Integrand size = 31, antiderivative size = 150 \[ \int \frac {\left (1-x^3\right )^{2/3} \left (-1+4 x^3\right )}{x^6 \left (-2+3 x^3\right )} \, dx=\frac {\left (1-x^3\right )^{2/3} \left (-4+29 x^3\right )}{40 x^5}-\frac {5 \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{2} \sqrt [3]{1-x^3}}\right )}{4\ 2^{2/3} \sqrt {3}}+\frac {5 \log \left (-x+\sqrt [3]{2} \sqrt [3]{1-x^3}\right )}{12\ 2^{2/3}}-\frac {5 \log \left (x^2+\sqrt [3]{2} x \sqrt [3]{1-x^3}+2^{2/3} \left (1-x^3\right )^{2/3}\right )}{24\ 2^{2/3}} \]
1/40*(-x^3+1)^(2/3)*(29*x^3-4)/x^5-5/24*arctan(3^(1/2)*x/(x+2*2^(1/3)*(-x^ 3+1)^(1/3)))*2^(1/3)*3^(1/2)+5/24*ln(-x+2^(1/3)*(-x^3+1)^(1/3))*2^(1/3)-5/ 48*ln(x^2+2^(1/3)*x*(-x^3+1)^(1/3)+2^(2/3)*(-x^3+1)^(2/3))*2^(1/3)
Time = 0.33 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.89 \[ \int \frac {\left (1-x^3\right )^{2/3} \left (-1+4 x^3\right )}{x^6 \left (-2+3 x^3\right )} \, dx=\frac {1}{240} \left (-\frac {24 \left (1-x^3\right )^{2/3}}{x^5}+\frac {174 \left (1-x^3\right )^{2/3}}{x^2}-50 \sqrt [3]{2} \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{2-2 x^3}}\right )+50 \sqrt [3]{2} \log \left (-x+\sqrt [3]{2-2 x^3}\right )-25 \sqrt [3]{2} \log \left (x^2+x \sqrt [3]{2-2 x^3}+\left (2-2 x^3\right )^{2/3}\right )\right ) \]
((-24*(1 - x^3)^(2/3))/x^5 + (174*(1 - x^3)^(2/3))/x^2 - 50*2^(1/3)*Sqrt[3 ]*ArcTan[(Sqrt[3]*x)/(x + 2*(2 - 2*x^3)^(1/3))] + 50*2^(1/3)*Log[-x + (2 - 2*x^3)^(1/3)] - 25*2^(1/3)*Log[x^2 + x*(2 - 2*x^3)^(1/3) + (2 - 2*x^3)^(2 /3)])/240
Time = 0.27 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.89, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {1050, 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.
| \(\displaystyle \int \frac {\left (1-x^3\right )^{2/3} \left (4 x^3-1\right )}{x^6 \left (3 x^3-2\right )} \, dx\) |
| \(\Big \downarrow \) 1050 |
| \(\displaystyle -\frac {1}{10} \int \frac {29-31 x^3}{x^3 \left (2-3 x^3\right ) \sqrt [3]{1-x^3}}dx-\frac {\left (1-x^3\right )^{2/3}}{10 x^5}\) |
| \(\Big \downarrow \) 1053 |
| \(\displaystyle \frac {1}{10} \left (\frac {1}{4} \int -\frac {50}{\left (2-3 x^3\right ) \sqrt [3]{1-x^3}}dx+\frac {29 \left (1-x^3\right )^{2/3}}{4 x^2}\right )-\frac {\left (1-x^3\right )^{2/3}}{10 x^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{10} \left (\frac {29 \left (1-x^3\right )^{2/3}}{4 x^2}-\frac {25}{2} \int \frac {1}{\left (2-3 x^3\right ) \sqrt [3]{1-x^3}}dx\right )-\frac {\left (1-x^3\right )^{2/3}}{10 x^5}\) |
| \(\Big \downarrow \) 901 |
| \(\displaystyle \frac {1}{10} \left (\frac {29 \left (1-x^3\right )^{2/3}}{4 x^2}-\frac {25}{2} \left (\frac {\arctan \left (\frac {\frac {2^{2/3} x}{\sqrt [3]{1-x^3}}+1}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3}}+\frac {\log \left (2-3 x^3\right )}{6\ 2^{2/3}}-\frac {\log \left (\frac {x}{\sqrt [3]{2}}-\sqrt [3]{1-x^3}\right )}{2\ 2^{2/3}}\right )\right )-\frac {\left (1-x^3\right )^{2/3}}{10 x^5}\) |
-1/10*(1 - x^3)^(2/3)/x^5 + ((29*(1 - x^3)^(2/3))/(4*x^2) - (25*(ArcTan[(1 + (2^(2/3)*x)/(1 - x^3)^(1/3))/Sqrt[3]]/(2^(2/3)*Sqrt[3]) + Log[2 - 3*x^3 ]/(6*2^(2/3)) - Log[x/2^(1/3) - (1 - x^3)^(1/3)]/(2*2^(2/3))))/2)/10
3.21.72.3.1 Defintions of rubi rules used
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 = 13.59 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.11
| method | result | size |
| pseudoelliptic | \(\frac {50 \sqrt {3}\, 2^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3}\, \left (x +2 \,2^{\frac {1}{3}} \left (-x^{3}+1\right )^{\frac {1}{3}}\right )}{3 x}\right ) x^{5}-25 \,2^{\frac {1}{3}} x^{5} \ln \left (2\right )+50 \,2^{\frac {1}{3}} x^{5} \ln \left (\frac {-2^{\frac {2}{3}} x +2 {\left (-\left (-1+x \right ) \left (x^{2}+x +1\right )\right )}^{\frac {1}{3}}}{x}\right )-25 \,2^{\frac {1}{3}} x^{5} \ln \left (\frac {2^{\frac {2}{3}} {\left (-\left (-1+x \right ) \left (x^{2}+x +1\right )\right )}^{\frac {1}{3}} x +2^{\frac {1}{3}} x^{2}+2 {\left (-\left (-1+x \right ) \left (x^{2}+x +1\right )\right )}^{\frac {2}{3}}}{x^{2}}\right )+174 \left (-x^{3}+1\right )^{\frac {2}{3}} x^{3}-24 \left (-x^{3}+1\right )^{\frac {2}{3}}}{240 x^{5}}\) | \(167\) |
| risch | \(\text {Expression too large to display}\) | \(908\) |
| trager | \(\text {Expression too large to display}\) | \(1127\) |
1/240*(50*3^(1/2)*2^(1/3)*arctan(1/3*3^(1/2)/x*(x+2*2^(1/3)*(-x^3+1)^(1/3) ))*x^5-25*2^(1/3)*x^5*ln(2)+50*2^(1/3)*x^5*ln((-2^(2/3)*x+2*(-(-1+x)*(x^2+ x+1))^(1/3))/x)-25*2^(1/3)*x^5*ln((2^(2/3)*(-(-1+x)*(x^2+x+1))^(1/3)*x+2^( 1/3)*x^2+2*(-(-1+x)*(x^2+x+1))^(2/3))/x^2)+174*(-x^3+1)^(2/3)*x^3-24*(-x^3 +1)^(2/3))/x^5
Leaf count of result is larger than twice the leaf count of optimal. 278 vs. \(2 (114) = 228\).
Time = 2.02 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.85 \[ \int \frac {\left (1-x^3\right )^{2/3} \left (-1+4 x^3\right )}{x^6 \left (-2+3 x^3\right )} \, dx=\frac {100 \cdot 4^{\frac {1}{6}} \sqrt {3} x^{5} \arctan \left (\frac {4^{\frac {1}{6}} {\left (12 \cdot 4^{\frac {2}{3}} \sqrt {3} {\left (3 \, x^{4} - 2 \, x\right )} {\left (-x^{3} + 1\right )}^{\frac {2}{3}} - 4^{\frac {1}{3}} \sqrt {3} {\left (27 \, x^{9} - 72 \, x^{6} + 36 \, x^{3} + 8\right )} + 12 \, \sqrt {3} {\left (9 \, x^{8} - 6 \, x^{5} - 4 \, x^{2}\right )} {\left (-x^{3} + 1\right )}^{\frac {1}{3}}\right )}}{6 \, {\left (27 \, x^{9} - 36 \, x^{3} + 8\right )}}\right ) + 50 \cdot 4^{\frac {2}{3}} x^{5} \log \left (-\frac {6 \cdot 4^{\frac {1}{3}} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} x^{2} - 4^{\frac {2}{3}} {\left (3 \, x^{3} - 2\right )} - 12 \, {\left (-x^{3} + 1\right )}^{\frac {2}{3}} x}{3 \, x^{3} - 2}\right ) - 25 \cdot 4^{\frac {2}{3}} x^{5} \log \left (\frac {6 \cdot 4^{\frac {2}{3}} {\left (-x^{3} + 1\right )}^{\frac {2}{3}} x - 4^{\frac {1}{3}} {\left (9 \, x^{6} - 6 \, x^{3} - 4\right )} - 6 \, {\left (3 \, x^{5} - 4 \, x^{2}\right )} {\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{9 \, x^{6} - 12 \, x^{3} + 4}\right ) + 36 \, {\left (29 \, x^{3} - 4\right )} {\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{1440 \, x^{5}} \]
1/1440*(100*4^(1/6)*sqrt(3)*x^5*arctan(1/6*4^(1/6)*(12*4^(2/3)*sqrt(3)*(3* x^4 - 2*x)*(-x^3 + 1)^(2/3) - 4^(1/3)*sqrt(3)*(27*x^9 - 72*x^6 + 36*x^3 + 8) + 12*sqrt(3)*(9*x^8 - 6*x^5 - 4*x^2)*(-x^3 + 1)^(1/3))/(27*x^9 - 36*x^3 + 8)) + 50*4^(2/3)*x^5*log(-(6*4^(1/3)*(-x^3 + 1)^(1/3)*x^2 - 4^(2/3)*(3* x^3 - 2) - 12*(-x^3 + 1)^(2/3)*x)/(3*x^3 - 2)) - 25*4^(2/3)*x^5*log((6*4^( 2/3)*(-x^3 + 1)^(2/3)*x - 4^(1/3)*(9*x^6 - 6*x^3 - 4) - 6*(3*x^5 - 4*x^2)* (-x^3 + 1)^(1/3))/(9*x^6 - 12*x^3 + 4)) + 36*(29*x^3 - 4)*(-x^3 + 1)^(2/3) )/x^5
\[ \int \frac {\left (1-x^3\right )^{2/3} \left (-1+4 x^3\right )}{x^6 \left (-2+3 x^3\right )} \, dx=\int \frac {\left (- \left (x - 1\right ) \left (x^{2} + x + 1\right )\right )^{\frac {2}{3}} \cdot \left (4 x^{3} - 1\right )}{x^{6} \cdot \left (3 x^{3} - 2\right )}\, dx \]
\[ \int \frac {\left (1-x^3\right )^{2/3} \left (-1+4 x^3\right )}{x^6 \left (-2+3 x^3\right )} \, dx=\int { \frac {{\left (4 \, x^{3} - 1\right )} {\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{{\left (3 \, x^{3} - 2\right )} x^{6}} \,d x } \]
\[ \int \frac {\left (1-x^3\right )^{2/3} \left (-1+4 x^3\right )}{x^6 \left (-2+3 x^3\right )} \, dx=\int { \frac {{\left (4 \, x^{3} - 1\right )} {\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{{\left (3 \, x^{3} - 2\right )} x^{6}} \,d x } \]
Timed out. \[ \int \frac {\left (1-x^3\right )^{2/3} \left (-1+4 x^3\right )}{x^6 \left (-2+3 x^3\right )} \, dx=\int \frac {{\left (1-x^3\right )}^{2/3}\,\left (4\,x^3-1\right )}{x^6\,\left (3\,x^3-2\right )} \,d x \]