Integrand size = 31, antiderivative size = 232 \[ \int \frac {x^4 \left (4+x^2+3 x^4+5 x^6\right )}{\left (3+2 x^2+x^4\right )^2} \, dx=-17 x+\frac {5 x^3}{3}-\frac {25 x \left (3+x^2\right )}{8 \left (3+2 x^2+x^4\right )}-\frac {1}{16} \sqrt {\frac {1}{2} \left (14395+26499 \sqrt {3}\right )} \arctan \left (\frac {\sqrt {2 \left (-1+\sqrt {3}\right )}-2 x}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )+\frac {1}{16} \sqrt {\frac {1}{2} \left (14395+26499 \sqrt {3}\right )} \arctan \left (\frac {\sqrt {2 \left (-1+\sqrt {3}\right )}+2 x}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )-\frac {1}{32} \sqrt {\frac {1}{2} \left (-14395+26499 \sqrt {3}\right )} \log \left (\sqrt {3}-\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2\right )+\frac {1}{32} \sqrt {\frac {1}{2} \left (-14395+26499 \sqrt {3}\right )} \log \left (\sqrt {3}+\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2\right ) \]
-17*x+5/3*x^3-25/8*x*(x^2+3)/(x^4+2*x^2+3)-1/64*ln(x^2+3^(1/2)-x*(-2+2*3^( 1/2))^(1/2))*(-28790+52998*3^(1/2))^(1/2)+1/64*ln(x^2+3^(1/2)+x*(-2+2*3^(1 /2))^(1/2))*(-28790+52998*3^(1/2))^(1/2)-1/32*arctan((-2*x+(-2+2*3^(1/2))^ (1/2))/(2+2*3^(1/2))^(1/2))*(28790+52998*3^(1/2))^(1/2)+1/32*arctan((2*x+( -2+2*3^(1/2))^(1/2))/(2+2*3^(1/2))^(1/2))*(28790+52998*3^(1/2))^(1/2)
Result contains complex when optimal does not.
Time = 0.11 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.56 \[ \int \frac {x^4 \left (4+x^2+3 x^4+5 x^6\right )}{\left (3+2 x^2+x^4\right )^2} \, dx=-17 x+\frac {5 x^3}{3}-\frac {25 x \left (3+x^2\right )}{8 \left (3+2 x^2+x^4\right )}+\frac {\left (-356 i+127 \sqrt {2}\right ) \arctan \left (\frac {x}{\sqrt {1-i \sqrt {2}}}\right )}{16 \sqrt {2-2 i \sqrt {2}}}+\frac {\left (356 i+127 \sqrt {2}\right ) \arctan \left (\frac {x}{\sqrt {1+i \sqrt {2}}}\right )}{16 \sqrt {2+2 i \sqrt {2}}} \]
-17*x + (5*x^3)/3 - (25*x*(3 + x^2))/(8*(3 + 2*x^2 + x^4)) + ((-356*I + 12 7*Sqrt[2])*ArcTan[x/Sqrt[1 - I*Sqrt[2]]])/(16*Sqrt[2 - (2*I)*Sqrt[2]]) + ( (356*I + 127*Sqrt[2])*ArcTan[x/Sqrt[1 + I*Sqrt[2]]])/(16*Sqrt[2 + (2*I)*Sq rt[2]])
Time = 0.50 (sec) , antiderivative size = 237, 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.
| \(\displaystyle \int \frac {x^4 \left (5 x^6+3 x^4+x^2+4\right )}{\left (x^4+2 x^2+3\right )^2} \, dx\) |
| \(\Big \downarrow \) 2197 |
| \(\displaystyle \frac {1}{48} \int \frac {6 \left (40 x^6-56 x^4-25 x^2+75\right )}{x^4+2 x^2+3}dx-\frac {25 x \left (x^2+3\right )}{8 \left (x^4+2 x^2+3\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{8} \int \frac {40 x^6-56 x^4-25 x^2+75}{x^4+2 x^2+3}dx-\frac {25 x \left (x^2+3\right )}{8 \left (x^4+2 x^2+3\right )}\) |
| \(\Big \downarrow \) 2205 |
| \(\displaystyle \frac {1}{8} \int \left (40 x^2+\frac {127 x^2+483}{x^4+2 x^2+3}-136\right )dx-\frac {25 x \left (x^2+3\right )}{8 \left (x^4+2 x^2+3\right )}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{8} \left (-\frac {1}{2} \sqrt {\frac {1}{2} \left (14395+26499 \sqrt {3}\right )} \arctan \left (\frac {\sqrt {2 \left (\sqrt {3}-1\right )}-2 x}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )+\frac {1}{2} \sqrt {\frac {1}{2} \left (14395+26499 \sqrt {3}\right )} \arctan \left (\frac {2 x+\sqrt {2 \left (\sqrt {3}-1\right )}}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )+\frac {40 x^3}{3}-\frac {1}{4} \sqrt {\frac {1}{2} \left (26499 \sqrt {3}-14395\right )} \log \left (x^2-\sqrt {2 \left (\sqrt {3}-1\right )} x+\sqrt {3}\right )+\frac {1}{4} \sqrt {\frac {1}{2} \left (26499 \sqrt {3}-14395\right )} \log \left (x^2+\sqrt {2 \left (\sqrt {3}-1\right )} x+\sqrt {3}\right )-136 x\right )-\frac {25 x \left (x^2+3\right )}{8 \left (x^4+2 x^2+3\right )}\) |
(-25*x*(3 + x^2))/(8*(3 + 2*x^2 + x^4)) + (-136*x + (40*x^3)/3 - (Sqrt[(14 395 + 26499*Sqrt[3])/2]*ArcTan[(Sqrt[2*(-1 + Sqrt[3])] - 2*x)/Sqrt[2*(1 + Sqrt[3])]])/2 + (Sqrt[(14395 + 26499*Sqrt[3])/2]*ArcTan[(Sqrt[2*(-1 + Sqrt [3])] + 2*x)/Sqrt[2*(1 + Sqrt[3])]])/2 - (Sqrt[(-14395 + 26499*Sqrt[3])/2] *Log[Sqrt[3] - Sqrt[2*(-1 + Sqrt[3])]*x + x^2])/4 + (Sqrt[(-14395 + 26499* Sqrt[3])/2]*Log[Sqrt[3] + Sqrt[2*(-1 + Sqrt[3])]*x + x^2])/4)/8
3.2.11.3.1 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 = 69, normalized size of antiderivative = 0.30
| method | result | size |
| risch | \(\frac {5 x^{3}}{3}-17 x +\frac {-\frac {25}{8} x^{3}-\frac {75}{8} x}{x^{4}+2 x^{2}+3}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}+2 \textit {\_Z}^{2}+3\right )}{\sum }\frac {\left (127 \textit {\_R}^{2}+483\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}+\textit {\_R}}\right )}{32}\) | \(69\) |
| default | \(\frac {5 x^{3}}{3}-17 x +\frac {-\frac {25}{8} x^{3}-\frac {75}{8} x}{x^{4}+2 x^{2}+3}+\frac {\left (-17 \sqrt {-2+2 \sqrt {3}}\, \sqrt {3}-178 \sqrt {-2+2 \sqrt {3}}\right ) \ln \left (x^{2}+\sqrt {3}-x \sqrt {-2+2 \sqrt {3}}\right )}{64}+\frac {\left (322 \sqrt {3}+\frac {\left (-17 \sqrt {-2+2 \sqrt {3}}\, \sqrt {3}-178 \sqrt {-2+2 \sqrt {3}}\right ) \sqrt {-2+2 \sqrt {3}}}{2}\right ) \arctan \left (\frac {2 x -\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{16 \sqrt {2+2 \sqrt {3}}}+\frac {\left (17 \sqrt {-2+2 \sqrt {3}}\, \sqrt {3}+178 \sqrt {-2+2 \sqrt {3}}\right ) \ln \left (x^{2}+\sqrt {3}+x \sqrt {-2+2 \sqrt {3}}\right )}{64}+\frac {\left (322 \sqrt {3}-\frac {\left (17 \sqrt {-2+2 \sqrt {3}}\, \sqrt {3}+178 \sqrt {-2+2 \sqrt {3}}\right ) \sqrt {-2+2 \sqrt {3}}}{2}\right ) \arctan \left (\frac {2 x +\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{16 \sqrt {2+2 \sqrt {3}}}\) | \(285\) |
5/3*x^3-17*x+(-25/8*x^3-75/8*x)/(x^4+2*x^2+3)+1/32*sum((127*_R^2+483)/(_R^ 3+_R)*ln(x-_R),_R=RootOf(_Z^4+2*_Z^2+3))
Result contains complex when optimal does not.
Time = 0.25 (sec) , antiderivative size = 205, normalized size of antiderivative = 0.88 \[ \int \frac {x^4 \left (4+x^2+3 x^4+5 x^6\right )}{\left (3+2 x^2+x^4\right )^2} \, dx=\frac {160 \, x^{7} - 1312 \, x^{5} - 3084 \, x^{3} + 3 \, {\left (x^{4} + 2 \, x^{2} + 3\right )} \sqrt {30817 i \, \sqrt {2} - 14395} \log \left (\sqrt {30817 i \, \sqrt {2} - 14395} {\left (17 i \, \sqrt {2} + 161\right )} + 26499 \, x\right ) - 3 \, {\left (x^{4} + 2 \, x^{2} + 3\right )} \sqrt {30817 i \, \sqrt {2} - 14395} \log \left (\sqrt {30817 i \, \sqrt {2} - 14395} {\left (-17 i \, \sqrt {2} - 161\right )} + 26499 \, x\right ) - 3 \, {\left (x^{4} + 2 \, x^{2} + 3\right )} \sqrt {-30817 i \, \sqrt {2} - 14395} \log \left ({\left (17 i \, \sqrt {2} - 161\right )} \sqrt {-30817 i \, \sqrt {2} - 14395} + 26499 \, x\right ) + 3 \, {\left (x^{4} + 2 \, x^{2} + 3\right )} \sqrt {-30817 i \, \sqrt {2} - 14395} \log \left ({\left (-17 i \, \sqrt {2} + 161\right )} \sqrt {-30817 i \, \sqrt {2} - 14395} + 26499 \, x\right ) - 5796 \, x}{96 \, {\left (x^{4} + 2 \, x^{2} + 3\right )}} \]
1/96*(160*x^7 - 1312*x^5 - 3084*x^3 + 3*(x^4 + 2*x^2 + 3)*sqrt(30817*I*sqr t(2) - 14395)*log(sqrt(30817*I*sqrt(2) - 14395)*(17*I*sqrt(2) + 161) + 264 99*x) - 3*(x^4 + 2*x^2 + 3)*sqrt(30817*I*sqrt(2) - 14395)*log(sqrt(30817*I *sqrt(2) - 14395)*(-17*I*sqrt(2) - 161) + 26499*x) - 3*(x^4 + 2*x^2 + 3)*s qrt(-30817*I*sqrt(2) - 14395)*log((17*I*sqrt(2) - 161)*sqrt(-30817*I*sqrt( 2) - 14395) + 26499*x) + 3*(x^4 + 2*x^2 + 3)*sqrt(-30817*I*sqrt(2) - 14395 )*log((-17*I*sqrt(2) + 161)*sqrt(-30817*I*sqrt(2) - 14395) + 26499*x) - 57 96*x)/(x^4 + 2*x^2 + 3)
Time = 0.33 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.26 \[ \int \frac {x^4 \left (4+x^2+3 x^4+5 x^6\right )}{\left (3+2 x^2+x^4\right )^2} \, dx=\frac {5 x^{3}}{3} - 17 x + \frac {- 25 x^{3} - 75 x}{8 x^{4} + 16 x^{2} + 24} + \operatorname {RootSum} {\left (1048576 t^{4} + 29480960 t^{2} + 2106591003, \left ( t \mapsto t \log {\left (\frac {557056 t^{3}}{816619683} + \frac {166600064 t}{816619683} + x \right )} \right )\right )} \]
5*x**3/3 - 17*x + (-25*x**3 - 75*x)/(8*x**4 + 16*x**2 + 24) + RootSum(1048 576*_t**4 + 29480960*_t**2 + 2106591003, Lambda(_t, _t*log(557056*_t**3/81 6619683 + 166600064*_t/816619683 + x)))
\[ \int \frac {x^4 \left (4+x^2+3 x^4+5 x^6\right )}{\left (3+2 x^2+x^4\right )^2} \, dx=\int { \frac {{\left (5 \, x^{6} + 3 \, x^{4} + x^{2} + 4\right )} x^{4}}{{\left (x^{4} + 2 \, x^{2} + 3\right )}^{2}} \,d x } \]
5/3*x^3 - 17*x - 25/8*(x^3 + 3*x)/(x^4 + 2*x^2 + 3) + 1/8*integrate((127*x ^2 + 483)/(x^4 + 2*x^2 + 3), x)
Leaf count of result is larger than twice the leaf count of optimal. 573 vs. \(2 (163) = 326\).
Time = 0.59 (sec) , antiderivative size = 573, normalized size of antiderivative = 2.47 \[ \int \frac {x^4 \left (4+x^2+3 x^4+5 x^6\right )}{\left (3+2 x^2+x^4\right )^2} \, dx=\frac {5}{3} \, x^{3} - \frac {1}{20736} \, \sqrt {2} {\left (127 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 2286 \cdot 3^{\frac {3}{4}} \sqrt {2} \sqrt {6 \, \sqrt {3} + 18} {\left (\sqrt {3} - 3\right )} - 2286 \cdot 3^{\frac {3}{4}} {\left (\sqrt {3} + 3\right )} \sqrt {-6 \, \sqrt {3} + 18} + 127 \cdot 3^{\frac {3}{4}} {\left (-6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} - 17388 \cdot 3^{\frac {1}{4}} \sqrt {2} \sqrt {6 \, \sqrt {3} + 18} + 17388 \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}{20736} \, \sqrt {2} {\left (127 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 2286 \cdot 3^{\frac {3}{4}} \sqrt {2} \sqrt {6 \, \sqrt {3} + 18} {\left (\sqrt {3} - 3\right )} - 2286 \cdot 3^{\frac {3}{4}} {\left (\sqrt {3} + 3\right )} \sqrt {-6 \, \sqrt {3} + 18} + 127 \cdot 3^{\frac {3}{4}} {\left (-6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} - 17388 \cdot 3^{\frac {1}{4}} \sqrt {2} \sqrt {6 \, \sqrt {3} + 18} + 17388 \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}{41472} \, \sqrt {2} {\left (2286 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (\sqrt {3} + 3\right )} \sqrt {-6 \, \sqrt {3} + 18} - 127 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (-6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 127 \cdot 3^{\frac {3}{4}} {\left (6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 2286 \cdot 3^{\frac {3}{4}} \sqrt {6 \, \sqrt {3} + 18} {\left (\sqrt {3} - 3\right )} - 17388 \cdot 3^{\frac {1}{4}} \sqrt {2} \sqrt {-6 \, \sqrt {3} + 18} - 17388 \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}{41472} \, \sqrt {2} {\left (2286 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (\sqrt {3} + 3\right )} \sqrt {-6 \, \sqrt {3} + 18} - 127 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (-6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 127 \cdot 3^{\frac {3}{4}} {\left (6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 2286 \cdot 3^{\frac {3}{4}} \sqrt {6 \, \sqrt {3} + 18} {\left (\sqrt {3} - 3\right )} - 17388 \cdot 3^{\frac {1}{4}} \sqrt {2} \sqrt {-6 \, \sqrt {3} + 18} - 17388 \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 ) - 17 \, x - \frac {25 \, {\left (x^{3} + 3 \, x\right )}}{8 \, {\left (x^{4} + 2 \, x^{2} + 3\right )}} \]
5/3*x^3 - 1/20736*sqrt(2)*(127*3^(3/4)*sqrt(2)*(6*sqrt(3) + 18)^(3/2) + 22 86*3^(3/4)*sqrt(2)*sqrt(6*sqrt(3) + 18)*(sqrt(3) - 3) - 2286*3^(3/4)*(sqrt (3) + 3)*sqrt(-6*sqrt(3) + 18) + 127*3^(3/4)*(-6*sqrt(3) + 18)^(3/2) - 173 88*3^(1/4)*sqrt(2)*sqrt(6*sqrt(3) + 18) + 17388*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*sq rt(3) + 1/2)) - 1/20736*sqrt(2)*(127*3^(3/4)*sqrt(2)*(6*sqrt(3) + 18)^(3/2 ) + 2286*3^(3/4)*sqrt(2)*sqrt(6*sqrt(3) + 18)*(sqrt(3) - 3) - 2286*3^(3/4) *(sqrt(3) + 3)*sqrt(-6*sqrt(3) + 18) + 127*3^(3/4)*(-6*sqrt(3) + 18)^(3/2) - 17388*3^(1/4)*sqrt(2)*sqrt(6*sqrt(3) + 18) + 17388*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)) - 1/41472*sqrt(2)*(2286*3^(3/4)*sqrt(2)*(sqrt(3) + 3)* sqrt(-6*sqrt(3) + 18) - 127*3^(3/4)*sqrt(2)*(-6*sqrt(3) + 18)^(3/2) + 127* 3^(3/4)*(6*sqrt(3) + 18)^(3/2) + 2286*3^(3/4)*sqrt(6*sqrt(3) + 18)*(sqrt(3 ) - 3) - 17388*3^(1/4)*sqrt(2)*sqrt(-6*sqrt(3) + 18) - 17388*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)) + 1/41472*sqrt(2)*(2286*3^(3/4)*sqrt(2)*(sqrt(3) + 3)*sqrt(-6*sqrt(3) + 1 8) - 127*3^(3/4)*sqrt(2)*(-6*sqrt(3) + 18)^(3/2) + 127*3^(3/4)*(6*sqrt(3) + 18)^(3/2) + 2286*3^(3/4)*sqrt(6*sqrt(3) + 18)*(sqrt(3) - 3) - 17388*3^(1 /4)*sqrt(2)*sqrt(-6*sqrt(3) + 18) - 17388*3^(1/4)*sqrt(6*sqrt(3) + 18))*lo g(x^2 - 2*3^(1/4)*x*sqrt(-1/6*sqrt(3) + 1/2) + sqrt(3)) - 17*x - 25/8*(...
Time = 0.07 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.70 \[ \int \frac {x^4 \left (4+x^2+3 x^4+5 x^6\right )}{\left (3+2 x^2+x^4\right )^2} \, dx=\frac {5\,x^3}{3}-\frac {\frac {25\,x^3}{8}+\frac {75\,x}{8}}{x^4+2\,x^2+3}-17\,x+\frac {\mathrm {atan}\left (\frac {x\,\sqrt {-14395-\sqrt {2}\,30817{}\mathrm {i}}\,30817{}\mathrm {i}}{64\,\left (-\frac {1571667}{64}+\frac {\sqrt {2}\,14884611{}\mathrm {i}}{128}\right )}-\frac {30817\,\sqrt {2}\,x\,\sqrt {-14395-\sqrt {2}\,30817{}\mathrm {i}}}{128\,\left (-\frac {1571667}{64}+\frac {\sqrt {2}\,14884611{}\mathrm {i}}{128}\right )}\right )\,\sqrt {-14395-\sqrt {2}\,30817{}\mathrm {i}}\,1{}\mathrm {i}}{16}-\frac {\mathrm {atan}\left (\frac {x\,\sqrt {-14395+\sqrt {2}\,30817{}\mathrm {i}}\,30817{}\mathrm {i}}{64\,\left (\frac {1571667}{64}+\frac {\sqrt {2}\,14884611{}\mathrm {i}}{128}\right )}+\frac {30817\,\sqrt {2}\,x\,\sqrt {-14395+\sqrt {2}\,30817{}\mathrm {i}}}{128\,\left (\frac {1571667}{64}+\frac {\sqrt {2}\,14884611{}\mathrm {i}}{128}\right )}\right )\,\sqrt {-14395+\sqrt {2}\,30817{}\mathrm {i}}\,1{}\mathrm {i}}{16} \]
(atan((x*(- 2^(1/2)*30817i - 14395)^(1/2)*30817i)/(64*((2^(1/2)*14884611i) /128 - 1571667/64)) - (30817*2^(1/2)*x*(- 2^(1/2)*30817i - 14395)^(1/2))/( 128*((2^(1/2)*14884611i)/128 - 1571667/64)))*(- 2^(1/2)*30817i - 14395)^(1 /2)*1i)/16 - ((75*x)/8 + (25*x^3)/8)/(2*x^2 + x^4 + 3) - 17*x - (atan((x*( 2^(1/2)*30817i - 14395)^(1/2)*30817i)/(64*((2^(1/2)*14884611i)/128 + 15716 67/64)) + (30817*2^(1/2)*x*(2^(1/2)*30817i - 14395)^(1/2))/(128*((2^(1/2)* 14884611i)/128 + 1571667/64)))*(2^(1/2)*30817i - 14395)^(1/2)*1i)/16 + (5* x^3)/3