Integrand size = 27, antiderivative size = 149 \[ \int \frac {\left (-1+x^3\right ) \left (1+3 x^3\right )^{2/3}}{x^6 \left (1+x^3\right )} \, dx=\frac {\left (1-2 x^3\right ) \left (1+3 x^3\right )^{2/3}}{5 x^5}+\frac {2\ 2^{2/3} \arctan \left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{1+3 x^3}}\right )}{\sqrt {3}}-\frac {2}{3} 2^{2/3} \log \left (-2 x+2^{2/3} \sqrt [3]{1+3 x^3}\right )+\frac {1}{3} 2^{2/3} \log \left (2 x^2+2^{2/3} x \sqrt [3]{1+3 x^3}+\sqrt [3]{2} \left (1+3 x^3\right )^{2/3}\right ) \]
1/5*(-2*x^3+1)*(3*x^3+1)^(2/3)/x^5+2/3*2^(2/3)*arctan(3^(1/2)*x/(x+2^(2/3) *(3*x^3+1)^(1/3)))*3^(1/2)-2/3*2^(2/3)*ln(-2*x+2^(2/3)*(3*x^3+1)^(1/3))+1/ 3*2^(2/3)*ln(2*x^2+2^(2/3)*x*(3*x^3+1)^(1/3)+2^(1/3)*(3*x^3+1)^(2/3))
Time = 0.34 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-1+x^3\right ) \left (1+3 x^3\right )^{2/3}}{x^6 \left (1+x^3\right )} \, dx=\frac {\left (1-2 x^3\right ) \left (1+3 x^3\right )^{2/3}}{5 x^5}+\frac {2\ 2^{2/3} \arctan \left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{1+3 x^3}}\right )}{\sqrt {3}}-\frac {2}{3} 2^{2/3} \log \left (-2 x+2^{2/3} \sqrt [3]{1+3 x^3}\right )+\frac {1}{3} 2^{2/3} \log \left (2 x^2+2^{2/3} x \sqrt [3]{1+3 x^3}+\sqrt [3]{2} \left (1+3 x^3\right )^{2/3}\right ) \]
((1 - 2*x^3)*(1 + 3*x^3)^(2/3))/(5*x^5) + (2*2^(2/3)*ArcTan[(Sqrt[3]*x)/(x + 2^(2/3)*(1 + 3*x^3)^(1/3))])/Sqrt[3] - (2*2^(2/3)*Log[-2*x + 2^(2/3)*(1 + 3*x^3)^(1/3)])/3 + (2^(2/3)*Log[2*x^2 + 2^(2/3)*x*(1 + 3*x^3)^(1/3) + 2 ^(1/3)*(1 + 3*x^3)^(2/3)])/3
Time = 0.27 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.87, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {1050, 27, 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.
| \(\displaystyle \int \frac {\left (x^3-1\right ) \left (3 x^3+1\right )^{2/3}}{x^6 \left (x^3+1\right )} \, dx\) |
| \(\Big \downarrow \) 1050 |
| \(\displaystyle \frac {1}{5} \int \frac {4 \left (6 x^3+1\right )}{x^3 \left (x^3+1\right ) \sqrt [3]{3 x^3+1}}dx+\frac {\left (3 x^3+1\right )^{2/3}}{5 x^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {4}{5} \int \frac {6 x^3+1}{x^3 \left (x^3+1\right ) \sqrt [3]{3 x^3+1}}dx+\frac {\left (3 x^3+1\right )^{2/3}}{5 x^5}\) |
| \(\Big \downarrow \) 1053 |
| \(\displaystyle \frac {4}{5} \left (-\frac {1}{2} \int -\frac {10}{\left (x^3+1\right ) \sqrt [3]{3 x^3+1}}dx-\frac {\left (3 x^3+1\right )^{2/3}}{2 x^2}\right )+\frac {\left (3 x^3+1\right )^{2/3}}{5 x^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {4}{5} \left (5 \int \frac {1}{\left (x^3+1\right ) \sqrt [3]{3 x^3+1}}dx-\frac {\left (3 x^3+1\right )^{2/3}}{2 x^2}\right )+\frac {\left (3 x^3+1\right )^{2/3}}{5 x^5}\) |
| \(\Big \downarrow \) 901 |
| \(\displaystyle \frac {4}{5} \left (5 \left (\frac {\arctan \left (\frac {\frac {2 \sqrt [3]{2} x}{\sqrt [3]{3 x^3+1}}+1}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3}}+\frac {\log \left (x^3+1\right )}{6 \sqrt [3]{2}}-\frac {\log \left (\sqrt [3]{2} x-\sqrt [3]{3 x^3+1}\right )}{2 \sqrt [3]{2}}\right )-\frac {\left (3 x^3+1\right )^{2/3}}{2 x^2}\right )+\frac {\left (3 x^3+1\right )^{2/3}}{5 x^5}\) |
(1 + 3*x^3)^(2/3)/(5*x^5) + (4*(-1/2*(1 + 3*x^3)^(2/3)/x^2 + 5*(ArcTan[(1 + (2*2^(1/3)*x)/(1 + 3*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 + 3*x^3)^(1/3)]/(2*2^(1/3)))))/5
3.21.63.3.1 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 = 13.80 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.94
| method | result | size |
| pseudoelliptic | \(\frac {-10 \sqrt {3}\, 2^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3}\, \left (x +2^{\frac {2}{3}} \left (3 x^{3}+1\right )^{\frac {1}{3}}\right )}{3 x}\right ) x^{5}-10 \,2^{\frac {2}{3}} \ln \left (\frac {-2^{\frac {1}{3}} x +\left (3 x^{3}+1\right )^{\frac {1}{3}}}{x}\right ) x^{5}+5 \,2^{\frac {2}{3}} \ln \left (\frac {2^{\frac {2}{3}} x^{2}+2^{\frac {1}{3}} \left (3 x^{3}+1\right )^{\frac {1}{3}} x +\left (3 x^{3}+1\right )^{\frac {2}{3}}}{x^{2}}\right ) x^{5}-6 \left (3 x^{3}+1\right )^{\frac {2}{3}} x^{3}+3 \left (3 x^{3}+1\right )^{\frac {2}{3}}}{15 x^{5}}\) | \(140\) |
| risch | \(\text {Expression too large to display}\) | \(958\) |
| trager | \(\text {Expression too large to display}\) | \(1163\) |
1/15*(-10*3^(1/2)*2^(2/3)*arctan(1/3*3^(1/2)/x*(x+2^(2/3)*(3*x^3+1)^(1/3)) )*x^5-10*2^(2/3)*ln((-2^(1/3)*x+(3*x^3+1)^(1/3))/x)*x^5+5*2^(2/3)*ln((2^(2 /3)*x^2+2^(1/3)*(3*x^3+1)^(1/3)*x+(3*x^3+1)^(2/3))/x^2)*x^5-6*(3*x^3+1)^(2 /3)*x^3+3*(3*x^3+1)^(2/3))/x^5
Leaf count of result is larger than twice the leaf count of optimal. 279 vs. \(2 (115) = 230\).
Time = 1.87 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.87 \[ \int \frac {\left (-1+x^3\right ) \left (1+3 x^3\right )^{2/3}}{x^6 \left (1+x^3\right )} \, dx=\frac {10 \, \sqrt {3} \left (-4\right )^{\frac {1}{3}} x^{5} \arctan \left (\frac {3 \, \sqrt {3} \left (-4\right )^{\frac {2}{3}} {\left (7 \, x^{7} + 8 \, x^{4} + x\right )} {\left (3 \, x^{3} + 1\right )}^{\frac {2}{3}} - 6 \, \sqrt {3} \left (-4\right )^{\frac {1}{3}} {\left (55 \, x^{8} + 20 \, x^{5} + x^{2}\right )} {\left (3 \, x^{3} + 1\right )}^{\frac {1}{3}} + \sqrt {3} {\left (433 \, x^{9} + 255 \, x^{6} + 39 \, x^{3} + 1\right )}}{3 \, {\left (323 \, x^{9} + 105 \, x^{6} - 3 \, x^{3} - 1\right )}}\right ) + 10 \, \left (-4\right )^{\frac {1}{3}} x^{5} \log \left (-\frac {3 \, \left (-4\right )^{\frac {2}{3}} {\left (3 \, x^{3} + 1\right )}^{\frac {1}{3}} x^{2} - 6 \, {\left (3 \, x^{3} + 1\right )}^{\frac {2}{3}} x - \left (-4\right )^{\frac {1}{3}} {\left (x^{3} + 1\right )}}{x^{3} + 1}\right ) - 5 \, \left (-4\right )^{\frac {1}{3}} x^{5} \log \left (-\frac {6 \, \left (-4\right )^{\frac {1}{3}} {\left (7 \, x^{4} + x\right )} {\left (3 \, x^{3} + 1\right )}^{\frac {2}{3}} - \left (-4\right )^{\frac {2}{3}} {\left (55 \, x^{6} + 20 \, x^{3} + 1\right )} - 24 \, {\left (4 \, x^{5} + x^{2}\right )} {\left (3 \, x^{3} + 1\right )}^{\frac {1}{3}}}{x^{6} + 2 \, x^{3} + 1}\right ) - 9 \, {\left (3 \, x^{3} + 1\right )}^{\frac {2}{3}} {\left (2 \, x^{3} - 1\right )}}{45 \, x^{5}} \]
1/45*(10*sqrt(3)*(-4)^(1/3)*x^5*arctan(1/3*(3*sqrt(3)*(-4)^(2/3)*(7*x^7 + 8*x^4 + x)*(3*x^3 + 1)^(2/3) - 6*sqrt(3)*(-4)^(1/3)*(55*x^8 + 20*x^5 + x^2 )*(3*x^3 + 1)^(1/3) + sqrt(3)*(433*x^9 + 255*x^6 + 39*x^3 + 1))/(323*x^9 + 105*x^6 - 3*x^3 - 1)) + 10*(-4)^(1/3)*x^5*log(-(3*(-4)^(2/3)*(3*x^3 + 1)^ (1/3)*x^2 - 6*(3*x^3 + 1)^(2/3)*x - (-4)^(1/3)*(x^3 + 1))/(x^3 + 1)) - 5*( -4)^(1/3)*x^5*log(-(6*(-4)^(1/3)*(7*x^4 + x)*(3*x^3 + 1)^(2/3) - (-4)^(2/3 )*(55*x^6 + 20*x^3 + 1) - 24*(4*x^5 + x^2)*(3*x^3 + 1)^(1/3))/(x^6 + 2*x^3 + 1)) - 9*(3*x^3 + 1)^(2/3)*(2*x^3 - 1))/x^5
\[ \int \frac {\left (-1+x^3\right ) \left (1+3 x^3\right )^{2/3}}{x^6 \left (1+x^3\right )} \, dx=\int \frac {\left (x - 1\right ) \left (3 x^{3} + 1\right )^{\frac {2}{3}} \left (x^{2} + x + 1\right )}{x^{6} \left (x + 1\right ) \left (x^{2} - x + 1\right )}\, dx \]
\[ \int \frac {\left (-1+x^3\right ) \left (1+3 x^3\right )^{2/3}}{x^6 \left (1+x^3\right )} \, dx=\int { \frac {{\left (3 \, x^{3} + 1\right )}^{\frac {2}{3}} {\left (x^{3} - 1\right )}}{{\left (x^{3} + 1\right )} x^{6}} \,d x } \]
\[ \int \frac {\left (-1+x^3\right ) \left (1+3 x^3\right )^{2/3}}{x^6 \left (1+x^3\right )} \, dx=\int { \frac {{\left (3 \, x^{3} + 1\right )}^{\frac {2}{3}} {\left (x^{3} - 1\right )}}{{\left (x^{3} + 1\right )} x^{6}} \,d x } \]
Timed out. \[ \int \frac {\left (-1+x^3\right ) \left (1+3 x^3\right )^{2/3}}{x^6 \left (1+x^3\right )} \, dx=\int \frac {\left (x^3-1\right )\,{\left (3\,x^3+1\right )}^{2/3}}{x^6\,\left (x^3+1\right )} \,d x \]