Integrand size = 32, antiderivative size = 325 \[ \int \frac {\left (1+x^3\right )^{2/3} \left (-1+x^6\right )}{x^6 \left (-1-2 x^3+2 x^6\right )} \, dx=\frac {\left (1+x^3\right )^{2/3} \left (-1+4 x^3\right )}{5 x^5}-\frac {1}{2} \sqrt [3]{\frac {1}{3} \left (45+26 \sqrt {3}\right )} \arctan \left (\frac {3^{2/3} x}{\sqrt [6]{3} x-2 \sqrt [3]{1+x^3}}\right )-\frac {\arctan \left (\frac {3^{2/3} x}{\sqrt [6]{3} x+2 \sqrt [3]{1+x^3}}\right )}{2 \sqrt [3]{45+26 \sqrt {3}}}+\frac {\log \left (-3 x+3^{5/6} \sqrt [3]{1+x^3}\right )}{2\ 3^{2/3} \sqrt [3]{26+15 \sqrt {3}}}+\frac {\sqrt [3]{26+15 \sqrt {3}} \log \left (3 x+3^{5/6} \sqrt [3]{1+x^3}\right )}{2\ 3^{2/3}}-\frac {\sqrt [3]{26+15 \sqrt {3}} \log \left (-3 x^2+3^{5/6} x \sqrt [3]{1+x^3}-3^{2/3} \left (1+x^3\right )^{2/3}\right )}{4\ 3^{2/3}}-\frac {\log \left (3 x^2+3^{5/6} x \sqrt [3]{1+x^3}+3^{2/3} \left (1+x^3\right )^{2/3}\right )}{4\ 3^{2/3} \sqrt [3]{26+15 \sqrt {3}}} \]
1/5*(x^3+1)^(2/3)*(4*x^3-1)/x^5-1/2*(15+26/3*3^(1/2))^(1/3)*arctan(3^(2/3) *x/(3^(1/6)*x-2*(x^3+1)^(1/3)))-1/2*arctan(3^(2/3)*x/(3^(1/6)*x+2*(x^3+1)^ (1/3)))/(45+26*3^(1/2))^(1/3)+1/6*ln(-3*x+3^(5/6)*(x^3+1)^(1/3))*3^(1/3)/( 26+15*3^(1/2))^(1/3)+1/6*(26+15*3^(1/2))^(1/3)*ln(3*x+3^(5/6)*(x^3+1)^(1/3 ))*3^(1/3)-1/12*(26+15*3^(1/2))^(1/3)*ln(-3*x^2+3^(5/6)*x*(x^3+1)^(1/3)-3^ (2/3)*(x^3+1)^(2/3))*3^(1/3)-1/12*ln(3*x^2+3^(5/6)*x*(x^3+1)^(1/3)+3^(2/3) *(x^3+1)^(2/3))*3^(1/3)/(26+15*3^(1/2))^(1/3)
Time = 1.83 (sec) , antiderivative size = 296, normalized size of antiderivative = 0.91 \[ \int \frac {\left (1+x^3\right )^{2/3} \left (-1+x^6\right )}{x^6 \left (-1-2 x^3+2 x^6\right )} \, dx=\frac {1}{60} \left (\frac {12 \left (1+x^3\right )^{2/3} \left (-1+4 x^3\right )}{x^5}-10\ 3^{2/3} \sqrt [3]{45+26 \sqrt {3}} \arctan \left (\frac {3^{2/3} x}{\sqrt [6]{3} x-2 \sqrt [3]{1+x^3}}\right )-\frac {30 \arctan \left (\frac {3^{2/3} x}{\sqrt [6]{3} x+2 \sqrt [3]{1+x^3}}\right )}{\sqrt [3]{45+26 \sqrt {3}}}+10 \sqrt [3]{78-45 \sqrt {3}} \log \left (-3 x+3^{5/6} \sqrt [3]{1+x^3}\right )+10 \sqrt [3]{78+45 \sqrt {3}} \log \left (3 x+3^{5/6} \sqrt [3]{1+x^3}\right )-5 \sqrt [3]{78+45 \sqrt {3}} \log \left (-3 x^2+3^{5/6} x \sqrt [3]{1+x^3}-3^{2/3} \left (1+x^3\right )^{2/3}\right )-5 \sqrt [3]{78-45 \sqrt {3}} \log \left (3 x^2+3^{5/6} x \sqrt [3]{1+x^3}+3^{2/3} \left (1+x^3\right )^{2/3}\right )\right ) \]
((12*(1 + x^3)^(2/3)*(-1 + 4*x^3))/x^5 - 10*3^(2/3)*(45 + 26*Sqrt[3])^(1/3 )*ArcTan[(3^(2/3)*x)/(3^(1/6)*x - 2*(1 + x^3)^(1/3))] - (30*ArcTan[(3^(2/3 )*x)/(3^(1/6)*x + 2*(1 + x^3)^(1/3))])/(45 + 26*Sqrt[3])^(1/3) + 10*(78 - 45*Sqrt[3])^(1/3)*Log[-3*x + 3^(5/6)*(1 + x^3)^(1/3)] + 10*(78 + 45*Sqrt[3 ])^(1/3)*Log[3*x + 3^(5/6)*(1 + x^3)^(1/3)] - 5*(78 + 45*Sqrt[3])^(1/3)*Lo g[-3*x^2 + 3^(5/6)*x*(1 + x^3)^(1/3) - 3^(2/3)*(1 + x^3)^(2/3)] - 5*(78 - 45*Sqrt[3])^(1/3)*Log[3*x^2 + 3^(5/6)*x*(1 + x^3)^(1/3) + 3^(2/3)*(1 + x^3 )^(2/3)])/60
Time = 1.17 (sec) , antiderivative size = 564, normalized size of antiderivative = 1.74, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.094, Rules used = {1388, 7279, 2009}
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.
| \(\displaystyle \int \frac {\left (x^3+1\right )^{2/3} \left (x^6-1\right )}{x^6 \left (2 x^6-2 x^3-1\right )} \, dx\) |
| \(\Big \downarrow \) 1388 |
| \(\displaystyle \int \frac {\left (x^3-1\right ) \left (x^3+1\right )^{5/3}}{x^6 \left (2 x^6-2 x^3-1\right )}dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \int \left (-\frac {3 \left (x^3+1\right )^{5/3}}{x^3}+\frac {2 \left (3 x^3-4\right ) \left (x^3+1\right )^{5/3}}{2 x^6-2 x^3-1}+\frac {\left (x^3+1\right )^{5/3}}{x^6}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{18} \left (19+12 \sqrt {3}\right ) \arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )-\frac {1}{18} \left (19-12 \sqrt {3}\right ) \arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )-\frac {4 \arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {\left (1+\sqrt {3}\right ) \arctan \left (\frac {1-\frac {2 \sqrt [6]{3} x}{\sqrt [3]{x^3+1}}}{\sqrt {3}}\right )}{2 \sqrt [6]{3} \left (1-\sqrt {3}\right )}+\frac {\left (1-\sqrt {3}\right ) \arctan \left (\frac {\frac {2 \sqrt [6]{3} x}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )}{2 \sqrt [6]{3} \left (1+\sqrt {3}\right )}+\frac {1}{18} \left (9+5 \sqrt {3}\right ) x \left (x^3+1\right )^{2/3}+\frac {1}{18} \left (9-5 \sqrt {3}\right ) x \left (x^3+1\right )^{2/3}-\frac {5}{2} x \left (x^3+1\right )^{2/3}+\frac {\left (1+\sqrt {3}\right ) \log \left (4 x^3-2 \left (1-\sqrt {3}\right )\right )}{4\ 3^{2/3} \left (1-\sqrt {3}\right )}+\frac {\left (1-\sqrt {3}\right ) \log \left (4 x^3-2 \left (1+\sqrt {3}\right )\right )}{4\ 3^{2/3} \left (1+\sqrt {3}\right )}-\frac {\sqrt [3]{3} \left (1+\sqrt {3}\right ) \log \left (-\sqrt [3]{x^3+1}-\sqrt [6]{3} x\right )}{4 \left (1-\sqrt {3}\right )}-\frac {\sqrt [3]{3} \left (1-\sqrt {3}\right ) \log \left (\sqrt [6]{3} x-\sqrt [3]{x^3+1}\right )}{4 \left (1+\sqrt {3}\right )}-\frac {1}{36} \left (36+19 \sqrt {3}\right ) \log \left (\sqrt [3]{x^3+1}-x\right )-\frac {1}{36} \left (36-19 \sqrt {3}\right ) \log \left (\sqrt [3]{x^3+1}-x\right )+2 \log \left (\sqrt [3]{x^3+1}-x\right )-\frac {\left (x^3+1\right )^{5/3}}{5 x^5}+\frac {3 \left (x^3+1\right )^{5/3}}{2 x^2}-\frac {\left (x^3+1\right )^{2/3}}{2 x^2}\) |
-1/2*(1 + x^3)^(2/3)/x^2 - (5*x*(1 + x^3)^(2/3))/2 + ((9 - 5*Sqrt[3])*x*(1 + x^3)^(2/3))/18 + ((9 + 5*Sqrt[3])*x*(1 + x^3)^(2/3))/18 - (1 + x^3)^(5/ 3)/(5*x^5) + (3*(1 + x^3)^(5/3))/(2*x^2) - (4*ArcTan[(1 + (2*x)/(1 + x^3)^ (1/3))/Sqrt[3]])/Sqrt[3] - ((19 - 12*Sqrt[3])*ArcTan[(1 + (2*x)/(1 + x^3)^ (1/3))/Sqrt[3]])/18 + ((19 + 12*Sqrt[3])*ArcTan[(1 + (2*x)/(1 + x^3)^(1/3) )/Sqrt[3]])/18 + ((1 + Sqrt[3])*ArcTan[(1 - (2*3^(1/6)*x)/(1 + x^3)^(1/3)) /Sqrt[3]])/(2*3^(1/6)*(1 - Sqrt[3])) + ((1 - Sqrt[3])*ArcTan[(1 + (2*3^(1/ 6)*x)/(1 + x^3)^(1/3))/Sqrt[3]])/(2*3^(1/6)*(1 + Sqrt[3])) + ((1 + Sqrt[3] )*Log[-2*(1 - Sqrt[3]) + 4*x^3])/(4*3^(2/3)*(1 - Sqrt[3])) + ((1 - Sqrt[3] )*Log[-2*(1 + Sqrt[3]) + 4*x^3])/(4*3^(2/3)*(1 + Sqrt[3])) - (3^(1/3)*(1 + Sqrt[3])*Log[-(3^(1/6)*x) - (1 + x^3)^(1/3)])/(4*(1 - Sqrt[3])) - (3^(1/3 )*(1 - Sqrt[3])*Log[3^(1/6)*x - (1 + x^3)^(1/3)])/(4*(1 + Sqrt[3])) + 2*Lo g[-x + (1 + x^3)^(1/3)] - ((36 - 19*Sqrt[3])*Log[-x + (1 + x^3)^(1/3)])/36 - ((36 + 19*Sqrt[3])*Log[-x + (1 + x^3)^(1/3)])/36
3.30.8.3.1 Defintions of rubi rules used
Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[u*(d + e*x^n)^(p + q)*(a/d + (c/e)*x^n)^p, x] /; FreeQ[{a, c, d, e, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[c*d^2 + a*e^2, 0] && (Integer Q[p] || (GtQ[a, 0] && GtQ[d, 0]))
Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[ {v = RationalFunctionExpand[u/(a + b*x^n + c*x^(2*n)), x]}, Int[v, x] /; Su mQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
Time = 111.85 (sec) , antiderivative size = 272, normalized size of antiderivative = 0.84
| method | result | size |
| pseudoelliptic | \(\frac {-5 x^{5} \left (3^{\frac {1}{3}}+\frac {3^{\frac {5}{6}}}{2}\right ) \ln \left (\frac {-3^{\frac {1}{6}} \left (x^{3}+1\right )^{\frac {1}{3}} x +3^{\frac {1}{3}} x^{2}+\left (x^{3}+1\right )^{\frac {2}{3}}}{x^{2}}\right )-5 x^{5} \left (-\frac {3^{\frac {5}{6}}}{2}+3^{\frac {1}{3}}\right ) \ln \left (\frac {3^{\frac {1}{6}} \left (x^{3}+1\right )^{\frac {1}{3}} x +3^{\frac {1}{3}} x^{2}+\left (x^{3}+1\right )^{\frac {2}{3}}}{x^{2}}\right )+15 x^{5} \left (3^{\frac {1}{3}}+\frac {2 \,3^{\frac {5}{6}}}{3}\right ) \arctan \left (\frac {x \sqrt {3}-2 \,3^{\frac {1}{3}} \left (x^{3}+1\right )^{\frac {1}{3}}}{3 x}\right )-15 x^{5} \left (3^{\frac {1}{3}}-\frac {2 \,3^{\frac {5}{6}}}{3}\right ) \arctan \left (\frac {x \sqrt {3}+2 \,3^{\frac {1}{3}} \left (x^{3}+1\right )^{\frac {1}{3}}}{3 x}\right )+10 x^{5} \left (-\frac {3^{\frac {5}{6}}}{2}+3^{\frac {1}{3}}\right ) \ln \left (\frac {-3^{\frac {1}{6}} x +\left (x^{3}+1\right )^{\frac {1}{3}}}{x}\right )+10 x^{5} \left (3^{\frac {1}{3}}+\frac {3^{\frac {5}{6}}}{2}\right ) \ln \left (\frac {3^{\frac {1}{6}} x +\left (x^{3}+1\right )^{\frac {1}{3}}}{x}\right )+10 \,3^{\frac {1}{3}} \ln \left (3\right ) x^{5}+24 x^{3} \left (x^{3}+1\right )^{\frac {2}{3}}-6 \left (x^{3}+1\right )^{\frac {2}{3}}}{30 x^{5}}\) | \(272\) |
| trager | \(\text {Expression too large to display}\) | \(7505\) |
| risch | \(\text {Expression too large to display}\) | \(11950\) |
1/30*(-5*x^5*(3^(1/3)+1/2*3^(5/6))*ln((-3^(1/6)*(x^3+1)^(1/3)*x+3^(1/3)*x^ 2+(x^3+1)^(2/3))/x^2)-5*x^5*(-1/2*3^(5/6)+3^(1/3))*ln((3^(1/6)*(x^3+1)^(1/ 3)*x+3^(1/3)*x^2+(x^3+1)^(2/3))/x^2)+15*x^5*(3^(1/3)+2/3*3^(5/6))*arctan(1 /3*(x*3^(1/2)-2*3^(1/3)*(x^3+1)^(1/3))/x)-15*x^5*(3^(1/3)-2/3*3^(5/6))*arc tan(1/3*(x*3^(1/2)+2*3^(1/3)*(x^3+1)^(1/3))/x)+10*x^5*(-1/2*3^(5/6)+3^(1/3 ))*ln((-3^(1/6)*x+(x^3+1)^(1/3))/x)+10*x^5*(3^(1/3)+1/2*3^(5/6))*ln((3^(1/ 6)*x+(x^3+1)^(1/3))/x)+10*3^(1/3)*ln(3)*x^5+24*x^3*(x^3+1)^(2/3)-6*(x^3+1) ^(2/3))/x^5
Exception generated. \[ \int \frac {\left (1+x^3\right )^{2/3} \left (-1+x^6\right )}{x^6 \left (-1-2 x^3+2 x^6\right )} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (trace 0)
Timed out. \[ \int \frac {\left (1+x^3\right )^{2/3} \left (-1+x^6\right )}{x^6 \left (-1-2 x^3+2 x^6\right )} \, dx=\text {Timed out} \]
\[ \int \frac {\left (1+x^3\right )^{2/3} \left (-1+x^6\right )}{x^6 \left (-1-2 x^3+2 x^6\right )} \, dx=\int { \frac {{\left (x^{6} - 1\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{{\left (2 \, x^{6} - 2 \, x^{3} - 1\right )} x^{6}} \,d x } \]
\[ \int \frac {\left (1+x^3\right )^{2/3} \left (-1+x^6\right )}{x^6 \left (-1-2 x^3+2 x^6\right )} \, dx=\int { \frac {{\left (x^{6} - 1\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{{\left (2 \, x^{6} - 2 \, x^{3} - 1\right )} x^{6}} \,d x } \]
Timed out. \[ \int \frac {\left (1+x^3\right )^{2/3} \left (-1+x^6\right )}{x^6 \left (-1-2 x^3+2 x^6\right )} \, dx=-\int \frac {{\left (x^3+1\right )}^{2/3}\,\left (x^6-1\right )}{x^6\,\left (-2\,x^6+2\,x^3+1\right )} \,d x \]