Integrand size = 31, antiderivative size = 225 \[ \int \frac {x^2 \left (4+x^2+3 x^4+5 x^6\right )}{\left (3+2 x^2+x^4\right )^2} \, dx=5 x+\frac {25 x \left (1+x^2\right )}{8 \left (3+2 x^2+x^4\right )}+\frac {1}{16} \sqrt {\frac {1}{6} \left (19291+12899 \sqrt {3}\right )} \arctan \left (\frac {\sqrt {2 \left (-1+\sqrt {3}\right )}-2 x}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )-\frac {1}{16} \sqrt {\frac {1}{6} \left (19291+12899 \sqrt {3}\right )} \arctan \left (\frac {\sqrt {2 \left (-1+\sqrt {3}\right )}+2 x}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )-\frac {1}{32} \sqrt {\frac {1}{6} \left (-19291+12899 \sqrt {3}\right )} \log \left (\sqrt {3}-\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2\right )+\frac {1}{32} \sqrt {\frac {1}{6} \left (-19291+12899 \sqrt {3}\right )} \log \left (\sqrt {3}+\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2\right ) \]
5*x+25/8*x*(x^2+1)/(x^4+2*x^2+3)-1/192*ln(x^2+3^(1/2)-x*(-2+2*3^(1/2))^(1/ 2))*(-115746+77394*3^(1/2))^(1/2)+1/192*ln(x^2+3^(1/2)+x*(-2+2*3^(1/2))^(1 /2))*(-115746+77394*3^(1/2))^(1/2)+1/96*arctan((-2*x+(-2+2*3^(1/2))^(1/2)) /(2+2*3^(1/2))^(1/2))*(115746+77394*3^(1/2))^(1/2)-1/96*arctan((2*x+(-2+2* 3^(1/2))^(1/2))/(2+2*3^(1/2))^(1/2))*(115746+77394*3^(1/2))^(1/2)
Result contains complex when optimal does not.
Time = 0.11 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.54 \[ \int \frac {x^2 \left (4+x^2+3 x^4+5 x^6\right )}{\left (3+2 x^2+x^4\right )^2} \, dx=5 x+\frac {25 \left (x+x^3\right )}{8 \left (3+2 x^2+x^4\right )}-\frac {\left (-34 i+111 \sqrt {2}\right ) \arctan \left (\frac {x}{\sqrt {1-i \sqrt {2}}}\right )}{16 \sqrt {2-2 i \sqrt {2}}}-\frac {\left (34 i+111 \sqrt {2}\right ) \arctan \left (\frac {x}{\sqrt {1+i \sqrt {2}}}\right )}{16 \sqrt {2+2 i \sqrt {2}}} \]
5*x + (25*(x + x^3))/(8*(3 + 2*x^2 + x^4)) - ((-34*I + 111*Sqrt[2])*ArcTan [x/Sqrt[1 - I*Sqrt[2]]])/(16*Sqrt[2 - (2*I)*Sqrt[2]]) - ((34*I + 111*Sqrt[ 2])*ArcTan[x/Sqrt[1 + I*Sqrt[2]]])/(16*Sqrt[2 + (2*I)*Sqrt[2]])
Time = 0.48 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.02, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {2197, 27, 2205, 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.
| \(\displaystyle \int \frac {x^2 \left (5 x^6+3 x^4+x^2+4\right )}{\left (x^4+2 x^2+3\right )^2} \, dx\) |
| \(\Big \downarrow \) 2197 |
| \(\displaystyle \frac {1}{48} \int -\frac {6 \left (-40 x^4+31 x^2+25\right )}{x^4+2 x^2+3}dx+\frac {25 x \left (x^2+1\right )}{8 \left (x^4+2 x^2+3\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {25 x \left (x^2+1\right )}{8 \left (x^4+2 x^2+3\right )}-\frac {1}{8} \int \frac {-40 x^4+31 x^2+25}{x^4+2 x^2+3}dx\) |
| \(\Big \downarrow \) 2205 |
| \(\displaystyle \frac {25 x \left (x^2+1\right )}{8 \left (x^4+2 x^2+3\right )}-\frac {1}{8} \int \left (\frac {111 x^2+145}{x^4+2 x^2+3}-40\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{2} \sqrt {\frac {1}{6} \left (19291+12899 \sqrt {3}\right )} \arctan \left (\frac {\sqrt {2 \left (\sqrt {3}-1\right )}-2 x}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )-\frac {1}{2} \sqrt {\frac {1}{6} \left (19291+12899 \sqrt {3}\right )} \arctan \left (\frac {2 x+\sqrt {2 \left (\sqrt {3}-1\right )}}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )-\frac {1}{4} \sqrt {\frac {1}{6} \left (12899 \sqrt {3}-19291\right )} \log \left (x^2-\sqrt {2 \left (\sqrt {3}-1\right )} x+\sqrt {3}\right )+\frac {1}{4} \sqrt {\frac {1}{6} \left (12899 \sqrt {3}-19291\right )} \log \left (x^2+\sqrt {2 \left (\sqrt {3}-1\right )} x+\sqrt {3}\right )+40 x\right )+\frac {25 x \left (x^2+1\right )}{8 \left (x^4+2 x^2+3\right )}\) |
(25*x*(1 + x^2))/(8*(3 + 2*x^2 + x^4)) + (40*x + (Sqrt[(19291 + 12899*Sqrt [3])/6]*ArcTan[(Sqrt[2*(-1 + Sqrt[3])] - 2*x)/Sqrt[2*(1 + Sqrt[3])]])/2 - (Sqrt[(19291 + 12899*Sqrt[3])/6]*ArcTan[(Sqrt[2*(-1 + Sqrt[3])] + 2*x)/Sqr t[2*(1 + Sqrt[3])]])/2 - (Sqrt[(-19291 + 12899*Sqrt[3])/6]*Log[Sqrt[3] - S qrt[2*(-1 + Sqrt[3])]*x + x^2])/4 + (Sqrt[(-19291 + 12899*Sqrt[3])/6]*Log[ Sqrt[3] + Sqrt[2*(-1 + Sqrt[3])]*x + x^2])/4)/8
3.2.12.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(Pq_)*(x_)^(m_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{Qx = PolynomialQuotient[x^m*Pq, a + b*x^2 + c*x^4, x], d = Coeff[Pol ynomialRemainder[x^m*Pq, a + b*x^2 + c*x^4, x], x, 0], e = Coeff[Polynomial Remainder[x^m*Pq, a + b*x^2 + c*x^4, x], x, 2]}, Simp[x*(a + b*x^2 + c*x^4) ^(p + 1)*((a*b*e - d*(b^2 - 2*a*c) - c*(b*d - 2*a*e)*x^2)/(2*a*(p + 1)*(b^2 - 4*a*c))), x] + Simp[1/(2*a*(p + 1)*(b^2 - 4*a*c)) Int[(a + b*x^2 + c*x ^4)^(p + 1)*ExpandToSum[2*a*(p + 1)*(b^2 - 4*a*c)*Qx + b^2*d*(2*p + 3) - 2* a*c*d*(4*p + 5) - a*b*e + c*(4*p + 7)*(b*d - 2*a*e)*x^2, x], x], x]] /; Fre eQ[{a, b, c}, x] && PolyQ[Pq, x^2] && GtQ[Expon[Pq, x^2], 1] && NeQ[b^2 - 4 *a*c, 0] && LtQ[p, -1] && IGtQ[m/2, 0]
Int[(Px_)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> Int[ExpandInte grand[Px/(a + b*x^2 + c*x^4), x], x] /; FreeQ[{a, b, c}, x] && PolyQ[Px, x^ 2] && Expon[Px, x^2] > 1
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.08 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.28
| method | result | size |
| risch | \(5 x +\frac {\frac {25}{8} x^{3}+\frac {25}{8} x}{x^{4}+2 x^{2}+3}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}+2 \textit {\_Z}^{2}+3\right )}{\sum }\frac {\left (-111 \textit {\_R}^{2}-145\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}+\textit {\_R}}\right )}{32}\) | \(64\) |
| default | \(5 x -\frac {-\frac {25}{8} x^{3}-\frac {25}{8} x}{x^{4}+2 x^{2}+3}-\frac {\left (94 \sqrt {-2+2 \sqrt {3}}\, \sqrt {3}-51 \sqrt {-2+2 \sqrt {3}}\right ) \ln \left (x^{2}+\sqrt {3}-x \sqrt {-2+2 \sqrt {3}}\right )}{192}-\frac {\left (290 \sqrt {3}+\frac {\left (94 \sqrt {-2+2 \sqrt {3}}\, \sqrt {3}-51 \sqrt {-2+2 \sqrt {3}}\right ) \sqrt {-2+2 \sqrt {3}}}{2}\right ) \arctan \left (\frac {2 x -\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{48 \sqrt {2+2 \sqrt {3}}}-\frac {\left (-94 \sqrt {-2+2 \sqrt {3}}\, \sqrt {3}+51 \sqrt {-2+2 \sqrt {3}}\right ) \ln \left (x^{2}+\sqrt {3}+x \sqrt {-2+2 \sqrt {3}}\right )}{192}-\frac {\left (290 \sqrt {3}-\frac {\left (-94 \sqrt {-2+2 \sqrt {3}}\, \sqrt {3}+51 \sqrt {-2+2 \sqrt {3}}\right ) \sqrt {-2+2 \sqrt {3}}}{2}\right ) \arctan \left (\frac {2 x +\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{48 \sqrt {2+2 \sqrt {3}}}\) | \(281\) |
5*x+(25/8*x^3+25/8*x)/(x^4+2*x^2+3)+1/32*sum((-111*_R^2-145)/(_R^3+_R)*ln( x-_R),_R=RootOf(_Z^4+2*_Z^2+3))
Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 222, normalized size of antiderivative = 0.99 \[ \int \frac {x^2 \left (4+x^2+3 x^4+5 x^6\right )}{\left (3+2 x^2+x^4\right )^2} \, dx=\frac {480 \, x^{5} + 1260 \, x^{3} - \sqrt {3} {\left (x^{4} + 2 \, x^{2} + 3\right )} \sqrt {7969 i \, \sqrt {2} - 19291} \log \left (\sqrt {3} \sqrt {7969 i \, \sqrt {2} - 19291} {\left (94 i \, \sqrt {2} + 145\right )} + 38697 \, x\right ) + \sqrt {3} {\left (x^{4} + 2 \, x^{2} + 3\right )} \sqrt {7969 i \, \sqrt {2} - 19291} \log \left (\sqrt {3} \sqrt {7969 i \, \sqrt {2} - 19291} {\left (-94 i \, \sqrt {2} - 145\right )} + 38697 \, x\right ) + \sqrt {3} {\left (x^{4} + 2 \, x^{2} + 3\right )} \sqrt {-7969 i \, \sqrt {2} - 19291} \log \left (\sqrt {3} {\left (94 i \, \sqrt {2} - 145\right )} \sqrt {-7969 i \, \sqrt {2} - 19291} + 38697 \, x\right ) - \sqrt {3} {\left (x^{4} + 2 \, x^{2} + 3\right )} \sqrt {-7969 i \, \sqrt {2} - 19291} \log \left (\sqrt {3} {\left (-94 i \, \sqrt {2} + 145\right )} \sqrt {-7969 i \, \sqrt {2} - 19291} + 38697 \, x\right ) + 1740 \, x}{96 \, {\left (x^{4} + 2 \, x^{2} + 3\right )}} \]
1/96*(480*x^5 + 1260*x^3 - sqrt(3)*(x^4 + 2*x^2 + 3)*sqrt(7969*I*sqrt(2) - 19291)*log(sqrt(3)*sqrt(7969*I*sqrt(2) - 19291)*(94*I*sqrt(2) + 145) + 38 697*x) + sqrt(3)*(x^4 + 2*x^2 + 3)*sqrt(7969*I*sqrt(2) - 19291)*log(sqrt(3 )*sqrt(7969*I*sqrt(2) - 19291)*(-94*I*sqrt(2) - 145) + 38697*x) + sqrt(3)* (x^4 + 2*x^2 + 3)*sqrt(-7969*I*sqrt(2) - 19291)*log(sqrt(3)*(94*I*sqrt(2) - 145)*sqrt(-7969*I*sqrt(2) - 19291) + 38697*x) - sqrt(3)*(x^4 + 2*x^2 + 3 )*sqrt(-7969*I*sqrt(2) - 19291)*log(sqrt(3)*(-94*I*sqrt(2) + 145)*sqrt(-79 69*I*sqrt(2) - 19291) + 38697*x) + 1740*x)/(x^4 + 2*x^2 + 3)
Time = 0.33 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.23 \[ \int \frac {x^2 \left (4+x^2+3 x^4+5 x^6\right )}{\left (3+2 x^2+x^4\right )^2} \, dx=5 x + \frac {25 x^{3} + 25 x}{8 x^{4} + 16 x^{2} + 24} + \operatorname {RootSum} {\left (3145728 t^{4} + 39507968 t^{2} + 166384201, \left ( t \mapsto t \log {\left (- \frac {9240576 t^{3}}{102792131} - \frac {95003488 t}{102792131} + x \right )} \right )\right )} \]
5*x + (25*x**3 + 25*x)/(8*x**4 + 16*x**2 + 24) + RootSum(3145728*_t**4 + 3 9507968*_t**2 + 166384201, Lambda(_t, _t*log(-9240576*_t**3/102792131 - 95 003488*_t/102792131 + x)))
\[ \int \frac {x^2 \left (4+x^2+3 x^4+5 x^6\right )}{\left (3+2 x^2+x^4\right )^2} \, dx=\int { \frac {{\left (5 \, x^{6} + 3 \, x^{4} + x^{2} + 4\right )} x^{2}}{{\left (x^{4} + 2 \, x^{2} + 3\right )}^{2}} \,d x } \]
5*x + 25/8*(x^3 + x)/(x^4 + 2*x^2 + 3) - 1/8*integrate((111*x^2 + 145)/(x^ 4 + 2*x^2 + 3), x)
Leaf count of result is larger than twice the leaf count of optimal. 566 vs. \(2 (158) = 316\).
Time = 0.61 (sec) , antiderivative size = 566, normalized size of antiderivative = 2.52 \[ \int \frac {x^2 \left (4+x^2+3 x^4+5 x^6\right )}{\left (3+2 x^2+x^4\right )^2} \, dx=\frac {1}{6912} \, \sqrt {2} {\left (37 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 666 \cdot 3^{\frac {3}{4}} \sqrt {2} \sqrt {6 \, \sqrt {3} + 18} {\left (\sqrt {3} - 3\right )} - 666 \cdot 3^{\frac {3}{4}} {\left (\sqrt {3} + 3\right )} \sqrt {-6 \, \sqrt {3} + 18} + 37 \cdot 3^{\frac {3}{4}} {\left (-6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} - 1740 \cdot 3^{\frac {1}{4}} \sqrt {2} \sqrt {6 \, \sqrt {3} + 18} + 1740 \cdot 3^{\frac {1}{4}} \sqrt {-6 \, \sqrt {3} + 18}\right )} \arctan \left (\frac {3^{\frac {3}{4}} {\left (x + 3^{\frac {1}{4}} \sqrt {-\frac {1}{6} \, \sqrt {3} + \frac {1}{2}}\right )}}{3 \, \sqrt {\frac {1}{6} \, \sqrt {3} + \frac {1}{2}}}\right ) + \frac {1}{6912} \, \sqrt {2} {\left (37 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 666 \cdot 3^{\frac {3}{4}} \sqrt {2} \sqrt {6 \, \sqrt {3} + 18} {\left (\sqrt {3} - 3\right )} - 666 \cdot 3^{\frac {3}{4}} {\left (\sqrt {3} + 3\right )} \sqrt {-6 \, \sqrt {3} + 18} + 37 \cdot 3^{\frac {3}{4}} {\left (-6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} - 1740 \cdot 3^{\frac {1}{4}} \sqrt {2} \sqrt {6 \, \sqrt {3} + 18} + 1740 \cdot 3^{\frac {1}{4}} \sqrt {-6 \, \sqrt {3} + 18}\right )} \arctan \left (\frac {3^{\frac {3}{4}} {\left (x - 3^{\frac {1}{4}} \sqrt {-\frac {1}{6} \, \sqrt {3} + \frac {1}{2}}\right )}}{3 \, \sqrt {\frac {1}{6} \, \sqrt {3} + \frac {1}{2}}}\right ) + \frac {1}{13824} \, \sqrt {2} {\left (666 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (\sqrt {3} + 3\right )} \sqrt {-6 \, \sqrt {3} + 18} - 37 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (-6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 37 \cdot 3^{\frac {3}{4}} {\left (6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 666 \cdot 3^{\frac {3}{4}} \sqrt {6 \, \sqrt {3} + 18} {\left (\sqrt {3} - 3\right )} - 1740 \cdot 3^{\frac {1}{4}} \sqrt {2} \sqrt {-6 \, \sqrt {3} + 18} - 1740 \cdot 3^{\frac {1}{4}} \sqrt {6 \, \sqrt {3} + 18}\right )} \log \left (x^{2} + 2 \cdot 3^{\frac {1}{4}} x \sqrt {-\frac {1}{6} \, \sqrt {3} + \frac {1}{2}} + \sqrt {3}\right ) - \frac {1}{13824} \, \sqrt {2} {\left (666 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (\sqrt {3} + 3\right )} \sqrt {-6 \, \sqrt {3} + 18} - 37 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (-6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 37 \cdot 3^{\frac {3}{4}} {\left (6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 666 \cdot 3^{\frac {3}{4}} \sqrt {6 \, \sqrt {3} + 18} {\left (\sqrt {3} - 3\right )} - 1740 \cdot 3^{\frac {1}{4}} \sqrt {2} \sqrt {-6 \, \sqrt {3} + 18} - 1740 \cdot 3^{\frac {1}{4}} \sqrt {6 \, \sqrt {3} + 18}\right )} \log \left (x^{2} - 2 \cdot 3^{\frac {1}{4}} x \sqrt {-\frac {1}{6} \, \sqrt {3} + \frac {1}{2}} + \sqrt {3}\right ) + 5 \, x + \frac {25 \, {\left (x^{3} + x\right )}}{8 \, {\left (x^{4} + 2 \, x^{2} + 3\right )}} \]
1/6912*sqrt(2)*(37*3^(3/4)*sqrt(2)*(6*sqrt(3) + 18)^(3/2) + 666*3^(3/4)*sq rt(2)*sqrt(6*sqrt(3) + 18)*(sqrt(3) - 3) - 666*3^(3/4)*(sqrt(3) + 3)*sqrt( -6*sqrt(3) + 18) + 37*3^(3/4)*(-6*sqrt(3) + 18)^(3/2) - 1740*3^(1/4)*sqrt( 2)*sqrt(6*sqrt(3) + 18) + 1740*3^(1/4)*sqrt(-6*sqrt(3) + 18))*arctan(1/3*3 ^(3/4)*(x + 3^(1/4)*sqrt(-1/6*sqrt(3) + 1/2))/sqrt(1/6*sqrt(3) + 1/2)) + 1 /6912*sqrt(2)*(37*3^(3/4)*sqrt(2)*(6*sqrt(3) + 18)^(3/2) + 666*3^(3/4)*sqr t(2)*sqrt(6*sqrt(3) + 18)*(sqrt(3) - 3) - 666*3^(3/4)*(sqrt(3) + 3)*sqrt(- 6*sqrt(3) + 18) + 37*3^(3/4)*(-6*sqrt(3) + 18)^(3/2) - 1740*3^(1/4)*sqrt(2 )*sqrt(6*sqrt(3) + 18) + 1740*3^(1/4)*sqrt(-6*sqrt(3) + 18))*arctan(1/3*3^ (3/4)*(x - 3^(1/4)*sqrt(-1/6*sqrt(3) + 1/2))/sqrt(1/6*sqrt(3) + 1/2)) + 1/ 13824*sqrt(2)*(666*3^(3/4)*sqrt(2)*(sqrt(3) + 3)*sqrt(-6*sqrt(3) + 18) - 3 7*3^(3/4)*sqrt(2)*(-6*sqrt(3) + 18)^(3/2) + 37*3^(3/4)*(6*sqrt(3) + 18)^(3 /2) + 666*3^(3/4)*sqrt(6*sqrt(3) + 18)*(sqrt(3) - 3) - 1740*3^(1/4)*sqrt(2 )*sqrt(-6*sqrt(3) + 18) - 1740*3^(1/4)*sqrt(6*sqrt(3) + 18))*log(x^2 + 2*3 ^(1/4)*x*sqrt(-1/6*sqrt(3) + 1/2) + sqrt(3)) - 1/13824*sqrt(2)*(666*3^(3/4 )*sqrt(2)*(sqrt(3) + 3)*sqrt(-6*sqrt(3) + 18) - 37*3^(3/4)*sqrt(2)*(-6*sqr t(3) + 18)^(3/2) + 37*3^(3/4)*(6*sqrt(3) + 18)^(3/2) + 666*3^(3/4)*sqrt(6* sqrt(3) + 18)*(sqrt(3) - 3) - 1740*3^(1/4)*sqrt(2)*sqrt(-6*sqrt(3) + 18) - 1740*3^(1/4)*sqrt(6*sqrt(3) + 18))*log(x^2 - 2*3^(1/4)*x*sqrt(-1/6*sqrt(3 ) + 1/2) + sqrt(3)) + 5*x + 25/8*(x^3 + x)/(x^4 + 2*x^2 + 3)
Time = 0.09 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.69 \[ \int \frac {x^2 \left (4+x^2+3 x^4+5 x^6\right )}{\left (3+2 x^2+x^4\right )^2} \, dx=5\,x+\frac {\frac {25\,x^3}{8}+\frac {25\,x}{8}}{x^4+2\,x^2+3}+\frac {\mathrm {atan}\left (\frac {x\,\sqrt {-57873-\sqrt {2}\,23907{}\mathrm {i}}\,7969{}\mathrm {i}}{576\,\left (-\frac {374543}{96}+\frac {\sqrt {2}\,1155505{}\mathrm {i}}{384}\right )}+\frac {7969\,\sqrt {2}\,x\,\sqrt {-57873-\sqrt {2}\,23907{}\mathrm {i}}}{1152\,\left (-\frac {374543}{96}+\frac {\sqrt {2}\,1155505{}\mathrm {i}}{384}\right )}\right )\,\sqrt {-57873-\sqrt {2}\,23907{}\mathrm {i}}\,1{}\mathrm {i}}{48}-\frac {\mathrm {atan}\left (\frac {x\,\sqrt {-57873+\sqrt {2}\,23907{}\mathrm {i}}\,7969{}\mathrm {i}}{576\,\left (\frac {374543}{96}+\frac {\sqrt {2}\,1155505{}\mathrm {i}}{384}\right )}-\frac {7969\,\sqrt {2}\,x\,\sqrt {-57873+\sqrt {2}\,23907{}\mathrm {i}}}{1152\,\left (\frac {374543}{96}+\frac {\sqrt {2}\,1155505{}\mathrm {i}}{384}\right )}\right )\,\sqrt {-57873+\sqrt {2}\,23907{}\mathrm {i}}\,1{}\mathrm {i}}{48} \]
5*x + ((25*x)/8 + (25*x^3)/8)/(2*x^2 + x^4 + 3) + (atan((x*(- 2^(1/2)*2390 7i - 57873)^(1/2)*7969i)/(576*((2^(1/2)*1155505i)/384 - 374543/96)) + (796 9*2^(1/2)*x*(- 2^(1/2)*23907i - 57873)^(1/2))/(1152*((2^(1/2)*1155505i)/38 4 - 374543/96)))*(- 2^(1/2)*23907i - 57873)^(1/2)*1i)/48 - (atan((x*(2^(1/ 2)*23907i - 57873)^(1/2)*7969i)/(576*((2^(1/2)*1155505i)/384 + 374543/96)) - (7969*2^(1/2)*x*(2^(1/2)*23907i - 57873)^(1/2))/(1152*((2^(1/2)*1155505 i)/384 + 374543/96)))*(2^(1/2)*23907i - 57873)^(1/2)*1i)/48