\(\int \frac {x^3}{(1-x^3)^{4/3} (1+x^3)} \, dx\) [861]

Optimal result

Integrand size = 22, antiderivative size = 106 \[ \int \frac {x^3}{\left (1-x^3\right )^{4/3} \left (1+x^3\right )} \, dx=\frac {x}{2 \sqrt [3]{1-x^3}}+\frac {\arctan \left (\frac {1-\frac {2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{2 \sqrt [3]{2} \sqrt {3}}+\frac {\log \left (1+x^3\right )}{12 \sqrt [3]{2}}-\frac {\log \left (-\sqrt [3]{2} x-\sqrt [3]{1-x^3}\right )}{4 \sqrt [3]{2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.35 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.31 \[ \int \frac {x^3}{\left (1-x^3\right )^{4/3} \left (1+x^3\right )} \, dx=\frac {1}{24} \left (\frac {12 x}{\sqrt [3]{1-x^3}}+2\ 2^{2/3} \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x-2^{2/3} \sqrt [3]{1-x^3}}\right )-2\ 2^{2/3} \log \left (2 x+2^{2/3} \sqrt [3]{1-x^3}\right )+2^{2/3} \log \left (-2 x^2+2^{2/3} x \sqrt [3]{1-x^3}-\sqrt [3]{2} \left (1-x^3\right )^{2/3}\right )\right ) \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.02, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {971, 901}

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.

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

\(\Big \downarrow \) 971

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

\(\Big \downarrow \) 901

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 901

Int[1/(((a_) + (b_.)*(x_)^3)^(1/3)*((c_) + (d_.)*(x_)^3)), x_Symbol] :> Wit 
h[{q = Rt[(b*c - a*d)/c, 3]}, Simp[ArcTan[(1 + (2*q*x)/(a + b*x^3)^(1/3))/S 
qrt[3]]/(Sqrt[3]*c*q), x] + (-Simp[Log[q*x - (a + b*x^3)^(1/3)]/(2*c*q), x] 
 + Simp[Log[c + d*x^3]/(6*c*q), x])] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - 
 a*d, 0]
 

rule 971

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

Maple [A] (verified)

Time = 7.83 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.33

method	result	size
pseudoelliptic	\(-\frac {\sqrt {3}\, 2^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3}\, \left (-2^{\frac {2}{3}} \left (-x^{3}+1\right )^{\frac {1}{3}}+x \right )}{3 x}\right ) \left (-x^{3}+1\right )^{\frac {1}{3}}-\frac {2^{\frac {2}{3}} \ln \left (\frac {2^{\frac {2}{3}} x^{2}-2^{\frac {1}{3}} \left (-x^{3}+1\right )^{\frac {1}{3}} x +\left (-x^{3}+1\right )^{\frac {2}{3}}}{x^{2}}\right ) \left (-x^{3}+1\right )^{\frac {1}{3}}}{2}+2^{\frac {2}{3}} \ln \left (\frac {2^{\frac {1}{3}} x +\left (-x^{3}+1\right )^{\frac {1}{3}}}{x}\right ) \left (-x^{3}+1\right )^{\frac {1}{3}}-6 x}{12 \left (-x^{3}+1\right )^{\frac {1}{3}}}\)	\(141\)
risch	\(\text {Expression too large to display}\)	\(627\)
trager	\(\text {Expression too large to display}\)	\(776\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 281 vs. \(2 (79) = 158\).

Time = 1.61 (sec) , antiderivative size = 281, normalized size of antiderivative = 2.65 \[ \int \frac {x^3}{\left (1-x^3\right )^{4/3} \left (1+x^3\right )} \, dx=\frac {12 \cdot 2^{\frac {1}{6}} \sqrt {\frac {1}{6}} {\left (x^{3} - 1\right )} \arctan \left (\frac {2^{\frac {1}{6}} \sqrt {\frac {1}{6}} {\left (6 \cdot 2^{\frac {2}{3}} {\left (5 \, x^{7} + 4 \, x^{4} - x\right )} {\left (-x^{3} + 1\right )}^{\frac {2}{3}} - 2^{\frac {1}{3}} {\left (71 \, x^{9} - 111 \, x^{6} + 33 \, x^{3} - 1\right )} + 12 \, {\left (19 \, x^{8} - 16 \, x^{5} + x^{2}\right )} {\left (-x^{3} + 1\right )}^{\frac {1}{3}}\right )}}{109 \, x^{9} - 105 \, x^{6} + 3 \, x^{3} + 1}\right ) - 2 \cdot 2^{\frac {2}{3}} {\left (x^{3} - 1\right )} \log \left (\frac {6 \cdot 2^{\frac {1}{3}} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} x^{2} + 2^{\frac {2}{3}} {\left (x^{3} + 1\right )} + 6 \, {\left (-x^{3} + 1\right )}^{\frac {2}{3}} x}{x^{3} + 1}\right ) + 2^{\frac {2}{3}} {\left (x^{3} - 1\right )} \log \left (\frac {3 \cdot 2^{\frac {2}{3}} {\left (5 \, x^{4} - x\right )} {\left (-x^{3} + 1\right )}^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (19 \, x^{6} - 16 \, x^{3} + 1\right )} - 12 \, {\left (2 \, x^{5} - x^{2}\right )} {\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{x^{6} + 2 \, x^{3} + 1}\right ) - 36 \, {\left (-x^{3} + 1\right )}^{\frac {2}{3}} x}{72 \, {\left (x^{3} - 1\right )}} \] Input:

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

Output:

1/72*(12*2^(1/6)*sqrt(1/6)*(x^3 - 1)*arctan(2^(1/6)*sqrt(1/6)*(6*2^(2/3)*( 
5*x^7 + 4*x^4 - x)*(-x^3 + 1)^(2/3) - 2^(1/3)*(71*x^9 - 111*x^6 + 33*x^3 - 
 1) + 12*(19*x^8 - 16*x^5 + x^2)*(-x^3 + 1)^(1/3))/(109*x^9 - 105*x^6 + 3* 
x^3 + 1)) - 2*2^(2/3)*(x^3 - 1)*log((6*2^(1/3)*(-x^3 + 1)^(1/3)*x^2 + 2^(2 
/3)*(x^3 + 1) + 6*(-x^3 + 1)^(2/3)*x)/(x^3 + 1)) + 2^(2/3)*(x^3 - 1)*log(( 
3*2^(2/3)*(5*x^4 - x)*(-x^3 + 1)^(2/3) + 2^(1/3)*(19*x^6 - 16*x^3 + 1) - 1 
2*(2*x^5 - x^2)*(-x^3 + 1)^(1/3))/(x^6 + 2*x^3 + 1)) - 36*(-x^3 + 1)^(2/3) 
*x)/(x^3 - 1)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

 - int(x**3/(( - x**3 + 1)**(1/3)*x**6 - ( - x**3 + 1)**(1/3)),x)