Integrand size = 31, antiderivative size = 238 \[ \int \frac {x^4 \left (4+x^2+3 x^4+5 x^6\right )}{\left (3+2 x^2+x^4\right )^3} \, dx=-\frac {25 x \left (3+x^2\right )}{16 \left (3+2 x^2+x^4\right )^2}+\frac {x \left (238-59 x^2\right )}{64 \left (3+2 x^2+x^4\right )}-\frac {1}{256} \sqrt {3 \left (-48835+32827 \sqrt {3}\right )} \arctan \left (\frac {\sqrt {2 \left (-1+\sqrt {3}\right )}-2 x}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )+\frac {1}{256} \sqrt {3 \left (-48835+32827 \sqrt {3}\right )} \arctan \left (\frac {\sqrt {2 \left (-1+\sqrt {3}\right )}+2 x}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )+\frac {1}{512} \sqrt {3 \left (48835+32827 \sqrt {3}\right )} \log \left (\sqrt {3}-\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2\right )-\frac {1}{512} \sqrt {3 \left (48835+32827 \sqrt {3}\right )} \log \left (\sqrt {3}+\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2\right ) \]
-25/16*x*(x^2+3)/(x^4+2*x^2+3)^2+1/64*x*(-59*x^2+238)/(x^4+2*x^2+3)-1/256* arctan((-2*x+(-2+2*3^(1/2))^(1/2))/(2+2*3^(1/2))^(1/2))*(-146505+98481*3^( 1/2))^(1/2)+1/256*arctan((2*x+(-2+2*3^(1/2))^(1/2))/(2+2*3^(1/2))^(1/2))*( -146505+98481*3^(1/2))^(1/2)+1/512*ln(x^2+3^(1/2)-x*(-2+2*3^(1/2))^(1/2))* (146505+98481*3^(1/2))^(1/2)-1/512*ln(x^2+3^(1/2)+x*(-2+2*3^(1/2))^(1/2))* (146505+98481*3^(1/2))^(1/2)
Result contains complex when optimal does not.
Time = 0.20 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.54 \[ \int \frac {x^4 \left (4+x^2+3 x^4+5 x^6\right )}{\left (3+2 x^2+x^4\right )^3} \, dx=\frac {1}{256} \left (\frac {4 x \left (414+199 x^2+120 x^4-59 x^6\right )}{\left (3+2 x^2+x^4\right )^2}+\frac {3 \left (174+133 i \sqrt {2}\right ) \arctan \left (\frac {x}{\sqrt {1-i \sqrt {2}}}\right )}{\sqrt {1-i \sqrt {2}}}+\frac {3 \left (174-133 i \sqrt {2}\right ) \arctan \left (\frac {x}{\sqrt {1+i \sqrt {2}}}\right )}{\sqrt {1+i \sqrt {2}}}\right ) \]
((4*x*(414 + 199*x^2 + 120*x^4 - 59*x^6))/(3 + 2*x^2 + x^4)^2 + (3*(174 + (133*I)*Sqrt[2])*ArcTan[x/Sqrt[1 - I*Sqrt[2]]])/Sqrt[1 - I*Sqrt[2]] + (3*( 174 - (133*I)*Sqrt[2])*ArcTan[x/Sqrt[1 + I*Sqrt[2]]])/Sqrt[1 + I*Sqrt[2]]) /256
Time = 0.57 (sec) , antiderivative size = 292, normalized size of antiderivative = 1.23, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.323, Rules used = {2197, 27, 2206, 27, 1483, 1142, 25, 1083, 217, 1103}
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.
| \(\displaystyle \int \frac {x^4 \left (5 x^6+3 x^4+x^2+4\right )}{\left (x^4+2 x^2+3\right )^3} \, dx\) |
| \(\Big \downarrow \) 2197 |
| \(\displaystyle \frac {1}{96} \int \frac {6 \left (80 x^6-112 x^4-125 x^2+75\right )}{\left (x^4+2 x^2+3\right )^2}dx-\frac {25 x \left (x^2+3\right )}{16 \left (x^4+2 x^2+3\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{16} \int \frac {80 x^6-112 x^4-125 x^2+75}{\left (x^4+2 x^2+3\right )^2}dx-\frac {25 x \left (x^2+3\right )}{16 \left (x^4+2 x^2+3\right )^2}\) |
| \(\Big \downarrow \) 2206 |
| \(\displaystyle \frac {1}{16} \left (\frac {1}{48} \int -\frac {36 \left (46-87 x^2\right )}{x^4+2 x^2+3}dx+\frac {x \left (238-59 x^2\right )}{4 \left (x^4+2 x^2+3\right )}\right )-\frac {25 x \left (x^2+3\right )}{16 \left (x^4+2 x^2+3\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{16} \left (\frac {x \left (238-59 x^2\right )}{4 \left (x^4+2 x^2+3\right )}-\frac {3}{4} \int \frac {46-87 x^2}{x^4+2 x^2+3}dx\right )-\frac {25 x \left (x^2+3\right )}{16 \left (x^4+2 x^2+3\right )^2}\) |
| \(\Big \downarrow \) 1483 |
| \(\displaystyle \frac {1}{16} \left (\frac {x \left (238-59 x^2\right )}{4 \left (x^4+2 x^2+3\right )}-\frac {3}{4} \left (\frac {\int \frac {46 \sqrt {2 \left (-1+\sqrt {3}\right )}-\left (46+87 \sqrt {3}\right ) x}{x^2-\sqrt {2 \left (-1+\sqrt {3}\right )} x+\sqrt {3}}dx}{2 \sqrt {6 \left (\sqrt {3}-1\right )}}+\frac {\int \frac {\left (46+87 \sqrt {3}\right ) x+46 \sqrt {2 \left (-1+\sqrt {3}\right )}}{x^2+\sqrt {2 \left (-1+\sqrt {3}\right )} x+\sqrt {3}}dx}{2 \sqrt {6 \left (\sqrt {3}-1\right )}}\right )\right )-\frac {25 x \left (x^2+3\right )}{16 \left (x^4+2 x^2+3\right )^2}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {1}{16} \left (\frac {x \left (238-59 x^2\right )}{4 \left (x^4+2 x^2+3\right )}-\frac {3}{4} \left (\frac {-\frac {1}{2} \sqrt {65654 \sqrt {3}-97670} \int \frac {1}{x^2-\sqrt {2 \left (-1+\sqrt {3}\right )} x+\sqrt {3}}dx-\frac {1}{2} \left (46+87 \sqrt {3}\right ) \int -\frac {\sqrt {2 \left (-1+\sqrt {3}\right )}-2 x}{x^2-\sqrt {2 \left (-1+\sqrt {3}\right )} x+\sqrt {3}}dx}{2 \sqrt {6 \left (\sqrt {3}-1\right )}}+\frac {\frac {1}{2} \left (46+87 \sqrt {3}\right ) \int \frac {2 x+\sqrt {2 \left (-1+\sqrt {3}\right )}}{x^2+\sqrt {2 \left (-1+\sqrt {3}\right )} x+\sqrt {3}}dx-\frac {1}{2} \sqrt {65654 \sqrt {3}-97670} \int \frac {1}{x^2+\sqrt {2 \left (-1+\sqrt {3}\right )} x+\sqrt {3}}dx}{2 \sqrt {6 \left (\sqrt {3}-1\right )}}\right )\right )-\frac {25 x \left (x^2+3\right )}{16 \left (x^4+2 x^2+3\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{16} \left (\frac {x \left (238-59 x^2\right )}{4 \left (x^4+2 x^2+3\right )}-\frac {3}{4} \left (\frac {\frac {1}{2} \left (46+87 \sqrt {3}\right ) \int \frac {\sqrt {2 \left (-1+\sqrt {3}\right )}-2 x}{x^2-\sqrt {2 \left (-1+\sqrt {3}\right )} x+\sqrt {3}}dx-\frac {1}{2} \sqrt {65654 \sqrt {3}-97670} \int \frac {1}{x^2-\sqrt {2 \left (-1+\sqrt {3}\right )} x+\sqrt {3}}dx}{2 \sqrt {6 \left (\sqrt {3}-1\right )}}+\frac {\frac {1}{2} \left (46+87 \sqrt {3}\right ) \int \frac {2 x+\sqrt {2 \left (-1+\sqrt {3}\right )}}{x^2+\sqrt {2 \left (-1+\sqrt {3}\right )} x+\sqrt {3}}dx-\frac {1}{2} \sqrt {65654 \sqrt {3}-97670} \int \frac {1}{x^2+\sqrt {2 \left (-1+\sqrt {3}\right )} x+\sqrt {3}}dx}{2 \sqrt {6 \left (\sqrt {3}-1\right )}}\right )\right )-\frac {25 x \left (x^2+3\right )}{16 \left (x^4+2 x^2+3\right )^2}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {1}{16} \left (\frac {x \left (238-59 x^2\right )}{4 \left (x^4+2 x^2+3\right )}-\frac {3}{4} \left (\frac {\frac {1}{2} \left (46+87 \sqrt {3}\right ) \int \frac {\sqrt {2 \left (-1+\sqrt {3}\right )}-2 x}{x^2-\sqrt {2 \left (-1+\sqrt {3}\right )} x+\sqrt {3}}dx+\sqrt {65654 \sqrt {3}-97670} \int \frac {1}{-\left (2 x-\sqrt {2 \left (-1+\sqrt {3}\right )}\right )^2-2 \left (1+\sqrt {3}\right )}d\left (2 x-\sqrt {2 \left (-1+\sqrt {3}\right )}\right )}{2 \sqrt {6 \left (\sqrt {3}-1\right )}}+\frac {\frac {1}{2} \left (46+87 \sqrt {3}\right ) \int \frac {2 x+\sqrt {2 \left (-1+\sqrt {3}\right )}}{x^2+\sqrt {2 \left (-1+\sqrt {3}\right )} x+\sqrt {3}}dx+\sqrt {65654 \sqrt {3}-97670} \int \frac {1}{-\left (2 x+\sqrt {2 \left (-1+\sqrt {3}\right )}\right )^2-2 \left (1+\sqrt {3}\right )}d\left (2 x+\sqrt {2 \left (-1+\sqrt {3}\right )}\right )}{2 \sqrt {6 \left (\sqrt {3}-1\right )}}\right )\right )-\frac {25 x \left (x^2+3\right )}{16 \left (x^4+2 x^2+3\right )^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{16} \left (\frac {x \left (238-59 x^2\right )}{4 \left (x^4+2 x^2+3\right )}-\frac {3}{4} \left (\frac {\frac {1}{2} \left (46+87 \sqrt {3}\right ) \int \frac {\sqrt {2 \left (-1+\sqrt {3}\right )}-2 x}{x^2-\sqrt {2 \left (-1+\sqrt {3}\right )} x+\sqrt {3}}dx-\sqrt {\frac {65654 \sqrt {3}-97670}{2 \left (1+\sqrt {3}\right )}} \arctan \left (\frac {2 x-\sqrt {2 \left (\sqrt {3}-1\right )}}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )}{2 \sqrt {6 \left (\sqrt {3}-1\right )}}+\frac {\frac {1}{2} \left (46+87 \sqrt {3}\right ) \int \frac {2 x+\sqrt {2 \left (-1+\sqrt {3}\right )}}{x^2+\sqrt {2 \left (-1+\sqrt {3}\right )} x+\sqrt {3}}dx-\sqrt {\frac {65654 \sqrt {3}-97670}{2 \left (1+\sqrt {3}\right )}} \arctan \left (\frac {2 x+\sqrt {2 \left (\sqrt {3}-1\right )}}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )}{2 \sqrt {6 \left (\sqrt {3}-1\right )}}\right )\right )-\frac {25 x \left (x^2+3\right )}{16 \left (x^4+2 x^2+3\right )^2}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {1}{16} \left (\frac {x \left (238-59 x^2\right )}{4 \left (x^4+2 x^2+3\right )}-\frac {3}{4} \left (\frac {-\sqrt {\frac {65654 \sqrt {3}-97670}{2 \left (1+\sqrt {3}\right )}} \arctan \left (\frac {2 x-\sqrt {2 \left (\sqrt {3}-1\right )}}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )-\frac {1}{2} \left (46+87 \sqrt {3}\right ) \log \left (x^2-\sqrt {2 \left (\sqrt {3}-1\right )} x+\sqrt {3}\right )}{2 \sqrt {6 \left (\sqrt {3}-1\right )}}+\frac {\frac {1}{2} \left (46+87 \sqrt {3}\right ) \log \left (x^2+\sqrt {2 \left (\sqrt {3}-1\right )} x+\sqrt {3}\right )-\sqrt {\frac {65654 \sqrt {3}-97670}{2 \left (1+\sqrt {3}\right )}} \arctan \left (\frac {2 x+\sqrt {2 \left (\sqrt {3}-1\right )}}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )}{2 \sqrt {6 \left (\sqrt {3}-1\right )}}\right )\right )-\frac {25 x \left (x^2+3\right )}{16 \left (x^4+2 x^2+3\right )^2}\) |
(-25*x*(3 + x^2))/(16*(3 + 2*x^2 + x^4)^2) + ((x*(238 - 59*x^2))/(4*(3 + 2 *x^2 + x^4)) - (3*((-(Sqrt[(-97670 + 65654*Sqrt[3])/(2*(1 + Sqrt[3]))]*Arc Tan[(-Sqrt[2*(-1 + Sqrt[3])] + 2*x)/Sqrt[2*(1 + Sqrt[3])]]) - ((46 + 87*Sq rt[3])*Log[Sqrt[3] - Sqrt[2*(-1 + Sqrt[3])]*x + x^2])/2)/(2*Sqrt[6*(-1 + S qrt[3])]) + (-(Sqrt[(-97670 + 65654*Sqrt[3])/(2*(1 + Sqrt[3]))]*ArcTan[(Sq rt[2*(-1 + Sqrt[3])] + 2*x)/Sqrt[2*(1 + Sqrt[3])]]) + ((46 + 87*Sqrt[3])*L og[Sqrt[3] + Sqrt[2*(-1 + Sqrt[3])]*x + x^2])/2)/(2*Sqrt[6*(-1 + Sqrt[3])] )))/4)/16
3.2.20.3.1 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[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : > With[{q = Rt[a/c, 2]}, With[{r = Rt[2*q - b/c, 2]}, Simp[1/(2*c*q*r) In t[(d*r - (d - e*q)*x)/(q - r*x + x^2), x], x] + Simp[1/(2*c*q*r) Int[(d*r + (d - e*q)*x)/(q + r*x + x^2), x], x]]] /; FreeQ[{a, b, c, d, e}, x] && N eQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NegQ[b^2 - 4*a*c]
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)^(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 = 71, normalized size of antiderivative = 0.30
| method | result | size |
| risch | \(\frac {-\frac {59}{64} x^{7}+\frac {15}{8} x^{5}+\frac {199}{64} x^{3}+\frac {207}{32} x}{\left (x^{4}+2 x^{2}+3\right )^{2}}+\frac {3 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}+2 \textit {\_Z}^{2}+3\right )}{\sum }\frac {\left (87 \textit {\_R}^{2}-46\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}+\textit {\_R}}\right )}{256}\) | \(71\) |
| default | \(\frac {-\frac {59}{64} x^{7}+\frac {15}{8} x^{5}+\frac {199}{64} x^{3}+\frac {207}{32} x}{\left (x^{4}+2 x^{2}+3\right )^{2}}+\frac {\left (307 \sqrt {-2+2 \sqrt {3}}\, \sqrt {3}+399 \sqrt {-2+2 \sqrt {3}}\right ) \ln \left (x^{2}+\sqrt {3}-x \sqrt {-2+2 \sqrt {3}}\right )}{1024}+\frac {\left (-184 \sqrt {3}+\frac {\left (307 \sqrt {-2+2 \sqrt {3}}\, \sqrt {3}+399 \sqrt {-2+2 \sqrt {3}}\right ) \sqrt {-2+2 \sqrt {3}}}{2}\right ) \arctan \left (\frac {2 x -\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{256 \sqrt {2+2 \sqrt {3}}}+\frac {\left (-307 \sqrt {-2+2 \sqrt {3}}\, \sqrt {3}-399 \sqrt {-2+2 \sqrt {3}}\right ) \ln \left (x^{2}+\sqrt {3}+x \sqrt {-2+2 \sqrt {3}}\right )}{1024}+\frac {\left (-184 \sqrt {3}-\frac {\left (-307 \sqrt {-2+2 \sqrt {3}}\, \sqrt {3}-399 \sqrt {-2+2 \sqrt {3}}\right ) \sqrt {-2+2 \sqrt {3}}}{2}\right ) \arctan \left (\frac {2 x +\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{256 \sqrt {2+2 \sqrt {3}}}\) | \(287\) |
(-59/64*x^7+15/8*x^5+199/64*x^3+207/32*x)/(x^4+2*x^2+3)^2+3/256*sum((87*_R ^2-46)/(_R^3+_R)*ln(x-_R),_R=RootOf(_Z^4+2*_Z^2+3))
Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 267, normalized size of antiderivative = 1.12 \[ \int \frac {x^4 \left (4+x^2+3 x^4+5 x^6\right )}{\left (3+2 x^2+x^4\right )^3} \, dx=-\frac {472 \, x^{7} - 960 \, x^{5} - 1592 \, x^{3} + \sqrt {2} {\left (x^{8} + 4 \, x^{6} + 10 \, x^{4} + 12 \, x^{2} + 9\right )} \sqrt {61773 i \, \sqrt {2} + 146505} \log \left ({\left (46 \, \sqrt {2} - 307 i\right )} \sqrt {61773 i \, \sqrt {2} + 146505} + 98481 \, x\right ) - \sqrt {2} {\left (x^{8} + 4 \, x^{6} + 10 \, x^{4} + 12 \, x^{2} + 9\right )} \sqrt {61773 i \, \sqrt {2} + 146505} \log \left (-{\left (46 \, \sqrt {2} - 307 i\right )} \sqrt {61773 i \, \sqrt {2} + 146505} + 98481 \, x\right ) + \sqrt {2} {\left (x^{8} + 4 \, x^{6} + 10 \, x^{4} + 12 \, x^{2} + 9\right )} \sqrt {-61773 i \, \sqrt {2} + 146505} \log \left ({\left (46 \, \sqrt {2} + 307 i\right )} \sqrt {-61773 i \, \sqrt {2} + 146505} + 98481 \, x\right ) - \sqrt {2} {\left (x^{8} + 4 \, x^{6} + 10 \, x^{4} + 12 \, x^{2} + 9\right )} \sqrt {-61773 i \, \sqrt {2} + 146505} \log \left (-{\left (46 \, \sqrt {2} + 307 i\right )} \sqrt {-61773 i \, \sqrt {2} + 146505} + 98481 \, x\right ) - 3312 \, x}{512 \, {\left (x^{8} + 4 \, x^{6} + 10 \, x^{4} + 12 \, x^{2} + 9\right )}} \]
-1/512*(472*x^7 - 960*x^5 - 1592*x^3 + sqrt(2)*(x^8 + 4*x^6 + 10*x^4 + 12* x^2 + 9)*sqrt(61773*I*sqrt(2) + 146505)*log((46*sqrt(2) - 307*I)*sqrt(6177 3*I*sqrt(2) + 146505) + 98481*x) - sqrt(2)*(x^8 + 4*x^6 + 10*x^4 + 12*x^2 + 9)*sqrt(61773*I*sqrt(2) + 146505)*log(-(46*sqrt(2) - 307*I)*sqrt(61773*I *sqrt(2) + 146505) + 98481*x) + sqrt(2)*(x^8 + 4*x^6 + 10*x^4 + 12*x^2 + 9 )*sqrt(-61773*I*sqrt(2) + 146505)*log((46*sqrt(2) + 307*I)*sqrt(-61773*I*s qrt(2) + 146505) + 98481*x) - sqrt(2)*(x^8 + 4*x^6 + 10*x^4 + 12*x^2 + 9)* sqrt(-61773*I*sqrt(2) + 146505)*log(-(46*sqrt(2) + 307*I)*sqrt(-61773*I*sq rt(2) + 146505) + 98481*x) - 3312*x)/(x^8 + 4*x^6 + 10*x^4 + 12*x^2 + 9)
Leaf count of result is larger than twice the leaf count of optimal. 1198 vs. \(2 (201) = 402\).
Time = 0.70 (sec) , antiderivative size = 1198, normalized size of antiderivative = 5.03 \[ \int \frac {x^4 \left (4+x^2+3 x^4+5 x^6\right )}{\left (3+2 x^2+x^4\right )^3} \, dx=\text {Too large to display} \]
(-59*x**7 + 120*x**5 + 199*x**3 + 414*x)/(64*x**8 + 256*x**6 + 640*x**4 + 768*x**2 + 576) - sqrt(146505/262144 + 98481*sqrt(3)/262144)*log(x**2 + x* (-307*sqrt(6)*sqrt(48835 + 32827*sqrt(3))*sqrt(1603106545*sqrt(3) + 280884 6506)/675940757 + 10626354*sqrt(3)*sqrt(48835 + 32827*sqrt(3))/675940757 + 1228*sqrt(48835 + 32827*sqrt(3))/20591) - 941929306825573*sqrt(2)*sqrt(16 03106545*sqrt(3) + 2808846506)/456895906973733049 - 47771215762*sqrt(6)*sq rt(1603106545*sqrt(3) + 2808846506)/41754888382161 + 97477949666790882353/ 456895906973733049 + 5200450130596150*sqrt(3)/41754888382161) + sqrt(14650 5/262144 + 98481*sqrt(3)/262144)*log(x**2 + x*(-1228*sqrt(48835 + 32827*sq rt(3))/20591 - 10626354*sqrt(3)*sqrt(48835 + 32827*sqrt(3))/675940757 + 30 7*sqrt(6)*sqrt(48835 + 32827*sqrt(3))*sqrt(1603106545*sqrt(3) + 2808846506 )/675940757) - 941929306825573*sqrt(2)*sqrt(1603106545*sqrt(3) + 280884650 6)/456895906973733049 - 47771215762*sqrt(6)*sqrt(1603106545*sqrt(3) + 2808 846506)/41754888382161 + 97477949666790882353/456895906973733049 + 5200450 130596150*sqrt(3)/41754888382161) + 2*sqrt(-3*sqrt(2)*sqrt(1603106545*sqrt (3) + 2808846506)/131072 + 146505/262144 + 295443*sqrt(3)/262144)*atan(135 1881514*sqrt(3)*x/(-1894372*sqrt(-2*sqrt(2)*sqrt(1603106545*sqrt(3) + 2808 846506) + 48835 + 98481*sqrt(3)) + 307*sqrt(2)*sqrt(1603106545*sqrt(3) + 2 808846506)*sqrt(-2*sqrt(2)*sqrt(1603106545*sqrt(3) + 2808846506) + 48835 + 98481*sqrt(3))) - 40311556*sqrt(3)*sqrt(48835 + 32827*sqrt(3))/(-18943...
\[ \int \frac {x^4 \left (4+x^2+3 x^4+5 x^6\right )}{\left (3+2 x^2+x^4\right )^3} \, dx=\int { \frac {{\left (5 \, x^{6} + 3 \, x^{4} + x^{2} + 4\right )} x^{4}}{{\left (x^{4} + 2 \, x^{2} + 3\right )}^{3}} \,d x } \]
-1/64*(59*x^7 - 120*x^5 - 199*x^3 - 414*x)/(x^8 + 4*x^6 + 10*x^4 + 12*x^2 + 9) + 3/64*integrate((87*x^2 - 46)/(x^4 + 2*x^2 + 3), x)
Leaf count of result is larger than twice the leaf count of optimal. 577 vs. \(2 (177) = 354\).
Time = 0.87 (sec) , antiderivative size = 577, normalized size of antiderivative = 2.42 \[ \int \frac {x^4 \left (4+x^2+3 x^4+5 x^6\right )}{\left (3+2 x^2+x^4\right )^3} \, dx=-\frac {1}{18432} \, \sqrt {2} {\left (29 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 522 \cdot 3^{\frac {3}{4}} \sqrt {2} \sqrt {6 \, \sqrt {3} + 18} {\left (\sqrt {3} - 3\right )} - 522 \cdot 3^{\frac {3}{4}} {\left (\sqrt {3} + 3\right )} \sqrt {-6 \, \sqrt {3} + 18} + 29 \cdot 3^{\frac {3}{4}} {\left (-6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 552 \cdot 3^{\frac {1}{4}} \sqrt {2} \sqrt {6 \, \sqrt {3} + 18} - 552 \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}{18432} \, \sqrt {2} {\left (29 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 522 \cdot 3^{\frac {3}{4}} \sqrt {2} \sqrt {6 \, \sqrt {3} + 18} {\left (\sqrt {3} - 3\right )} - 522 \cdot 3^{\frac {3}{4}} {\left (\sqrt {3} + 3\right )} \sqrt {-6 \, \sqrt {3} + 18} + 29 \cdot 3^{\frac {3}{4}} {\left (-6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 552 \cdot 3^{\frac {1}{4}} \sqrt {2} \sqrt {6 \, \sqrt {3} + 18} - 552 \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}{36864} \, \sqrt {2} {\left (522 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (\sqrt {3} + 3\right )} \sqrt {-6 \, \sqrt {3} + 18} - 29 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (-6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 29 \cdot 3^{\frac {3}{4}} {\left (6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 522 \cdot 3^{\frac {3}{4}} \sqrt {6 \, \sqrt {3} + 18} {\left (\sqrt {3} - 3\right )} + 552 \cdot 3^{\frac {1}{4}} \sqrt {2} \sqrt {-6 \, \sqrt {3} + 18} + 552 \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}{36864} \, \sqrt {2} {\left (522 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (\sqrt {3} + 3\right )} \sqrt {-6 \, \sqrt {3} + 18} - 29 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (-6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 29 \cdot 3^{\frac {3}{4}} {\left (6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 522 \cdot 3^{\frac {3}{4}} \sqrt {6 \, \sqrt {3} + 18} {\left (\sqrt {3} - 3\right )} + 552 \cdot 3^{\frac {1}{4}} \sqrt {2} \sqrt {-6 \, \sqrt {3} + 18} + 552 \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 {59 \, x^{7} - 120 \, x^{5} - 199 \, x^{3} - 414 \, x}{64 \, {\left (x^{4} + 2 \, x^{2} + 3\right )}^{2}} \]
-1/18432*sqrt(2)*(29*3^(3/4)*sqrt(2)*(6*sqrt(3) + 18)^(3/2) + 522*3^(3/4)* sqrt(2)*sqrt(6*sqrt(3) + 18)*(sqrt(3) - 3) - 522*3^(3/4)*(sqrt(3) + 3)*sqr t(-6*sqrt(3) + 18) + 29*3^(3/4)*(-6*sqrt(3) + 18)^(3/2) + 552*3^(1/4)*sqrt (2)*sqrt(6*sqrt(3) + 18) - 552*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 /18432*sqrt(2)*(29*3^(3/4)*sqrt(2)*(6*sqrt(3) + 18)^(3/2) + 522*3^(3/4)*sq rt(2)*sqrt(6*sqrt(3) + 18)*(sqrt(3) - 3) - 522*3^(3/4)*(sqrt(3) + 3)*sqrt( -6*sqrt(3) + 18) + 29*3^(3/4)*(-6*sqrt(3) + 18)^(3/2) + 552*3^(1/4)*sqrt(2 )*sqrt(6*sqrt(3) + 18) - 552*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)) - 1/3 6864*sqrt(2)*(522*3^(3/4)*sqrt(2)*(sqrt(3) + 3)*sqrt(-6*sqrt(3) + 18) - 29 *3^(3/4)*sqrt(2)*(-6*sqrt(3) + 18)^(3/2) + 29*3^(3/4)*(6*sqrt(3) + 18)^(3/ 2) + 522*3^(3/4)*sqrt(6*sqrt(3) + 18)*(sqrt(3) - 3) + 552*3^(1/4)*sqrt(2)* sqrt(-6*sqrt(3) + 18) + 552*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)) + 1/36864*sqrt(2)*(522*3^(3/4)*s qrt(2)*(sqrt(3) + 3)*sqrt(-6*sqrt(3) + 18) - 29*3^(3/4)*sqrt(2)*(-6*sqrt(3 ) + 18)^(3/2) + 29*3^(3/4)*(6*sqrt(3) + 18)^(3/2) + 522*3^(3/4)*sqrt(6*sqr t(3) + 18)*(sqrt(3) - 3) + 552*3^(1/4)*sqrt(2)*sqrt(-6*sqrt(3) + 18) + 552 *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)) - 1/64*(59*x^7 - 120*x^5 - 199*x^3 - 414*x)/(x^4 + 2*x^2...
Time = 0.13 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.73 \[ \int \frac {x^4 \left (4+x^2+3 x^4+5 x^6\right )}{\left (3+2 x^2+x^4\right )^3} \, dx=\frac {-\frac {59\,x^7}{64}+\frac {15\,x^5}{8}+\frac {199\,x^3}{64}+\frac {207\,x}{32}}{x^8+4\,x^6+10\,x^4+12\,x^2+9}+\frac {\mathrm {atan}\left (\frac {x\,\sqrt {293010-\sqrt {2}\,123546{}\mathrm {i}}\,61773{}\mathrm {i}}{131072\,\left (\frac {56892933}{131072}+\frac {\sqrt {2}\,4262337{}\mathrm {i}}{65536}\right )}+\frac {61773\,\sqrt {2}\,x\,\sqrt {293010-\sqrt {2}\,123546{}\mathrm {i}}}{262144\,\left (\frac {56892933}{131072}+\frac {\sqrt {2}\,4262337{}\mathrm {i}}{65536}\right )}\right )\,\sqrt {293010-\sqrt {2}\,123546{}\mathrm {i}}\,1{}\mathrm {i}}{256}-\frac {\mathrm {atan}\left (\frac {x\,\sqrt {293010+\sqrt {2}\,123546{}\mathrm {i}}\,61773{}\mathrm {i}}{131072\,\left (-\frac {56892933}{131072}+\frac {\sqrt {2}\,4262337{}\mathrm {i}}{65536}\right )}-\frac {61773\,\sqrt {2}\,x\,\sqrt {293010+\sqrt {2}\,123546{}\mathrm {i}}}{262144\,\left (-\frac {56892933}{131072}+\frac {\sqrt {2}\,4262337{}\mathrm {i}}{65536}\right )}\right )\,\sqrt {293010+\sqrt {2}\,123546{}\mathrm {i}}\,1{}\mathrm {i}}{256} \]
((207*x)/32 + (199*x^3)/64 + (15*x^5)/8 - (59*x^7)/64)/(12*x^2 + 10*x^4 + 4*x^6 + x^8 + 9) + (atan((x*(293010 - 2^(1/2)*123546i)^(1/2)*61773i)/(1310 72*((2^(1/2)*4262337i)/65536 + 56892933/131072)) + (61773*2^(1/2)*x*(29301 0 - 2^(1/2)*123546i)^(1/2))/(262144*((2^(1/2)*4262337i)/65536 + 56892933/1 31072)))*(293010 - 2^(1/2)*123546i)^(1/2)*1i)/256 - (atan((x*(2^(1/2)*1235 46i + 293010)^(1/2)*61773i)/(131072*((2^(1/2)*4262337i)/65536 - 56892933/1 31072)) - (61773*2^(1/2)*x*(2^(1/2)*123546i + 293010)^(1/2))/(262144*((2^( 1/2)*4262337i)/65536 - 56892933/131072)))*(2^(1/2)*123546i + 293010)^(1/2) *1i)/256