Integrand size = 28, antiderivative size = 116 \[ \int \frac {2-x+x^2}{\sqrt [3]{-1+x^2} \left (3+4 x+x^2\right )} \, dx=\frac {3 \left (-1+x^2\right )^{2/3}}{1+x}-\frac {7}{4} \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{-1+x^2}}{1-x+\sqrt [3]{-1+x^2}}\right )-\frac {7}{4} \log \left (-1+x+2 \sqrt [3]{-1+x^2}\right )+\frac {7}{8} \log \left (1-2 x+x^2+(2-2 x) \sqrt [3]{-1+x^2}+4 \left (-1+x^2\right )^{2/3}\right ) \]
3*(x^2-1)^(2/3)/(1+x)-7/4*3^(1/2)*arctan(3^(1/2)*(x^2-1)^(1/3)/(1-x+(x^2-1 )^(1/3)))-7/4*ln(-1+x+2*(x^2-1)^(1/3))+7/8*ln(1-2*x+x^2+(2-2*x)*(x^2-1)^(1 /3)+4*(x^2-1)^(2/3))
Time = 0.28 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.00 \[ \int \frac {2-x+x^2}{\sqrt [3]{-1+x^2} \left (3+4 x+x^2\right )} \, dx=\frac {3 \left (-1+x^2\right )^{2/3}}{1+x}-\frac {7}{4} \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{-1+x^2}}{1-x+\sqrt [3]{-1+x^2}}\right )-\frac {7}{4} \log \left (-1+x+2 \sqrt [3]{-1+x^2}\right )+\frac {7}{8} \log \left (1-2 x+x^2+(2-2 x) \sqrt [3]{-1+x^2}+4 \left (-1+x^2\right )^{2/3}\right ) \]
(3*(-1 + x^2)^(2/3))/(1 + x) - (7*Sqrt[3]*ArcTan[(Sqrt[3]*(-1 + x^2)^(1/3) )/(1 - x + (-1 + x^2)^(1/3))])/4 - (7*Log[-1 + x + 2*(-1 + x^2)^(1/3)])/4 + (7*Log[1 - 2*x + x^2 + (2 - 2*x)*(-1 + x^2)^(1/3) + 4*(-1 + x^2)^(2/3)]) /8
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.
| \(\displaystyle \int \frac {x^2-x+2}{\sqrt [3]{x^2-1} \left (x^2+4 x+3\right )} \, dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \int \left (\frac {1}{\sqrt [3]{x^2-1}}-\frac {5 x+1}{\sqrt [3]{x^2-1} \left (x^2+4 x+3\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\int \frac {5 x+1}{\sqrt [3]{x^2-1} \left (x^2+4 x+3\right )}dx+\frac {\sqrt {2} 3^{3/4} \left (\sqrt [3]{x^2-1}+1\right ) \sqrt {\frac {\left (x^2-1\right )^{2/3}-\sqrt [3]{x^2-1}+1}{\left (\sqrt [3]{x^2-1}+\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{x^2-1}-\sqrt {3}+1}{\sqrt [3]{x^2-1}+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{\sqrt {\frac {\sqrt [3]{x^2-1}+1}{\left (\sqrt [3]{x^2-1}+\sqrt {3}+1\right )^2}} x}-\frac {3 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \left (\sqrt [3]{x^2-1}+1\right ) \sqrt {\frac {\left (x^2-1\right )^{2/3}-\sqrt [3]{x^2-1}+1}{\left (\sqrt [3]{x^2-1}+\sqrt {3}+1\right )^2}} E\left (\arcsin \left (\frac {\sqrt [3]{x^2-1}-\sqrt {3}+1}{\sqrt [3]{x^2-1}+\sqrt {3}+1}\right )|-7-4 \sqrt {3}\right )}{2 \sqrt {\frac {\sqrt [3]{x^2-1}+1}{\left (\sqrt [3]{x^2-1}+\sqrt {3}+1\right )^2}} x}+\frac {3 x}{\sqrt [3]{x^2-1}+\sqrt {3}+1}\) |
3.18.14.3.1 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 = 2.06 (sec) , antiderivative size = 396, normalized size of antiderivative = 3.41
| method | result | size |
| risch | \(\frac {-3+3 x}{\left (x^{2}-1\right )^{\frac {1}{3}}}-\frac {7 \ln \left (-\frac {448 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right )^{2} x^{2}+864 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) \left (x^{2}-1\right )^{\frac {2}{3}}-516 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) \left (x^{2}-1\right )^{\frac {1}{3}} x -1344 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right )^{2} x -468 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) x^{2}-948 \left (x^{2}-1\right )^{\frac {2}{3}}+516 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) \left (x^{2}-1\right )^{\frac {1}{3}}-216 x \left (x^{2}-1\right )^{\frac {1}{3}}+3000 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) x +119 x^{2}+216 \left (x^{2}-1\right )^{\frac {1}{3}}+1596 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right )-1326 x -969}{\left (3+x \right )^{2}}\right )}{4}+\frac {7 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) \ln \left (\frac {272 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right )^{2} x^{2}+432 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) \left (x^{2}-1\right )^{\frac {2}{3}}+474 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) \left (x^{2}-1\right )^{\frac {1}{3}} x -816 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right )^{2} x -129 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) x^{2}+258 \left (x^{2}-1\right )^{\frac {2}{3}}-474 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) \left (x^{2}-1\right )^{\frac {1}{3}}-108 x \left (x^{2}-1\right )^{\frac {1}{3}}-582 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) x +7 x^{2}+108 \left (x^{2}-1\right )^{\frac {1}{3}}-969 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right )+378 x +399}{\left (3+x \right )^{2}}\right )}{2}\) | \(396\) |
| trager | \(\frac {3 \left (x^{2}-1\right )^{\frac {2}{3}}}{1+x}+\frac {7 \ln \left (\frac {-864 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )^{2} x^{2}+2592 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) \left (x^{2}-1\right )^{\frac {2}{3}}-1296 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) \left (x^{2}-1\right )^{\frac {1}{3}} x +2592 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )^{2} x +834 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) x^{2}+516 \left (x^{2}-1\right )^{\frac {2}{3}}+1296 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) \left (x^{2}-1\right )^{\frac {1}{3}}-258 x \left (x^{2}-1\right )^{\frac {1}{3}}-1476 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) x -17 x^{2}+258 \left (x^{2}-1\right )^{\frac {1}{3}}+1026 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )-918 x -969}{\left (3+x \right )^{2}}\right )}{4}-\frac {21 \ln \left (\frac {-864 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )^{2} x^{2}+2592 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) \left (x^{2}-1\right )^{\frac {2}{3}}-1296 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) \left (x^{2}-1\right )^{\frac {1}{3}} x +2592 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )^{2} x +834 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) x^{2}+516 \left (x^{2}-1\right )^{\frac {2}{3}}+1296 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) \left (x^{2}-1\right )^{\frac {1}{3}}-258 x \left (x^{2}-1\right )^{\frac {1}{3}}-1476 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) x -17 x^{2}+258 \left (x^{2}-1\right )^{\frac {1}{3}}+1026 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )-918 x -969}{\left (3+x \right )^{2}}\right ) \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )}{2}+\frac {21 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) \ln \left (-\frac {432 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )^{2} x^{2}+1296 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) \left (x^{2}-1\right )^{\frac {2}{3}}-648 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) \left (x^{2}-1\right )^{\frac {1}{3}} x -1296 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )^{2} x +273 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) x^{2}-474 \left (x^{2}-1\right )^{\frac {2}{3}}+648 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) \left (x^{2}-1\right )^{\frac {1}{3}}+237 x \left (x^{2}-1\right )^{\frac {1}{3}}-306 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) x -49 x^{2}-237 \left (x^{2}-1\right )^{\frac {1}{3}}+513 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )+546 x +399}{\left (3+x \right )^{2}}\right )}{2}\) | \(594\) |
3*(-1+x)/(x^2-1)^(1/3)-7/4*ln(-(448*RootOf(4*_Z^2-2*_Z+1)^2*x^2+864*RootOf (4*_Z^2-2*_Z+1)*(x^2-1)^(2/3)-516*RootOf(4*_Z^2-2*_Z+1)*(x^2-1)^(1/3)*x-13 44*RootOf(4*_Z^2-2*_Z+1)^2*x-468*RootOf(4*_Z^2-2*_Z+1)*x^2-948*(x^2-1)^(2/ 3)+516*RootOf(4*_Z^2-2*_Z+1)*(x^2-1)^(1/3)-216*x*(x^2-1)^(1/3)+3000*RootOf (4*_Z^2-2*_Z+1)*x+119*x^2+216*(x^2-1)^(1/3)+1596*RootOf(4*_Z^2-2*_Z+1)-132 6*x-969)/(3+x)^2)+7/2*RootOf(4*_Z^2-2*_Z+1)*ln((272*RootOf(4*_Z^2-2*_Z+1)^ 2*x^2+432*RootOf(4*_Z^2-2*_Z+1)*(x^2-1)^(2/3)+474*RootOf(4*_Z^2-2*_Z+1)*(x ^2-1)^(1/3)*x-816*RootOf(4*_Z^2-2*_Z+1)^2*x-129*RootOf(4*_Z^2-2*_Z+1)*x^2+ 258*(x^2-1)^(2/3)-474*RootOf(4*_Z^2-2*_Z+1)*(x^2-1)^(1/3)-108*x*(x^2-1)^(1 /3)-582*RootOf(4*_Z^2-2*_Z+1)*x+7*x^2+108*(x^2-1)^(1/3)-969*RootOf(4*_Z^2- 2*_Z+1)+378*x+399)/(3+x)^2)
Time = 0.53 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.09 \[ \int \frac {2-x+x^2}{\sqrt [3]{-1+x^2} \left (3+4 x+x^2\right )} \, dx=\frac {14 \, \sqrt {3} {\left (x + 1\right )} \arctan \left (\frac {286273 \, \sqrt {3} {\left (x^{2} - 1\right )}^{\frac {1}{3}} {\left (x - 1\right )} + \sqrt {3} {\left (66978 \, x^{2} + 434719 \, x + 635653\right )} + 539695 \, \sqrt {3} {\left (x^{2} - 1\right )}^{\frac {2}{3}}}{226981 \, x^{2} - 1974837 \, x - 1293894}\right ) - 7 \, {\left (x + 1\right )} \log \left (\frac {x^{2} + 6 \, {\left (x^{2} - 1\right )}^{\frac {1}{3}} {\left (x - 1\right )} + 6 \, x + 12 \, {\left (x^{2} - 1\right )}^{\frac {2}{3}} + 9}{x^{2} + 6 \, x + 9}\right ) + 24 \, {\left (x^{2} - 1\right )}^{\frac {2}{3}}}{8 \, {\left (x + 1\right )}} \]
1/8*(14*sqrt(3)*(x + 1)*arctan((286273*sqrt(3)*(x^2 - 1)^(1/3)*(x - 1) + s qrt(3)*(66978*x^2 + 434719*x + 635653) + 539695*sqrt(3)*(x^2 - 1)^(2/3))/( 226981*x^2 - 1974837*x - 1293894)) - 7*(x + 1)*log((x^2 + 6*(x^2 - 1)^(1/3 )*(x - 1) + 6*x + 12*(x^2 - 1)^(2/3) + 9)/(x^2 + 6*x + 9)) + 24*(x^2 - 1)^ (2/3))/(x + 1)
\[ \int \frac {2-x+x^2}{\sqrt [3]{-1+x^2} \left (3+4 x+x^2\right )} \, dx=\int \frac {x^{2} - x + 2}{\sqrt [3]{\left (x - 1\right ) \left (x + 1\right )} \left (x + 1\right ) \left (x + 3\right )}\, dx \]
\[ \int \frac {2-x+x^2}{\sqrt [3]{-1+x^2} \left (3+4 x+x^2\right )} \, dx=\int { \frac {x^{2} - x + 2}{{\left (x^{2} + 4 \, x + 3\right )} {\left (x^{2} - 1\right )}^{\frac {1}{3}}} \,d x } \]
\[ \int \frac {2-x+x^2}{\sqrt [3]{-1+x^2} \left (3+4 x+x^2\right )} \, dx=\int { \frac {x^{2} - x + 2}{{\left (x^{2} + 4 \, x + 3\right )} {\left (x^{2} - 1\right )}^{\frac {1}{3}}} \,d x } \]
Timed out. \[ \int \frac {2-x+x^2}{\sqrt [3]{-1+x^2} \left (3+4 x+x^2\right )} \, dx=\int \frac {x^2-x+2}{{\left (x^2-1\right )}^{1/3}\,\left (x^2+4\,x+3\right )} \,d x \]