Integrand size = 16, antiderivative size = 185 \[ \int \frac {d+e x}{\left (1+x^2+x^4\right )^3} \, dx=\frac {d x \left (1-x^2\right )}{12 \left (1+x^2+x^4\right )^2}+\frac {e \left (1+2 x^2\right )}{12 \left (1+x^2+x^4\right )^2}+\frac {d x \left (2-7 x^2\right )}{24 \left (1+x^2+x^4\right )}+\frac {e \left (1+2 x^2\right )}{6 \left (1+x^2+x^4\right )}-\frac {13 d \arctan \left (\frac {1-2 x}{\sqrt {3}}\right )}{48 \sqrt {3}}+\frac {13 d \arctan \left (\frac {1+2 x}{\sqrt {3}}\right )}{48 \sqrt {3}}+\frac {2 e \arctan \left (\frac {1+2 x^2}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {9}{32} d \log \left (1-x+x^2\right )+\frac {9}{32} d \log \left (1+x+x^2\right ) \]
1/12*d*x*(-x^2+1)/(x^4+x^2+1)^2+1/12*e*(2*x^2+1)/(x^4+x^2+1)^2+1/24*d*x*(- 7*x^2+2)/(x^4+x^2+1)+1/6*e*(2*x^2+1)/(x^4+x^2+1)-9/32*d*ln(x^2-x+1)+9/32*d *ln(x^2+x+1)-13/144*d*arctan(1/3*(1-2*x)*3^(1/2))*3^(1/2)+13/144*d*arctan( 1/3*(1+2*x)*3^(1/2))*3^(1/2)+2/9*e*arctan(1/3*(2*x^2+1)*3^(1/2))*3^(1/2)
Result contains complex when optimal does not.
Time = 0.53 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.01 \[ \int \frac {d+e x}{\left (1+x^2+x^4\right )^3} \, dx=\frac {1}{144} \left (\frac {12 \left (e+d x+2 e x^2-d x^3\right )}{\left (1+x^2+x^4\right )^2}+\frac {6 \left (d x \left (2-7 x^2\right )+e \left (4+8 x^2\right )\right )}{1+x^2+x^4}-\frac {\left (-47 i+7 \sqrt {3}\right ) d \arctan \left (\frac {1}{2} \left (-i+\sqrt {3}\right ) x\right )}{\sqrt {\frac {1}{6} \left (1+i \sqrt {3}\right )}}-\frac {\left (47 i+7 \sqrt {3}\right ) d \arctan \left (\frac {1}{2} \left (i+\sqrt {3}\right ) x\right )}{\sqrt {\frac {1}{6} \left (1-i \sqrt {3}\right )}}-32 \sqrt {3} e \arctan \left (\frac {\sqrt {3}}{1+2 x^2}\right )\right ) \]
((12*(e + d*x + 2*e*x^2 - d*x^3))/(1 + x^2 + x^4)^2 + (6*(d*x*(2 - 7*x^2) + e*(4 + 8*x^2)))/(1 + x^2 + x^4) - ((-47*I + 7*Sqrt[3])*d*ArcTan[((-I + S qrt[3])*x)/2])/Sqrt[(1 + I*Sqrt[3])/6] - ((47*I + 7*Sqrt[3])*d*ArcTan[((I + Sqrt[3])*x)/2])/Sqrt[(1 - I*Sqrt[3])/6] - 32*Sqrt[3]*e*ArcTan[Sqrt[3]/(1 + 2*x^2)])/144
Time = 0.48 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.09, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {2202, 27, 1405, 1432, 1086, 1086, 1083, 217, 1492, 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.
| \(\displaystyle \int \frac {d+e x}{\left (x^4+x^2+1\right )^3} \, dx\) |
| \(\Big \downarrow \) 2202 |
| \(\displaystyle \int \frac {d}{\left (x^4+x^2+1\right )^3}dx+\int \frac {e x}{\left (x^4+x^2+1\right )^3}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle d \int \frac {1}{\left (x^4+x^2+1\right )^3}dx+e \int \frac {x}{\left (x^4+x^2+1\right )^3}dx\) |
| \(\Big \downarrow \) 1405 |
| \(\displaystyle d \left (\frac {1}{12} \int \frac {11-5 x^2}{\left (x^4+x^2+1\right )^2}dx+\frac {x \left (1-x^2\right )}{12 \left (x^4+x^2+1\right )^2}\right )+e \int \frac {x}{\left (x^4+x^2+1\right )^3}dx\) |
| \(\Big \downarrow \) 1432 |
| \(\displaystyle d \left (\frac {1}{12} \int \frac {11-5 x^2}{\left (x^4+x^2+1\right )^2}dx+\frac {x \left (1-x^2\right )}{12 \left (x^4+x^2+1\right )^2}\right )+\frac {1}{2} e \int \frac {1}{\left (x^4+x^2+1\right )^3}dx^2\) |
| \(\Big \downarrow \) 1086 |
| \(\displaystyle d \left (\frac {1}{12} \int \frac {11-5 x^2}{\left (x^4+x^2+1\right )^2}dx+\frac {x \left (1-x^2\right )}{12 \left (x^4+x^2+1\right )^2}\right )+\frac {1}{2} e \left (\int \frac {1}{\left (x^4+x^2+1\right )^2}dx^2+\frac {2 x^2+1}{6 \left (x^4+x^2+1\right )^2}\right )\) |
| \(\Big \downarrow \) 1086 |
| \(\displaystyle d \left (\frac {1}{12} \int \frac {11-5 x^2}{\left (x^4+x^2+1\right )^2}dx+\frac {x \left (1-x^2\right )}{12 \left (x^4+x^2+1\right )^2}\right )+\frac {1}{2} e \left (\frac {2}{3} \int \frac {1}{x^4+x^2+1}dx^2+\frac {2 x^2+1}{3 \left (x^4+x^2+1\right )}+\frac {2 x^2+1}{6 \left (x^4+x^2+1\right )^2}\right )\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle d \left (\frac {1}{12} \int \frac {11-5 x^2}{\left (x^4+x^2+1\right )^2}dx+\frac {x \left (1-x^2\right )}{12 \left (x^4+x^2+1\right )^2}\right )+\frac {1}{2} e \left (-\frac {4}{3} \int \frac {1}{-x^4-3}d\left (2 x^2+1\right )+\frac {2 x^2+1}{3 \left (x^4+x^2+1\right )}+\frac {2 x^2+1}{6 \left (x^4+x^2+1\right )^2}\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle d \left (\frac {1}{12} \int \frac {11-5 x^2}{\left (x^4+x^2+1\right )^2}dx+\frac {x \left (1-x^2\right )}{12 \left (x^4+x^2+1\right )^2}\right )+\frac {1}{2} e \left (\frac {4 \arctan \left (\frac {2 x^2+1}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {2 x^2+1}{3 \left (x^4+x^2+1\right )}+\frac {2 x^2+1}{6 \left (x^4+x^2+1\right )^2}\right )\) |
| \(\Big \downarrow \) 1492 |
| \(\displaystyle d \left (\frac {1}{12} \left (\frac {1}{6} \int \frac {3 \left (20-7 x^2\right )}{x^4+x^2+1}dx+\frac {x \left (2-7 x^2\right )}{2 \left (x^4+x^2+1\right )}\right )+\frac {x \left (1-x^2\right )}{12 \left (x^4+x^2+1\right )^2}\right )+\frac {1}{2} e \left (\frac {4 \arctan \left (\frac {2 x^2+1}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {2 x^2+1}{3 \left (x^4+x^2+1\right )}+\frac {2 x^2+1}{6 \left (x^4+x^2+1\right )^2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle d \left (\frac {1}{12} \left (\frac {1}{2} \int \frac {20-7 x^2}{x^4+x^2+1}dx+\frac {x \left (2-7 x^2\right )}{2 \left (x^4+x^2+1\right )}\right )+\frac {x \left (1-x^2\right )}{12 \left (x^4+x^2+1\right )^2}\right )+\frac {1}{2} e \left (\frac {4 \arctan \left (\frac {2 x^2+1}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {2 x^2+1}{3 \left (x^4+x^2+1\right )}+\frac {2 x^2+1}{6 \left (x^4+x^2+1\right )^2}\right )\) |
| \(\Big \downarrow \) 1483 |
| \(\displaystyle d \left (\frac {1}{12} \left (\frac {1}{2} \left (\frac {1}{2} \int \frac {20-27 x}{x^2-x+1}dx+\frac {1}{2} \int \frac {27 x+20}{x^2+x+1}dx\right )+\frac {x \left (2-7 x^2\right )}{2 \left (x^4+x^2+1\right )}\right )+\frac {x \left (1-x^2\right )}{12 \left (x^4+x^2+1\right )^2}\right )+\frac {1}{2} e \left (\frac {4 \arctan \left (\frac {2 x^2+1}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {2 x^2+1}{3 \left (x^4+x^2+1\right )}+\frac {2 x^2+1}{6 \left (x^4+x^2+1\right )^2}\right )\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle d \left (\frac {1}{12} \left (\frac {1}{2} \left (\frac {1}{2} \left (\frac {13}{2} \int \frac {1}{x^2-x+1}dx-\frac {27}{2} \int -\frac {1-2 x}{x^2-x+1}dx\right )+\frac {1}{2} \left (\frac {13}{2} \int \frac {1}{x^2+x+1}dx+\frac {27}{2} \int \frac {2 x+1}{x^2+x+1}dx\right )\right )+\frac {x \left (2-7 x^2\right )}{2 \left (x^4+x^2+1\right )}\right )+\frac {x \left (1-x^2\right )}{12 \left (x^4+x^2+1\right )^2}\right )+\frac {1}{2} e \left (\frac {4 \arctan \left (\frac {2 x^2+1}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {2 x^2+1}{3 \left (x^4+x^2+1\right )}+\frac {2 x^2+1}{6 \left (x^4+x^2+1\right )^2}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle d \left (\frac {1}{12} \left (\frac {1}{2} \left (\frac {1}{2} \left (\frac {13}{2} \int \frac {1}{x^2-x+1}dx+\frac {27}{2} \int \frac {1-2 x}{x^2-x+1}dx\right )+\frac {1}{2} \left (\frac {13}{2} \int \frac {1}{x^2+x+1}dx+\frac {27}{2} \int \frac {2 x+1}{x^2+x+1}dx\right )\right )+\frac {x \left (2-7 x^2\right )}{2 \left (x^4+x^2+1\right )}\right )+\frac {x \left (1-x^2\right )}{12 \left (x^4+x^2+1\right )^2}\right )+\frac {1}{2} e \left (\frac {4 \arctan \left (\frac {2 x^2+1}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {2 x^2+1}{3 \left (x^4+x^2+1\right )}+\frac {2 x^2+1}{6 \left (x^4+x^2+1\right )^2}\right )\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle d \left (\frac {1}{12} \left (\frac {1}{2} \left (\frac {1}{2} \left (\frac {27}{2} \int \frac {1-2 x}{x^2-x+1}dx-13 \int \frac {1}{-(2 x-1)^2-3}d(2 x-1)\right )+\frac {1}{2} \left (\frac {27}{2} \int \frac {2 x+1}{x^2+x+1}dx-13 \int \frac {1}{-(2 x+1)^2-3}d(2 x+1)\right )\right )+\frac {x \left (2-7 x^2\right )}{2 \left (x^4+x^2+1\right )}\right )+\frac {x \left (1-x^2\right )}{12 \left (x^4+x^2+1\right )^2}\right )+\frac {1}{2} e \left (\frac {4 \arctan \left (\frac {2 x^2+1}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {2 x^2+1}{3 \left (x^4+x^2+1\right )}+\frac {2 x^2+1}{6 \left (x^4+x^2+1\right )^2}\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle d \left (\frac {1}{12} \left (\frac {1}{2} \left (\frac {1}{2} \left (\frac {27}{2} \int \frac {1-2 x}{x^2-x+1}dx+\frac {13 \arctan \left (\frac {2 x-1}{\sqrt {3}}\right )}{\sqrt {3}}\right )+\frac {1}{2} \left (\frac {27}{2} \int \frac {2 x+1}{x^2+x+1}dx+\frac {13 \arctan \left (\frac {2 x+1}{\sqrt {3}}\right )}{\sqrt {3}}\right )\right )+\frac {x \left (2-7 x^2\right )}{2 \left (x^4+x^2+1\right )}\right )+\frac {x \left (1-x^2\right )}{12 \left (x^4+x^2+1\right )^2}\right )+\frac {1}{2} e \left (\frac {4 \arctan \left (\frac {2 x^2+1}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {2 x^2+1}{3 \left (x^4+x^2+1\right )}+\frac {2 x^2+1}{6 \left (x^4+x^2+1\right )^2}\right )\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle d \left (\frac {1}{12} \left (\frac {1}{2} \left (\frac {1}{2} \left (\frac {13 \arctan \left (\frac {2 x-1}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {27}{2} \log \left (x^2-x+1\right )\right )+\frac {1}{2} \left (\frac {13 \arctan \left (\frac {2 x+1}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {27}{2} \log \left (x^2+x+1\right )\right )\right )+\frac {x \left (2-7 x^2\right )}{2 \left (x^4+x^2+1\right )}\right )+\frac {x \left (1-x^2\right )}{12 \left (x^4+x^2+1\right )^2}\right )+\frac {1}{2} e \left (\frac {4 \arctan \left (\frac {2 x^2+1}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {2 x^2+1}{3 \left (x^4+x^2+1\right )}+\frac {2 x^2+1}{6 \left (x^4+x^2+1\right )^2}\right )\) |
(e*((1 + 2*x^2)/(6*(1 + x^2 + x^4)^2) + (1 + 2*x^2)/(3*(1 + x^2 + x^4)) + (4*ArcTan[(1 + 2*x^2)/Sqrt[3]])/(3*Sqrt[3])))/2 + d*((x*(1 - x^2))/(12*(1 + x^2 + x^4)^2) + ((x*(2 - 7*x^2))/(2*(1 + x^2 + x^4)) + (((13*ArcTan[(-1 + 2*x)/Sqrt[3]])/Sqrt[3] - (27*Log[1 - x + x^2])/2)/2 + ((13*ArcTan[(1 + 2 *x)/Sqrt[3]])/Sqrt[3] + (27*Log[1 + x + x^2])/2)/2)/2)/12)
3.1.47.3.1 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[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x) *((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] - Simp[2*c*((2*p + 3)/((p + 1)*(b^2 - 4*a*c))) Int[(a + b*x + c*x^2)^(p + 1), x], x] /; Fre eQ[{a, b, c}, x] && ILtQ[p, -1]
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[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[(-x)*(b^2 - 2*a*c + b*c*x^2)*((a + b*x^2 + c*x^4)^(p + 1)/(2*a*(p + 1)*(b^2 - 4*a*c)) ), x] + Simp[1/(2*a*(p + 1)*(b^2 - 4*a*c)) Int[(b^2 - 2*a*c + 2*(p + 1)*( b^2 - 4*a*c) + b*c*(4*p + 7)*x^2)*(a + b*x^2 + c*x^4)^(p + 1), x], x] /; Fr eeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && IntegerQ[2*p]
Int[(x_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[1/2 Subst[Int[(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, p}, 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[((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symb ol] :> Simp[x*(a*b*e - d*(b^2 - 2*a*c) - c*(b*d - 2*a*e)*x^2)*((a + b*x^2 + c*x^4)^(p + 1)/(2*a*(p + 1)*(b^2 - 4*a*c))), x] + Simp[1/(2*a*(p + 1)*(b^2 - 4*a*c)) Int[Simp[(2*p + 3)*d*b^2 - a*b*e - 2*a*c*d*(4*p + 5) + (4*p + 7)*(d*b - 2*a*e)*c*x^2, x]*(a + b*x^2 + c*x^4)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && IntegerQ[2*p]
Int[(Pn_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Module[{n = Expon[Pn, x], k}, Int[Sum[Coeff[Pn, x, 2*k]*x^(2*k), {k, 0, n/2}]*(a + b *x^2 + c*x^4)^p, x] + Int[x*Sum[Coeff[Pn, x, 2*k + 1]*x^(2*k), {k, 0, (n - 1)/2}]*(a + b*x^2 + c*x^4)^p, x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pn, x] && !PolyQ[Pn, x^2]
Time = 0.18 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.85
| method | result | size |
| default | \(-\frac {\left (\frac {7 d}{3}-\frac {4 e}{3}\right ) x^{3}-6 d \,x^{2}+\left (\frac {20 d}{3}+\frac {e}{3}\right ) x -4 d -2 e}{16 \left (x^{2}-x +1\right )^{2}}-\frac {9 d \ln \left (x^{2}-x +1\right )}{32}-\frac {\left (-\frac {13 d}{2}-16 e \right ) \sqrt {3}\, \arctan \left (\frac {\left (2 x -1\right ) \sqrt {3}}{3}\right )}{72}+\frac {\left (-\frac {7 d}{3}-\frac {4 e}{3}\right ) x^{3}-6 d \,x^{2}+\left (-\frac {20 d}{3}+\frac {e}{3}\right ) x -4 d +2 e}{16 \left (x^{2}+x +1\right )^{2}}+\frac {9 d \ln \left (x^{2}+x +1\right )}{32}+\frac {\left (\frac {13 d}{2}-16 e \right ) \arctan \left (\frac {\left (1+2 x \right ) \sqrt {3}}{3}\right ) \sqrt {3}}{72}\) | \(158\) |
| risch | \(-\frac {9 d \ln \left (15457716 d^{2} x^{2}+7237632 e^{2} x^{2}-15457716 d^{2} x -7237632 e^{2} x +15457716 d^{2}+7237632 e^{2}\right )}{32}+\frac {13 \sqrt {3}\, d \arctan \left (\frac {1458 d^{2} x \sqrt {3}}{2187 d^{2}+1024 e^{2}}+\frac {2048 e^{2} x \sqrt {3}}{3 \left (2187 d^{2}+1024 e^{2}\right )}+\frac {729 \sqrt {3}\, d^{2}}{2187 d^{2}+1024 e^{2}}+\frac {1024 e^{2} \sqrt {3}}{3 \left (2187 d^{2}+1024 e^{2}\right )}\right )}{144}-\frac {13 \sqrt {3}\, d \arctan \left (\frac {1269 \sqrt {3}\, d}{567 d +1504 e}-\frac {224 \sqrt {3}\, e}{567 d +1504 e}\right )}{144}-\frac {2 \sqrt {3}\, e \arctan \left (\frac {1458 d^{2} x \sqrt {3}}{2187 d^{2}+1024 e^{2}}+\frac {2048 e^{2} x \sqrt {3}}{3 \left (2187 d^{2}+1024 e^{2}\right )}+\frac {729 \sqrt {3}\, d^{2}}{2187 d^{2}+1024 e^{2}}+\frac {1024 e^{2} \sqrt {3}}{3 \left (2187 d^{2}+1024 e^{2}\right )}\right )}{9}+\frac {2 \sqrt {3}\, e \arctan \left (\frac {1269 \sqrt {3}\, d}{567 d +1504 e}-\frac {224 \sqrt {3}\, e}{567 d +1504 e}\right )}{9}+\frac {13 \sqrt {3}\, d \arctan \left (\frac {1458 d^{2} x \sqrt {3}}{2187 d^{2}+1024 e^{2}}+\frac {2048 e^{2} x \sqrt {3}}{3 \left (2187 d^{2}+1024 e^{2}\right )}-\frac {729 \sqrt {3}\, d^{2}}{2187 d^{2}+1024 e^{2}}-\frac {1024 e^{2} \sqrt {3}}{3 \left (2187 d^{2}+1024 e^{2}\right )}\right )}{144}+\frac {13 \sqrt {3}\, d \arctan \left (\frac {1269 \sqrt {3}\, d}{567 d -1504 e}+\frac {224 \sqrt {3}\, e}{567 d -1504 e}\right )}{144}+\frac {2 \sqrt {3}\, e \arctan \left (\frac {1458 d^{2} x \sqrt {3}}{2187 d^{2}+1024 e^{2}}+\frac {2048 e^{2} x \sqrt {3}}{3 \left (2187 d^{2}+1024 e^{2}\right )}-\frac {729 \sqrt {3}\, d^{2}}{2187 d^{2}+1024 e^{2}}-\frac {1024 e^{2} \sqrt {3}}{3 \left (2187 d^{2}+1024 e^{2}\right )}\right )}{9}+\frac {2 \sqrt {3}\, e \arctan \left (\frac {1269 \sqrt {3}\, d}{567 d -1504 e}+\frac {224 \sqrt {3}\, e}{567 d -1504 e}\right )}{9}+\frac {9 d \ln \left (15457716 d^{2} x^{2}+7237632 e^{2} x^{2}+15457716 d^{2} x +7237632 e^{2} x +15457716 d^{2}+7237632 e^{2}\right )}{32}+\frac {-\frac {7}{24} d \,x^{7}+\frac {1}{3} e \,x^{6}-\frac {5}{24} x^{5} d +\frac {1}{2} e \,x^{4}-\frac {7}{24} x^{3} d +\frac {2}{3} e \,x^{2}+\frac {1}{6} d x +\frac {1}{4} e}{\left (x^{4}+x^{2}+1\right )^{2}}\) | \(671\) |
-1/16*((7/3*d-4/3*e)*x^3-6*d*x^2+(20/3*d+1/3*e)*x-4*d-2*e)/(x^2-x+1)^2-9/3 2*d*ln(x^2-x+1)-1/72*(-13/2*d-16*e)*3^(1/2)*arctan(1/3*(2*x-1)*3^(1/2))+1/ 16*((-7/3*d-4/3*e)*x^3-6*d*x^2+(-20/3*d+1/3*e)*x-4*d+2*e)/(x^2+x+1)^2+9/32 *d*ln(x^2+x+1)+1/72*(13/2*d-16*e)*arctan(1/3*(1+2*x)*3^(1/2))*3^(1/2)
Time = 0.29 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.50 \[ \int \frac {d+e x}{\left (1+x^2+x^4\right )^3} \, dx=-\frac {84 \, d x^{7} - 96 \, e x^{6} + 60 \, d x^{5} - 144 \, e x^{4} + 84 \, d x^{3} - 192 \, e x^{2} - 2 \, \sqrt {3} {\left ({\left (13 \, d - 32 \, e\right )} x^{8} + 2 \, {\left (13 \, d - 32 \, e\right )} x^{6} + 3 \, {\left (13 \, d - 32 \, e\right )} x^{4} + 2 \, {\left (13 \, d - 32 \, e\right )} x^{2} + 13 \, d - 32 \, e\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) - 2 \, \sqrt {3} {\left ({\left (13 \, d + 32 \, e\right )} x^{8} + 2 \, {\left (13 \, d + 32 \, e\right )} x^{6} + 3 \, {\left (13 \, d + 32 \, e\right )} x^{4} + 2 \, {\left (13 \, d + 32 \, e\right )} x^{2} + 13 \, d + 32 \, e\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) - 48 \, d x - 81 \, {\left (d x^{8} + 2 \, d x^{6} + 3 \, d x^{4} + 2 \, d x^{2} + d\right )} \log \left (x^{2} + x + 1\right ) + 81 \, {\left (d x^{8} + 2 \, d x^{6} + 3 \, d x^{4} + 2 \, d x^{2} + d\right )} \log \left (x^{2} - x + 1\right ) - 72 \, e}{288 \, {\left (x^{8} + 2 \, x^{6} + 3 \, x^{4} + 2 \, x^{2} + 1\right )}} \]
-1/288*(84*d*x^7 - 96*e*x^6 + 60*d*x^5 - 144*e*x^4 + 84*d*x^3 - 192*e*x^2 - 2*sqrt(3)*((13*d - 32*e)*x^8 + 2*(13*d - 32*e)*x^6 + 3*(13*d - 32*e)*x^4 + 2*(13*d - 32*e)*x^2 + 13*d - 32*e)*arctan(1/3*sqrt(3)*(2*x + 1)) - 2*sq rt(3)*((13*d + 32*e)*x^8 + 2*(13*d + 32*e)*x^6 + 3*(13*d + 32*e)*x^4 + 2*( 13*d + 32*e)*x^2 + 13*d + 32*e)*arctan(1/3*sqrt(3)*(2*x - 1)) - 48*d*x - 8 1*(d*x^8 + 2*d*x^6 + 3*d*x^4 + 2*d*x^2 + d)*log(x^2 + x + 1) + 81*(d*x^8 + 2*d*x^6 + 3*d*x^4 + 2*d*x^2 + d)*log(x^2 - x + 1) - 72*e)/(x^8 + 2*x^6 + 3*x^4 + 2*x^2 + 1)
Result contains complex when optimal does not.
Time = 2.19 (sec) , antiderivative size = 1103, normalized size of antiderivative = 5.96 \[ \int \frac {d+e x}{\left (1+x^2+x^4\right )^3} \, dx=\text {Too large to display} \]
(-9*d/32 - sqrt(3)*I*(13*d + 32*e)/288)*log(x + (-1025428432*d**4*e - 3347 52912*d**4*(-9*d/32 - sqrt(3)*I*(13*d + 32*e)/288) - 431308800*d**2*e**3 - 3143688192*d**2*e**2*(-9*d/32 - sqrt(3)*I*(13*d + 32*e)/288) + 9917005824 *d**2*e*(-9*d/32 - sqrt(3)*I*(13*d + 32*e)/288)**2 + 11878244352*d**2*(-9* d/32 - sqrt(3)*I*(13*d + 32*e)/288)**3 + 142606336*e**5 + 754974720*e**4*( -9*d/32 - sqrt(3)*I*(13*d + 32*e)/288) + 3850371072*e**3*(-9*d/32 - sqrt(3 )*I*(13*d + 32*e)/288)**2 + 20384317440*e**2*(-9*d/32 - sqrt(3)*I*(13*d + 32*e)/288)**3)/(217696167*d**5 - 1217128448*d**3*e**2 - 617611264*d*e**4)) + (-9*d/32 + sqrt(3)*I*(13*d + 32*e)/288)*log(x + (-1025428432*d**4*e - 3 34752912*d**4*(-9*d/32 + sqrt(3)*I*(13*d + 32*e)/288) - 431308800*d**2*e** 3 - 3143688192*d**2*e**2*(-9*d/32 + sqrt(3)*I*(13*d + 32*e)/288) + 9917005 824*d**2*e*(-9*d/32 + sqrt(3)*I*(13*d + 32*e)/288)**2 + 11878244352*d**2*( -9*d/32 + sqrt(3)*I*(13*d + 32*e)/288)**3 + 142606336*e**5 + 754974720*e** 4*(-9*d/32 + sqrt(3)*I*(13*d + 32*e)/288) + 3850371072*e**3*(-9*d/32 + sqr t(3)*I*(13*d + 32*e)/288)**2 + 20384317440*e**2*(-9*d/32 + sqrt(3)*I*(13*d + 32*e)/288)**3)/(217696167*d**5 - 1217128448*d**3*e**2 - 617611264*d*e** 4)) + (9*d/32 - sqrt(3)*I*(13*d - 32*e)/288)*log(x + (-1025428432*d**4*e - 334752912*d**4*(9*d/32 - sqrt(3)*I*(13*d - 32*e)/288) - 431308800*d**2*e* *3 - 3143688192*d**2*e**2*(9*d/32 - sqrt(3)*I*(13*d - 32*e)/288) + 9917005 824*d**2*e*(9*d/32 - sqrt(3)*I*(13*d - 32*e)/288)**2 + 11878244352*d**2...
Time = 0.27 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.74 \[ \int \frac {d+e x}{\left (1+x^2+x^4\right )^3} \, dx=\frac {1}{144} \, \sqrt {3} {\left (13 \, d - 32 \, e\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) + \frac {1}{144} \, \sqrt {3} {\left (13 \, d + 32 \, e\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + \frac {9}{32} \, d \log \left (x^{2} + x + 1\right ) - \frac {9}{32} \, d \log \left (x^{2} - x + 1\right ) - \frac {7 \, d x^{7} - 8 \, e x^{6} + 5 \, d x^{5} - 12 \, e x^{4} + 7 \, d x^{3} - 16 \, e x^{2} - 4 \, d x - 6 \, e}{24 \, {\left (x^{8} + 2 \, x^{6} + 3 \, x^{4} + 2 \, x^{2} + 1\right )}} \]
1/144*sqrt(3)*(13*d - 32*e)*arctan(1/3*sqrt(3)*(2*x + 1)) + 1/144*sqrt(3)* (13*d + 32*e)*arctan(1/3*sqrt(3)*(2*x - 1)) + 9/32*d*log(x^2 + x + 1) - 9/ 32*d*log(x^2 - x + 1) - 1/24*(7*d*x^7 - 8*e*x^6 + 5*d*x^5 - 12*e*x^4 + 7*d *x^3 - 16*e*x^2 - 4*d*x - 6*e)/(x^8 + 2*x^6 + 3*x^4 + 2*x^2 + 1)
Time = 0.29 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.68 \[ \int \frac {d+e x}{\left (1+x^2+x^4\right )^3} \, dx=\frac {1}{144} \, \sqrt {3} {\left (13 \, d - 32 \, e\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) + \frac {1}{144} \, \sqrt {3} {\left (13 \, d + 32 \, e\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + \frac {9}{32} \, d \log \left (x^{2} + x + 1\right ) - \frac {9}{32} \, d \log \left (x^{2} - x + 1\right ) - \frac {7 \, d x^{7} - 8 \, e x^{6} + 5 \, d x^{5} - 12 \, e x^{4} + 7 \, d x^{3} - 16 \, e x^{2} - 4 \, d x - 6 \, e}{24 \, {\left (x^{4} + x^{2} + 1\right )}^{2}} \]
1/144*sqrt(3)*(13*d - 32*e)*arctan(1/3*sqrt(3)*(2*x + 1)) + 1/144*sqrt(3)* (13*d + 32*e)*arctan(1/3*sqrt(3)*(2*x - 1)) + 9/32*d*log(x^2 + x + 1) - 9/ 32*d*log(x^2 - x + 1) - 1/24*(7*d*x^7 - 8*e*x^6 + 5*d*x^5 - 12*e*x^4 + 7*d *x^3 - 16*e*x^2 - 4*d*x - 6*e)/(x^4 + x^2 + 1)^2
Time = 0.15 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.00 \[ \int \frac {d+e x}{\left (1+x^2+x^4\right )^3} \, dx=\frac {-\frac {7\,d\,x^7}{24}+\frac {e\,x^6}{3}-\frac {5\,d\,x^5}{24}+\frac {e\,x^4}{2}-\frac {7\,d\,x^3}{24}+\frac {2\,e\,x^2}{3}+\frac {d\,x}{6}+\frac {e}{4}}{x^8+2\,x^6+3\,x^4+2\,x^2+1}-\ln \left (x-\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {9\,d}{32}+\frac {\sqrt {3}\,d\,13{}\mathrm {i}}{288}+\frac {\sqrt {3}\,e\,1{}\mathrm {i}}{9}\right )+\ln \left (x+\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {9\,d}{32}-\frac {\sqrt {3}\,d\,13{}\mathrm {i}}{288}+\frac {\sqrt {3}\,e\,1{}\mathrm {i}}{9}\right )+\ln \left (x-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-\frac {9\,d}{32}+\frac {\sqrt {3}\,d\,13{}\mathrm {i}}{288}+\frac {\sqrt {3}\,e\,1{}\mathrm {i}}{9}\right )+\ln \left (x+\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {9\,d}{32}+\frac {\sqrt {3}\,d\,13{}\mathrm {i}}{288}-\frac {\sqrt {3}\,e\,1{}\mathrm {i}}{9}\right ) \]
(e/4 + (d*x)/6 - (7*d*x^3)/24 - (5*d*x^5)/24 - (7*d*x^7)/24 + (2*e*x^2)/3 + (e*x^4)/2 + (e*x^6)/3)/(2*x^2 + 3*x^4 + 2*x^6 + x^8 + 1) - log(x - (3^(1 /2)*1i)/2 - 1/2)*((9*d)/32 + (3^(1/2)*d*13i)/288 + (3^(1/2)*e*1i)/9) + log (x - (3^(1/2)*1i)/2 + 1/2)*((9*d)/32 - (3^(1/2)*d*13i)/288 + (3^(1/2)*e*1i )/9) + log(x + (3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*d*13i)/288 - (9*d)/32 + (3^ (1/2)*e*1i)/9) + log(x + (3^(1/2)*1i)/2 + 1/2)*((9*d)/32 + (3^(1/2)*d*13i) /288 - (3^(1/2)*e*1i)/9)