Integrand size = 36, antiderivative size = 254 \[ \int \frac {\left (-1+x^2\right ) \sqrt {1-x-x^2+x^3+x^4}}{x^2 \left (1+x^2\right )} \, dx=\frac {\sqrt {1-x-x^2+x^3+x^4}}{x}+\frac {\log (x)}{2}-\frac {1}{2} \log \left (2-x-2 x^2+2 \sqrt {1-x-x^2+x^3+x^4}\right )+\text {RootSum}\left [9+12 \text {$\#$1}+62 \text {$\#$1}^2-4 \text {$\#$1}^3+\text {$\#$1}^4\&,\frac {9 \log (x)-9 \log \left (2-x-2 x^2+2 \sqrt {1-x-x^2+x^3+x^4}+x \text {$\#$1}\right )-10 \log (x) \text {$\#$1}+10 \log \left (2-x-2 x^2+2 \sqrt {1-x-x^2+x^3+x^4}+x \text {$\#$1}\right ) \text {$\#$1}-3 \log (x) \text {$\#$1}^2+3 \log \left (2-x-2 x^2+2 \sqrt {1-x-x^2+x^3+x^4}+x \text {$\#$1}\right ) \text {$\#$1}^2}{3+31 \text {$\#$1}-3 \text {$\#$1}^2+\text {$\#$1}^3}\&\right ] \]
Time = 0.00 (sec) , antiderivative size = 229, normalized size of antiderivative = 0.90 \[ \int \frac {\left (-1+x^2\right ) \sqrt {1-x-x^2+x^3+x^4}}{x^2 \left (1+x^2\right )} \, dx=\frac {1}{2} \left (\frac {2 \sqrt {1-x-x^2+x^3+x^4}}{x}+\log (x)-\log \left (2-x-2 x^2+2 \sqrt {1-x-x^2+x^3+x^4}\right )-\text {RootSum}\left [5-16 \text {$\#$1}+14 \text {$\#$1}^2+\text {$\#$1}^4\&,\frac {-\log (x)+\log \left (1-x^2+\sqrt {1-x-x^2+x^3+x^4}-x \text {$\#$1}\right )+8 \log (x) \text {$\#$1}-8 \log \left (1-x^2+\sqrt {1-x-x^2+x^3+x^4}-x \text {$\#$1}\right ) \text {$\#$1}-3 \log (x) \text {$\#$1}^2+3 \log \left (1-x^2+\sqrt {1-x-x^2+x^3+x^4}-x \text {$\#$1}\right ) \text {$\#$1}^2}{-4+7 \text {$\#$1}+\text {$\#$1}^3}\&\right ]\right ) \]
((2*Sqrt[1 - x - x^2 + x^3 + x^4])/x + Log[x] - Log[2 - x - 2*x^2 + 2*Sqrt [1 - x - x^2 + x^3 + x^4]] - RootSum[5 - 16*#1 + 14*#1^2 + #1^4 & , (-Log[ x] + Log[1 - x^2 + Sqrt[1 - x - x^2 + x^3 + x^4] - x*#1] + 8*Log[x]*#1 - 8 *Log[1 - x^2 + Sqrt[1 - x - x^2 + x^3 + x^4] - x*#1]*#1 - 3*Log[x]*#1^2 + 3*Log[1 - x^2 + Sqrt[1 - x - x^2 + x^3 + x^4] - x*#1]*#1^2)/(-4 + 7*#1 + # 1^3) & ])/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.
| \(\displaystyle \int \frac {\left (x^2-1\right ) \sqrt {x^4+x^3-x^2-x+1}}{x^2 \left (x^2+1\right )} \, dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \int \left (\frac {2 \sqrt {x^4+x^3-x^2-x+1}}{x^2+1}-\frac {\sqrt {x^4+x^3-x^2-x+1}}{x^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle i \int \frac {\sqrt {x^4+x^3-x^2-x+1}}{i-x}dx-\int \frac {\sqrt {x^4+x^3-x^2-x+1}}{x^2}dx+i \int \frac {\sqrt {x^4+x^3-x^2-x+1}}{x+i}dx\) |
3.28.40.3.1 Defintions of rubi rules used
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
Time = 4.09 (sec) , antiderivative size = 378, normalized size of antiderivative = 1.49
| method | result | size |
| risch | \(\frac {\sqrt {x^{4}+x^{3}-x^{2}-x +1}}{x}+\frac {-4 \left (\left (x^{2}-3 x -1\right ) \sqrt {13}+13 x \right ) \sqrt {\frac {x^{4}+x^{3}-x^{2}-x +1}{\left (\sqrt {13}\, x +x^{2}-3 x -1\right )^{2}}}\, \operatorname {arctanh}\left (\frac {4 \sqrt {\frac {x^{4}+x^{3}-x^{2}-x +1}{\left (\sqrt {13}\, x +x^{2}-3 x -1\right )^{2}}}}{\sqrt {6+2 \sqrt {13}}}\right )+\sqrt {78+26 \sqrt {13}}\, \left (\frac {\sqrt {-2+\sqrt {13}}\, \arctan \left (\frac {35 \sqrt {-2+\sqrt {13}}\, \sqrt {13}\, \left (\sqrt {13}\, x +x^{2}-3 x -1\right ) \sqrt {\frac {x^{4}+x^{3}-x^{2}-x +1}{\left (\sqrt {13}\, x +x^{2}-3 x -1\right )^{2}}}\, \left (\sqrt {13}+\frac {19}{5}\right ) \left (\sqrt {13}-\frac {26}{7}\right ) \left (-\sqrt {13}\, x +x^{2}-3 x -1\right )}{468 \left (x^{4}+x^{3}-x^{2}-x +1\right )}\right ) \left (\left (x^{2}+2 x -1\right ) \sqrt {13}+5 x^{2}-2 x -5\right ) \sqrt {\frac {x^{4}+x^{3}-x^{2}-x +1}{\left (\sqrt {13}\, x +x^{2}-3 x -1\right )^{2}}}}{3}+\operatorname {arcsinh}\left (\frac {\sqrt {3}\, \left (2 x^{2}+x -2\right )}{3 x}\right ) \sqrt {\frac {x^{4}+x^{3}-x^{2}-x +1}{x^{2}}}\, x \right )}{2 \sqrt {78+26 \sqrt {13}}\, \sqrt {\frac {x^{4}+x^{3}-x^{2}-x +1}{x^{2}}}\, x}\) | \(378\) |
| default | \(\frac {104 \left (\left (x^{2}-\frac {13}{2} x -1\right ) \sqrt {13}-\frac {7 x^{2}}{2}+\frac {47 x}{2}+\frac {7}{2}\right ) \sqrt {\frac {x^{4}+x^{3}-x^{2}-x +1}{\left (\sqrt {13}\, x +x^{2}-3 x -1\right )^{2}}}\, \operatorname {arctanh}\left (\frac {4 \sqrt {\frac {x^{4}+x^{3}-x^{2}-x +1}{\left (\sqrt {13}\, x +x^{2}-3 x -1\right )^{2}}}}{\sqrt {6+2 \sqrt {13}}}\right )+\sqrt {78+26 \sqrt {13}}\, \left (\sqrt {-2+\sqrt {13}}\, \arctan \left (\frac {35 \sqrt {-2+\sqrt {13}}\, \sqrt {13}\, \left (\sqrt {13}\, x +x^{2}-3 x -1\right ) \sqrt {\frac {x^{4}+x^{3}-x^{2}-x +1}{\left (\sqrt {13}\, x +x^{2}-3 x -1\right )^{2}}}\, \left (\sqrt {13}+\frac {19}{5}\right ) \left (\sqrt {13}-\frac {26}{7}\right ) \left (-\sqrt {13}\, x +x^{2}-3 x -1\right )}{468 \left (x^{4}+x^{3}-x^{2}-x +1\right )}\right ) \left (\left (3 x^{2}-22 x -3\right ) \sqrt {13}-13 x^{2}+78 x +13\right ) \sqrt {\frac {x^{4}+x^{3}-x^{2}-x +1}{\left (\sqrt {13}\, x +x^{2}-3 x -1\right )^{2}}}+7 \sqrt {\frac {x^{4}+x^{3}-x^{2}-x +1}{x^{2}}}\, \left (\operatorname {arcsinh}\left (\frac {\sqrt {3}\, \left (2 x^{2}+x -2\right )}{3 x}\right ) x +2 \sqrt {x^{4}+x^{3}-x^{2}-x +1}\right ) \left (\sqrt {13}-\frac {26}{7}\right )\right )}{2 \sqrt {\frac {x^{4}+x^{3}-x^{2}-x +1}{x^{2}}}\, \sqrt {78+26 \sqrt {13}}\, x \left (7 \sqrt {13}-26\right )}\) | \(399\) |
| pseudoelliptic | \(\frac {104 \left (\left (x^{2}-\frac {13}{2} x -1\right ) \sqrt {13}-\frac {7 x^{2}}{2}+\frac {47 x}{2}+\frac {7}{2}\right ) \sqrt {\frac {x^{4}+x^{3}-x^{2}-x +1}{\left (\sqrt {13}\, x +x^{2}-3 x -1\right )^{2}}}\, \operatorname {arctanh}\left (\frac {4 \sqrt {\frac {x^{4}+x^{3}-x^{2}-x +1}{\left (\sqrt {13}\, x +x^{2}-3 x -1\right )^{2}}}}{\sqrt {6+2 \sqrt {13}}}\right )+\sqrt {78+26 \sqrt {13}}\, \left (\sqrt {-2+\sqrt {13}}\, \arctan \left (\frac {35 \sqrt {-2+\sqrt {13}}\, \sqrt {13}\, \left (\sqrt {13}\, x +x^{2}-3 x -1\right ) \sqrt {\frac {x^{4}+x^{3}-x^{2}-x +1}{\left (\sqrt {13}\, x +x^{2}-3 x -1\right )^{2}}}\, \left (\sqrt {13}+\frac {19}{5}\right ) \left (\sqrt {13}-\frac {26}{7}\right ) \left (-\sqrt {13}\, x +x^{2}-3 x -1\right )}{468 \left (x^{4}+x^{3}-x^{2}-x +1\right )}\right ) \left (\left (3 x^{2}-22 x -3\right ) \sqrt {13}-13 x^{2}+78 x +13\right ) \sqrt {\frac {x^{4}+x^{3}-x^{2}-x +1}{\left (\sqrt {13}\, x +x^{2}-3 x -1\right )^{2}}}+7 \sqrt {\frac {x^{4}+x^{3}-x^{2}-x +1}{x^{2}}}\, \left (\operatorname {arcsinh}\left (\frac {\sqrt {3}\, \left (2 x^{2}+x -2\right )}{3 x}\right ) x +2 \sqrt {x^{4}+x^{3}-x^{2}-x +1}\right ) \left (\sqrt {13}-\frac {26}{7}\right )\right )}{2 \sqrt {\frac {x^{4}+x^{3}-x^{2}-x +1}{x^{2}}}\, \sqrt {78+26 \sqrt {13}}\, x \left (7 \sqrt {13}-26\right )}\) | \(399\) |
| trager | \(\frac {\sqrt {x^{4}+x^{3}-x^{2}-x +1}}{x}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+6 \textit {\_Z}^{2}+13\right ) \ln \left (-\frac {-51 \operatorname {RootOf}\left (\textit {\_Z}^{4}+6 \textit {\_Z}^{2}+13\right )^{5} x^{2}+51 \operatorname {RootOf}\left (\textit {\_Z}^{4}+6 \textit {\_Z}^{2}+13\right )^{5}-362 x^{2} \operatorname {RootOf}\left (\textit {\_Z}^{4}+6 \textit {\_Z}^{2}+13\right )^{3}-136 x \operatorname {RootOf}\left (\textit {\_Z}^{4}+6 \textit {\_Z}^{2}+13\right )^{3}+448 \sqrt {x^{4}+x^{3}-x^{2}-x +1}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}+6 \textit {\_Z}^{2}+13\right )^{2}+362 \operatorname {RootOf}\left (\textit {\_Z}^{4}+6 \textit {\_Z}^{2}+13\right )^{3}-247 \operatorname {RootOf}\left (\textit {\_Z}^{4}+6 \textit {\_Z}^{2}+13\right ) x^{2}-104 \operatorname {RootOf}\left (\textit {\_Z}^{4}+6 \textit {\_Z}^{2}+13\right ) x +1872 \sqrt {x^{4}+x^{3}-x^{2}-x +1}+247 \operatorname {RootOf}\left (\textit {\_Z}^{4}+6 \textit {\_Z}^{2}+13\right )}{{\left (x \operatorname {RootOf}\left (\textit {\_Z}^{4}+6 \textit {\_Z}^{2}+13\right )^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{4}+6 \textit {\_Z}^{2}+13\right )^{2}+5 x -1\right )}^{2}}\right )}{2}-\frac {\operatorname {RootOf}\left (\operatorname {RootOf}\left (\textit {\_Z}^{4}+6 \textit {\_Z}^{2}+13\right )^{2}+\textit {\_Z}^{2}+6\right ) \ln \left (\frac {51 \operatorname {RootOf}\left (\operatorname {RootOf}\left (\textit {\_Z}^{4}+6 \textit {\_Z}^{2}+13\right )^{2}+\textit {\_Z}^{2}+6\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}+6 \textit {\_Z}^{2}+13\right )^{4} x^{2}-51 \operatorname {RootOf}\left (\textit {\_Z}^{4}+6 \textit {\_Z}^{2}+13\right )^{4} \operatorname {RootOf}\left (\operatorname {RootOf}\left (\textit {\_Z}^{4}+6 \textit {\_Z}^{2}+13\right )^{2}+\textit {\_Z}^{2}+6\right )+250 \operatorname {RootOf}\left (\operatorname {RootOf}\left (\textit {\_Z}^{4}+6 \textit {\_Z}^{2}+13\right )^{2}+\textit {\_Z}^{2}+6\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}+6 \textit {\_Z}^{2}+13\right )^{2} x^{2}-136 \operatorname {RootOf}\left (\operatorname {RootOf}\left (\textit {\_Z}^{4}+6 \textit {\_Z}^{2}+13\right )^{2}+\textit {\_Z}^{2}+6\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}+6 \textit {\_Z}^{2}+13\right )^{2} x +448 \sqrt {x^{4}+x^{3}-x^{2}-x +1}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}+6 \textit {\_Z}^{2}+13\right )^{2}-250 \operatorname {RootOf}\left (\textit {\_Z}^{4}+6 \textit {\_Z}^{2}+13\right )^{2} \operatorname {RootOf}\left (\operatorname {RootOf}\left (\textit {\_Z}^{4}+6 \textit {\_Z}^{2}+13\right )^{2}+\textit {\_Z}^{2}+6\right )-89 \operatorname {RootOf}\left (\operatorname {RootOf}\left (\textit {\_Z}^{4}+6 \textit {\_Z}^{2}+13\right )^{2}+\textit {\_Z}^{2}+6\right ) x^{2}-712 \operatorname {RootOf}\left (\operatorname {RootOf}\left (\textit {\_Z}^{4}+6 \textit {\_Z}^{2}+13\right )^{2}+\textit {\_Z}^{2}+6\right ) x +816 \sqrt {x^{4}+x^{3}-x^{2}-x +1}+89 \operatorname {RootOf}\left (\operatorname {RootOf}\left (\textit {\_Z}^{4}+6 \textit {\_Z}^{2}+13\right )^{2}+\textit {\_Z}^{2}+6\right )}{{\left (x \operatorname {RootOf}\left (\textit {\_Z}^{4}+6 \textit {\_Z}^{2}+13\right )^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{4}+6 \textit {\_Z}^{2}+13\right )^{2}+x -5\right )}^{2}}\right )}{2}-\frac {\ln \left (-\frac {-2 x^{2}+2 \sqrt {x^{4}+x^{3}-x^{2}-x +1}-x +2}{x}\right )}{2}\) | \(650\) |
| elliptic | \(\text {Expression too large to display}\) | \(286046\) |
(x^4+x^3-x^2-x+1)^(1/2)/x+1/2/(78+26*13^(1/2))^(1/2)*(-4*((x^2-3*x-1)*13^( 1/2)+13*x)*((x^4+x^3-x^2-x+1)/(13^(1/2)*x+x^2-3*x-1)^2)^(1/2)*arctanh(4*(( x^4+x^3-x^2-x+1)/(13^(1/2)*x+x^2-3*x-1)^2)^(1/2)/(6+2*13^(1/2))^(1/2))+(78 +26*13^(1/2))^(1/2)*(1/3*(-2+13^(1/2))^(1/2)*arctan(35/468*(-2+13^(1/2))^( 1/2)*13^(1/2)*(13^(1/2)*x+x^2-3*x-1)*((x^4+x^3-x^2-x+1)/(13^(1/2)*x+x^2-3* x-1)^2)^(1/2)*(13^(1/2)+19/5)*(13^(1/2)-26/7)*(-13^(1/2)*x+x^2-3*x-1)/(x^4 +x^3-x^2-x+1))*((x^2+2*x-1)*13^(1/2)+5*x^2-2*x-5)*((x^4+x^3-x^2-x+1)/(13^( 1/2)*x+x^2-3*x-1)^2)^(1/2)+arcsinh(1/3*3^(1/2)*(2*x^2+x-2)/x)*((x^4+x^3-x^ 2-x+1)/x^2)^(1/2)*x))/((x^4+x^3-x^2-x+1)/x^2)^(1/2)/x
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.36 (sec) , antiderivative size = 361, normalized size of antiderivative = 1.42 \[ \int \frac {\left (-1+x^2\right ) \sqrt {1-x-x^2+x^3+x^4}}{x^2 \left (1+x^2\right )} \, dx=-\frac {\sqrt {8 i - 12} x \log \left (\frac {\sqrt {8 i - 12} {\left (\left (73 i + 14\right ) \, x^{4} + \left (66 i - 112\right ) \, x^{3} - \left (78 i + 104\right ) \, x^{2} - \left (66 i - 112\right ) \, x + 73 i + 14\right )} - 4 \, \sqrt {x^{4} + x^{3} - x^{2} - x + 1} {\left (-\left (33 i - 56\right ) \, x^{2} + \left (112 i + 66\right ) \, x + 33 i - 56\right )}}{x^{4} + 2 \, x^{2} + 1}\right ) + \sqrt {-8 i - 12} x \log \left (\frac {\sqrt {-8 i - 12} {\left (-\left (73 i - 14\right ) \, x^{4} - \left (66 i + 112\right ) \, x^{3} + \left (78 i - 104\right ) \, x^{2} + \left (66 i + 112\right ) \, x - 73 i + 14\right )} - 4 \, \sqrt {x^{4} + x^{3} - x^{2} - x + 1} {\left (\left (33 i + 56\right ) \, x^{2} - \left (112 i - 66\right ) \, x - 33 i - 56\right )}}{x^{4} + 2 \, x^{2} + 1}\right ) - \sqrt {-8 i - 12} x \log \left (\frac {\sqrt {-8 i - 12} {\left (\left (73 i - 14\right ) \, x^{4} + \left (66 i + 112\right ) \, x^{3} - \left (78 i - 104\right ) \, x^{2} - \left (66 i + 112\right ) \, x + 73 i - 14\right )} - 4 \, \sqrt {x^{4} + x^{3} - x^{2} - x + 1} {\left (\left (33 i + 56\right ) \, x^{2} - \left (112 i - 66\right ) \, x - 33 i - 56\right )}}{x^{4} + 2 \, x^{2} + 1}\right ) - \sqrt {8 i - 12} x \log \left (\frac {\sqrt {8 i - 12} {\left (-\left (73 i + 14\right ) \, x^{4} - \left (66 i - 112\right ) \, x^{3} + \left (78 i + 104\right ) \, x^{2} + \left (66 i - 112\right ) \, x - 73 i - 14\right )} - 4 \, \sqrt {x^{4} + x^{3} - x^{2} - x + 1} {\left (-\left (33 i - 56\right ) \, x^{2} + \left (112 i + 66\right ) \, x + 33 i - 56\right )}}{x^{4} + 2 \, x^{2} + 1}\right ) - 4 \, x \log \left (-\frac {2 \, x^{2} + x + 2 \, \sqrt {x^{4} + x^{3} - x^{2} - x + 1} - 2}{x}\right ) - 8 \, \sqrt {x^{4} + x^{3} - x^{2} - x + 1}}{8 \, x} \]
-1/8*(sqrt(8*I - 12)*x*log((sqrt(8*I - 12)*((73*I + 14)*x^4 + (66*I - 112) *x^3 - (78*I + 104)*x^2 - (66*I - 112)*x + 73*I + 14) - 4*sqrt(x^4 + x^3 - x^2 - x + 1)*(-(33*I - 56)*x^2 + (112*I + 66)*x + 33*I - 56))/(x^4 + 2*x^ 2 + 1)) + sqrt(-8*I - 12)*x*log((sqrt(-8*I - 12)*(-(73*I - 14)*x^4 - (66*I + 112)*x^3 + (78*I - 104)*x^2 + (66*I + 112)*x - 73*I + 14) - 4*sqrt(x^4 + x^3 - x^2 - x + 1)*((33*I + 56)*x^2 - (112*I - 66)*x - 33*I - 56))/(x^4 + 2*x^2 + 1)) - sqrt(-8*I - 12)*x*log((sqrt(-8*I - 12)*((73*I - 14)*x^4 + (66*I + 112)*x^3 - (78*I - 104)*x^2 - (66*I + 112)*x + 73*I - 14) - 4*sqrt (x^4 + x^3 - x^2 - x + 1)*((33*I + 56)*x^2 - (112*I - 66)*x - 33*I - 56))/ (x^4 + 2*x^2 + 1)) - sqrt(8*I - 12)*x*log((sqrt(8*I - 12)*(-(73*I + 14)*x^ 4 - (66*I - 112)*x^3 + (78*I + 104)*x^2 + (66*I - 112)*x - 73*I - 14) - 4* sqrt(x^4 + x^3 - x^2 - x + 1)*(-(33*I - 56)*x^2 + (112*I + 66)*x + 33*I - 56))/(x^4 + 2*x^2 + 1)) - 4*x*log(-(2*x^2 + x + 2*sqrt(x^4 + x^3 - x^2 - x + 1) - 2)/x) - 8*sqrt(x^4 + x^3 - x^2 - x + 1))/x
Not integrable
Time = 2.23 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.13 \[ \int \frac {\left (-1+x^2\right ) \sqrt {1-x-x^2+x^3+x^4}}{x^2 \left (1+x^2\right )} \, dx=\int \frac {\left (x - 1\right ) \left (x + 1\right ) \sqrt {x^{4} + x^{3} - x^{2} - x + 1}}{x^{2} \left (x^{2} + 1\right )}\, dx \]
Not integrable
Time = 0.30 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.14 \[ \int \frac {\left (-1+x^2\right ) \sqrt {1-x-x^2+x^3+x^4}}{x^2 \left (1+x^2\right )} \, dx=\int { \frac {\sqrt {x^{4} + x^{3} - x^{2} - x + 1} {\left (x^{2} - 1\right )}}{{\left (x^{2} + 1\right )} x^{2}} \,d x } \]
Not integrable
Time = 0.31 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.14 \[ \int \frac {\left (-1+x^2\right ) \sqrt {1-x-x^2+x^3+x^4}}{x^2 \left (1+x^2\right )} \, dx=\int { \frac {\sqrt {x^{4} + x^{3} - x^{2} - x + 1} {\left (x^{2} - 1\right )}}{{\left (x^{2} + 1\right )} x^{2}} \,d x } \]
Not integrable
Time = 0.00 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.14 \[ \int \frac {\left (-1+x^2\right ) \sqrt {1-x-x^2+x^3+x^4}}{x^2 \left (1+x^2\right )} \, dx=\int \frac {\left (x^2-1\right )\,\sqrt {x^4+x^3-x^2-x+1}}{x^2\,\left (x^2+1\right )} \,d x \]