3}-\frac {1-x}{\sqrt [3]{1-x^3}}}{\frac {(1-x)^2}{\left (1-x^3\right )^{2/3}}+\frac {2^{2/3} (1-x)}{\sqrt [3]{1-x^3}}+2 \sqrt [3]{2}}d\frac {1-x}{\sqrt [3]{1-x^3}}}{3\ 2^{2/3}}\right )\right )\)

\(\Big \downarrow \) 16

\(\displaystyle -9 \left (\frac {2}{9} \left (\frac {\int \frac {\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}{\frac {2^{2/3} (1-x)^2}{\left (1-x^3\right )^{2/3}}-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}d\frac {1-x}{\sqrt [3]{1-x^3}}}{3 \sqrt [3]{2}}-\frac {\log \left (\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )}{3\ 2^{2/3}}\right )+\frac {1}{9} \left (-\frac {\int \frac {2^{2/3}-\frac {1-x}{\sqrt [3]{1-x^3}}}{\frac {(1-x)^2}{\left (1-x^3\right )^{2/3}}+\frac {2^{2/3} (1-x)}{\sqrt [3]{1-x^3}}+2 \sqrt [3]{2}}d\frac {1-x}{\sqrt [3]{1-x^3}}}{3\ 2^{2/3}}-\frac {\log \left (2^{2/3}-\frac {1-x}{\sqrt [3]{1-x^3}}\right )}{3\ 2^{2/3}}\right )\right )\)

\(\Big \downarrow \) 1142

\(\displaystyle -9 \left (\frac {2}{9} \left (\frac {\frac {3}{2} \int \frac {1}{\frac {2^{2/3} (1-x)^2}{\left (1-x^3\right )^{2/3}}-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}d\frac {1-x}{\sqrt [3]{1-x^3}}+\frac {\int -\frac {\sqrt [3]{2} \left (1-\frac {2 \sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}\right )}{\frac {2^{2/3} (1-x)^2}{\left (1-x^3\right )^{2/3}}-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}d\frac {1-x}{\sqrt [3]{1-x^3}}}{2 \sqrt [3]{2}}}{3 \sqrt [3]{2}}-\frac {\log \left (\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )}{3\ 2^{2/3}}\right )+\frac {1}{9} \left (-\frac {\frac {3 \int \frac {1}{\frac {(1-x)^2}{\left (1-x^3\right )^{2/3}}+\frac {2^{2/3} (1-x)}{\sqrt [3]{1-x^3}}+2 \sqrt [3]{2}}d\frac {1-x}{\sqrt [3]{1-x^3}}}{\sqrt [3]{2}}-\frac {1}{2} \int \frac {2^{2/3} \left (\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )}{\frac {(1-x)^2}{\left (1-x^3\right )^{2/3}}+\frac {2^{2/3} (1-x)}{\sqrt [3]{1-x^3}}+2 \sqrt [3]{2}}d\frac {1-x}{\sqrt [3]{1-x^3}}}{3\ 2^{2/3}}-\frac {\log \left (2^{2/3}-\frac {1-x}{\sqrt [3]{1-x^3}}\right )}{3\ 2^{2/3}}\right )\right )\)

\(\Big \downarrow \) 25

\(\displaystyle -9 \left (\frac {2}{9} \left (\frac {\frac {3}{2} \int \frac {1}{\frac {2^{2/3} (1-x)^2}{\left (1-x^3\right )^{2/3}}-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}d\frac {1-x}{\sqrt [3]{1-x^3}}-\frac {\int \frac {\sqrt [3]{2} \left (1-\frac {2 \sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}\right )}{\frac {2^{2/3} (1-x)^2}{\left (1-x^3\right )^{2/3}}-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}d\frac {1-x}{\sqrt [3]{1-x^3}}}{2 \sqrt [3]{2}}}{3 \sqrt [3]{2}}-\frac {\log \left (\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )}{3\ 2^{2/3}}\right )+\frac {1}{9} \left (-\frac {\frac {3 \int \frac {1}{\frac {(1-x)^2}{\left (1-x^3\right )^{2/3}}+\frac {2^{2/3} (1-x)}{\sqrt [3]{1-x^3}}+2 \sqrt [3]{2}}d\frac {1-x}{\sqrt [3]{1-x^3}}}{\sqrt [3]{2}}-\frac {1}{2} \int \frac {2^{2/3} \left (\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )}{\frac {(1-x)^2}{\left (1-x^3\right )^{2/3}}+\frac {2^{2/3} (1-x)}{\sqrt [3]{1-x^3}}+2 \sqrt [3]{2}}d\frac {1-x}{\sqrt [3]{1-x^3}}}{3\ 2^{2/3}}-\frac {\log \left (2^{2/3}-\frac {1-x}{\sqrt [3]{1-x^3}}\right )}{3\ 2^{2/3}}\right )\right )\)

\(\Big \downarrow \) 27

\(\displaystyle -9 \left (\frac {2}{9} \left (\frac {\frac {3}{2} \int \frac {1}{\frac {2^{2/3} (1-x)^2}{\left (1-x^3\right )^{2/3}}-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}d\frac {1-x}{\sqrt [3]{1-x^3}}-\frac {1}{2} \int \frac {1-\frac {2 \sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}}{\frac {2^{2/3} (1-x)^2}{\left (1-x^3\right )^{2/3}}-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}d\frac {1-x}{\sqrt [3]{1-x^3}}}{3 \sqrt [3]{2}}-\frac {\log \left (\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )}{3\ 2^{2/3}}\right )+\frac {1}{9} \left (-\frac {\frac {3 \int \frac {1}{\frac {(1-x)^2}{\left (1-x^3\right )^{2/3}}+\frac {2^{2/3} (1-x)}{\sqrt [3]{1-x^3}}+2 \sqrt [3]{2}}d\frac {1-x}{\sqrt [3]{1-x^3}}}{\sqrt [3]{2}}-\frac {\int \frac {\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}{\frac {(1-x)^2}{\left (1-x^3\right )^{2/3}}+\frac {2^{2/3} (1-x)}{\sqrt [3]{1-x^3}}+2 \sqrt [3]{2}}d\frac {1-x}{\sqrt [3]{1-x^3}}}{\sqrt [3]{2}}}{3\ 2^{2/3}}-\frac {\log \left (2^{2/3}-\frac {1-x}{\sqrt [3]{1-x^3}}\right )}{3\ 2^{2/3}}\right )\right )\)

\(\Big \downarrow \) 1082

\(\displaystyle -9 \left (\frac {2}{9} \left (\frac {\frac {3 \int \frac {1}{-\frac {(1-x)^2}{\left (1-x^3\right )^{2/3}}-3}d\left (1-\frac {2 \sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}\right )}{\sqrt [3]{2}}-\frac {1}{2} \int \frac {1-\frac {2 \sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}}{\frac {2^{2/3} (1-x)^2}{\left (1-x^3\right )^{2/3}}-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}d\frac {1-x}{\sqrt [3]{1-x^3}}}{3 \sqrt [3]{2}}-\frac {\log \left (\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )}{3\ 2^{2/3}}\right )+\frac {1}{9} \left (-\frac {-3 \int \frac {1}{-\frac {(1-x)^2}{\left (1-x^3\right )^{2/3}}-3}d\left (\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )-\frac {\int \frac {\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}{\frac {(1-x)^2}{\left (1-x^3\right )^{2/3}}+\frac {2^{2/3} (1-x)}{\sqrt [3]{1-x^3}}+2 \sqrt [3]{2}}d\frac {1-x}{\sqrt [3]{1-x^3}}}{\sqrt [3]{2}}}{3\ 2^{2/3}}-\frac {\log \left (2^{2/3}-\frac {1-x}{\sqrt [3]{1-x^3}}\right )}{3\ 2^{2/3}}\right )\right )\)

\(\Big \downarrow \) 217

\(\displaystyle -9 \left (\frac {2}{9} \left (\frac {-\frac {1}{2} \int \frac {1-\frac {2 \sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}}{\frac {2^{2/3} (1-x)^2}{\left (1-x^3\right )^{2/3}}-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}d\frac {1-x}{\sqrt [3]{1-x^3}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt [3]{2}}}{3 \sqrt [3]{2}}-\frac {\log \left (\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )}{3\ 2^{2/3}}\right )+\frac {1}{9} \left (-\frac {\sqrt {3} \arctan \left (\frac {\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}{\sqrt {3}}\right )-\frac {\int \frac {\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}{\frac {(1-x)^2}{\left (1-x^3\right )^{2/3}}+\frac {2^{2/3} (1-x)}{\sqrt [3]{1-x^3}}+2 \sqrt [3]{2}}d\frac {1-x}{\sqrt [3]{1-x^3}}}{\sqrt [3]{2}}}{3\ 2^{2/3}}-\frac {\log \left (2^{2/3}-\frac {1-x}{\sqrt [3]{1-x^3}}\right )}{3\ 2^{2/3}}\right )\right )\)

\(\Big \downarrow \) 1103

\(\displaystyle -9 \left (\frac {2}{9} \left (\frac {\frac {\log \left (\frac {2^{2/3} (1-x)^2}{\left (1-x^3\right )^{2/3}}-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )}{2 \sqrt [3]{2}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt [3]{2}}}{3 \sqrt [3]{2}}-\frac {\log \left (\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )}{3\ 2^{2/3}}\right )+\frac {1}{9} \left (-\frac {\sqrt {3} \arctan \left (\frac {\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}{\sqrt {3}}\right )-\frac {1}{2} \log \left (\frac {(1-x)^2}{\left (1-x^3\right )^{2/3}}+\frac {2^{2/3} (1-x)}{\sqrt [3]{1-x^3}}+2 \sqrt [3]{2}\right )}{3\ 2^{2/3}}-\frac {\log \left (2^{2/3}-\frac {1-x}{\sqrt [3]{1-x^3}}\right )}{3\ 2^{2/3}}\right )\right )\)

Input:

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

Output:

-9*((2*((-((Sqrt[3]*ArcTan[(1 - (2*2^(1/3)*(1 - x))/(1 - x^3)^(1/3))/Sqrt[ 
3]])/2^(1/3)) + Log[1 + (2^(2/3)*(1 - x)^2)/(1 - x^3)^(2/3) - (2^(1/3)*(1 
- x))/(1 - x^3)^(1/3)]/(2*2^(1/3)))/(3*2^(1/3)) - Log[1 + (2^(1/3)*(1 - x) 
)/(1 - x^3)^(1/3)]/(3*2^(2/3))))/9 + (-1/3*Log[2^(2/3) - (1 - x)/(1 - x^3) 
^(1/3)]/2^(2/3) - (Sqrt[3]*ArcTan[(1 + (2^(1/3)*(1 - x))/(1 - x^3)^(1/3))/ 
Sqrt[3]] - Log[2*2^(1/3) + (1 - x)^2/(1 - x^3)^(2/3) + (2^(2/3)*(1 - x))/( 
1 - x^3)^(1/3)]/2)/(3*2^(2/3)))/9)
 

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 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, 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 821

Int[(x_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> Simp[-(3*Rt[a, 3]*Rt[b, 3])^(- 
1)   Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Simp[1/(3*Rt[a, 3]*Rt[b, 3]) 
 Int[(Rt[a, 3] + Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2 
*x^2), x], x] /; FreeQ[{a, b}, x]
 

rule 927

Int[((a_) + (b_.)*(x_)^3)^(1/3)/((c_) + (d_.)*(x_)^3), x_Symbol] :> With[{q 
 = Rt[b/a, 3]}, Simp[9*(a/(c*q))   Subst[Int[x/((4 - a*x^3)*(1 + 2*a*x^3)), 
 x], x, (1 + q*x)/(a + b*x^3)^(1/3)], x]] /; FreeQ[{a, b, c, d}, x] && NeQ[ 
b*c - a*d, 0] && EqQ[b*c + a*d, 0]
 

rule 982

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

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 1103

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, 
e}, x] && EqQ[2*c*d - b*e, 0]
 

rule 1142

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[(2*c*d - b*e)/(2*c)   Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) 
Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
 

Maple [C] (warning: unable to verify)

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

Time = 6.40 (sec) , antiderivative size = 1151, normalized size of antiderivative = 4.23

method	result	size
trager	\(\text {Expression too large to display}\)	\(1151\)

Input:

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

Output:

1/6*RootOf(_Z^3-2)*ln((6*RootOf(RootOf(_Z^3-2)^2+3*_Z*RootOf(_Z^3-2)+9*_Z^ 
2)*RootOf(_Z^3-2)^4*x^3-18*RootOf(RootOf(_Z^3-2)^2+3*_Z*RootOf(_Z^3-2)+9*_ 
Z^2)^2*RootOf(_Z^3-2)^3*x^3+x^6*RootOf(_Z^3-2)^2-3*RootOf(RootOf(_Z^3-2)^2 
+3*_Z*RootOf(_Z^3-2)+9*_Z^2)*RootOf(_Z^3-2)*x^6+18*RootOf(_Z^3-2)^2*(-x^3+ 
1)^(2/3)*RootOf(RootOf(_Z^3-2)^2+3*_Z*RootOf(_Z^3-2)+9*_Z^2)*x^2-6*RootOf( 
_Z^3-2)*(-x^3+1)^(1/3)*x^4-2*RootOf(_Z^3-2)^2*x^3+6*RootOf(RootOf(_Z^3-2)^ 
2+3*_Z*RootOf(_Z^3-2)+9*_Z^2)*RootOf(_Z^3-2)*x^3+12*(-x^3+1)^(2/3)*x^2+6*R 
ootOf(_Z^3-2)*(-x^3+1)^(1/3)*x+RootOf(_Z^3-2)^2-3*RootOf(RootOf(_Z^3-2)^2+ 
3*_Z*RootOf(_Z^3-2)+9*_Z^2)*RootOf(_Z^3-2))/(1+x)^2/(x^2-x+1)^2)-1/6*ln((6 
*RootOf(RootOf(_Z^3-2)^2+3*_Z*RootOf(_Z^3-2)+9*_Z^2)*RootOf(_Z^3-2)^4*x^3- 
18*RootOf(RootOf(_Z^3-2)^2+3*_Z*RootOf(_Z^3-2)+9*_Z^2)^2*RootOf(_Z^3-2)^3* 
x^3-x^6*RootOf(_Z^3-2)^2+3*RootOf(RootOf(_Z^3-2)^2+3*_Z*RootOf(_Z^3-2)+9*_ 
Z^2)*RootOf(_Z^3-2)*x^6-18*RootOf(_Z^3-2)^2*(-x^3+1)^(2/3)*RootOf(RootOf(_ 
Z^3-2)^2+3*_Z*RootOf(_Z^3-2)+9*_Z^2)*x^2+6*RootOf(_Z^3-2)*(-x^3+1)^(1/3)*x 
^4+18*(-x^3+1)^(1/3)*RootOf(RootOf(_Z^3-2)^2+3*_Z*RootOf(_Z^3-2)+9*_Z^2)*x 
^4+6*RootOf(_Z^3-2)^2*x^3-18*RootOf(RootOf(_Z^3-2)^2+3*_Z*RootOf(_Z^3-2)+9 
*_Z^2)*RootOf(_Z^3-2)*x^3-6*RootOf(_Z^3-2)*(-x^3+1)^(1/3)*x-18*(-x^3+1)^(1 
/3)*RootOf(RootOf(_Z^3-2)^2+3*_Z*RootOf(_Z^3-2)+9*_Z^2)*x-RootOf(_Z^3-2)^2 
+3*RootOf(RootOf(_Z^3-2)^2+3*_Z*RootOf(_Z^3-2)+9*_Z^2)*RootOf(_Z^3-2))/(1+ 
x)^2/(x^2-x+1)^2)*RootOf(_Z^3-2)-1/2*ln((6*RootOf(RootOf(_Z^3-2)^2+3*_Z...
 

Fricas [A] (verification not implemented)

Time = 1.61 (sec) , antiderivative size = 341, normalized size of antiderivative = 1.25 \[ \int \frac {\sqrt [3]{1-x^3}}{1+x^3} \, dx=\frac {1}{18} \, \sqrt {3} 2^{\frac {1}{3}} \arctan \left (-\frac {6 \, \sqrt {3} 2^{\frac {2}{3}} {\left (x^{16} - 33 \, x^{13} + 110 \, x^{10} - 110 \, x^{7} + 33 \, x^{4} - x\right )} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} - 24 \, \sqrt {3} 2^{\frac {1}{3}} {\left (x^{14} - 2 \, x^{11} - 6 \, x^{8} - 2 \, x^{5} + x^{2}\right )} {\left (-x^{3} + 1\right )}^{\frac {2}{3}} - \sqrt {3} {\left (x^{18} + 42 \, x^{15} - 417 \, x^{12} + 812 \, x^{9} - 417 \, x^{6} + 42 \, x^{3} + 1\right )}}{3 \, {\left (x^{18} - 102 \, x^{15} + 447 \, x^{12} - 628 \, x^{9} + 447 \, x^{6} - 102 \, x^{3} + 1\right )}}\right ) + \frac {1}{18} \cdot 2^{\frac {1}{3}} \log \left (-\frac {12 \, {\left (-x^{3} + 1\right )}^{\frac {2}{3}} x^{2} + 2^{\frac {2}{3}} {\left (x^{6} + 2 \, x^{3} + 1\right )} - 6 \cdot 2^{\frac {1}{3}} {\left (x^{4} - x\right )} {\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{x^{6} + 2 \, x^{3} + 1}\right ) - \frac {1}{36} \cdot 2^{\frac {1}{3}} \log \left (\frac {12 \cdot 2^{\frac {2}{3}} {\left (x^{8} - 4 \, x^{5} + x^{2}\right )} {\left (-x^{3} + 1\right )}^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (x^{12} - 32 \, x^{9} + 78 \, x^{6} - 32 \, x^{3} + 1\right )} + 6 \, {\left (x^{10} - 11 \, x^{7} + 11 \, x^{4} - x\right )} {\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{x^{12} + 4 \, x^{9} + 6 \, x^{6} + 4 \, x^{3} + 1}\right ) \] Input:

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

Output:

1/18*sqrt(3)*2^(1/3)*arctan(-1/3*(6*sqrt(3)*2^(2/3)*(x^16 - 33*x^13 + 110* 
x^10 - 110*x^7 + 33*x^4 - x)*(-x^3 + 1)^(1/3) - 24*sqrt(3)*2^(1/3)*(x^14 - 
 2*x^11 - 6*x^8 - 2*x^5 + x^2)*(-x^3 + 1)^(2/3) - sqrt(3)*(x^18 + 42*x^15 
- 417*x^12 + 812*x^9 - 417*x^6 + 42*x^3 + 1))/(x^18 - 102*x^15 + 447*x^12 
- 628*x^9 + 447*x^6 - 102*x^3 + 1)) + 1/18*2^(1/3)*log(-(12*(-x^3 + 1)^(2/ 
3)*x^2 + 2^(2/3)*(x^6 + 2*x^3 + 1) - 6*2^(1/3)*(x^4 - x)*(-x^3 + 1)^(1/3)) 
/(x^6 + 2*x^3 + 1)) - 1/36*2^(1/3)*log((12*2^(2/3)*(x^8 - 4*x^5 + x^2)*(-x 
^3 + 1)^(2/3) + 2^(1/3)*(x^12 - 32*x^9 + 78*x^6 - 32*x^3 + 1) + 6*(x^10 - 
11*x^7 + 11*x^4 - x)*(-x^3 + 1)^(1/3))/(x^12 + 4*x^9 + 6*x^6 + 4*x^3 + 1))
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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