Integrand size = 33, antiderivative size = 72 \[ \int \frac {\left (-1+x^2\right ) \sqrt {1+x^4}}{\left (1-x+x^2\right ) \left (1+x+x^2\right )^2} \, dx=\frac {\sqrt {1+x^4}}{2 \left (1+x+x^2\right )}+\frac {1}{2} \arctan \left (\frac {x}{1-x+x^2+\sqrt {1+x^4}}\right )-\frac {3}{2} \arctan \left (\frac {x}{1+x+x^2+\sqrt {1+x^4}}\right ) \]
(x^4+1)^(1/2)/(2*x^2+2*x+2)+1/2*arctan(x/(1-x+x^2+(x^4+1)^(1/2)))-3/2*arct an(x/(1+x+x^2+(x^4+1)^(1/2)))
Time = 1.43 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.93 \[ \int \frac {\left (-1+x^2\right ) \sqrt {1+x^4}}{\left (1-x+x^2\right ) \left (1+x+x^2\right )^2} \, dx=\frac {1}{2} \left (\frac {\sqrt {1+x^4}}{1+x+x^2}+\arctan \left (\frac {x}{1-x+x^2+\sqrt {1+x^4}}\right )-3 \arctan \left (\frac {x}{1+x+x^2+\sqrt {1+x^4}}\right )\right ) \]
(Sqrt[1 + x^4]/(1 + x + x^2) + ArcTan[x/(1 - x + x^2 + Sqrt[1 + x^4])] - 3 *ArcTan[x/(1 + x + x^2 + Sqrt[1 + x^4])])/2
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 10.26 (sec) , antiderivative size = 2585, normalized size of antiderivative = 35.90, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {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.
| \(\displaystyle \int \frac {\left (x^2-1\right ) \sqrt {x^4+1}}{\left (x^2-x+1\right ) \left (x^2+x+1\right )^2} \, dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \int \left (\frac {\sqrt {x^4+1} (-2 x-1)}{2 \left (x^2+x+1\right )^2}+\frac {(x+1) \sqrt {x^4+1}}{4 \left (x^2-x+1\right )}+\frac {(-x-3) \sqrt {x^4+1}}{4 \left (x^2+x+1\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\left (3+2 i \sqrt {3}\right ) \sqrt {x^4+1} x}{12 \left (x^2+1\right )}-\frac {\left (3-2 i \sqrt {3}\right ) \sqrt {x^4+1} x}{12 \left (x^2+1\right )}-\frac {\left (1+i \sqrt {3}\right ) \sqrt {x^4+1} x}{3 \left (x^2+1\right )}-\frac {\left (1-i \sqrt {3}\right ) \sqrt {x^4+1} x}{3 \left (x^2+1\right )}+\frac {7 \sqrt {x^4+1} x}{6 \left (x^2+1\right )}+\frac {1}{48} \left (9+i \sqrt {3}\right ) \text {arcsinh}\left (x^2\right )-\frac {1}{32} \left (1+i \sqrt {3}\right )^2 \text {arcsinh}\left (x^2\right )-\frac {3}{16} \left (1+i \sqrt {3}\right ) \text {arcsinh}\left (x^2\right )+\frac {1}{48} \left (9-i \sqrt {3}\right ) \text {arcsinh}\left (x^2\right )-\frac {1}{4} \left (1-i \sqrt {3}\right ) \text {arcsinh}\left (x^2\right )-\frac {\left (i-\sqrt {3}\right ) \arctan \left (\frac {x}{\sqrt {x^4+1}}\right )}{3 \left (i+\sqrt {3}\right )}-\frac {1}{24} \left (3+2 i \sqrt {3}\right ) \arctan \left (\frac {x}{\sqrt {x^4+1}}\right )-\frac {1}{24} \left (3-2 i \sqrt {3}\right ) \arctan \left (\frac {x}{\sqrt {x^4+1}}\right )+\frac {1}{6} \left (1+i \sqrt {3}\right ) \arctan \left (\frac {x}{\sqrt {x^4+1}}\right )-\frac {\arctan \left (\frac {x}{\sqrt {x^4+1}}\right )}{3 \left (1+i \sqrt {3}\right )}-\frac {\arctan \left (\frac {x}{\sqrt {x^4+1}}\right )}{3 \left (1-i \sqrt {3}\right )}-\frac {5}{12} \arctan \left (\frac {x}{\sqrt {x^4+1}}\right )+\frac {1}{8} i \text {arctanh}\left (\frac {2-\left (1-i \sqrt {3}\right ) x^2}{\sqrt {2 \left (1-i \sqrt {3}\right )} \sqrt {x^4+1}}\right )-\frac {\left (i+\sqrt {3}\right ) \text {arctanh}\left (\frac {2-\left (1+i \sqrt {3}\right ) x^2}{\sqrt {2 \left (1+i \sqrt {3}\right )} \sqrt {x^4+1}}\right )}{6 \left (1+i \sqrt {3}\right )}-\frac {1}{24} \left (3 i-2 \sqrt {3}\right ) \text {arctanh}\left (\frac {2-\left (1+i \sqrt {3}\right ) x^2}{\sqrt {2 \left (1+i \sqrt {3}\right )} \sqrt {x^4+1}}\right )-\frac {11}{24} i \text {arctanh}\left (\frac {2-\left (1+i \sqrt {3}\right ) x^2}{\sqrt {2 \left (1+i \sqrt {3}\right )} \sqrt {x^4+1}}\right )+\frac {1}{24} \left (3 i+2 \sqrt {3}\right ) \text {arctanh}\left (\frac {\left (1+i \sqrt {3}\right )^2 x^2+4}{2 \sqrt {2 \left (1-i \sqrt {3}\right )} \sqrt {x^4+1}}\right )-\frac {\left (1+i \sqrt {3}\right ) \text {arctanh}\left (\frac {\left (1+i \sqrt {3}\right )^2 x^2+4}{2 \sqrt {2 \left (1-i \sqrt {3}\right )} \sqrt {x^4+1}}\right )}{6 \left (i+\sqrt {3}\right )}+\frac {1}{3} i \text {arctanh}\left (\frac {\left (1+i \sqrt {3}\right )^2 x^2+4}{2 \sqrt {2 \left (1-i \sqrt {3}\right )} \sqrt {x^4+1}}\right )+\frac {\left (3+2 i \sqrt {3}\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} E\left (2 \arctan (x)\left |\frac {1}{2}\right .\right )}{12 \sqrt {x^4+1}}+\frac {\left (3-2 i \sqrt {3}\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} E\left (2 \arctan (x)\left |\frac {1}{2}\right .\right )}{12 \sqrt {x^4+1}}+\frac {\left (1+i \sqrt {3}\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} E\left (2 \arctan (x)\left |\frac {1}{2}\right .\right )}{3 \sqrt {x^4+1}}+\frac {\left (1-i \sqrt {3}\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} E\left (2 \arctan (x)\left |\frac {1}{2}\right .\right )}{3 \sqrt {x^4+1}}-\frac {7 \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} E\left (2 \arctan (x)\left |\frac {1}{2}\right .\right )}{6 \sqrt {x^4+1}}-\frac {\left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{3 \left (i+\sqrt {3}\right )^2 \sqrt {x^4+1}}-\frac {\left (9+i \sqrt {3}\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{24 \sqrt {x^4+1}}-\frac {\left (1+i \sqrt {3}\right )^2 \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{16 \sqrt {x^4+1}}-\frac {5 \left (1+i \sqrt {3}\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{12 \sqrt {x^4+1}}-\frac {\left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{3 \left (1+i \sqrt {3}\right ) \sqrt {x^4+1}}-\frac {\left (9-i \sqrt {3}\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{24 \sqrt {x^4+1}}-\frac {13 \left (1-i \sqrt {3}\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{24 \sqrt {x^4+1}}-\frac {\left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{3 \left (1-i \sqrt {3}\right ) \sqrt {x^4+1}}-\frac {\left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{3 \left (i-\sqrt {3}\right )^2 \sqrt {x^4+1}}+\frac {11 \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{6 \sqrt {x^4+1}}-\frac {\left (3 i+\sqrt {3}\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticPi}\left (\frac {1}{4},2 \arctan (x),\frac {1}{2}\right )}{6 \left (i+\sqrt {3}\right ) \sqrt {x^4+1}}+\frac {\left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticPi}\left (\frac {1}{4},2 \arctan (x),\frac {1}{2}\right )}{2 \sqrt {3} \left (i+\sqrt {3}\right ) \sqrt {x^4+1}}+\frac {\left (2+i \sqrt {3}\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticPi}\left (\frac {1}{4},2 \arctan (x),\frac {1}{2}\right )}{16 \sqrt {x^4+1}}-\frac {\left (3-i \sqrt {3}\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticPi}\left (\frac {1}{4},2 \arctan (x),\frac {1}{2}\right )}{12 \sqrt {x^4+1}}+\frac {\left (2-i \sqrt {3}\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticPi}\left (\frac {1}{4},2 \arctan (x),\frac {1}{2}\right )}{16 \sqrt {x^4+1}}-\frac {\left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticPi}\left (\frac {1}{4},2 \arctan (x),\frac {1}{2}\right )}{2 \sqrt {3} \left (i-\sqrt {3}\right ) \sqrt {x^4+1}}-\frac {\left (i-\sqrt {3}\right ) \sqrt {x^4+1}}{6 \left (-2 i x-\sqrt {3}+i\right )}+\frac {i \sqrt {x^4+1}}{6 \left (-2 i x-\sqrt {3}+i\right )}-\frac {\left (i+\sqrt {3}\right ) \sqrt {x^4+1}}{6 \left (-2 i x+\sqrt {3}+i\right )}+\frac {i \sqrt {x^4+1}}{6 \left (-2 i x+\sqrt {3}+i\right )}+\frac {\left (i-\sqrt {3}\right ) \sqrt {x^4+1}}{6 \left (2 i x-\sqrt {3}+i\right )}-\frac {i \sqrt {x^4+1}}{6 \left (2 i x-\sqrt {3}+i\right )}+\frac {\left (i+\sqrt {3}\right ) \sqrt {x^4+1}}{6 \left (2 i x+\sqrt {3}+i\right )}-\frac {i \sqrt {x^4+1}}{6 \left (2 i x+\sqrt {3}+i\right )}+\frac {\left (1+i \sqrt {3}\right ) \sqrt {x^4+1}}{6 \left (2 x^2-i \sqrt {3}+1\right )}+\frac {\left (1-i \sqrt {3}\right ) \sqrt {x^4+1}}{3 \left (2 x^2-i \sqrt {3}+1\right )}+\frac {\left (1+i \sqrt {3}\right ) \sqrt {x^4+1}}{3 \left (2 x^2+i \sqrt {3}+1\right )}+\frac {\left (1-i \sqrt {3}\right ) \sqrt {x^4+1}}{6 \left (2 x^2+i \sqrt {3}+1\right )}-\frac {1}{48} \left (3+5 i \sqrt {3}\right ) \sqrt {x^4+1}-\frac {1}{48} \left (3-5 i \sqrt {3}\right ) \sqrt {x^4+1}+\frac {1}{16} \left (1+i \sqrt {3}\right ) \sqrt {x^4+1}+\frac {1}{16} \left (1-i \sqrt {3}\right ) \sqrt {x^4+1}\) |
((1 - I*Sqrt[3])*Sqrt[1 + x^4])/16 + ((1 + I*Sqrt[3])*Sqrt[1 + x^4])/16 - ((3 - (5*I)*Sqrt[3])*Sqrt[1 + x^4])/48 - ((3 + (5*I)*Sqrt[3])*Sqrt[1 + x^4 ])/48 + ((I/6)*Sqrt[1 + x^4])/(I - Sqrt[3] - (2*I)*x) - ((I - Sqrt[3])*Sqr t[1 + x^4])/(6*(I - Sqrt[3] - (2*I)*x)) + ((I/6)*Sqrt[1 + x^4])/(I + Sqrt[ 3] - (2*I)*x) - ((I + Sqrt[3])*Sqrt[1 + x^4])/(6*(I + Sqrt[3] - (2*I)*x)) - ((I/6)*Sqrt[1 + x^4])/(I - Sqrt[3] + (2*I)*x) + ((I - Sqrt[3])*Sqrt[1 + x^4])/(6*(I - Sqrt[3] + (2*I)*x)) - ((I/6)*Sqrt[1 + x^4])/(I + Sqrt[3] + ( 2*I)*x) + ((I + Sqrt[3])*Sqrt[1 + x^4])/(6*(I + Sqrt[3] + (2*I)*x)) + (7*x *Sqrt[1 + x^4])/(6*(1 + x^2)) - ((1 - I*Sqrt[3])*x*Sqrt[1 + x^4])/(3*(1 + x^2)) - ((1 + I*Sqrt[3])*x*Sqrt[1 + x^4])/(3*(1 + x^2)) - ((3 - (2*I)*Sqrt [3])*x*Sqrt[1 + x^4])/(12*(1 + x^2)) - ((3 + (2*I)*Sqrt[3])*x*Sqrt[1 + x^4 ])/(12*(1 + x^2)) + ((1 - I*Sqrt[3])*Sqrt[1 + x^4])/(3*(1 - I*Sqrt[3] + 2* x^2)) + ((1 + I*Sqrt[3])*Sqrt[1 + x^4])/(6*(1 - I*Sqrt[3] + 2*x^2)) + ((1 - I*Sqrt[3])*Sqrt[1 + x^4])/(6*(1 + I*Sqrt[3] + 2*x^2)) + ((1 + I*Sqrt[3]) *Sqrt[1 + x^4])/(3*(1 + I*Sqrt[3] + 2*x^2)) - ((1 - I*Sqrt[3])*ArcSinh[x^2 ])/4 + ((9 - I*Sqrt[3])*ArcSinh[x^2])/48 - (3*(1 + I*Sqrt[3])*ArcSinh[x^2] )/16 - ((1 + I*Sqrt[3])^2*ArcSinh[x^2])/32 + ((9 + I*Sqrt[3])*ArcSinh[x^2] )/48 - (5*ArcTan[x/Sqrt[1 + x^4]])/12 - ArcTan[x/Sqrt[1 + x^4]]/(3*(1 - I* Sqrt[3])) - ArcTan[x/Sqrt[1 + x^4]]/(3*(1 + I*Sqrt[3])) + ((1 + I*Sqrt[3]) *ArcTan[x/Sqrt[1 + x^4]])/6 - ((3 - (2*I)*Sqrt[3])*ArcTan[x/Sqrt[1 + x^...
3.10.47.3.1 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]
Time = 9.93 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.71
| method | result | size |
| risch | \(\frac {\sqrt {x^{4}+1}}{2 x^{2}+2 x +2}-\frac {\arctan \left (\frac {\left (x -1\right )^{2}}{\sqrt {x^{4}+1}}\right )}{4}-\frac {3 \arctan \left (\frac {\left (1+x \right )^{2}}{\sqrt {x^{4}+1}}\right )}{4}\) | \(51\) |
| default | \(\frac {\left (-x^{2}-x -1\right ) \arctan \left (\frac {\left (x -1\right )^{2}}{\sqrt {x^{4}+1}}\right )+\left (-3 x^{2}-3 x -3\right ) \arctan \left (\frac {\left (1+x \right )^{2}}{\sqrt {x^{4}+1}}\right )+2 \sqrt {x^{4}+1}}{4 x^{2}+4 x +4}\) | \(74\) |
| pseudoelliptic | \(\frac {\left (-x^{2}-x -1\right ) \arctan \left (\frac {\left (x -1\right )^{2}}{\sqrt {x^{4}+1}}\right )+\left (-3 x^{2}-3 x -3\right ) \arctan \left (\frac {\left (1+x \right )^{2}}{\sqrt {x^{4}+1}}\right )+2 \sqrt {x^{4}+1}}{4 x^{2}+4 x +4}\) | \(74\) |
| trager | \(\frac {\sqrt {x^{4}+1}}{2 x^{2}+2 x +2}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{8}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{7}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{6}+2 \sqrt {x^{4}+1}\, x^{5}+4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{5}+2 x^{4} \sqrt {x^{4}+1}+5 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{4}+2 \sqrt {x^{4}+1}\, x^{3}+4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{3}+2 \sqrt {x^{4}+1}\, x^{2}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{2}+2 \sqrt {x^{4}+1}\, x +2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x +\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )}{\left (x^{2}-x +1\right ) \left (x^{2}+x +1\right )^{3}}\right )}{4}\) | \(197\) |
| elliptic | \(\frac {\left (\frac {3 \sqrt {3}\, \operatorname {arctanh}\left (\frac {\sqrt {3}\, \sqrt {\frac {2 \left (x^{2}+1\right )^{2}}{\left (-x^{2}+1\right )^{2}}+2}}{2}\right ) \left (x^{2}+1\right )^{2}}{\left (-x^{2}+1\right )^{2}}-\frac {9 \arctan \left (\frac {\sqrt {\frac {2 \left (x^{2}+1\right )^{2}}{\left (-x^{2}+1\right )^{2}}+2}\, \left (x^{2}+1\right )}{\left (\frac {\left (x^{2}+1\right )^{2}}{\left (-x^{2}+1\right )^{2}}+1\right ) \left (-x^{2}+1\right )}\right ) \left (x^{2}+1\right )^{2}}{\left (-x^{2}+1\right )^{2}}+\sqrt {3}\, \operatorname {arctanh}\left (\frac {\sqrt {3}\, \sqrt {\frac {2 \left (x^{2}+1\right )^{2}}{\left (-x^{2}+1\right )^{2}}+2}}{2}\right )+\frac {6 \sqrt {\frac {2 \left (x^{2}+1\right )^{2}}{\left (-x^{2}+1\right )^{2}}+2}\, \left (x^{2}+1\right )}{-x^{2}+1}-3 \arctan \left (\frac {\sqrt {\frac {2 \left (x^{2}+1\right )^{2}}{\left (-x^{2}+1\right )^{2}}+2}\, \left (x^{2}+1\right )}{\left (\frac {\left (x^{2}+1\right )^{2}}{\left (-x^{2}+1\right )^{2}}+1\right ) \left (-x^{2}+1\right )}\right )\right ) \sqrt {\frac {2 \left (x^{2}+1\right )^{2}}{\left (-x^{2}+1\right )^{2}}+2}\, \sqrt {2}}{12 \left (\frac {3 \left (x^{2}+1\right )^{2}}{\left (-x^{2}+1\right )^{2}}+1\right ) \left (\frac {x^{2}+1}{-x^{2}+1}+1\right ) \sqrt {\frac {\frac {\left (x^{2}+1\right )^{2}}{\left (-x^{2}+1\right )^{2}}+1}{\left (\frac {x^{2}+1}{-x^{2}+1}+1\right )^{2}}}}-\frac {\sqrt {2}\, \sqrt {\frac {2 \left (x^{2}+1\right )^{2}}{\left (-x^{2}+1\right )^{2}}+2}\, \left (\sqrt {3}\, \operatorname {arctanh}\left (\frac {\sqrt {3}\, \sqrt {\frac {2 \left (x^{2}+1\right )^{2}}{\left (-x^{2}+1\right )^{2}}+2}}{2}\right )-9 \arctan \left (\frac {\sqrt {\frac {2 \left (x^{2}+1\right )^{2}}{\left (-x^{2}+1\right )^{2}}+2}\, \left (x^{2}+1\right )}{\left (\frac {\left (x^{2}+1\right )^{2}}{\left (-x^{2}+1\right )^{2}}+1\right ) \left (-x^{2}+1\right )}\right )\right )}{12 \left (\frac {x^{2}+1}{-x^{2}+1}+1\right ) \sqrt {\frac {\frac {\left (x^{2}+1\right )^{2}}{\left (-x^{2}+1\right )^{2}}+1}{\left (\frac {x^{2}+1}{-x^{2}+1}+1\right )^{2}}}}+\frac {\left (-\frac {\sqrt {2}\, \sqrt {x^{4}+1}}{4 x \left (\frac {x^{4}+1}{2 x^{2}}+\frac {1}{2}\right )}+\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {x^{4}+1}}{x}\right )}{2}\right ) \sqrt {2}}{2}\) | \(602\) |
1/2*(x^4+1)^(1/2)/(x^2+x+1)-1/4*arctan((x-1)^2/(x^4+1)^(1/2))-3/4*arctan(( 1+x)^2/(x^4+1)^(1/2))
Time = 0.29 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.01 \[ \int \frac {\left (-1+x^2\right ) \sqrt {1+x^4}}{\left (1-x+x^2\right ) \left (1+x+x^2\right )^2} \, dx=\frac {3 \, {\left (x^{2} + x + 1\right )} \arctan \left (\frac {\sqrt {x^{4} + 1}}{x^{2} + 2 \, x + 1}\right ) + {\left (x^{2} + x + 1\right )} \arctan \left (\frac {\sqrt {x^{4} + 1}}{x^{2} - 2 \, x + 1}\right ) + 2 \, \sqrt {x^{4} + 1}}{4 \, {\left (x^{2} + x + 1\right )}} \]
1/4*(3*(x^2 + x + 1)*arctan(sqrt(x^4 + 1)/(x^2 + 2*x + 1)) + (x^2 + x + 1) *arctan(sqrt(x^4 + 1)/(x^2 - 2*x + 1)) + 2*sqrt(x^4 + 1))/(x^2 + x + 1)
\[ \int \frac {\left (-1+x^2\right ) \sqrt {1+x^4}}{\left (1-x+x^2\right ) \left (1+x+x^2\right )^2} \, dx=\int \frac {\left (x - 1\right ) \left (x + 1\right ) \sqrt {x^{4} + 1}}{\left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )^{2}}\, dx \]
\[ \int \frac {\left (-1+x^2\right ) \sqrt {1+x^4}}{\left (1-x+x^2\right ) \left (1+x+x^2\right )^2} \, dx=\int { \frac {\sqrt {x^{4} + 1} {\left (x^{2} - 1\right )}}{{\left (x^{2} + x + 1\right )}^{2} {\left (x^{2} - x + 1\right )}} \,d x } \]
\[ \int \frac {\left (-1+x^2\right ) \sqrt {1+x^4}}{\left (1-x+x^2\right ) \left (1+x+x^2\right )^2} \, dx=\int { \frac {\sqrt {x^{4} + 1} {\left (x^{2} - 1\right )}}{{\left (x^{2} + x + 1\right )}^{2} {\left (x^{2} - x + 1\right )}} \,d x } \]
Timed out. \[ \int \frac {\left (-1+x^2\right ) \sqrt {1+x^4}}{\left (1-x+x^2\right ) \left (1+x+x^2\right )^2} \, dx=\int \frac {\left (x^2-1\right )\,\sqrt {x^4+1}}{\left (x^2-x+1\right )\,{\left (x^2+x+1\right )}^2} \,d x \]