Integrand size = 31, antiderivative size = 235 \[ \int \frac {x^6 \left (4+x^2+3 x^4+5 x^6\right )}{\left (3+2 x^2+x^4\right )^3} \, dx=5 x+\frac {25 x \left (3-x^2\right )}{16 \left (3+2 x^2+x^4\right )^2}+\frac {7 x \left (11+58 x^2\right )}{64 \left (3+2 x^2+x^4\right )}+\frac {1}{256} \sqrt {827621+1176531 \sqrt {3}} \arctan \left (\frac {\sqrt {2 \left (-1+\sqrt {3}\right )}-2 x}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )-\frac {1}{256} \sqrt {827621+1176531 \sqrt {3}} \arctan \left (\frac {\sqrt {2 \left (-1+\sqrt {3}\right )}+2 x}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )-\frac {1}{512} \sqrt {-827621+1176531 \sqrt {3}} \log \left (\sqrt {3}-\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2\right )+\frac {1}{512} \sqrt {-827621+1176531 \sqrt {3}} \log \left (\sqrt {3}+\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2\right ) \]
5*x+25/16*x*(-x^2+3)/(x^4+2*x^2+3)^2+7/64*x*(58*x^2+11)/(x^4+2*x^2+3)-1/51 2*ln(x^2+3^(1/2)-x*(-2+2*3^(1/2))^(1/2))*(-827621+1176531*3^(1/2))^(1/2)+1 /512*ln(x^2+3^(1/2)+x*(-2+2*3^(1/2))^(1/2))*(-827621+1176531*3^(1/2))^(1/2 )+1/256*arctan((-2*x+(-2+2*3^(1/2))^(1/2))/(2+2*3^(1/2))^(1/2))*(827621+11 76531*3^(1/2))^(1/2)-1/256*arctan((2*x+(-2+2*3^(1/2))^(1/2))/(2+2*3^(1/2)) ^(1/2))*(827621+1176531*3^(1/2))^(1/2)
Result contains complex when optimal does not.
Time = 0.22 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.59 \[ \int \frac {x^6 \left (4+x^2+3 x^4+5 x^6\right )}{\left (3+2 x^2+x^4\right )^3} \, dx=\frac {1}{256} \left (\frac {4 x \left (3411+5112 x^2+4089 x^4+1686 x^6+320 x^8\right )}{\left (3+2 x^2+x^4\right )^2}-\frac {i \left (-2644 i+185 \sqrt {2}\right ) \arctan \left (\frac {x}{\sqrt {1-i \sqrt {2}}}\right )}{\sqrt {1-i \sqrt {2}}}+\frac {i \left (2644 i+185 \sqrt {2}\right ) \arctan \left (\frac {x}{\sqrt {1+i \sqrt {2}}}\right )}{\sqrt {1+i \sqrt {2}}}\right ) \]
((4*x*(3411 + 5112*x^2 + 4089*x^4 + 1686*x^6 + 320*x^8))/(3 + 2*x^2 + x^4) ^2 - (I*(-2644*I + 185*Sqrt[2])*ArcTan[x/Sqrt[1 - I*Sqrt[2]]])/Sqrt[1 - I* Sqrt[2]] + (I*(2644*I + 185*Sqrt[2])*ArcTan[x/Sqrt[1 + I*Sqrt[2]]])/Sqrt[1 + I*Sqrt[2]])/256
Time = 0.58 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.04, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {2197, 27, 2206, 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.
| \(\displaystyle \int \frac {x^6 \left (5 x^6+3 x^4+x^2+4\right )}{\left (x^4+2 x^2+3\right )^3} \, dx\) |
| \(\Big \downarrow \) 2197 |
| \(\displaystyle \frac {1}{96} \int -\frac {6 \left (-80 x^8+112 x^6-275 x^2+75\right )}{\left (x^4+2 x^2+3\right )^2}dx+\frac {25 x \left (3-x^2\right )}{16 \left (x^4+2 x^2+3\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {25 x \left (3-x^2\right )}{16 \left (x^4+2 x^2+3\right )^2}-\frac {1}{16} \int \frac {-80 x^8+112 x^6-275 x^2+75}{\left (x^4+2 x^2+3\right )^2}dx\) |
| \(\Big \downarrow \) 2206 |
| \(\displaystyle \frac {1}{16} \left (\frac {7 x \left (58 x^2+11\right )}{4 \left (x^4+2 x^2+3\right )}-\frac {1}{48} \int \frac {12 \left (-320 x^4+682 x^2+177\right )}{x^4+2 x^2+3}dx\right )+\frac {25 x \left (3-x^2\right )}{16 \left (x^4+2 x^2+3\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{16} \left (\frac {7 x \left (58 x^2+11\right )}{4 \left (x^4+2 x^2+3\right )}-\frac {1}{4} \int \frac {-320 x^4+682 x^2+177}{x^4+2 x^2+3}dx\right )+\frac {25 x \left (3-x^2\right )}{16 \left (x^4+2 x^2+3\right )^2}\) |
| \(\Big \downarrow \) 2205 |
| \(\displaystyle \frac {1}{16} \left (\frac {7 x \left (58 x^2+11\right )}{4 \left (x^4+2 x^2+3\right )}-\frac {1}{4} \int \left (\frac {1322 x^2+1137}{x^4+2 x^2+3}-320\right )dx\right )+\frac {25 x \left (3-x^2\right )}{16 \left (x^4+2 x^2+3\right )^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{16} \left (\frac {1}{4} \left (\frac {1}{4} \sqrt {827621+1176531 \sqrt {3}} \arctan \left (\frac {\sqrt {2 \left (\sqrt {3}-1\right )}-2 x}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )-\frac {1}{4} \sqrt {827621+1176531 \sqrt {3}} \arctan \left (\frac {2 x+\sqrt {2 \left (\sqrt {3}-1\right )}}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )-\frac {1}{8} \sqrt {1176531 \sqrt {3}-827621} \log \left (x^2-\sqrt {2 \left (\sqrt {3}-1\right )} x+\sqrt {3}\right )+\frac {1}{8} \sqrt {1176531 \sqrt {3}-827621} \log \left (x^2+\sqrt {2 \left (\sqrt {3}-1\right )} x+\sqrt {3}\right )+320 x\right )+\frac {7 x \left (58 x^2+11\right )}{4 \left (x^4+2 x^2+3\right )}\right )+\frac {25 x \left (3-x^2\right )}{16 \left (x^4+2 x^2+3\right )^2}\) |
(25*x*(3 - x^2))/(16*(3 + 2*x^2 + x^4)^2) + ((7*x*(11 + 58*x^2))/(4*(3 + 2 *x^2 + x^4)) + (320*x + (Sqrt[827621 + 1176531*Sqrt[3]]*ArcTan[(Sqrt[2*(-1 + Sqrt[3])] - 2*x)/Sqrt[2*(1 + Sqrt[3])]])/4 - (Sqrt[827621 + 1176531*Sqr t[3]]*ArcTan[(Sqrt[2*(-1 + Sqrt[3])] + 2*x)/Sqrt[2*(1 + Sqrt[3])]])/4 - (S qrt[-827621 + 1176531*Sqrt[3]]*Log[Sqrt[3] - Sqrt[2*(-1 + Sqrt[3])]*x + x^ 2])/8 + (Sqrt[-827621 + 1176531*Sqrt[3]]*Log[Sqrt[3] + Sqrt[2*(-1 + Sqrt[3 ])]*x + x^2])/8)/4)/16
3.2.19.3.1 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
Int[(Px_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{d = Coeff[PolynomialRemainder[Px, a + b*x^2 + c*x^4, x], x, 0], e = Coeff[Poly nomialRemainder[Px, 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)*PolynomialQuotient[Px, a + b*x^2 + c*x^4, x] + 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]] /; FreeQ[{a, b, c}, x] && PolyQ[Px, x ^2] && Expon[Px, x^2] > 1 && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.08 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.31
| method | result | size |
| risch | \(5 x +\frac {\frac {203}{32} x^{7}+\frac {889}{64} x^{5}+\frac {159}{8} x^{3}+\frac {531}{64} x}{\left (x^{4}+2 x^{2}+3\right )^{2}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}+2 \textit {\_Z}^{2}+3\right )}{\sum }\frac {\left (-1322 \textit {\_R}^{2}-1137\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}+\textit {\_R}}\right )}{256}\) | \(74\) |
| default | \(5 x -\frac {-\frac {203}{32} x^{7}-\frac {889}{64} x^{5}-\frac {159}{8} x^{3}-\frac {531}{64} x}{\left (x^{4}+2 x^{2}+3\right )^{2}}-\frac {\left (943 \sqrt {-2+2 \sqrt {3}}\, \sqrt {3}+185 \sqrt {-2+2 \sqrt {3}}\right ) \ln \left (x^{2}+\sqrt {3}-x \sqrt {-2+2 \sqrt {3}}\right )}{1024}-\frac {\left (1516 \sqrt {3}+\frac {\left (943 \sqrt {-2+2 \sqrt {3}}\, \sqrt {3}+185 \sqrt {-2+2 \sqrt {3}}\right ) \sqrt {-2+2 \sqrt {3}}}{2}\right ) \arctan \left (\frac {2 x -\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{256 \sqrt {2+2 \sqrt {3}}}-\frac {\left (-943 \sqrt {-2+2 \sqrt {3}}\, \sqrt {3}-185 \sqrt {-2+2 \sqrt {3}}\right ) \ln \left (x^{2}+\sqrt {3}+x \sqrt {-2+2 \sqrt {3}}\right )}{1024}-\frac {\left (1516 \sqrt {3}-\frac {\left (-943 \sqrt {-2+2 \sqrt {3}}\, \sqrt {3}-185 \sqrt {-2+2 \sqrt {3}}\right ) \sqrt {-2+2 \sqrt {3}}}{2}\right ) \arctan \left (\frac {2 x +\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{256 \sqrt {2+2 \sqrt {3}}}\) | \(291\) |
5*x+(203/32*x^7+889/64*x^5+159/8*x^3+531/64*x)/(x^4+2*x^2+3)^2+1/256*sum(( -1322*_R^2-1137)/(_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 = 272, normalized size of antiderivative = 1.16 \[ \int \frac {x^6 \left (4+x^2+3 x^4+5 x^6\right )}{\left (3+2 x^2+x^4\right )^3} \, dx=\frac {2560 \, x^{9} + 13488 \, x^{7} + 32712 \, x^{5} + 40896 \, x^{3} - \sqrt {2} {\left (x^{8} + 4 \, x^{6} + 10 \, x^{4} + 12 \, x^{2} + 9\right )} \sqrt {1316761 i \, \sqrt {2} - 827621} \log \left ({\left (379 \, \sqrt {2} + 943 i\right )} \sqrt {1316761 i \, \sqrt {2} - 827621} + 1176531 \, x\right ) + \sqrt {2} {\left (x^{8} + 4 \, x^{6} + 10 \, x^{4} + 12 \, x^{2} + 9\right )} \sqrt {1316761 i \, \sqrt {2} - 827621} \log \left (-{\left (379 \, \sqrt {2} + 943 i\right )} \sqrt {1316761 i \, \sqrt {2} - 827621} + 1176531 \, x\right ) - \sqrt {2} {\left (x^{8} + 4 \, x^{6} + 10 \, x^{4} + 12 \, x^{2} + 9\right )} \sqrt {-1316761 i \, \sqrt {2} - 827621} \log \left ({\left (379 \, \sqrt {2} - 943 i\right )} \sqrt {-1316761 i \, \sqrt {2} - 827621} + 1176531 \, x\right ) + \sqrt {2} {\left (x^{8} + 4 \, x^{6} + 10 \, x^{4} + 12 \, x^{2} + 9\right )} \sqrt {-1316761 i \, \sqrt {2} - 827621} \log \left (-{\left (379 \, \sqrt {2} - 943 i\right )} \sqrt {-1316761 i \, \sqrt {2} - 827621} + 1176531 \, x\right ) + 27288 \, x}{512 \, {\left (x^{8} + 4 \, x^{6} + 10 \, x^{4} + 12 \, x^{2} + 9\right )}} \]
1/512*(2560*x^9 + 13488*x^7 + 32712*x^5 + 40896*x^3 - sqrt(2)*(x^8 + 4*x^6 + 10*x^4 + 12*x^2 + 9)*sqrt(1316761*I*sqrt(2) - 827621)*log((379*sqrt(2) + 943*I)*sqrt(1316761*I*sqrt(2) - 827621) + 1176531*x) + sqrt(2)*(x^8 + 4* x^6 + 10*x^4 + 12*x^2 + 9)*sqrt(1316761*I*sqrt(2) - 827621)*log(-(379*sqrt (2) + 943*I)*sqrt(1316761*I*sqrt(2) - 827621) + 1176531*x) - sqrt(2)*(x^8 + 4*x^6 + 10*x^4 + 12*x^2 + 9)*sqrt(-1316761*I*sqrt(2) - 827621)*log((379* sqrt(2) - 943*I)*sqrt(-1316761*I*sqrt(2) - 827621) + 1176531*x) + sqrt(2)* (x^8 + 4*x^6 + 10*x^4 + 12*x^2 + 9)*sqrt(-1316761*I*sqrt(2) - 827621)*log( -(379*sqrt(2) - 943*I)*sqrt(-1316761*I*sqrt(2) - 827621) + 1176531*x) + 27 288*x)/(x^8 + 4*x^6 + 10*x^4 + 12*x^2 + 9)
Time = 0.34 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.30 \[ \int \frac {x^6 \left (4+x^2+3 x^4+5 x^6\right )}{\left (3+2 x^2+x^4\right )^3} \, dx=5 x + \frac {406 x^{7} + 889 x^{5} + 1272 x^{3} + 531 x}{64 x^{8} + 256 x^{6} + 640 x^{4} + 768 x^{2} + 576} + \operatorname {RootSum} {\left (17179869184 t^{4} + 216955879424 t^{2} + 4152675581883, \left ( t \mapsto t \log {\left (- \frac {31641829376 t^{3}}{1549210136091} - \frac {455309168896 t}{1549210136091} + x \right )} \right )\right )} \]
5*x + (406*x**7 + 889*x**5 + 1272*x**3 + 531*x)/(64*x**8 + 256*x**6 + 640* x**4 + 768*x**2 + 576) + RootSum(17179869184*_t**4 + 216955879424*_t**2 + 4152675581883, Lambda(_t, _t*log(-31641829376*_t**3/1549210136091 - 455309 168896*_t/1549210136091 + x)))
\[ \int \frac {x^6 \left (4+x^2+3 x^4+5 x^6\right )}{\left (3+2 x^2+x^4\right )^3} \, dx=\int { \frac {{\left (5 \, x^{6} + 3 \, x^{4} + x^{2} + 4\right )} x^{6}}{{\left (x^{4} + 2 \, x^{2} + 3\right )}^{3}} \,d x } \]
5*x + 1/64*(406*x^7 + 889*x^5 + 1272*x^3 + 531*x)/(x^8 + 4*x^6 + 10*x^4 + 12*x^2 + 9) - 1/64*integrate((1322*x^2 + 1137)/(x^4 + 2*x^2 + 3), x)
Leaf count of result is larger than twice the leaf count of optimal. 580 vs. \(2 (180) = 360\).
Time = 0.72 (sec) , antiderivative size = 580, normalized size of antiderivative = 2.47 \[ \int \frac {x^6 \left (4+x^2+3 x^4+5 x^6\right )}{\left (3+2 x^2+x^4\right )^3} \, dx=\frac {1}{82944} \, \sqrt {2} {\left (661 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 11898 \cdot 3^{\frac {3}{4}} \sqrt {2} \sqrt {6 \, \sqrt {3} + 18} {\left (\sqrt {3} - 3\right )} - 11898 \cdot 3^{\frac {3}{4}} {\left (\sqrt {3} + 3\right )} \sqrt {-6 \, \sqrt {3} + 18} + 661 \cdot 3^{\frac {3}{4}} {\left (-6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} - 20466 \cdot 3^{\frac {1}{4}} \sqrt {2} \sqrt {6 \, \sqrt {3} + 18} + 20466 \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}{82944} \, \sqrt {2} {\left (661 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 11898 \cdot 3^{\frac {3}{4}} \sqrt {2} \sqrt {6 \, \sqrt {3} + 18} {\left (\sqrt {3} - 3\right )} - 11898 \cdot 3^{\frac {3}{4}} {\left (\sqrt {3} + 3\right )} \sqrt {-6 \, \sqrt {3} + 18} + 661 \cdot 3^{\frac {3}{4}} {\left (-6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} - 20466 \cdot 3^{\frac {1}{4}} \sqrt {2} \sqrt {6 \, \sqrt {3} + 18} + 20466 \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}{165888} \, \sqrt {2} {\left (11898 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (\sqrt {3} + 3\right )} \sqrt {-6 \, \sqrt {3} + 18} - 661 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (-6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 661 \cdot 3^{\frac {3}{4}} {\left (6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 11898 \cdot 3^{\frac {3}{4}} \sqrt {6 \, \sqrt {3} + 18} {\left (\sqrt {3} - 3\right )} - 20466 \cdot 3^{\frac {1}{4}} \sqrt {2} \sqrt {-6 \, \sqrt {3} + 18} - 20466 \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}{165888} \, \sqrt {2} {\left (11898 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (\sqrt {3} + 3\right )} \sqrt {-6 \, \sqrt {3} + 18} - 661 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (-6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 661 \cdot 3^{\frac {3}{4}} {\left (6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 11898 \cdot 3^{\frac {3}{4}} \sqrt {6 \, \sqrt {3} + 18} {\left (\sqrt {3} - 3\right )} - 20466 \cdot 3^{\frac {1}{4}} \sqrt {2} \sqrt {-6 \, \sqrt {3} + 18} - 20466 \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 {406 \, x^{7} + 889 \, x^{5} + 1272 \, x^{3} + 531 \, x}{64 \, {\left (x^{4} + 2 \, x^{2} + 3\right )}^{2}} \]
1/82944*sqrt(2)*(661*3^(3/4)*sqrt(2)*(6*sqrt(3) + 18)^(3/2) + 11898*3^(3/4 )*sqrt(2)*sqrt(6*sqrt(3) + 18)*(sqrt(3) - 3) - 11898*3^(3/4)*(sqrt(3) + 3) *sqrt(-6*sqrt(3) + 18) + 661*3^(3/4)*(-6*sqrt(3) + 18)^(3/2) - 20466*3^(1/ 4)*sqrt(2)*sqrt(6*sqrt(3) + 18) + 20466*3^(1/4)*sqrt(-6*sqrt(3) + 18))*arc tan(1/3*3^(3/4)*(x + 3^(1/4)*sqrt(-1/6*sqrt(3) + 1/2))/sqrt(1/6*sqrt(3) + 1/2)) + 1/82944*sqrt(2)*(661*3^(3/4)*sqrt(2)*(6*sqrt(3) + 18)^(3/2) + 1189 8*3^(3/4)*sqrt(2)*sqrt(6*sqrt(3) + 18)*(sqrt(3) - 3) - 11898*3^(3/4)*(sqrt (3) + 3)*sqrt(-6*sqrt(3) + 18) + 661*3^(3/4)*(-6*sqrt(3) + 18)^(3/2) - 204 66*3^(1/4)*sqrt(2)*sqrt(6*sqrt(3) + 18) + 20466*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)) + 1/165888*sqrt(2)*(11898*3^(3/4)*sqrt(2)*(sqrt(3) + 3)*sqrt (-6*sqrt(3) + 18) - 661*3^(3/4)*sqrt(2)*(-6*sqrt(3) + 18)^(3/2) + 661*3^(3 /4)*(6*sqrt(3) + 18)^(3/2) + 11898*3^(3/4)*sqrt(6*sqrt(3) + 18)*(sqrt(3) - 3) - 20466*3^(1/4)*sqrt(2)*sqrt(-6*sqrt(3) + 18) - 20466*3^(1/4)*sqrt(6*s qrt(3) + 18))*log(x^2 + 2*3^(1/4)*x*sqrt(-1/6*sqrt(3) + 1/2) + sqrt(3)) - 1/165888*sqrt(2)*(11898*3^(3/4)*sqrt(2)*(sqrt(3) + 3)*sqrt(-6*sqrt(3) + 18 ) - 661*3^(3/4)*sqrt(2)*(-6*sqrt(3) + 18)^(3/2) + 661*3^(3/4)*(6*sqrt(3) + 18)^(3/2) + 11898*3^(3/4)*sqrt(6*sqrt(3) + 18)*(sqrt(3) - 3) - 20466*3^(1 /4)*sqrt(2)*sqrt(-6*sqrt(3) + 18) - 20466*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)) + 5*x + 1/64*(4...
Time = 8.85 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.75 \[ \int \frac {x^6 \left (4+x^2+3 x^4+5 x^6\right )}{\left (3+2 x^2+x^4\right )^3} \, dx=5\,x+\frac {\frac {203\,x^7}{32}+\frac {889\,x^5}{64}+\frac {159\,x^3}{8}+\frac {531\,x}{64}}{x^8+4\,x^6+10\,x^4+12\,x^2+9}+\frac {\mathrm {atan}\left (\frac {x\,\sqrt {-1655242-\sqrt {2}\,2633522{}\mathrm {i}}\,1316761{}\mathrm {i}}{131072\,\left (-\frac {3725116869}{131072}+\frac {\sqrt {2}\,1497157257{}\mathrm {i}}{131072}\right )}+\frac {1316761\,\sqrt {2}\,x\,\sqrt {-1655242-\sqrt {2}\,2633522{}\mathrm {i}}}{262144\,\left (-\frac {3725116869}{131072}+\frac {\sqrt {2}\,1497157257{}\mathrm {i}}{131072}\right )}\right )\,\sqrt {-1655242-\sqrt {2}\,2633522{}\mathrm {i}}\,1{}\mathrm {i}}{256}-\frac {\mathrm {atan}\left (\frac {x\,\sqrt {-1655242+\sqrt {2}\,2633522{}\mathrm {i}}\,1316761{}\mathrm {i}}{131072\,\left (\frac {3725116869}{131072}+\frac {\sqrt {2}\,1497157257{}\mathrm {i}}{131072}\right )}-\frac {1316761\,\sqrt {2}\,x\,\sqrt {-1655242+\sqrt {2}\,2633522{}\mathrm {i}}}{262144\,\left (\frac {3725116869}{131072}+\frac {\sqrt {2}\,1497157257{}\mathrm {i}}{131072}\right )}\right )\,\sqrt {-1655242+\sqrt {2}\,2633522{}\mathrm {i}}\,1{}\mathrm {i}}{256} \]
5*x + (atan((x*(- 2^(1/2)*2633522i - 1655242)^(1/2)*1316761i)/(131072*((2^ (1/2)*1497157257i)/131072 - 3725116869/131072)) + (1316761*2^(1/2)*x*(- 2^ (1/2)*2633522i - 1655242)^(1/2))/(262144*((2^(1/2)*1497157257i)/131072 - 3 725116869/131072)))*(- 2^(1/2)*2633522i - 1655242)^(1/2)*1i)/256 - (atan(( x*(2^(1/2)*2633522i - 1655242)^(1/2)*1316761i)/(131072*((2^(1/2)*149715725 7i)/131072 + 3725116869/131072)) - (1316761*2^(1/2)*x*(2^(1/2)*2633522i - 1655242)^(1/2))/(262144*((2^(1/2)*1497157257i)/131072 + 3725116869/131072) ))*(2^(1/2)*2633522i - 1655242)^(1/2)*1i)/256 + ((531*x)/64 + (159*x^3)/8 + (889*x^5)/64 + (203*x^7)/32)/(12*x^2 + 10*x^4 + 4*x^6 + x^8 + 9)