Integrand size = 23, antiderivative size = 210 \[ \int \frac {1+x^2}{\left (-1+x+x^2\right ) \sqrt [3]{-1+x^6}} \, dx=-\frac {\arctan \left (\frac {\sqrt {3} \sqrt [3]{-1+x^6}}{-2^{2/3}-2^{2/3} x+2^{2/3} x^2+\sqrt [3]{-1+x^6}}\right )}{2^{2/3} \sqrt {3}}-\frac {\log \left (-2^{2/3}-2^{2/3} x+2^{2/3} x^2-2 \sqrt [3]{-1+x^6}\right )}{3\ 2^{2/3}}+\frac {\log \left (\sqrt [3]{2}+2 \sqrt [3]{2} x-\sqrt [3]{2} x^2-2 \sqrt [3]{2} x^3+\sqrt [3]{2} x^4+\left (-2^{2/3}-2^{2/3} x+2^{2/3} x^2\right ) \sqrt [3]{-1+x^6}+2 \left (-1+x^6\right )^{2/3}\right )}{6\ 2^{2/3}} \]
-1/6*arctan(3^(1/2)*(x^6-1)^(1/3)/(-2^(2/3)-2^(2/3)*x+2^(2/3)*x^2+(x^6-1)^ (1/3)))*2^(1/3)*3^(1/2)-1/6*ln(-2^(2/3)-2^(2/3)*x+2^(2/3)*x^2-2*(x^6-1)^(1 /3))*2^(1/3)+1/12*ln(2^(1/3)+2*2^(1/3)*x-2^(1/3)*x^2-2*2^(1/3)*x^3+2^(1/3) *x^4+(-2^(2/3)-2^(2/3)*x+2^(2/3)*x^2)*(x^6-1)^(1/3)+2*(x^6-1)^(2/3))*2^(1/ 3)
Time = 4.51 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.89 \[ \int \frac {1+x^2}{\left (-1+x+x^2\right ) \sqrt [3]{-1+x^6}} \, dx=\frac {2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{-1+x^6}}{2^{2/3}+2^{2/3} x-2^{2/3} x^2-\sqrt [3]{-1+x^6}}\right )-2 \log \left (-2^{2/3}-2^{2/3} x+2^{2/3} x^2-2 \sqrt [3]{-1+x^6}\right )+\log \left (\sqrt [3]{2}+2 \sqrt [3]{2} x-\sqrt [3]{2} x^2-2 \sqrt [3]{2} x^3+\sqrt [3]{2} x^4+2^{2/3} \left (-1-x+x^2\right ) \sqrt [3]{-1+x^6}+2 \left (-1+x^6\right )^{2/3}\right )}{6\ 2^{2/3}} \]
(2*Sqrt[3]*ArcTan[(Sqrt[3]*(-1 + x^6)^(1/3))/(2^(2/3) + 2^(2/3)*x - 2^(2/3 )*x^2 - (-1 + x^6)^(1/3))] - 2*Log[-2^(2/3) - 2^(2/3)*x + 2^(2/3)*x^2 - 2* (-1 + x^6)^(1/3)] + Log[2^(1/3) + 2*2^(1/3)*x - 2^(1/3)*x^2 - 2*2^(1/3)*x^ 3 + 2^(1/3)*x^4 + 2^(2/3)*(-1 - x + x^2)*(-1 + x^6)^(1/3) + 2*(-1 + x^6)^( 2/3)])/(6*2^(2/3))
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.
\(\displaystyle \int \frac {x^2+1}{\left (x^2+x-1\right ) \sqrt [3]{x^6-1}} \, dx\) |
\(\Big \downarrow \) 7279 |
\(\displaystyle \int \left (\frac {1}{\sqrt [3]{x^6-1}}+\frac {2-x}{\left (x^2+x-1\right ) \sqrt [3]{x^6-1}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\left (1-\sqrt {5}\right ) \int \frac {1}{\left (2 x-\sqrt {5}+1\right ) \sqrt [3]{x^6-1}}dx-\left (1+\sqrt {5}\right ) \int \frac {1}{\left (2 x+\sqrt {5}+1\right ) \sqrt [3]{x^6-1}}dx+\frac {x \sqrt [3]{1-x^6} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{3},\frac {7}{6},x^6\right )}{\sqrt [3]{x^6-1}}\) |
3.26.17.3.1 Defintions of rubi rules used
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]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 17.69 (sec) , antiderivative size = 2714, normalized size of antiderivative = 12.92
-1/6*ln(-(60*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)^2*RootOf( _Z^3+2)^2*x^5+6*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*RootOf (_Z^3+2)^3*x^5-100*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)^2*R ootOf(_Z^3+2)^2*x^3-10*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2) *RootOf(_Z^3+2)^3*x^3+60*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^ 2)^2*RootOf(_Z^3+2)^2*x+6*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z ^2)*RootOf(_Z^3+2)^3*x+70*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z ^2)+18*x^2*(x^6-1)^(2/3)-48*RootOf(_Z^3+2)^2*RootOf(RootOf(_Z^3+2)^2+2*_Z* RootOf(_Z^3+2)+4*_Z^2)*(x^6-1)^(2/3)*x^2-5*RootOf(_Z^3+2)*x^3+3*RootOf(_Z^ 3+2)*x-50*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*x^3+30*RootO f(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*x-7*RootOf(_Z^3+2)*x^6+3*Ro otOf(_Z^3+2)*x^5-70*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*x^ 6+30*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*x^5-18*x*(x^6-1)^ (2/3)-18*(x^6-1)^(2/3)+7*RootOf(_Z^3+2)+9*RootOf(_Z^3+2)^2*(x^6-1)^(1/3)+4 8*RootOf(_Z^3+2)*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*(x^6- 1)^(1/3)*x^4+48*RootOf(_Z^3+2)^2*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+ 2)+4*_Z^2)*(x^6-1)^(2/3)*x-96*RootOf(_Z^3+2)*RootOf(RootOf(_Z^3+2)^2+2*_Z* RootOf(_Z^3+2)+4*_Z^2)*(x^6-1)^(1/3)*x^3-48*RootOf(_Z^3+2)*RootOf(RootOf(_ Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*(x^6-1)^(1/3)*x^2+96*RootOf(_Z^3+2)*R ootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*(x^6-1)^(1/3)*x+9*Ro...
Time = 11.98 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.20 \[ \int \frac {1+x^2}{\left (-1+x+x^2\right ) \sqrt [3]{-1+x^6}} \, dx=-\frac {1}{6} \cdot 4^{\frac {1}{6}} \sqrt {3} \left (-1\right )^{\frac {1}{3}} \arctan \left (\frac {4^{\frac {1}{6}} \sqrt {3} {\left (2 \cdot 4^{\frac {2}{3}} \left (-1\right )^{\frac {2}{3}} {\left (x^{6} - 1\right )}^{\frac {2}{3}} {\left (x^{2} - x - 1\right )} + 4 \, \left (-1\right )^{\frac {1}{3}} {\left (x^{6} - 1\right )}^{\frac {1}{3}} {\left (x^{4} - 2 \, x^{3} - x^{2} + 2 \, x + 1\right )} - 4^{\frac {1}{3}} {\left (x^{6} + 3 \, x^{5} - 5 \, x^{3} + 3 \, x - 1\right )}\right )}}{6 \, {\left (3 \, x^{6} - 3 \, x^{5} + 5 \, x^{3} - 3 \, x - 3\right )}}\right ) - \frac {1}{24} \cdot 4^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} \log \left (\frac {4^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (x^{4} - 2 \, x^{3} - x^{2} + 2 \, x + 1\right )} - 4^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (x^{6} - 1\right )}^{\frac {2}{3}} + 2 \, {\left (x^{6} - 1\right )}^{\frac {1}{3}} {\left (x^{2} - x - 1\right )}}{x^{4} + 2 \, x^{3} - x^{2} - 2 \, x + 1}\right ) + \frac {1}{12} \cdot 4^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} \log \left (\frac {4^{\frac {1}{3}} {\left (x^{2} - x - 1\right )} + 2 \, \left (-1\right )^{\frac {1}{3}} {\left (x^{6} - 1\right )}^{\frac {1}{3}}}{x^{2} + x - 1}\right ) \]
-1/6*4^(1/6)*sqrt(3)*(-1)^(1/3)*arctan(1/6*4^(1/6)*sqrt(3)*(2*4^(2/3)*(-1) ^(2/3)*(x^6 - 1)^(2/3)*(x^2 - x - 1) + 4*(-1)^(1/3)*(x^6 - 1)^(1/3)*(x^4 - 2*x^3 - x^2 + 2*x + 1) - 4^(1/3)*(x^6 + 3*x^5 - 5*x^3 + 3*x - 1))/(3*x^6 - 3*x^5 + 5*x^3 - 3*x - 3)) - 1/24*4^(2/3)*(-1)^(1/3)*log((4^(1/3)*(-1)^(2 /3)*(x^4 - 2*x^3 - x^2 + 2*x + 1) - 4^(2/3)*(-1)^(1/3)*(x^6 - 1)^(2/3) + 2 *(x^6 - 1)^(1/3)*(x^2 - x - 1))/(x^4 + 2*x^3 - x^2 - 2*x + 1)) + 1/12*4^(2 /3)*(-1)^(1/3)*log((4^(1/3)*(x^2 - x - 1) + 2*(-1)^(1/3)*(x^6 - 1)^(1/3))/ (x^2 + x - 1))
\[ \int \frac {1+x^2}{\left (-1+x+x^2\right ) \sqrt [3]{-1+x^6}} \, dx=\int \frac {x^{2} + 1}{\sqrt [3]{\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )} \left (x^{2} + x - 1\right )}\, dx \]
Integral((x**2 + 1)/(((x - 1)*(x + 1)*(x**2 - x + 1)*(x**2 + x + 1))**(1/3 )*(x**2 + x - 1)), x)
\[ \int \frac {1+x^2}{\left (-1+x+x^2\right ) \sqrt [3]{-1+x^6}} \, dx=\int { \frac {x^{2} + 1}{{\left (x^{6} - 1\right )}^{\frac {1}{3}} {\left (x^{2} + x - 1\right )}} \,d x } \]
\[ \int \frac {1+x^2}{\left (-1+x+x^2\right ) \sqrt [3]{-1+x^6}} \, dx=\int { \frac {x^{2} + 1}{{\left (x^{6} - 1\right )}^{\frac {1}{3}} {\left (x^{2} + x - 1\right )}} \,d x } \]
Timed out. \[ \int \frac {1+x^2}{\left (-1+x+x^2\right ) \sqrt [3]{-1+x^6}} \, dx=\int \frac {x^2+1}{{\left (x^6-1\right )}^{1/3}\,\left (x^2+x-1\right )} \,d x \]