Integrand size = 25, antiderivative size = 173 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (1+x^3\right )}{x^6 \left (2+x^3\right )} \, dx=\frac {\left (-4-x^3\right ) \left (-1+x^3\right )^{2/3}}{40 x^5}+\frac {\sqrt [6]{3} \arctan \left (\frac {3^{5/6} x}{\sqrt [3]{3} x+2 \sqrt [3]{2} \sqrt [3]{-1+x^3}}\right )}{4\ 2^{2/3}}-\frac {\log \left (-3 x+\sqrt [3]{2} 3^{2/3} \sqrt [3]{-1+x^3}\right )}{4\ 2^{2/3} \sqrt [3]{3}}+\frac {\log \left (3 x^2+\sqrt [3]{2} 3^{2/3} x \sqrt [3]{-1+x^3}+2^{2/3} \sqrt [3]{3} \left (-1+x^3\right )^{2/3}\right )}{8\ 2^{2/3} \sqrt [3]{3}} \] Output:
1/40*(-x^3-4)*(x^3-1)^(2/3)/x^5+1/8*2^(1/3)*3^(1/6)*arctan(3^(5/6)*x/(3^(1 /3)*x+2*2^(1/3)*(x^3-1)^(1/3)))-1/24*2^(1/3)*3^(2/3)*ln(-3*x+2^(1/3)*3^(2/ 3)*(x^3-1)^(1/3))+1/48*ln(3*x^2+2^(1/3)*3^(2/3)*x*(x^3-1)^(1/3)+2^(2/3)*3^ (1/3)*(x^3-1)^(2/3))*2^(1/3)*3^(2/3)
Time = 0.39 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.97 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (1+x^3\right )}{x^6 \left (2+x^3\right )} \, dx=\frac {1}{240} \left (-\frac {6 \left (-1+x^3\right )^{2/3} \left (4+x^3\right )}{x^5}+30 \sqrt [3]{2} \sqrt [6]{3} \arctan \left (\frac {3^{5/6} x}{\sqrt [3]{3} x+2 \sqrt [3]{2} \sqrt [3]{-1+x^3}}\right )-10 \sqrt [3]{2} 3^{2/3} \log \left (-3 x+\sqrt [3]{2} 3^{2/3} \sqrt [3]{-1+x^3}\right )+5 \sqrt [3]{2} 3^{2/3} \log \left (3 x^2+\sqrt [3]{2} 3^{2/3} x \sqrt [3]{-1+x^3}+2^{2/3} \sqrt [3]{3} \left (-1+x^3\right )^{2/3}\right )\right ) \] Input:
Integrate[((-1 + x^3)^(2/3)*(1 + x^3))/(x^6*(2 + x^3)),x]
Output:
((-6*(-1 + x^3)^(2/3)*(4 + x^3))/x^5 + 30*2^(1/3)*3^(1/6)*ArcTan[(3^(5/6)* x)/(3^(1/3)*x + 2*2^(1/3)*(-1 + x^3)^(1/3))] - 10*2^(1/3)*3^(2/3)*Log[-3*x + 2^(1/3)*3^(2/3)*(-1 + x^3)^(1/3)] + 5*2^(1/3)*3^(2/3)*Log[3*x^2 + 2^(1/ 3)*3^(2/3)*x*(-1 + x^3)^(1/3) + 2^(2/3)*3^(1/3)*(-1 + x^3)^(2/3)])/240
Time = 0.27 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.81, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1050, 25, 1053, 27, 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.
| \(\displaystyle \int \frac {\left (x^3-1\right )^{2/3} \left (x^3+1\right )}{x^6 \left (x^3+2\right )} \, dx\) |
| \(\Big \downarrow \) 1050 |
| \(\displaystyle \frac {1}{10} \int -\frac {1-7 x^3}{x^3 \sqrt [3]{x^3-1} \left (x^3+2\right )}dx-\frac {\left (x^3-1\right )^{2/3}}{10 x^5}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {1}{10} \int \frac {1-7 x^3}{x^3 \sqrt [3]{x^3-1} \left (x^3+2\right )}dx-\frac {\left (x^3-1\right )^{2/3}}{10 x^5}\) |
| \(\Big \downarrow \) 1053 |
| \(\displaystyle \frac {1}{10} \left (-\frac {1}{4} \int -\frac {30}{\sqrt [3]{x^3-1} \left (x^3+2\right )}dx-\frac {\left (x^3-1\right )^{2/3}}{4 x^2}\right )-\frac {\left (x^3-1\right )^{2/3}}{10 x^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{10} \left (\frac {15}{2} \int \frac {1}{\sqrt [3]{x^3-1} \left (x^3+2\right )}dx-\frac {\left (x^3-1\right )^{2/3}}{4 x^2}\right )-\frac {\left (x^3-1\right )^{2/3}}{10 x^5}\) |
| \(\Big \downarrow \) 901 |
| \(\displaystyle \frac {1}{10} \left (\frac {15}{2} \left (\frac {\arctan \left (\frac {\frac {2^{2/3} \sqrt [3]{3} x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{2^{2/3} 3^{5/6}}+\frac {\log \left (x^3+2\right )}{6\ 2^{2/3} \sqrt [3]{3}}-\frac {\log \left (\sqrt [3]{\frac {3}{2}} x-\sqrt [3]{x^3-1}\right )}{2\ 2^{2/3} \sqrt [3]{3}}\right )-\frac {\left (x^3-1\right )^{2/3}}{4 x^2}\right )-\frac {\left (x^3-1\right )^{2/3}}{10 x^5}\) |
Input:
Int[((-1 + x^3)^(2/3)*(1 + x^3))/(x^6*(2 + x^3)),x]
Output:
-1/10*(-1 + x^3)^(2/3)/x^5 + (-1/4*(-1 + x^3)^(2/3)/x^2 + (15*(ArcTan[(1 + (2^(2/3)*3^(1/3)*x)/(-1 + x^3)^(1/3))/Sqrt[3]]/(2^(2/3)*3^(5/6)) + Log[2 + x^3]/(6*2^(2/3)*3^(1/3)) - Log[(3/2)^(1/3)*x - (-1 + x^3)^(1/3)]/(2*2^(2 /3)*3^(1/3))))/2)/10
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_ ))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b *x^n)^(p + 1)*((c + d*x^n)^q/(a*g*(m + 1))), x] - Simp[1/(a*g^n*(m + 1)) Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^(q - 1)*Simp[c*(b*e - a*f)*(m + 1) + e*n*(b*c*(p + 1) + a*d*q) + d*((b*e - a*f)*(m + 1) + b*e*n*(p + q + 1 ))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && IGtQ[n, 0] && G tQ[q, 0] && LtQ[m, -1] && !(EqQ[q, 1] && SimplerQ[e + f*x^n, c + d*x^n])
Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_ ))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b *x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*c*g*(m + 1))), x] + Simp[1/(a*c*g^n*( m + 1)) Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*f*c*(m + 1) - e*(b*c + a*d)*(m + n + 1) - e*n*(b*c*p + a*d*q) - b*e*d*(m + n*(p + q + 2 ) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] && IGtQ[n, 0] && LtQ[m, -1]
Time = 13.41 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.89
| method | result | size |
| pseudoelliptic | \(\frac {\left (-6 x^{3}-24\right ) \left (x^{3}-1\right )^{\frac {2}{3}}-10 \,2^{\frac {1}{3}} \left (\left (\ln \left (\frac {-2^{\frac {2}{3}} 3^{\frac {1}{3}} x +2 {\left (\left (-1+x \right ) \left (x^{2}+x +1\right )\right )}^{\frac {1}{3}}}{x}\right )-\frac {\ln \left (\frac {2^{\frac {2}{3}} 3^{\frac {1}{3}} {\left (\left (-1+x \right ) \left (x^{2}+x +1\right )\right )}^{\frac {1}{3}} x +2^{\frac {1}{3}} 3^{\frac {2}{3}} x^{2}+2 {\left (\left (-1+x \right ) \left (x^{2}+x +1\right )\right )}^{\frac {2}{3}}}{x^{2}}\right )}{2}-\frac {\ln \left (2\right )}{2}\right ) 3^{\frac {2}{3}}+3 \arctan \left (\frac {\sqrt {3}\, \left (2 \,2^{\frac {1}{3}} 3^{\frac {2}{3}} \left (x^{3}-1\right )^{\frac {1}{3}}+3 x \right )}{9 x}\right ) 3^{\frac {1}{6}}\right ) x^{5}}{240 x^{5}}\) | \(154\) |
| risch | \(\text {Expression too large to display}\) | \(740\) |
| trager | \(\text {Expression too large to display}\) | \(742\) |
Input:
int((x^3-1)^(2/3)*(x^3+1)/x^6/(x^3+2),x,method=_RETURNVERBOSE)
Output:
1/240*((-6*x^3-24)*(x^3-1)^(2/3)-10*2^(1/3)*((ln((-2^(2/3)*3^(1/3)*x+2*((- 1+x)*(x^2+x+1))^(1/3))/x)-1/2*ln((2^(2/3)*3^(1/3)*((-1+x)*(x^2+x+1))^(1/3) *x+2^(1/3)*3^(2/3)*x^2+2*((-1+x)*(x^2+x+1))^(2/3))/x^2)-1/2*ln(2))*3^(2/3) +3*arctan(1/9*3^(1/2)*(2*2^(1/3)*3^(2/3)*(x^3-1)^(1/3)+3*x)/x)*3^(1/6))*x^ 5)/x^5
Leaf count of result is larger than twice the leaf count of optimal. 261 vs. \(2 (123) = 246\).
Time = 1.94 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.51 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (1+x^3\right )}{x^6 \left (2+x^3\right )} \, dx=-\frac {10 \cdot 12^{\frac {2}{3}} x^{5} \log \left (-\frac {18 \cdot 12^{\frac {1}{3}} {\left (x^{3} - 1\right )}^{\frac {1}{3}} x^{2} - 12^{\frac {2}{3}} {\left (x^{3} + 2\right )} - 36 \, {\left (x^{3} - 1\right )}^{\frac {2}{3}} x}{x^{3} + 2}\right ) - 5 \cdot 12^{\frac {2}{3}} x^{5} \log \left (\frac {6 \cdot 12^{\frac {2}{3}} {\left (4 \, x^{4} - x\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}} + 12^{\frac {1}{3}} {\left (55 \, x^{6} - 50 \, x^{3} + 4\right )} + 18 \, {\left (7 \, x^{5} - 4 \, x^{2}\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x^{6} + 4 \, x^{3} + 4}\right ) - 60 \cdot 12^{\frac {1}{6}} x^{5} \arctan \left (\frac {12^{\frac {1}{6}} {\left (12 \cdot 12^{\frac {2}{3}} {\left (4 \, x^{7} + 7 \, x^{4} - 2 \, x\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}} - 12^{\frac {1}{3}} {\left (377 \, x^{9} - 600 \, x^{6} + 204 \, x^{3} - 8\right )} - 36 \, {\left (55 \, x^{8} - 50 \, x^{5} + 4 \, x^{2}\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}}\right )}}{6 \, {\left (487 \, x^{9} - 480 \, x^{6} + 12 \, x^{3} + 8\right )}}\right ) + 36 \, {\left (x^{3} + 4\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{1440 \, x^{5}} \] Input:
integrate((x^3-1)^(2/3)*(x^3+1)/x^6/(x^3+2),x, algorithm="fricas")
Output:
-1/1440*(10*12^(2/3)*x^5*log(-(18*12^(1/3)*(x^3 - 1)^(1/3)*x^2 - 12^(2/3)* (x^3 + 2) - 36*(x^3 - 1)^(2/3)*x)/(x^3 + 2)) - 5*12^(2/3)*x^5*log((6*12^(2 /3)*(4*x^4 - x)*(x^3 - 1)^(2/3) + 12^(1/3)*(55*x^6 - 50*x^3 + 4) + 18*(7*x ^5 - 4*x^2)*(x^3 - 1)^(1/3))/(x^6 + 4*x^3 + 4)) - 60*12^(1/6)*x^5*arctan(1 /6*12^(1/6)*(12*12^(2/3)*(4*x^7 + 7*x^4 - 2*x)*(x^3 - 1)^(2/3) - 12^(1/3)* (377*x^9 - 600*x^6 + 204*x^3 - 8) - 36*(55*x^8 - 50*x^5 + 4*x^2)*(x^3 - 1) ^(1/3))/(487*x^9 - 480*x^6 + 12*x^3 + 8)) + 36*(x^3 + 4)*(x^3 - 1)^(2/3))/ x^5
\[ \int \frac {\left (-1+x^3\right )^{2/3} \left (1+x^3\right )}{x^6 \left (2+x^3\right )} \, dx=\int \frac {\left (\left (x - 1\right ) \left (x^{2} + x + 1\right )\right )^{\frac {2}{3}} \left (x + 1\right ) \left (x^{2} - x + 1\right )}{x^{6} \left (x^{3} + 2\right )}\, dx \] Input:
integrate((x**3-1)**(2/3)*(x**3+1)/x**6/(x**3+2),x)
Output:
Integral(((x - 1)*(x**2 + x + 1))**(2/3)*(x + 1)*(x**2 - x + 1)/(x**6*(x** 3 + 2)), x)
\[ \int \frac {\left (-1+x^3\right )^{2/3} \left (1+x^3\right )}{x^6 \left (2+x^3\right )} \, dx=\int { \frac {{\left (x^{3} + 1\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (x^{3} + 2\right )} x^{6}} \,d x } \] Input:
integrate((x^3-1)^(2/3)*(x^3+1)/x^6/(x^3+2),x, algorithm="maxima")
Output:
integrate((x^3 + 1)*(x^3 - 1)^(2/3)/((x^3 + 2)*x^6), x)
\[ \int \frac {\left (-1+x^3\right )^{2/3} \left (1+x^3\right )}{x^6 \left (2+x^3\right )} \, dx=\int { \frac {{\left (x^{3} + 1\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (x^{3} + 2\right )} x^{6}} \,d x } \] Input:
integrate((x^3-1)^(2/3)*(x^3+1)/x^6/(x^3+2),x, algorithm="giac")
Output:
integrate((x^3 + 1)*(x^3 - 1)^(2/3)/((x^3 + 2)*x^6), x)
Timed out. \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (1+x^3\right )}{x^6 \left (2+x^3\right )} \, dx=\int \frac {{\left (x^3-1\right )}^{2/3}\,\left (x^3+1\right )}{x^6\,\left (x^3+2\right )} \,d x \] Input:
int(((x^3 - 1)^(2/3)*(x^3 + 1))/(x^6*(x^3 + 2)),x)
Output:
int(((x^3 - 1)^(2/3)*(x^3 + 1))/(x^6*(x^3 + 2)), x)
\[ \int \frac {\left (-1+x^3\right )^{2/3} \left (1+x^3\right )}{x^6 \left (2+x^3\right )} \, dx=\frac {7 \left (x^{3}-1\right )^{\frac {2}{3}} x^{3}-2 \left (x^{3}-1\right )^{\frac {2}{3}}-30 \left (\int \frac {\left (x^{3}-1\right )^{\frac {2}{3}}}{x^{9}+x^{6}-2 x^{3}}d x \right ) x^{5}}{20 x^{5}} \] Input:
int((x^3-1)^(2/3)*(x^3+1)/x^6/(x^3+2),x)
Output:
(7*(x**3 - 1)**(2/3)*x**3 - 2*(x**3 - 1)**(2/3) - 30*int((x**3 - 1)**(2/3) /(x**9 + x**6 - 2*x**3),x)*x**5)/(20*x**5)