\(\int \frac {(1-x) (1-x^3)^{2/3}}{1+x^3} \, dx\) [115]

Optimal result

Integrand size = 24, antiderivative size = 383 \[ \int \frac {(1-x) \left (1-x^3\right )^{2/3}}{1+x^3} \, dx=-\frac {2^{2/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\arctan \left (\frac {1+\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3}}+\frac {\arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {2^{2/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{2} x^2 \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},x^3\right )-\frac {\log \left ((1-x) (1+x)^2\right )}{6 \sqrt [3]{2}}-\frac {\log \left (1+x^3\right )}{3 \sqrt [3]{2}}-\frac {\log \left (1+\frac {2^{2/3} (1-x)^2}{\left (1-x^3\right )^{2/3}}-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}\right )}{3 \sqrt [3]{2}}+\frac {1}{3} 2^{2/3} \log \left (1+\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}\right )+\frac {\log \left (-\sqrt [3]{2} x-\sqrt [3]{1-x^3}\right )}{\sqrt [3]{2}}-\frac {1}{2} \log \left (x+\sqrt [3]{1-x^3}\right )+\frac {\log \left (-1+x+2^{2/3} \sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2}} \] Output:

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

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.

Time = 15.09 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.36 \[ \int \frac {(1-x) \left (1-x^3\right )^{2/3}}{1+x^3} \, dx=-\frac {1}{2} x^2 \operatorname {AppellF1}\left (\frac {2}{3},-\frac {2}{3},1,\frac {5}{3},x^3,-x^3\right )-\frac {4 x \left (1-x^3\right )^{2/3} \operatorname {AppellF1}\left (\frac {1}{3},-\frac {2}{3},1,\frac {4}{3},x^3,-x^3\right )}{\left (1+x^3\right ) \left (-4 \operatorname {AppellF1}\left (\frac {1}{3},-\frac {2}{3},1,\frac {4}{3},x^3,-x^3\right )+x^3 \left (3 \operatorname {AppellF1}\left (\frac {4}{3},-\frac {2}{3},2,\frac {7}{3},x^3,-x^3\right )+2 \operatorname {AppellF1}\left (\frac {4}{3},\frac {1}{3},1,\frac {7}{3},x^3,-x^3\right )\right )\right )} \] Input:

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

Output:

-1/2*(x^2*AppellF1[2/3, -2/3, 1, 5/3, x^3, -x^3]) - (4*x*(1 - x^3)^(2/3)*A 
ppellF1[1/3, -2/3, 1, 4/3, x^3, -x^3])/((1 + x^3)*(-4*AppellF1[1/3, -2/3, 
1, 4/3, x^3, -x^3] + x^3*(3*AppellF1[4/3, -2/3, 2, 7/3, x^3, -x^3] + 2*App 
ellF1[4/3, 1/3, 1, 7/3, x^3, -x^3])))
 

Rubi [A] (verified)

Time = 1.14 (sec) , antiderivative size = 655, normalized size of antiderivative = 1.71, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {7276, 2009}

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.

\(\displaystyle \int \frac {(1-x) \left (1-x^3\right )^{2/3}}{x^3+1} \, dx\)

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {2^{2/3} \arctan \left (\frac {\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {\left (1+(-1)^{2/3}\right ) \arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {\left (1-\sqrt [3]{-1}\right ) \arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {2 \arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {\left (1-\sqrt [3]{-1}\right ) \arctan \left (\frac {1-\frac {\sqrt [3]{2} \left (x+\sqrt [3]{-1}\right )}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3}}-\frac {\left (1+(-1)^{2/3}\right ) \arctan \left (\frac {\frac {(-1)^{2/3} \sqrt [3]{2} \left (\sqrt [3]{-1} x+1\right )}{\sqrt [3]{1-x^3}}+1}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3}}+\frac {1}{6} (-1)^{2/3} \left (1-\sqrt [3]{-1}\right ) x^2 \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},x^3\right )+\frac {1}{6} \left (1-\sqrt [3]{-1}\right ) x^2 \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},x^3\right )+\frac {1}{3} x^2 \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},x^3\right )-\frac {1}{6} \left (1+(-1)^{2/3}\right ) \log \left (\sqrt [3]{1-x^3}+x\right )-\frac {1}{6} \left (1-\sqrt [3]{-1}\right ) \log \left (\sqrt [3]{1-x^3}+x\right )-\frac {1}{3} \log \left (\sqrt [3]{1-x^3}+x\right )+\frac {\left (1-\sqrt [3]{-1}\right ) \log \left (-(-2)^{2/3} \sqrt [3]{1-x^3}-(-1)^{2/3} x+1\right )}{2 \sqrt [3]{2}}+\frac {\log \left (-2^{2/3} \sqrt [3]{1-x^3}-x+1\right )}{\sqrt [3]{2}}+\frac {\left (1+(-1)^{2/3}\right ) \log \left (\sqrt [3]{-1} 2^{2/3} \sqrt [3]{1-x^3}+\sqrt [3]{-1} x+1\right )}{2 \sqrt [3]{2}}-\frac {\log \left (-\left ((1-x) (x+1)^2\right )\right )}{3 \sqrt [3]{2}}-\frac {\left (1+(-1)^{2/3}\right ) \log \left (-(-1)^{2/3} \left (x+(-1)^{2/3}\right )^2 \left (\sqrt [3]{-1} x+1\right )\right )}{6 \sqrt [3]{2}}-\frac {\left (1-\sqrt [3]{-1}\right ) \log \left ((-1)^{2/3} \left (x+\sqrt [3]{-1}\right ) \left ((-1)^{2/3} x+1\right )^2\right )}{6 \sqrt [3]{2}}\)

Input:

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

Output:

-((2^(2/3)*ArcTan[(1 + (2^(1/3)*(1 - x))/(1 - x^3)^(1/3))/Sqrt[3]])/Sqrt[3 
]) + (2*ArcTan[(1 - (2*x)/(1 - x^3)^(1/3))/Sqrt[3]])/(3*Sqrt[3]) + ((1 - ( 
-1)^(1/3))*ArcTan[(1 - (2*x)/(1 - x^3)^(1/3))/Sqrt[3]])/(3*Sqrt[3]) + ((1 
+ (-1)^(2/3))*ArcTan[(1 - (2*x)/(1 - x^3)^(1/3))/Sqrt[3]])/(3*Sqrt[3]) - ( 
(1 - (-1)^(1/3))*ArcTan[(1 - (2^(1/3)*((-1)^(1/3) + x))/(1 - x^3)^(1/3))/S 
qrt[3]])/(2^(1/3)*Sqrt[3]) - ((1 + (-1)^(2/3))*ArcTan[(1 + ((-1)^(2/3)*2^( 
1/3)*(1 + (-1)^(1/3)*x))/(1 - x^3)^(1/3))/Sqrt[3]])/(2^(1/3)*Sqrt[3]) + (x 
^2*Hypergeometric2F1[1/3, 2/3, 5/3, x^3])/3 + ((1 - (-1)^(1/3))*x^2*Hyperg 
eometric2F1[1/3, 2/3, 5/3, x^3])/6 + ((-1)^(2/3)*(1 - (-1)^(1/3))*x^2*Hype 
rgeometric2F1[1/3, 2/3, 5/3, x^3])/6 - Log[-((1 - x)*(1 + x)^2)]/(3*2^(1/3 
)) - ((1 + (-1)^(2/3))*Log[-((-1)^(2/3)*((-1)^(2/3) + x)^2*(1 + (-1)^(1/3) 
*x))])/(6*2^(1/3)) - ((1 - (-1)^(1/3))*Log[(-1)^(2/3)*((-1)^(1/3) + x)*(1 
+ (-1)^(2/3)*x)^2])/(6*2^(1/3)) - Log[x + (1 - x^3)^(1/3)]/3 - ((1 - (-1)^ 
(1/3))*Log[x + (1 - x^3)^(1/3)])/6 - ((1 + (-1)^(2/3))*Log[x + (1 - x^3)^( 
1/3)])/6 + ((1 - (-1)^(1/3))*Log[1 - (-1)^(2/3)*x - (-2)^(2/3)*(1 - x^3)^( 
1/3)])/(2*2^(1/3)) + Log[1 - x - 2^(2/3)*(1 - x^3)^(1/3)]/2^(1/3) + ((1 + 
(-1)^(2/3))*Log[1 + (-1)^(1/3)*x + (-1)^(1/3)*2^(2/3)*(1 - x^3)^(1/3)])/(2 
*2^(1/3))
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 7276

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE 
xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ 
[n, 0]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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