\(\int \frac {x^7}{(1-x^3)^{2/3} (1+x^3)} \, dx\) [843]

Optimal result

Integrand size = 22, antiderivative size = 160 \[ \int \frac {x^7}{\left (1-x^3\right )^{2/3} \left (1+x^3\right )} \, dx=-\frac {1}{3} x^2 \sqrt [3]{1-x^3}+\frac {\arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {\arctan \left (\frac {1-\frac {2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3}}+\frac {\log \left (1+x^3\right )}{6\ 2^{2/3}}+\frac {1}{6} \log \left (-x-\sqrt [3]{1-x^3}\right )-\frac {\log \left (-\sqrt [3]{2} x-\sqrt [3]{1-x^3}\right )}{2\ 2^{2/3}} \] Output:

-1/3*x^2*(-x^3+1)^(1/3)+1/9*arctan(1/3*(1-2*x/(-x^3+1)^(1/3))*3^(1/2))*3^( 
1/2)-1/6*arctan(1/3*(1-2*2^(1/3)*x/(-x^3+1)^(1/3))*3^(1/2))*2^(1/3)*3^(1/2 
)+1/12*ln(x^3+1)*2^(1/3)+1/6*ln(-x-(-x^3+1)^(1/3))-1/4*ln(-2^(1/3)*x-(-x^3 
+1)^(1/3))*2^(1/3)
 

Mathematica [A] (verified)

Time = 0.51 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.39 \[ \int \frac {x^7}{\left (1-x^3\right )^{2/3} \left (1+x^3\right )} \, dx=\frac {1}{36} \left (-12 x^2 \sqrt [3]{1-x^3}+4 \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x-2 \sqrt [3]{1-x^3}}\right )-6 \sqrt [3]{2} \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x-2^{2/3} \sqrt [3]{1-x^3}}\right )+4 \log \left (x+\sqrt [3]{1-x^3}\right )-6 \sqrt [3]{2} \log \left (2 x+2^{2/3} \sqrt [3]{1-x^3}\right )-2 \log \left (x^2-x \sqrt [3]{1-x^3}+\left (1-x^3\right )^{2/3}\right )+3 \sqrt [3]{2} \log \left (-2 x^2+2^{2/3} x \sqrt [3]{1-x^3}-\sqrt [3]{2} \left (1-x^3\right )^{2/3}\right )\right ) \] Input:

Integrate[x^7/((1 - x^3)^(2/3)*(1 + x^3)),x]
 

Output:

(-12*x^2*(1 - x^3)^(1/3) + 4*Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x - 2*(1 - x^3)^( 
1/3))] - 6*2^(1/3)*Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x - 2^(2/3)*(1 - x^3)^(1/3) 
)] + 4*Log[x + (1 - x^3)^(1/3)] - 6*2^(1/3)*Log[2*x + 2^(2/3)*(1 - x^3)^(1 
/3)] - 2*Log[x^2 - x*(1 - x^3)^(1/3) + (1 - x^3)^(2/3)] + 3*2^(1/3)*Log[-2 
*x^2 + 2^(2/3)*x*(1 - x^3)^(1/3) - 2^(1/3)*(1 - x^3)^(2/3)])/36
 

Rubi [A] (verified)

Time = 0.47 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.01, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {979, 1054, 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.

\(\displaystyle \int \frac {x^7}{\left (1-x^3\right )^{2/3} \left (x^3+1\right )} \, dx\)

\(\Big \downarrow \) 979

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

\(\Big \downarrow \) 1054

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

\(\Big \downarrow \) 2009

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

Input:

Int[x^7/((1 - x^3)^(2/3)*(1 + x^3)),x]
 

Output:

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

Defintions of rubi rules used

rule 979

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ 
))^(q_), x_Symbol] :> Simp[e^(2*n - 1)*(e*x)^(m - 2*n + 1)*(a + b*x^n)^(p + 
 1)*((c + d*x^n)^(q + 1)/(b*d*(m + n*(p + q) + 1))), x] - Simp[e^(2*n)/(b*d 
*(m + n*(p + q) + 1))   Int[(e*x)^(m - 2*n)*(a + b*x^n)^p*(c + d*x^n)^q*Sim 
p[a*c*(m - 2*n + 1) + (a*d*(m + n*(q - 1) + 1) + b*c*(m + n*(p - 1) + 1))*x 
^n, x], x], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && I 
GtQ[n, 0] && GtQ[m - n + 1, n] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x 
]
 

rule 1054

Int[(((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((e_) + (f_.)*(x_)^(n 
_)))/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegrand[(g*x)^m*(a 
+ b*x^n)^p*((e + f*x^n)/(c + d*x^n)), x], x] /; FreeQ[{a, b, c, d, e, f, g, 
 m, p}, x] && IGtQ[n, 0]
 

rule 2009

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

Maple [A] (verified)

Time = 2.99 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.46

method	result	size
pseudoelliptic	\(\frac {3 \,2^{\frac {1}{3}} \ln \left (\frac {2^{\frac {2}{3}} x^{2}-2^{\frac {1}{3}} \left (-x^{3}+1\right )^{\frac {1}{3}} x +\left (-x^{3}+1\right )^{\frac {2}{3}}}{x^{2}}\right )-2 \ln \left (\frac {\left (-x^{3}+1\right )^{\frac {2}{3}}-\left (-x^{3}+1\right )^{\frac {1}{3}} x +x^{2}}{x^{2}}\right )-6 \,2^{\frac {1}{3}} \ln \left (\frac {2^{\frac {1}{3}} x +\left (-x^{3}+1\right )^{\frac {1}{3}}}{x}\right )+4 \ln \left (\frac {x +\left (-x^{3}+1\right )^{\frac {1}{3}}}{x}\right )-12 x^{2} \left (-x^{3}+1\right )^{\frac {1}{3}}+\left (6 \arctan \left (\frac {\sqrt {3}\, \left (-2^{\frac {2}{3}} \left (-x^{3}+1\right )^{\frac {1}{3}}+x \right )}{3 x}\right ) 2^{\frac {1}{3}}-4 \arctan \left (\frac {\left (-2 \left (-x^{3}+1\right )^{\frac {1}{3}}+x \right ) \sqrt {3}}{3 x}\right )\right ) \sqrt {3}}{36 \left (x +\left (-x^{3}+1\right )^{\frac {1}{3}}\right ) \left (\left (-x^{3}+1\right )^{\frac {2}{3}}+x \left (x -\left (-x^{3}+1\right )^{\frac {1}{3}}\right )\right )}\)	\(234\)

Input:

int(x^7/(-x^3+1)^(2/3)/(x^3+1),x,method=_RETURNVERBOSE)
 

Output:

1/36*(3*2^(1/3)*ln((2^(2/3)*x^2-2^(1/3)*(-x^3+1)^(1/3)*x+(-x^3+1)^(2/3))/x 
^2)-2*ln(((-x^3+1)^(2/3)-(-x^3+1)^(1/3)*x+x^2)/x^2)-6*2^(1/3)*ln((2^(1/3)* 
x+(-x^3+1)^(1/3))/x)+4*ln((x+(-x^3+1)^(1/3))/x)-12*x^2*(-x^3+1)^(1/3)+(6*a 
rctan(1/3*3^(1/2)*(-2^(2/3)*(-x^3+1)^(1/3)+x)/x)*2^(1/3)-4*arctan(1/3*(-2* 
(-x^3+1)^(1/3)+x)*3^(1/2)/x))*3^(1/2))/(x+(-x^3+1)^(1/3))/((-x^3+1)^(2/3)+ 
x*(x-(-x^3+1)^(1/3)))
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.30 \[ \int \frac {x^7}{\left (1-x^3\right )^{2/3} \left (1+x^3\right )} \, dx=-\frac {1}{3} \, {\left (-x^{3} + 1\right )}^{\frac {1}{3}} x^{2} - \frac {1}{2} \cdot 4^{\frac {1}{6}} \sqrt {\frac {1}{3}} \arctan \left (-\frac {4^{\frac {1}{6}} \sqrt {\frac {1}{3}} {\left (4^{\frac {1}{3}} x - 4^{\frac {2}{3}} {\left (-x^{3} + 1\right )}^{\frac {1}{3}}\right )}}{2 \, x}\right ) - \frac {1}{12} \cdot 4^{\frac {2}{3}} \log \left (\frac {4^{\frac {2}{3}} x + 2 \, {\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{x}\right ) + \frac {1}{24} \cdot 4^{\frac {2}{3}} \log \left (\frac {2 \cdot 4^{\frac {1}{3}} x^{2} - 4^{\frac {2}{3}} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} x + 2 \, {\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{x^{2}}\right ) + \frac {1}{9} \, \sqrt {3} \arctan \left (-\frac {\sqrt {3} x - 2 \, \sqrt {3} {\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{3 \, x}\right ) + \frac {1}{9} \, \log \left (\frac {x + {\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{x}\right ) - \frac {1}{18} \, \log \left (\frac {x^{2} - {\left (-x^{3} + 1\right )}^{\frac {1}{3}} x + {\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{x^{2}}\right ) \] Input:

integrate(x^7/(-x^3+1)^(2/3)/(x^3+1),x, algorithm="fricas")
 

Output:

-1/3*(-x^3 + 1)^(1/3)*x^2 - 1/2*4^(1/6)*sqrt(1/3)*arctan(-1/2*4^(1/6)*sqrt 
(1/3)*(4^(1/3)*x - 4^(2/3)*(-x^3 + 1)^(1/3))/x) - 1/12*4^(2/3)*log((4^(2/3 
)*x + 2*(-x^3 + 1)^(1/3))/x) + 1/24*4^(2/3)*log((2*4^(1/3)*x^2 - 4^(2/3)*( 
-x^3 + 1)^(1/3)*x + 2*(-x^3 + 1)^(2/3))/x^2) + 1/9*sqrt(3)*arctan(-1/3*(sq 
rt(3)*x - 2*sqrt(3)*(-x^3 + 1)^(1/3))/x) + 1/9*log((x + (-x^3 + 1)^(1/3))/ 
x) - 1/18*log((x^2 - (-x^3 + 1)^(1/3)*x + (-x^3 + 1)^(2/3))/x^2)
 

Sympy [F]

\[ \int \frac {x^7}{\left (1-x^3\right )^{2/3} \left (1+x^3\right )} \, dx=\int \frac {x^{7}}{\left (- \left (x - 1\right ) \left (x^{2} + x + 1\right )\right )^{\frac {2}{3}} \left (x + 1\right ) \left (x^{2} - x + 1\right )}\, dx \] Input:

integrate(x**7/(-x**3+1)**(2/3)/(x**3+1),x)
 

Output:

Integral(x**7/((-(x - 1)*(x**2 + x + 1))**(2/3)*(x + 1)*(x**2 - x + 1)), x 
)
 

Maxima [F]

\[ \int \frac {x^7}{\left (1-x^3\right )^{2/3} \left (1+x^3\right )} \, dx=\int { \frac {x^{7}}{{\left (x^{3} + 1\right )} {\left (-x^{3} + 1\right )}^{\frac {2}{3}}} \,d x } \] Input:

integrate(x^7/(-x^3+1)^(2/3)/(x^3+1),x, algorithm="maxima")
 

Output:

integrate(x^7/((x^3 + 1)*(-x^3 + 1)^(2/3)), x)
 

Giac [F]

\[ \int \frac {x^7}{\left (1-x^3\right )^{2/3} \left (1+x^3\right )} \, dx=\int { \frac {x^{7}}{{\left (x^{3} + 1\right )} {\left (-x^{3} + 1\right )}^{\frac {2}{3}}} \,d x } \] Input:

integrate(x^7/(-x^3+1)^(2/3)/(x^3+1),x, algorithm="giac")
 

Output:

integrate(x^7/((x^3 + 1)*(-x^3 + 1)^(2/3)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^7}{\left (1-x^3\right )^{2/3} \left (1+x^3\right )} \, dx=\int \frac {x^7}{{\left (1-x^3\right )}^{2/3}\,\left (x^3+1\right )} \,d x \] Input:

int(x^7/((1 - x^3)^(2/3)*(x^3 + 1)),x)
 

Output:

int(x^7/((1 - x^3)^(2/3)*(x^3 + 1)), x)
 

Reduce [F]

\[ \int \frac {x^7}{\left (1-x^3\right )^{2/3} \left (1+x^3\right )} \, dx=\int \frac {x^{7}}{\left (-x^{3}+1\right )^{\frac {2}{3}} x^{3}+\left (-x^{3}+1\right )^{\frac {2}{3}}}d x \] Input:

int(x^7/(-x^3+1)^(2/3)/(x^3+1),x)
                                                                                    
                                                                                    
 

Output:

int(x**7/(( - x**3 + 1)**(2/3)*x**3 + ( - x**3 + 1)**(2/3)),x)