Integrand size = 48, antiderivative size = 121 \[ \int \frac {\left (2+x^2\right ) \left (-2-2 x+x^2\right ) \sqrt {4-3 x^2+x^4}}{x^2 \left (-2+x^2\right ) \left (-4+x+2 x^2\right )} \, dx=\frac {\sqrt {4-3 x^2+x^4}}{2 x}+4 \text {arctanh}\left (\frac {x}{-2+x^2+\sqrt {4-3 x^2+x^4}}\right )-\frac {5}{2} \sqrt {5} \text {arctanh}\left (\frac {\sqrt {5} x}{-4+x+2 x^2+2 \sqrt {4-3 x^2+x^4}}\right )+\frac {5 \log (x)}{4}-\frac {5}{4} \log \left (-2+x^2+\sqrt {4-3 x^2+x^4}\right ) \]
1/2*(x^4-3*x^2+4)^(1/2)/x+4*arctanh(x/(-2+x^2+(x^4-3*x^2+4)^(1/2)))-5/2*5^ (1/2)*arctanh(5^(1/2)*x/(-4+x+2*x^2+2*(x^4-3*x^2+4)^(1/2)))+5/4*ln(x)-5/4* ln(-2+x^2+(x^4-3*x^2+4)^(1/2))
Time = 1.01 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.97 \[ \int \frac {\left (2+x^2\right ) \left (-2-2 x+x^2\right ) \sqrt {4-3 x^2+x^4}}{x^2 \left (-2+x^2\right ) \left (-4+x+2 x^2\right )} \, dx=\frac {1}{4} \left (\frac {2 \sqrt {4-3 x^2+x^4}}{x}+16 \text {arctanh}\left (\frac {x}{-2+x^2+\sqrt {4-3 x^2+x^4}}\right )-10 \sqrt {5} \text {arctanh}\left (\frac {\sqrt {5} x}{-4+x+2 x^2+2 \sqrt {4-3 x^2+x^4}}\right )+5 \log (x)-5 \log \left (-2+x^2+\sqrt {4-3 x^2+x^4}\right )\right ) \]
Integrate[((2 + x^2)*(-2 - 2*x + x^2)*Sqrt[4 - 3*x^2 + x^4])/(x^2*(-2 + x^ 2)*(-4 + x + 2*x^2)),x]
((2*Sqrt[4 - 3*x^2 + x^4])/x + 16*ArcTanh[x/(-2 + x^2 + Sqrt[4 - 3*x^2 + x ^4])] - 10*Sqrt[5]*ArcTanh[(Sqrt[5]*x)/(-4 + x + 2*x^2 + 2*Sqrt[4 - 3*x^2 + x^4])] + 5*Log[x] - 5*Log[-2 + x^2 + Sqrt[4 - 3*x^2 + x^4]])/4
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 3.23 (sec) , antiderivative size = 831, normalized size of antiderivative = 6.87, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {7276, 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.
| \(\displaystyle \int \frac {\left (x^2+2\right ) \left (x^2-2 x-2\right ) \sqrt {x^4-3 x^2+4}}{x^2 \left (x^2-2\right ) \left (2 x^2+x-4\right )} \, dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \int \left (\frac {5 \sqrt {x^4-3 x^2+4} (2 x+17)}{8 \left (2 x^2+x-4\right )}-\frac {5 \sqrt {x^4-3 x^2+4}}{8 x}-\frac {4 \sqrt {x^4-3 x^2+4}}{x^2-2}-\frac {\sqrt {x^4-3 x^2+4}}{2 x^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {5}{64} \left (7+\sqrt {33}\right ) \text {arcsinh}\left (\frac {3-2 x^2}{\sqrt {7}}\right )+\frac {5}{64} \left (7-\sqrt {33}\right ) \text {arcsinh}\left (\frac {3-2 x^2}{\sqrt {7}}\right )-\frac {15}{32} \text {arcsinh}\left (\frac {3-2 x^2}{\sqrt {7}}\right )+2 \text {arctanh}\left (\frac {x}{\sqrt {x^4-3 x^2+4}}\right )-\frac {5}{4} \sqrt {5} \text {arctanh}\left (\frac {\sqrt {5} x}{2 \sqrt {x^4-3 x^2+4}}\right )+\frac {5}{8} \text {arctanh}\left (\frac {8-3 x^2}{4 \sqrt {x^4-3 x^2+4}}\right )-\frac {5}{8} \sqrt {5} \text {arctanh}\left (\frac {2 \left (5-\sqrt {33}\right ) x^2+3 \sqrt {33}+13}{2 \sqrt {10 \left (17-\sqrt {33}\right )} \sqrt {x^4-3 x^2+4}}\right )+\frac {5}{8} \sqrt {5} \text {arctanh}\left (\frac {2 \left (5+\sqrt {33}\right ) x^2-3 \sqrt {33}+13}{2 \sqrt {10 \left (17+\sqrt {33}\right )} \sqrt {x^4-3 x^2+4}}\right )+\frac {5 \left (5+\sqrt {33}\right ) \left (x^2+2\right ) \sqrt {\frac {x^4-3 x^2+4}{\left (x^2+2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {x}{\sqrt {2}}\right ),\frac {7}{8}\right )}{\sqrt {2} \left (33+\sqrt {33}\right ) \sqrt {x^4-3 x^2+4}}+\frac {5 \left (5-\sqrt {33}\right ) \left (x^2+2\right ) \sqrt {\frac {x^4-3 x^2+4}{\left (x^2+2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {x}{\sqrt {2}}\right ),\frac {7}{8}\right )}{\sqrt {2} \left (33-\sqrt {33}\right ) \sqrt {x^4-3 x^2+4}}-\frac {4 \sqrt {2} \left (x^2+2\right ) \sqrt {\frac {x^4-3 x^2+4}{\left (x^2+2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {x}{\sqrt {2}}\right ),\frac {7}{8}\right )}{\sqrt {x^4-3 x^2+4}}+\frac {27 \left (x^2+2\right ) \sqrt {\frac {x^4-3 x^2+4}{\left (x^2+2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {x}{\sqrt {2}}\right ),\frac {7}{8}\right )}{4 \sqrt {2} \sqrt {x^4-3 x^2+4}}+\frac {25 \left (17+\sqrt {33}\right ) \left (x^2+2\right ) \sqrt {\frac {x^4-3 x^2+4}{\left (x^2+2\right )^2}} \operatorname {EllipticPi}\left (\frac {33}{32},2 \arctan \left (\frac {x}{\sqrt {2}}\right ),\frac {7}{8}\right )}{32 \sqrt {2} \left (33+17 \sqrt {33}\right ) \sqrt {x^4-3 x^2+4}}+\frac {25 \left (17-\sqrt {33}\right ) \left (x^2+2\right ) \sqrt {\frac {x^4-3 x^2+4}{\left (x^2+2\right )^2}} \operatorname {EllipticPi}\left (\frac {33}{32},2 \arctan \left (\frac {x}{\sqrt {2}}\right ),\frac {7}{8}\right )}{32 \sqrt {2} \left (33-17 \sqrt {33}\right ) \sqrt {x^4-3 x^2+4}}+\frac {\sqrt {x^4-3 x^2+4}}{2 x}+\frac {5}{32} \left (1+\sqrt {33}\right ) \sqrt {x^4-3 x^2+4}+\frac {5}{32} \left (1-\sqrt {33}\right ) \sqrt {x^4-3 x^2+4}-\frac {5}{16} \sqrt {x^4-3 x^2+4}\) |
(-5*Sqrt[4 - 3*x^2 + x^4])/16 + (5*(1 - Sqrt[33])*Sqrt[4 - 3*x^2 + x^4])/3 2 + (5*(1 + Sqrt[33])*Sqrt[4 - 3*x^2 + x^4])/32 + Sqrt[4 - 3*x^2 + x^4]/(2 *x) - (15*ArcSinh[(3 - 2*x^2)/Sqrt[7]])/32 + (5*(7 - Sqrt[33])*ArcSinh[(3 - 2*x^2)/Sqrt[7]])/64 + (5*(7 + Sqrt[33])*ArcSinh[(3 - 2*x^2)/Sqrt[7]])/64 + 2*ArcTanh[x/Sqrt[4 - 3*x^2 + x^4]] - (5*Sqrt[5]*ArcTanh[(Sqrt[5]*x)/(2* Sqrt[4 - 3*x^2 + x^4])])/4 + (5*ArcTanh[(8 - 3*x^2)/(4*Sqrt[4 - 3*x^2 + x^ 4])])/8 - (5*Sqrt[5]*ArcTanh[(13 + 3*Sqrt[33] + 2*(5 - Sqrt[33])*x^2)/(2*S qrt[10*(17 - Sqrt[33])]*Sqrt[4 - 3*x^2 + x^4])])/8 + (5*Sqrt[5]*ArcTanh[(1 3 - 3*Sqrt[33] + 2*(5 + Sqrt[33])*x^2)/(2*Sqrt[10*(17 + Sqrt[33])]*Sqrt[4 - 3*x^2 + x^4])])/8 + (27*(2 + x^2)*Sqrt[(4 - 3*x^2 + x^4)/(2 + x^2)^2]*El lipticF[2*ArcTan[x/Sqrt[2]], 7/8])/(4*Sqrt[2]*Sqrt[4 - 3*x^2 + x^4]) - (4* Sqrt[2]*(2 + x^2)*Sqrt[(4 - 3*x^2 + x^4)/(2 + x^2)^2]*EllipticF[2*ArcTan[x /Sqrt[2]], 7/8])/Sqrt[4 - 3*x^2 + x^4] + (5*(5 - Sqrt[33])*(2 + x^2)*Sqrt[ (4 - 3*x^2 + x^4)/(2 + x^2)^2]*EllipticF[2*ArcTan[x/Sqrt[2]], 7/8])/(Sqrt[ 2]*(33 - Sqrt[33])*Sqrt[4 - 3*x^2 + x^4]) + (5*(5 + Sqrt[33])*(2 + x^2)*Sq rt[(4 - 3*x^2 + x^4)/(2 + x^2)^2]*EllipticF[2*ArcTan[x/Sqrt[2]], 7/8])/(Sq rt[2]*(33 + Sqrt[33])*Sqrt[4 - 3*x^2 + x^4]) + (25*(17 - Sqrt[33])*(2 + x^ 2)*Sqrt[(4 - 3*x^2 + x^4)/(2 + x^2)^2]*EllipticPi[33/32, 2*ArcTan[x/Sqrt[2 ]], 7/8])/(32*Sqrt[2]*(33 - 17*Sqrt[33])*Sqrt[4 - 3*x^2 + x^4]) + (25*(17 + Sqrt[33])*(2 + x^2)*Sqrt[(4 - 3*x^2 + x^4)/(2 + x^2)^2]*EllipticPi[33...
3.18.89.3.1 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 = 2.79 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.65
| method | result | size |
| risch | \(\frac {\sqrt {x^{4}-3 x^{2}+4}}{2 x}+2 \,\operatorname {arctanh}\left (\frac {x}{\sqrt {x^{4}-3 x^{2}+4}}\right )-\frac {5 \,\operatorname {arcsinh}\left (\frac {x^{2}-2}{x}\right )}{4}+\frac {5 \sqrt {5}\, \operatorname {arctanh}\left (\frac {\left (x^{2}-2 x -2\right ) \sqrt {5}}{5 \sqrt {x^{4}-3 x^{2}+4}}\right )}{4}\) | \(79\) |
| default | \(\frac {5 \sqrt {5}\, \operatorname {arctanh}\left (\frac {\left (x^{2}-2 x -2\right ) \sqrt {5}}{5 \sqrt {x^{4}-3 x^{2}+4}}\right ) x +8 \,\operatorname {arctanh}\left (\frac {x}{\sqrt {x^{4}-3 x^{2}+4}}\right ) x -5 \,\operatorname {arcsinh}\left (\frac {x^{2}-2}{x}\right ) x +2 \sqrt {x^{4}-3 x^{2}+4}}{4 x}\) | \(84\) |
| pseudoelliptic | \(\frac {5 \sqrt {5}\, \operatorname {arctanh}\left (\frac {\left (x^{2}-2 x -2\right ) \sqrt {5}}{5 \sqrt {x^{4}-3 x^{2}+4}}\right ) x +8 \,\operatorname {arctanh}\left (\frac {x}{\sqrt {x^{4}-3 x^{2}+4}}\right ) x -5 \,\operatorname {arcsinh}\left (\frac {x^{2}-2}{x}\right ) x +2 \sqrt {x^{4}-3 x^{2}+4}}{4 x}\) | \(84\) |
| elliptic | \(-\frac {5 \,\operatorname {arcsinh}\left (\frac {2 \sqrt {7}\, \left (x^{2}-\frac {3}{2}\right )}{7}\right )}{8}-\frac {160 \,\operatorname {arctanh}\left (\frac {-3 x^{2}+8}{4 \sqrt {x^{4}-3 x^{2}+4}}\right )}{\left (17+\sqrt {33}\right ) \left (-17+\sqrt {33}\right )}-\frac {25 \left (-297+25 \sqrt {33}\right ) \sqrt {33}\, \operatorname {arctanh}\left (\frac {\frac {85}{4}-\frac {5 \sqrt {33}}{4}+4 \left (\frac {5}{4}-\frac {\sqrt {33}}{4}\right ) \left (x^{2}-\frac {17}{8}+\frac {\sqrt {33}}{8}\right )}{\left (\frac {\sqrt {165}}{8}-\frac {\sqrt {5}}{8}\right ) \sqrt {64 \left (x^{2}-\frac {17}{8}+\frac {\sqrt {33}}{8}\right )^{2}+64 \left (\frac {5}{4}-\frac {\sqrt {33}}{4}\right ) \left (x^{2}-\frac {17}{8}+\frac {\sqrt {33}}{8}\right )+170-10 \sqrt {33}}}\right )}{16 \left (-1122+66 \sqrt {33}\right ) \left (\frac {\sqrt {165}}{8}-\frac {\sqrt {5}}{8}\right )}+\frac {25 \left (297+25 \sqrt {33}\right ) \sqrt {33}\, \operatorname {arctanh}\left (\frac {\frac {85}{4}+\frac {5 \sqrt {33}}{4}+4 \left (\frac {5}{4}+\frac {\sqrt {33}}{4}\right ) \left (x^{2}-\frac {17}{8}-\frac {\sqrt {33}}{8}\right )}{\left (\frac {\sqrt {165}}{8}+\frac {\sqrt {5}}{8}\right ) \sqrt {64 \left (x^{2}-\frac {17}{8}-\frac {\sqrt {33}}{8}\right )^{2}+64 \left (\frac {5}{4}+\frac {\sqrt {33}}{4}\right ) \left (x^{2}-\frac {17}{8}-\frac {\sqrt {33}}{8}\right )+170+10 \sqrt {33}}}\right )}{16 \left (1122+66 \sqrt {33}\right ) \left (\frac {\sqrt {165}}{8}+\frac {\sqrt {5}}{8}\right )}+\frac {\left (\frac {\sqrt {x^{4}-3 x^{2}+4}\, \sqrt {2}}{2 x}-\frac {5 \sqrt {10}\, \operatorname {arctanh}\left (\frac {\sqrt {10}\, \sqrt {x^{4}-3 x^{2}+4}\, \sqrt {2}}{5 x}\right )}{4}+2 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {x^{4}-3 x^{2}+4}}{x}\right )\right ) \sqrt {2}}{2}\) | \(364\) |
| trager | \(\frac {\sqrt {x^{4}-3 x^{2}+4}}{2 x}+\frac {\ln \left (\frac {131072+524288 x -30198 x^{20}+4373760 x^{10}+1296128 x^{9}-2095616 x^{7}+65536 \sqrt {x^{4}-3 x^{2}+4}+154096 x^{18}+1437984 x^{14}+9128 x^{21}+3507 x^{22}-32744 x^{19}-156064 x^{15}-1568 x^{23}+81008 x^{17}+2336768 x^{5}+3865344 x^{6}+237568 x^{2}-1605632 x^{3}-4931072 x^{8}-1795584 x^{4}-546720 x^{16}+292096 x^{13}-2875968 x^{12}-624256 x^{11}-16 x^{26}+128 x^{25}-116 x^{24}+16 \sqrt {x^{4}-3 x^{2}+4}\, x^{24}-128 \sqrt {x^{4}-3 x^{2}+4}\, x^{23}+140 \sqrt {x^{4}-3 x^{2}+4}\, x^{22}+1376 \sqrt {x^{4}-3 x^{2}+4}\, x^{21}-3311 \sqrt {x^{4}-3 x^{2}+4}\, x^{20}-6952 \sqrt {x^{4}-3 x^{2}+4}\, x^{19}+25088 \sqrt {x^{4}-3 x^{2}+4}\, x^{18}+21280 \sqrt {x^{4}-3 x^{2}+4}\, x^{17}-113776 \sqrt {x^{4}-3 x^{2}+4}\, x^{16}-44608 \sqrt {x^{4}-3 x^{2}+4}\, x^{15}+358208 \sqrt {x^{4}-3 x^{2}+4}\, x^{14}+77632 \sqrt {x^{4}-3 x^{2}+4}\, x^{13}-828960 \sqrt {x^{4}-3 x^{2}+4}\, x^{12}-155264 \sqrt {x^{4}-3 x^{2}+4}\, x^{11}+1432832 \sqrt {x^{4}-3 x^{2}+4}\, x^{10}+356864 \sqrt {x^{4}-3 x^{2}+4}\, x^{9}-1820416 \sqrt {x^{4}-3 x^{2}+4}\, x^{8}-680960 \sqrt {x^{4}-3 x^{2}+4}\, x^{7}+1605632 \sqrt {x^{4}-3 x^{2}+4}\, x^{6}+889856 \sqrt {x^{4}-3 x^{2}+4}\, x^{5}-847616 \sqrt {x^{4}-3 x^{2}+4}\, x^{4}-704512 x^{3} \sqrt {x^{4}-3 x^{2}+4}+143360 \sqrt {x^{4}-3 x^{2}+4}\, x^{2}+262144 x \sqrt {x^{4}-3 x^{2}+4}}{\left (x^{2}-2\right )^{8} x^{5}}\right )}{4}-\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) x^{2}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) x -2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right )-5 \sqrt {x^{4}-3 x^{2}+4}}{2 x^{2}+x -4}\right )}{4}\) | \(645\) |
1/2*(x^4-3*x^2+4)^(1/2)/x+2*arctanh(1/(x^4-3*x^2+4)^(1/2)*x)-5/4*arcsinh(( x^2-2)/x)+5/4*5^(1/2)*arctanh(1/5*(x^2-2*x-2)*5^(1/2)/(x^4-3*x^2+4)^(1/2))
Time = 0.38 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.24 \[ \int \frac {\left (2+x^2\right ) \left (-2-2 x+x^2\right ) \sqrt {4-3 x^2+x^4}}{x^2 \left (-2+x^2\right ) \left (-4+x+2 x^2\right )} \, dx=\frac {5 \, \sqrt {5} x \log \left (\frac {6 \, x^{4} - 4 \, x^{3} + 2 \, \sqrt {5} \sqrt {x^{4} - 3 \, x^{2} + 4} {\left (x^{2} - 2 \, x - 2\right )} - 15 \, x^{2} + 8 \, x + 24}{4 \, x^{4} + 4 \, x^{3} - 15 \, x^{2} - 8 \, x + 16}\right ) + 16 \, x \log \left (-\frac {x + \sqrt {x^{4} - 3 \, x^{2} + 4}}{x^{2} - 2}\right ) + 10 \, x \log \left (-\frac {x^{2} - \sqrt {x^{4} - 3 \, x^{2} + 4} - 2}{x}\right ) + 4 \, \sqrt {x^{4} - 3 \, x^{2} + 4}}{8 \, x} \]
integrate((x^2+2)*(x^2-2*x-2)*(x^4-3*x^2+4)^(1/2)/x^2/(x^2-2)/(2*x^2+x-4), x, algorithm="fricas")
1/8*(5*sqrt(5)*x*log((6*x^4 - 4*x^3 + 2*sqrt(5)*sqrt(x^4 - 3*x^2 + 4)*(x^2 - 2*x - 2) - 15*x^2 + 8*x + 24)/(4*x^4 + 4*x^3 - 15*x^2 - 8*x + 16)) + 16 *x*log(-(x + sqrt(x^4 - 3*x^2 + 4))/(x^2 - 2)) + 10*x*log(-(x^2 - sqrt(x^4 - 3*x^2 + 4) - 2)/x) + 4*sqrt(x^4 - 3*x^2 + 4))/x
\[ \int \frac {\left (2+x^2\right ) \left (-2-2 x+x^2\right ) \sqrt {4-3 x^2+x^4}}{x^2 \left (-2+x^2\right ) \left (-4+x+2 x^2\right )} \, dx=\int \frac {\left (x^{2} + 2\right ) \left (x^{2} - 2 x - 2\right ) \sqrt {x^{4} - 3 x^{2} + 4}}{x^{2} \left (x^{2} - 2\right ) \left (2 x^{2} + x - 4\right )}\, dx \]
Integral((x**2 + 2)*(x**2 - 2*x - 2)*sqrt(x**4 - 3*x**2 + 4)/(x**2*(x**2 - 2)*(2*x**2 + x - 4)), x)
\[ \int \frac {\left (2+x^2\right ) \left (-2-2 x+x^2\right ) \sqrt {4-3 x^2+x^4}}{x^2 \left (-2+x^2\right ) \left (-4+x+2 x^2\right )} \, dx=\int { \frac {\sqrt {x^{4} - 3 \, x^{2} + 4} {\left (x^{2} - 2 \, x - 2\right )} {\left (x^{2} + 2\right )}}{{\left (2 \, x^{2} + x - 4\right )} {\left (x^{2} - 2\right )} x^{2}} \,d x } \]
integrate((x^2+2)*(x^2-2*x-2)*(x^4-3*x^2+4)^(1/2)/x^2/(x^2-2)/(2*x^2+x-4), x, algorithm="maxima")
integrate(sqrt(x^4 - 3*x^2 + 4)*(x^2 - 2*x - 2)*(x^2 + 2)/((2*x^2 + x - 4) *(x^2 - 2)*x^2), x)
\[ \int \frac {\left (2+x^2\right ) \left (-2-2 x+x^2\right ) \sqrt {4-3 x^2+x^4}}{x^2 \left (-2+x^2\right ) \left (-4+x+2 x^2\right )} \, dx=\int { \frac {\sqrt {x^{4} - 3 \, x^{2} + 4} {\left (x^{2} - 2 \, x - 2\right )} {\left (x^{2} + 2\right )}}{{\left (2 \, x^{2} + x - 4\right )} {\left (x^{2} - 2\right )} x^{2}} \,d x } \]
integrate((x^2+2)*(x^2-2*x-2)*(x^4-3*x^2+4)^(1/2)/x^2/(x^2-2)/(2*x^2+x-4), x, algorithm="giac")
integrate(sqrt(x^4 - 3*x^2 + 4)*(x^2 - 2*x - 2)*(x^2 + 2)/((2*x^2 + x - 4) *(x^2 - 2)*x^2), x)
Timed out. \[ \int \frac {\left (2+x^2\right ) \left (-2-2 x+x^2\right ) \sqrt {4-3 x^2+x^4}}{x^2 \left (-2+x^2\right ) \left (-4+x+2 x^2\right )} \, dx=\int -\frac {\left (x^2+2\right )\,\left (-x^2+2\,x+2\right )\,\sqrt {x^4-3\,x^2+4}}{x^2\,\left (x^2-2\right )\,\left (2\,x^2+x-4\right )} \,d x \]