\(\int \frac {1}{x^2 (1-x^3)^{2/3} (1+x^3)} \, dx\) [846]

Optimal result

Integrand size = 22, antiderivative size = 103 \[ \int \frac {1}{x^2 \left (1-x^3\right )^{2/3} \left (1+x^3\right )} \, dx=-\frac {\sqrt [3]{1-x^3}}{x}+\frac {\arctan \left (\frac {1-\frac {2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3}}-\frac {\log \left (1+x^3\right )}{6\ 2^{2/3}}+\frac {\log \left (-\sqrt [3]{2} x-\sqrt [3]{1-x^3}\right )}{2\ 2^{2/3}} \] Output:

-(-x^3+1)^(1/3)/x+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)+1/4*ln(-2^(1/3)*x-(-x^3+1)^(1/3))*2^ 
(1/3)
 

Mathematica [A] (verified)

Time = 0.36 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.39 \[ \int \frac {1}{x^2 \left (1-x^3\right )^{2/3} \left (1+x^3\right )} \, dx=-\frac {\sqrt [3]{1-x^3}}{x}-\frac {\arctan \left (\frac {\sqrt {3} x}{-x+2^{2/3} \sqrt [3]{1-x^3}}\right )}{2^{2/3} \sqrt {3}}+\frac {\log \left (2 x+2^{2/3} \sqrt [3]{1-x^3}\right )}{3\ 2^{2/3}}-\frac {\log \left (-2 x^2+2^{2/3} x \sqrt [3]{1-x^3}-\sqrt [3]{2} \left (1-x^3\right )^{2/3}\right )}{6\ 2^{2/3}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {980, 25, 992}

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.

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

\(\Big \downarrow \) 980

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 992

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 25

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

rule 980

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

rule 992

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

Maple [A] (verified)

Time = 26.87 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.17

method	result	size
pseudoelliptic	\(\frac {-2 \,2^{\frac {1}{3}} \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (-2^{\frac {2}{3}} \left (-x^{3}+1\right )^{\frac {1}{3}}+x \right )}{3 x}\right ) x +2 \,2^{\frac {1}{3}} \ln \left (\frac {2^{\frac {1}{3}} x +\left (-x^{3}+1\right )^{\frac {1}{3}}}{x}\right ) x -2^{\frac {1}{3}} \ln \left (\frac {2^{\frac {2}{3}} x^{2}-2^{\frac {1}{3}} \left (-x^{3}+1\right )^{\frac {1}{3}} x +\left (-x^{3}+1\right )^{\frac {2}{3}}}{x^{2}}\right ) x -12 \left (-x^{3}+1\right )^{\frac {1}{3}}}{12 x}\)	\(121\)
trager	\(\text {Expression too large to display}\)	\(1183\)
risch	\(\text {Expression too large to display}\)	\(1386\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 266 vs. \(2 (81) = 162\).

Time = 1.13 (sec) , antiderivative size = 266, normalized size of antiderivative = 2.58 \[ \int \frac {1}{x^2 \left (1-x^3\right )^{2/3} \left (1+x^3\right )} \, dx=\frac {12 \cdot 4^{\frac {1}{6}} \sqrt {\frac {1}{3}} x \arctan \left (\frac {4^{\frac {1}{6}} \sqrt {\frac {1}{3}} {\left (6 \cdot 4^{\frac {2}{3}} {\left (19 \, x^{8} - 16 \, x^{5} + x^{2}\right )} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + 12 \, {\left (5 \, x^{7} + 4 \, x^{4} - x\right )} {\left (-x^{3} + 1\right )}^{\frac {2}{3}} - 4^{\frac {1}{3}} {\left (71 \, x^{9} - 111 \, x^{6} + 33 \, x^{3} - 1\right )}\right )}}{2 \, {\left (109 \, x^{9} - 105 \, x^{6} + 3 \, x^{3} + 1\right )}}\right ) + 2 \cdot 4^{\frac {2}{3}} x \log \left (\frac {3 \cdot 4^{\frac {2}{3}} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} x^{2} + 6 \, {\left (-x^{3} + 1\right )}^{\frac {2}{3}} x + 4^{\frac {1}{3}} {\left (x^{3} + 1\right )}}{x^{3} + 1}\right ) - 4^{\frac {2}{3}} x \log \left (\frac {6 \cdot 4^{\frac {1}{3}} {\left (5 \, x^{4} - x\right )} {\left (-x^{3} + 1\right )}^{\frac {2}{3}} + 4^{\frac {2}{3}} {\left (19 \, x^{6} - 16 \, x^{3} + 1\right )} - 24 \, {\left (2 \, x^{5} - x^{2}\right )} {\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{x^{6} + 2 \, x^{3} + 1}\right ) - 72 \, {\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{72 \, x} \] Input:

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

Output:

1/72*(12*4^(1/6)*sqrt(1/3)*x*arctan(1/2*4^(1/6)*sqrt(1/3)*(6*4^(2/3)*(19*x 
^8 - 16*x^5 + x^2)*(-x^3 + 1)^(1/3) + 12*(5*x^7 + 4*x^4 - x)*(-x^3 + 1)^(2 
/3) - 4^(1/3)*(71*x^9 - 111*x^6 + 33*x^3 - 1))/(109*x^9 - 105*x^6 + 3*x^3 
+ 1)) + 2*4^(2/3)*x*log((3*4^(2/3)*(-x^3 + 1)^(1/3)*x^2 + 6*(-x^3 + 1)^(2/ 
3)*x + 4^(1/3)*(x^3 + 1))/(x^3 + 1)) - 4^(2/3)*x*log((6*4^(1/3)*(5*x^4 - x 
)*(-x^3 + 1)^(2/3) + 4^(2/3)*(19*x^6 - 16*x^3 + 1) - 24*(2*x^5 - x^2)*(-x^ 
3 + 1)^(1/3))/(x^6 + 2*x^3 + 1)) - 72*(-x^3 + 1)^(1/3))/x
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {1}{x^2 \left (1-x^3\right )^{2/3} \left (1+x^3\right )} \, dx=\int \frac {1}{\left (-x^{3}+1\right )^{\frac {2}{3}} x^{5}+\left (-x^{3}+1\right )^{\frac {2}{3}} x^{2}}d x \] Input:

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

Output:

int(1/(( - x**3 + 1)**(2/3)*x**5 + ( - x**3 + 1)**(2/3)*x**2),x)