Integrand size = 27, antiderivative size = 134 \[ \int \frac {\left (-2+x^3\right ) \left (1+x^3\right )^{2/3}}{x^6 \left (-1+2 x^3\right )} \, dx=\frac {\left (-4-19 x^3\right ) \left (1+x^3\right )^{2/3}}{10 x^5}+3 \sqrt [6]{3} \arctan \left (\frac {3^{5/6} x}{\sqrt [3]{3} x+2 \sqrt [3]{1+x^3}}\right )-3^{2/3} \log \left (-3 x+3^{2/3} \sqrt [3]{1+x^3}\right )+\frac {1}{2} 3^{2/3} \log \left (3 x^2+3^{2/3} x \sqrt [3]{1+x^3}+\sqrt [3]{3} \left (1+x^3\right )^{2/3}\right ) \]
1/10*(-19*x^3-4)*(x^3+1)^(2/3)/x^5+3*3^(1/6)*arctan(3^(5/6)*x/(3^(1/3)*x+2 *(x^3+1)^(1/3)))-ln(-3*x+3^(2/3)*(x^3+1)^(1/3))*3^(2/3)+1/2*ln(3*x^2+3^(2/ 3)*x*(x^3+1)^(1/3)+3^(1/3)*(x^3+1)^(2/3))*3^(2/3)
Time = 0.32 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-2+x^3\right ) \left (1+x^3\right )^{2/3}}{x^6 \left (-1+2 x^3\right )} \, dx=\frac {\left (-4-19 x^3\right ) \left (1+x^3\right )^{2/3}}{10 x^5}+3 \sqrt [6]{3} \arctan \left (\frac {3^{5/6} x}{\sqrt [3]{3} x+2 \sqrt [3]{1+x^3}}\right )-3^{2/3} \log \left (-3 x+3^{2/3} \sqrt [3]{1+x^3}\right )+\frac {1}{2} 3^{2/3} \log \left (3 x^2+3^{2/3} x \sqrt [3]{1+x^3}+\sqrt [3]{3} \left (1+x^3\right )^{2/3}\right ) \]
((-4 - 19*x^3)*(1 + x^3)^(2/3))/(10*x^5) + 3*3^(1/6)*ArcTan[(3^(5/6)*x)/(3 ^(1/3)*x + 2*(1 + x^3)^(1/3))] - 3^(2/3)*Log[-3*x + 3^(2/3)*(1 + x^3)^(1/3 )] + (3^(2/3)*Log[3*x^2 + 3^(2/3)*x*(1 + x^3)^(1/3) + 3^(1/3)*(1 + x^3)^(2 /3)])/2
Time = 0.27 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.89, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, 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.
| \(\displaystyle \int \frac {\left (x^3-2\right ) \left (x^3+1\right )^{2/3}}{x^6 \left (2 x^3-1\right )} \, dx\) |
| \(\Big \downarrow \) 1050 |
| \(\displaystyle -\frac {1}{5} \int -\frac {7 x^3+19}{x^3 \left (1-2 x^3\right ) \sqrt [3]{x^3+1}}dx-\frac {2 \left (x^3+1\right )^{2/3}}{5 x^5}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{5} \int \frac {7 x^3+19}{x^3 \left (1-2 x^3\right ) \sqrt [3]{x^3+1}}dx-\frac {2 \left (x^3+1\right )^{2/3}}{5 x^5}\) |
| \(\Big \downarrow \) 1053 |
| \(\displaystyle \frac {1}{5} \left (-\frac {1}{2} \int -\frac {90}{\left (1-2 x^3\right ) \sqrt [3]{x^3+1}}dx-\frac {19 \left (x^3+1\right )^{2/3}}{2 x^2}\right )-\frac {2 \left (x^3+1\right )^{2/3}}{5 x^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{5} \left (45 \int \frac {1}{\left (1-2 x^3\right ) \sqrt [3]{x^3+1}}dx-\frac {19 \left (x^3+1\right )^{2/3}}{2 x^2}\right )-\frac {2 \left (x^3+1\right )^{2/3}}{5 x^5}\) |
| \(\Big \downarrow \) 901 |
| \(\displaystyle \frac {1}{5} \left (45 \left (\frac {\arctan \left (\frac {\frac {2 \sqrt [3]{3} x}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )}{3^{5/6}}+\frac {\log \left (1-2 x^3\right )}{6 \sqrt [3]{3}}-\frac {\log \left (\sqrt [3]{3} x-\sqrt [3]{x^3+1}\right )}{2 \sqrt [3]{3}}\right )-\frac {19 \left (x^3+1\right )^{2/3}}{2 x^2}\right )-\frac {2 \left (x^3+1\right )^{2/3}}{5 x^5}\) |
(-2*(1 + x^3)^(2/3))/(5*x^5) + ((-19*(1 + x^3)^(2/3))/(2*x^2) + 45*(ArcTan [(1 + (2*3^(1/3)*x)/(1 + x^3)^(1/3))/Sqrt[3]]/3^(5/6) + Log[1 - 2*x^3]/(6* 3^(1/3)) - Log[3^(1/3)*x - (1 + x^3)^(1/3)]/(2*3^(1/3))))/5
3.20.25.3.1 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 = 13.49 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.96
| method | result | size |
| pseudoelliptic | \(\frac {-10 \,3^{\frac {2}{3}} \ln \left (\frac {-3^{\frac {1}{3}} x +\left (x^{3}+1\right )^{\frac {1}{3}}}{x}\right ) x^{5}+5 \,3^{\frac {2}{3}} \ln \left (\frac {3^{\frac {2}{3}} x^{2}+3^{\frac {1}{3}} \left (x^{3}+1\right )^{\frac {1}{3}} x +\left (x^{3}+1\right )^{\frac {2}{3}}}{x^{2}}\right ) x^{5}-30 \,3^{\frac {1}{6}} \arctan \left (\frac {\sqrt {3}\, \left (2 \,3^{\frac {2}{3}} \left (x^{3}+1\right )^{\frac {1}{3}}+3 x \right )}{9 x}\right ) x^{5}-19 x^{3} \left (x^{3}+1\right )^{\frac {2}{3}}-4 \left (x^{3}+1\right )^{\frac {2}{3}}}{10 x^{5}}\) | \(128\) |
| risch | \(\text {Expression too large to display}\) | \(612\) |
| trager | \(\text {Expression too large to display}\) | \(747\) |
1/10*(-10*3^(2/3)*ln((-3^(1/3)*x+(x^3+1)^(1/3))/x)*x^5+5*3^(2/3)*ln((3^(2/ 3)*x^2+3^(1/3)*(x^3+1)^(1/3)*x+(x^3+1)^(2/3))/x^2)*x^5-30*3^(1/6)*arctan(1 /9*3^(1/2)*(2*3^(2/3)*(x^3+1)^(1/3)+3*x)/x)*x^5-19*x^3*(x^3+1)^(2/3)-4*(x^ 3+1)^(2/3))/x^5
Leaf count of result is larger than twice the leaf count of optimal. 274 vs. \(2 (104) = 208\).
Time = 1.55 (sec) , antiderivative size = 274, normalized size of antiderivative = 2.04 \[ \int \frac {\left (-2+x^3\right ) \left (1+x^3\right )^{2/3}}{x^6 \left (-1+2 x^3\right )} \, dx=-\frac {10 \, \sqrt {3} \left (-9\right )^{\frac {1}{3}} x^{5} \arctan \left (\frac {2 \, \sqrt {3} \left (-9\right )^{\frac {2}{3}} {\left (14 \, x^{7} - 5 \, x^{4} - x\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}} + 6 \, \sqrt {3} \left (-9\right )^{\frac {1}{3}} {\left (31 \, x^{8} + 23 \, x^{5} + x^{2}\right )} {\left (x^{3} + 1\right )}^{\frac {1}{3}} - \sqrt {3} {\left (127 \, x^{9} + 201 \, x^{6} + 48 \, x^{3} + 1\right )}}{3 \, {\left (251 \, x^{9} + 231 \, x^{6} + 6 \, x^{3} - 1\right )}}\right ) - 10 \, \left (-9\right )^{\frac {1}{3}} x^{5} \log \left (\frac {3 \, \left (-9\right )^{\frac {2}{3}} {\left (x^{3} + 1\right )}^{\frac {1}{3}} x^{2} - 9 \, {\left (x^{3} + 1\right )}^{\frac {2}{3}} x + \left (-9\right )^{\frac {1}{3}} {\left (2 \, x^{3} - 1\right )}}{2 \, x^{3} - 1}\right ) + 5 \, \left (-9\right )^{\frac {1}{3}} x^{5} \log \left (-\frac {9 \, \left (-9\right )^{\frac {1}{3}} {\left (7 \, x^{4} + x\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}} - \left (-9\right )^{\frac {2}{3}} {\left (31 \, x^{6} + 23 \, x^{3} + 1\right )} - 27 \, {\left (5 \, x^{5} + 2 \, x^{2}\right )} {\left (x^{3} + 1\right )}^{\frac {1}{3}}}{4 \, x^{6} - 4 \, x^{3} + 1}\right ) + 3 \, {\left (19 \, x^{3} + 4\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{30 \, x^{5}} \]
-1/30*(10*sqrt(3)*(-9)^(1/3)*x^5*arctan(1/3*(2*sqrt(3)*(-9)^(2/3)*(14*x^7 - 5*x^4 - x)*(x^3 + 1)^(2/3) + 6*sqrt(3)*(-9)^(1/3)*(31*x^8 + 23*x^5 + x^2 )*(x^3 + 1)^(1/3) - sqrt(3)*(127*x^9 + 201*x^6 + 48*x^3 + 1))/(251*x^9 + 2 31*x^6 + 6*x^3 - 1)) - 10*(-9)^(1/3)*x^5*log((3*(-9)^(2/3)*(x^3 + 1)^(1/3) *x^2 - 9*(x^3 + 1)^(2/3)*x + (-9)^(1/3)*(2*x^3 - 1))/(2*x^3 - 1)) + 5*(-9) ^(1/3)*x^5*log(-(9*(-9)^(1/3)*(7*x^4 + x)*(x^3 + 1)^(2/3) - (-9)^(2/3)*(31 *x^6 + 23*x^3 + 1) - 27*(5*x^5 + 2*x^2)*(x^3 + 1)^(1/3))/(4*x^6 - 4*x^3 + 1)) + 3*(19*x^3 + 4)*(x^3 + 1)^(2/3))/x^5
\[ \int \frac {\left (-2+x^3\right ) \left (1+x^3\right )^{2/3}}{x^6 \left (-1+2 x^3\right )} \, dx=\int \frac {\left (\left (x + 1\right ) \left (x^{2} - x + 1\right )\right )^{\frac {2}{3}} \left (x^{3} - 2\right )}{x^{6} \cdot \left (2 x^{3} - 1\right )}\, dx \]
\[ \int \frac {\left (-2+x^3\right ) \left (1+x^3\right )^{2/3}}{x^6 \left (-1+2 x^3\right )} \, dx=\int { \frac {{\left (x^{3} + 1\right )}^{\frac {2}{3}} {\left (x^{3} - 2\right )}}{{\left (2 \, x^{3} - 1\right )} x^{6}} \,d x } \]
\[ \int \frac {\left (-2+x^3\right ) \left (1+x^3\right )^{2/3}}{x^6 \left (-1+2 x^3\right )} \, dx=\int { \frac {{\left (x^{3} + 1\right )}^{\frac {2}{3}} {\left (x^{3} - 2\right )}}{{\left (2 \, x^{3} - 1\right )} x^{6}} \,d x } \]
Timed out. \[ \int \frac {\left (-2+x^3\right ) \left (1+x^3\right )^{2/3}}{x^6 \left (-1+2 x^3\right )} \, dx=\int \frac {{\left (x^3+1\right )}^{2/3}\,\left (x^3-2\right )}{x^6\,\left (2\,x^3-1\right )} \,d x \]