Integrand size = 20, antiderivative size = 88 \[ \int \frac {x}{\left (1-x^3\right )^{2/3} \left (1+x^3\right )} \, dx=-\frac {\arctan \left (\frac {1-\frac {2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3}}+\frac {\log \left (1+x^3\right )}{6\ 2^{2/3}}-\frac {\log \left (-\sqrt [3]{2} x-\sqrt [3]{1-x^3}\right )}{2\ 2^{2/3}} \] Output:
-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)
Time = 0.26 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.30 \[ \int \frac {x}{\left (1-x^3\right )^{2/3} \left (1+x^3\right )} \, dx=\frac {-2 \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x-2^{2/3} \sqrt [3]{1-x^3}}\right )-2 \log \left (2 x+2^{2/3} \sqrt [3]{1-x^3}\right )+\log \left (-2 x^2+2^{2/3} x \sqrt [3]{1-x^3}-\sqrt [3]{2} \left (1-x^3\right )^{2/3}\right )}{6\ 2^{2/3}} \] Input:
Integrate[x/((1 - x^3)^(2/3)*(1 + x^3)),x]
Output:
(-2*Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x - 2^(2/3)*(1 - x^3)^(1/3))] - 2*Log[2*x + 2^(2/3)*(1 - x^3)^(1/3)] + 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))
Time = 0.30 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {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.
\(\displaystyle \int \frac {x}{\left (1-x^3\right )^{2/3} \left (x^3+1\right )} \, dx\) |
\(\Big \downarrow \) 992 |
\(\displaystyle -\frac {\arctan \left (\frac {1-\frac {2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3}}+\frac {\log \left (x^3+1\right )}{6\ 2^{2/3}}-\frac {\log \left (-\sqrt [3]{1-x^3}-\sqrt [3]{2} x\right )}{2\ 2^{2/3}}\) |
Input:
Int[x/((1 - x^3)^(2/3)*(1 + x^3)),x]
Output:
-(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) )
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]
Time = 3.84 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.09
method | result | size |
pseudoelliptic | \(\frac {2^{\frac {1}{3}} \left (2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (-2^{\frac {2}{3}} \left (-x^{3}+1\right )^{\frac {1}{3}}+x \right )}{3 x}\right )-2 \ln \left (\frac {2^{\frac {1}{3}} x +\left (-x^{3}+1\right )^{\frac {1}{3}}}{x}\right )+\ln \left (\frac {2^{\frac {2}{3}} x^{2}-2^{\frac {1}{3}} \left (-x^{3}+1\right )^{\frac {1}{3}} x +\left (-x^{3}+1\right )^{\frac {2}{3}}}{x^{2}}\right )\right )}{12}\) | \(96\) |
trager | \(\text {Expression too large to display}\) | \(954\) |
Input:
int(x/(-x^3+1)^(2/3)/(x^3+1),x,method=_RETURNVERBOSE)
Output:
1/12*2^(1/3)*(2*3^(1/2)*arctan(1/3*3^(1/2)*(-2^(2/3)*(-x^3+1)^(1/3)+x)/x)- 2*ln((2^(1/3)*x+(-x^3+1)^(1/3))/x)+ln((2^(2/3)*x^2-2^(1/3)*(-x^3+1)^(1/3)* x+(-x^3+1)^(2/3))/x^2))
Leaf count of result is larger than twice the leaf count of optimal. 247 vs. \(2 (67) = 134\).
Time = 1.29 (sec) , antiderivative size = 247, normalized size of antiderivative = 2.81 \[ \int \frac {x}{\left (1-x^3\right )^{2/3} \left (1+x^3\right )} \, dx=-\frac {1}{6} \cdot 4^{\frac {1}{6}} \sqrt {\frac {1}{3}} \arctan \left (\frac {4^{\frac {1}{6}} \sqrt {\frac {1}{3}} {\left (6 \cdot 4^{\frac {2}{3}} {\left (19 \, x^{8} - 16 \, x^{5} + x^{2}\right )} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + 12 \, {\left (5 \, x^{7} + 4 \, x^{4} - x\right )} {\left (-x^{3} + 1\right )}^{\frac {2}{3}} - 4^{\frac {1}{3}} {\left (71 \, x^{9} - 111 \, x^{6} + 33 \, x^{3} - 1\right )}\right )}}{2 \, {\left (109 \, x^{9} - 105 \, x^{6} + 3 \, x^{3} + 1\right )}}\right ) - \frac {1}{36} \cdot 4^{\frac {2}{3}} \log \left (\frac {3 \cdot 4^{\frac {2}{3}} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} x^{2} + 6 \, {\left (-x^{3} + 1\right )}^{\frac {2}{3}} x + 4^{\frac {1}{3}} {\left (x^{3} + 1\right )}}{x^{3} + 1}\right ) + \frac {1}{72} \cdot 4^{\frac {2}{3}} \log \left (\frac {6 \cdot 4^{\frac {1}{3}} {\left (5 \, x^{4} - x\right )} {\left (-x^{3} + 1\right )}^{\frac {2}{3}} + 4^{\frac {2}{3}} {\left (19 \, x^{6} - 16 \, x^{3} + 1\right )} - 24 \, {\left (2 \, x^{5} - x^{2}\right )} {\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{x^{6} + 2 \, x^{3} + 1}\right ) \] Input:
integrate(x/(-x^3+1)^(2/3)/(x^3+1),x, algorithm="fricas")
Output:
-1/6*4^(1/6)*sqrt(1/3)*arctan(1/2*4^(1/6)*sqrt(1/3)*(6*4^(2/3)*(19*x^8 - 1 6*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)) - 1/36*4^(2/3)*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)) + 1/72*4^(2/3)*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))
\[ \int \frac {x}{\left (1-x^3\right )^{2/3} \left (1+x^3\right )} \, dx=\int \frac {x}{\left (- \left (x - 1\right ) \left (x^{2} + x + 1\right )\right )^{\frac {2}{3}} \left (x + 1\right ) \left (x^{2} - x + 1\right )}\, dx \] Input:
integrate(x/(-x**3+1)**(2/3)/(x**3+1),x)
Output:
Integral(x/((-(x - 1)*(x**2 + x + 1))**(2/3)*(x + 1)*(x**2 - x + 1)), x)
\[ \int \frac {x}{\left (1-x^3\right )^{2/3} \left (1+x^3\right )} \, dx=\int { \frac {x}{{\left (x^{3} + 1\right )} {\left (-x^{3} + 1\right )}^{\frac {2}{3}}} \,d x } \] Input:
integrate(x/(-x^3+1)^(2/3)/(x^3+1),x, algorithm="maxima")
Output:
integrate(x/((x^3 + 1)*(-x^3 + 1)^(2/3)), x)
\[ \int \frac {x}{\left (1-x^3\right )^{2/3} \left (1+x^3\right )} \, dx=\int { \frac {x}{{\left (x^{3} + 1\right )} {\left (-x^{3} + 1\right )}^{\frac {2}{3}}} \,d x } \] Input:
integrate(x/(-x^3+1)^(2/3)/(x^3+1),x, algorithm="giac")
Output:
integrate(x/((x^3 + 1)*(-x^3 + 1)^(2/3)), x)
Timed out. \[ \int \frac {x}{\left (1-x^3\right )^{2/3} \left (1+x^3\right )} \, dx=\int \frac {x}{{\left (1-x^3\right )}^{2/3}\,\left (x^3+1\right )} \,d x \] Input:
int(x/((1 - x^3)^(2/3)*(x^3 + 1)),x)
Output:
int(x/((1 - x^3)^(2/3)*(x^3 + 1)), x)
\[ \int \frac {x}{\left (1-x^3\right )^{2/3} \left (1+x^3\right )} \, dx=\int \frac {x}{\left (-x^{3}+1\right )^{\frac {2}{3}} x^{3}+\left (-x^{3}+1\right )^{\frac {2}{3}}}d x \] Input:
int(x/(-x^3+1)^(2/3)/(x^3+1),x)
Output:
int(x/(( - x**3 + 1)**(2/3)*x**3 + ( - x**3 + 1)**(2/3)),x)