Integrand size = 26, antiderivative size = 92 \[ \int \frac {(-1+x) (3+x)}{\left (-1+x^2\right )^{2/3} \left (2-x+x^2\right )} \, dx=-\sqrt {3} \arctan \left (\frac {\sqrt {3} \left (-1+x^2\right )^{2/3}}{-2-2 x+\left (-1+x^2\right )^{2/3}}\right )-\log \left (1+x+\left (-1+x^2\right )^{2/3}\right )+\frac {1}{2} \log \left (1+2 x+x^2+(-1-x) \left (-1+x^2\right )^{2/3}+\left (-1+x^2\right )^{4/3}\right ) \]
-3^(1/2)*arctan(3^(1/2)*(x^2-1)^(2/3)/(-2-2*x+(x^2-1)^(2/3)))-ln(1+x+(x^2- 1)^(2/3))+1/2*ln(1+2*x+x^2+(-1-x)*(x^2-1)^(2/3)+(x^2-1)^(4/3))
Time = 0.13 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.00 \[ \int \frac {(-1+x) (3+x)}{\left (-1+x^2\right )^{2/3} \left (2-x+x^2\right )} \, dx=-\sqrt {3} \arctan \left (\frac {\sqrt {3} \left (-1+x^2\right )^{2/3}}{-2-2 x+\left (-1+x^2\right )^{2/3}}\right )-\log \left (1+x+\left (-1+x^2\right )^{2/3}\right )+\frac {1}{2} \log \left (1+2 x+x^2+(-1-x) \left (-1+x^2\right )^{2/3}+\left (-1+x^2\right )^{4/3}\right ) \]
-(Sqrt[3]*ArcTan[(Sqrt[3]*(-1 + x^2)^(2/3))/(-2 - 2*x + (-1 + x^2)^(2/3))] ) - Log[1 + x + (-1 + x^2)^(2/3)] + Log[1 + 2*x + x^2 + (-1 - x)*(-1 + x^2 )^(2/3) + (-1 + x^2)^(4/3)]/2
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.
| \(\displaystyle \int \frac {(x-1) (x+3)}{\left (x^2-1\right )^{2/3} \left (x^2-x+2\right )} \, dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \int \left (\frac {1}{\left (x^2-1\right )^{2/3}}-\frac {5-3 x}{\left (x^2-1\right )^{2/3} \left (x^2-x+2\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3^{3/4} \sqrt {2+\sqrt {3}} \left (\sqrt [3]{x^2-1}+1\right ) \sqrt {\frac {\left (x^2-1\right )^{2/3}-\sqrt [3]{x^2-1}+1}{\left (\sqrt [3]{x^2-1}+\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{x^2-1}-\sqrt {3}+1}{\sqrt [3]{x^2-1}+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{x \sqrt {\frac {\sqrt [3]{x^2-1}+1}{\left (\sqrt [3]{x^2-1}+\sqrt {3}+1\right )^2}}}-\int \frac {5-3 x}{\left (x^2-1\right )^{2/3} \left (x^2-x+2\right )}dx\) |
3.13.60.3.1 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 = 6.19 (sec) , antiderivative size = 362, normalized size of antiderivative = 3.93
| method | result | size |
| trager | \(\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \ln \left (\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{2}-6 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x -3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}+7 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x -3 \left (x^{2}-1\right )^{\frac {2}{3}}+3 x \left (x^{2}-1\right )^{\frac {1}{3}}+x^{2}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-3 \left (x^{2}-1\right )^{\frac {1}{3}}-2 x +1}{x^{2}-x +2}\right )-\ln \left (\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{2}-6 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x -\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}+5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x -3 \left (x^{2}-1\right )^{\frac {2}{3}}+3 x \left (x^{2}-1\right )^{\frac {1}{3}}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-3 \left (x^{2}-1\right )^{\frac {1}{3}}-x -1}{x^{2}-x +2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+\ln \left (\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{2}-6 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x -\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}+5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x -3 \left (x^{2}-1\right )^{\frac {2}{3}}+3 x \left (x^{2}-1\right )^{\frac {1}{3}}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-3 \left (x^{2}-1\right )^{\frac {1}{3}}-x -1}{x^{2}-x +2}\right )\) | \(362\) |
RootOf(_Z^2-_Z+1)*ln((2*RootOf(_Z^2-_Z+1)^2*x^2-6*RootOf(_Z^2-_Z+1)^2*x-3* RootOf(_Z^2-_Z+1)*x^2+7*RootOf(_Z^2-_Z+1)*x-3*(x^2-1)^(2/3)+3*x*(x^2-1)^(1 /3)+x^2-2*RootOf(_Z^2-_Z+1)-3*(x^2-1)^(1/3)-2*x+1)/(x^2-x+2))-ln((2*RootOf (_Z^2-_Z+1)^2*x^2-6*RootOf(_Z^2-_Z+1)^2*x-RootOf(_Z^2-_Z+1)*x^2+5*RootOf(_ Z^2-_Z+1)*x-3*(x^2-1)^(2/3)+3*x*(x^2-1)^(1/3)+2*RootOf(_Z^2-_Z+1)-3*(x^2-1 )^(1/3)-x-1)/(x^2-x+2))*RootOf(_Z^2-_Z+1)+ln((2*RootOf(_Z^2-_Z+1)^2*x^2-6* RootOf(_Z^2-_Z+1)^2*x-RootOf(_Z^2-_Z+1)*x^2+5*RootOf(_Z^2-_Z+1)*x-3*(x^2-1 )^(2/3)+3*x*(x^2-1)^(1/3)+2*RootOf(_Z^2-_Z+1)-3*(x^2-1)^(1/3)-x-1)/(x^2-x+ 2))
Time = 0.54 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.07 \[ \int \frac {(-1+x) (3+x)}{\left (-1+x^2\right )^{2/3} \left (2-x+x^2\right )} \, dx=-\sqrt {3} \arctan \left (\frac {4 \, \sqrt {3} {\left (x^{2} - 1\right )}^{\frac {1}{3}} {\left (x - 1\right )} + \sqrt {3} {\left (x + 1\right )} + 2 \, \sqrt {3} {\left (x^{2} - 1\right )}^{\frac {2}{3}}}{8 \, x^{2} - 17 \, x + 7}\right ) - \frac {1}{2} \, \log \left (\frac {x^{2} + 3 \, {\left (x^{2} - 1\right )}^{\frac {1}{3}} {\left (x - 1\right )} - x + 3 \, {\left (x^{2} - 1\right )}^{\frac {2}{3}} + 2}{x^{2} - x + 2}\right ) \]
-sqrt(3)*arctan((4*sqrt(3)*(x^2 - 1)^(1/3)*(x - 1) + sqrt(3)*(x + 1) + 2*s qrt(3)*(x^2 - 1)^(2/3))/(8*x^2 - 17*x + 7)) - 1/2*log((x^2 + 3*(x^2 - 1)^( 1/3)*(x - 1) - x + 3*(x^2 - 1)^(2/3) + 2)/(x^2 - x + 2))
\[ \int \frac {(-1+x) (3+x)}{\left (-1+x^2\right )^{2/3} \left (2-x+x^2\right )} \, dx=\int \frac {\left (x - 1\right ) \left (x + 3\right )}{\left (\left (x - 1\right ) \left (x + 1\right )\right )^{\frac {2}{3}} \left (x^{2} - x + 2\right )}\, dx \]
\[ \int \frac {(-1+x) (3+x)}{\left (-1+x^2\right )^{2/3} \left (2-x+x^2\right )} \, dx=\int { \frac {{\left (x + 3\right )} {\left (x - 1\right )}}{{\left (x^{2} - x + 2\right )} {\left (x^{2} - 1\right )}^{\frac {2}{3}}} \,d x } \]
\[ \int \frac {(-1+x) (3+x)}{\left (-1+x^2\right )^{2/3} \left (2-x+x^2\right )} \, dx=\int { \frac {{\left (x + 3\right )} {\left (x - 1\right )}}{{\left (x^{2} - x + 2\right )} {\left (x^{2} - 1\right )}^{\frac {2}{3}}} \,d x } \]
Timed out. \[ \int \frac {(-1+x) (3+x)}{\left (-1+x^2\right )^{2/3} \left (2-x+x^2\right )} \, dx=\int \frac {\left (x-1\right )\,\left (x+3\right )}{{\left (x^2-1\right )}^{2/3}\,\left (x^2-x+2\right )} \,d x \]