\(\int \frac {d+e x+f x^2}{(1+x^2+x^4)^2} \, dx\) [29]

Optimal result

Integrand size = 21, antiderivative size = 148 \[ \int \frac {d+e x+f x^2}{\left (1+x^2+x^4\right )^2} \, dx=\frac {e \left (1+2 x^2\right )}{6 \left (1+x^2+x^4\right )}+\frac {x \left (d+f-(d-2 f) x^2\right )}{6 \left (1+x^2+x^4\right )}-\frac {(4 d+f) \arctan \left (\frac {1-2 x}{\sqrt {3}}\right )}{12 \sqrt {3}}+\frac {(4 d+f) \arctan \left (\frac {1+2 x}{\sqrt {3}}\right )}{12 \sqrt {3}}+\frac {2 e \arctan \left (\frac {1+2 x^2}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {1}{4} (2 d-f) \text {arctanh}\left (\frac {x}{1+x^2}\right ) \] Output:

e*(2*x^2+1)/(6*x^4+6*x^2+6)+x*(d+f-(d-2*f)*x^2)/(6*x^4+6*x^2+6)-1/36*(4*d+ 
f)*arctan(1/3*(1-2*x)*3^(1/2))*3^(1/2)+1/36*(4*d+f)*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)+1/4*(2*d-f)*arct 
anh(x/(x^2+1))
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.52 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.26 \[ \int \frac {d+e x+f x^2}{\left (1+x^2+x^4\right )^2} \, dx=\frac {1}{36} \left (\frac {6 \left (e+2 e x^2+x \left (d+f-d x^2+2 f x^2\right )\right )}{1+x^2+x^4}-\frac {\left (\left (-11 i+\sqrt {3}\right ) d-2 \left (-2 i+\sqrt {3}\right ) f\right ) \arctan \left (\frac {1}{2} \left (-i+\sqrt {3}\right ) x\right )}{\sqrt {\frac {1}{6} \left (1+i \sqrt {3}\right )}}-\frac {\left (\left (11 i+\sqrt {3}\right ) d-2 \left (2 i+\sqrt {3}\right ) f\right ) \arctan \left (\frac {1}{2} \left (i+\sqrt {3}\right ) x\right )}{\sqrt {\frac {1}{6} \left (1-i \sqrt {3}\right )}}-8 \sqrt {3} e \arctan \left (\frac {\sqrt {3}}{1+2 x^2}\right )\right ) \] Input:

Integrate[(d + e*x + f*x^2)/(1 + x^2 + x^4)^2,x]
 

Output:

((6*(e + 2*e*x^2 + x*(d + f - d*x^2 + 2*f*x^2)))/(1 + x^2 + x^4) - (((-11* 
I + Sqrt[3])*d - 2*(-2*I + Sqrt[3])*f)*ArcTan[((-I + Sqrt[3])*x)/2])/Sqrt[ 
(1 + I*Sqrt[3])/6] - (((11*I + Sqrt[3])*d - 2*(2*I + Sqrt[3])*f)*ArcTan[(( 
I + Sqrt[3])*x)/2])/Sqrt[(1 - I*Sqrt[3])/6] - 8*Sqrt[3]*e*ArcTan[Sqrt[3]/( 
1 + 2*x^2)])/36
 

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.20, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.619, Rules used = {2202, 27, 1432, 1086, 1083, 217, 1492, 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.

\(\displaystyle \int \frac {d+e x+f x^2}{\left (x^4+x^2+1\right )^2} \, dx\)

\(\Big \downarrow \) 2202

\(\displaystyle \int \frac {f x^2+d}{\left (x^4+x^2+1\right )^2}dx+\int \frac {e x}{\left (x^4+x^2+1\right )^2}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \int \frac {f x^2+d}{\left (x^4+x^2+1\right )^2}dx+e \int \frac {x}{\left (x^4+x^2+1\right )^2}dx\)

\(\Big \downarrow \) 1432

\(\displaystyle \int \frac {f x^2+d}{\left (x^4+x^2+1\right )^2}dx+\frac {1}{2} e \int \frac {1}{\left (x^4+x^2+1\right )^2}dx^2\)

\(\Big \downarrow \) 1086

\(\displaystyle \int \frac {f x^2+d}{\left (x^4+x^2+1\right )^2}dx+\frac {1}{2} e \left (\frac {2}{3} \int \frac {1}{x^4+x^2+1}dx^2+\frac {2 x^2+1}{3 \left (x^4+x^2+1\right )}\right )\)

\(\Big \downarrow \) 1083

\(\displaystyle \int \frac {f x^2+d}{\left (x^4+x^2+1\right )^2}dx+\frac {1}{2} e \left (\frac {2 x^2+1}{3 \left (x^4+x^2+1\right )}-\frac {4}{3} \int \frac {1}{-x^4-3}d\left (2 x^2+1\right )\right )\)

\(\Big \downarrow \) 217

\(\displaystyle \int \frac {f x^2+d}{\left (x^4+x^2+1\right )^2}dx+\frac {1}{2} e \left (\frac {4 \arctan \left (\frac {2 x^2+1}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {2 x^2+1}{3 \left (x^4+x^2+1\right )}\right )\)

\(\Big \downarrow \) 1492

\(\displaystyle \frac {1}{6} \int \frac {-\left ((d-2 f) x^2\right )+5 d-f}{x^4+x^2+1}dx+\frac {1}{2} e \left (\frac {4 \arctan \left (\frac {2 x^2+1}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {2 x^2+1}{3 \left (x^4+x^2+1\right )}\right )+\frac {x \left (-\left (x^2 (d-2 f)\right )+d+f\right )}{6 \left (x^4+x^2+1\right )}\)

\(\Big \downarrow \) 1483

\(\displaystyle \frac {1}{6} \left (\frac {1}{2} \int \frac {5 d-f-3 (2 d-f) x}{x^2-x+1}dx+\frac {1}{2} \int \frac {5 d-f+3 (2 d-f) x}{x^2+x+1}dx\right )+\frac {1}{2} e \left (\frac {4 \arctan \left (\frac {2 x^2+1}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {2 x^2+1}{3 \left (x^4+x^2+1\right )}\right )+\frac {x \left (-\left (x^2 (d-2 f)\right )+d+f\right )}{6 \left (x^4+x^2+1\right )}\)

\(\Big \downarrow \) 1142

\(\displaystyle \frac {1}{6} \left (\frac {1}{2} \left (\frac {1}{2} (4 d+f) \int \frac {1}{x^2-x+1}dx-\frac {3}{2} (2 d-f) \int -\frac {1-2 x}{x^2-x+1}dx\right )+\frac {1}{2} \left (\frac {1}{2} (4 d+f) \int \frac {1}{x^2+x+1}dx+\frac {3}{2} (2 d-f) \int \frac {2 x+1}{x^2+x+1}dx\right )\right )+\frac {1}{2} e \left (\frac {4 \arctan \left (\frac {2 x^2+1}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {2 x^2+1}{3 \left (x^4+x^2+1\right )}\right )+\frac {x \left (-\left (x^2 (d-2 f)\right )+d+f\right )}{6 \left (x^4+x^2+1\right )}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{6} \left (\frac {1}{2} \left (\frac {1}{2} (4 d+f) \int \frac {1}{x^2-x+1}dx+\frac {3}{2} (2 d-f) \int \frac {1-2 x}{x^2-x+1}dx\right )+\frac {1}{2} \left (\frac {1}{2} (4 d+f) \int \frac {1}{x^2+x+1}dx+\frac {3}{2} (2 d-f) \int \frac {2 x+1}{x^2+x+1}dx\right )\right )+\frac {1}{2} e \left (\frac {4 \arctan \left (\frac {2 x^2+1}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {2 x^2+1}{3 \left (x^4+x^2+1\right )}\right )+\frac {x \left (-\left (x^2 (d-2 f)\right )+d+f\right )}{6 \left (x^4+x^2+1\right )}\)

\(\Big \downarrow \) 1083

\(\displaystyle \frac {1}{6} \left (\frac {1}{2} \left (\frac {3}{2} (2 d-f) \int \frac {1-2 x}{x^2-x+1}dx-(4 d+f) \int \frac {1}{-(2 x-1)^2-3}d(2 x-1)\right )+\frac {1}{2} \left (\frac {3}{2} (2 d-f) \int \frac {2 x+1}{x^2+x+1}dx-(4 d+f) \int \frac {1}{-(2 x+1)^2-3}d(2 x+1)\right )\right )+\frac {1}{2} e \left (\frac {4 \arctan \left (\frac {2 x^2+1}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {2 x^2+1}{3 \left (x^4+x^2+1\right )}\right )+\frac {x \left (-\left (x^2 (d-2 f)\right )+d+f\right )}{6 \left (x^4+x^2+1\right )}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {1}{6} \left (\frac {1}{2} \left (\frac {3}{2} (2 d-f) \int \frac {1-2 x}{x^2-x+1}dx+\frac {\arctan \left (\frac {2 x-1}{\sqrt {3}}\right ) (4 d+f)}{\sqrt {3}}\right )+\frac {1}{2} \left (\frac {3}{2} (2 d-f) \int \frac {2 x+1}{x^2+x+1}dx+\frac {\arctan \left (\frac {2 x+1}{\sqrt {3}}\right ) (4 d+f)}{\sqrt {3}}\right )\right )+\frac {1}{2} e \left (\frac {4 \arctan \left (\frac {2 x^2+1}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {2 x^2+1}{3 \left (x^4+x^2+1\right )}\right )+\frac {x \left (-\left (x^2 (d-2 f)\right )+d+f\right )}{6 \left (x^4+x^2+1\right )}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {1}{6} \left (\frac {1}{2} \left (\frac {\arctan \left (\frac {2 x-1}{\sqrt {3}}\right ) (4 d+f)}{\sqrt {3}}-\frac {3}{2} (2 d-f) \log \left (x^2-x+1\right )\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {2 x+1}{\sqrt {3}}\right ) (4 d+f)}{\sqrt {3}}+\frac {3}{2} (2 d-f) \log \left (x^2+x+1\right )\right )\right )+\frac {1}{2} e \left (\frac {4 \arctan \left (\frac {2 x^2+1}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {2 x^2+1}{3 \left (x^4+x^2+1\right )}\right )+\frac {x \left (-\left (x^2 (d-2 f)\right )+d+f\right )}{6 \left (x^4+x^2+1\right )}\)

Input:

Int[(d + e*x + f*x^2)/(1 + x^2 + x^4)^2,x]
 

Output:

(x*(d + f - (d - 2*f)*x^2))/(6*(1 + x^2 + x^4)) + (e*((1 + 2*x^2)/(3*(1 + 
x^2 + x^4)) + (4*ArcTan[(1 + 2*x^2)/Sqrt[3]])/(3*Sqrt[3])))/2 + ((((4*d + 
f)*ArcTan[(-1 + 2*x)/Sqrt[3]])/Sqrt[3] - (3*(2*d - f)*Log[1 - x + x^2])/2) 
/2 + (((4*d + f)*ArcTan[(1 + 2*x)/Sqrt[3]])/Sqrt[3] + (3*(2*d - f)*Log[1 + 
 x + x^2])/2)/2)/6
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 217

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])
 

rule 1083

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]
 

rule 1086

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]
 

rule 1103

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]
 

rule 1142

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]
 

rule 1432

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]
 

rule 1483

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]
 

rule 1492

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]
 

rule 2202

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]
 

Maple [A] (verified)

Time = 0.18 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.04

method	result	size
default	\(\frac {\left (-\frac {d}{3}-\frac {e}{3}+\frac {2 f}{3}\right ) x -\frac {2 d}{3}+\frac {e}{3}+\frac {f}{3}}{4 x^{2}+4 x +4}+\frac {\left (6 d -3 f \right ) \ln \left (x^{2}+x +1\right )}{24}+\frac {\left (2 d -4 e +\frac {f}{2}\right ) \arctan \left (\frac {\left (1+2 x \right ) \sqrt {3}}{3}\right ) \sqrt {3}}{18}-\frac {\left (\frac {d}{3}-\frac {e}{3}-\frac {2 f}{3}\right ) x -\frac {2 d}{3}-\frac {e}{3}+\frac {f}{3}}{4 \left (x^{2}-x +1\right )}-\frac {\left (6 d -3 f \right ) \ln \left (x^{2}-x +1\right )}{24}-\frac {\left (-2 d -4 e -\frac {f}{2}\right ) \sqrt {3}\, \arctan \left (\frac {\left (2 x -1\right ) \sqrt {3}}{3}\right )}{18}\)	\(154\)
risch	\(\text {Expression too large to display}\)	\(8243\)

Input:

int((f*x^2+e*x+d)/(x^4+x^2+1)^2,x,method=_RETURNVERBOSE)
 

Output:

1/4*((-1/3*d-1/3*e+2/3*f)*x-2/3*d+1/3*e+1/3*f)/(x^2+x+1)+1/24*(6*d-3*f)*ln 
(x^2+x+1)+1/18*(2*d-4*e+1/2*f)*arctan(1/3*(1+2*x)*3^(1/2))*3^(1/2)-1/4*((1 
/3*d-1/3*e-2/3*f)*x-2/3*d-1/3*e+1/3*f)/(x^2-x+1)-1/24*(6*d-3*f)*ln(x^2-x+1 
)-1/18*(-2*d-4*e-1/2*f)*3^(1/2)*arctan(1/3*(2*x-1)*3^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.43 \[ \int \frac {d+e x+f x^2}{\left (1+x^2+x^4\right )^2} \, dx=-\frac {12 \, {\left (d - 2 \, f\right )} x^{3} - 24 \, e x^{2} - 2 \, \sqrt {3} {\left ({\left (4 \, d - 8 \, e + f\right )} x^{4} + {\left (4 \, d - 8 \, e + f\right )} x^{2} + 4 \, d - 8 \, e + f\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) - 2 \, \sqrt {3} {\left ({\left (4 \, d + 8 \, e + f\right )} x^{4} + {\left (4 \, d + 8 \, e + f\right )} x^{2} + 4 \, d + 8 \, e + f\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) - 12 \, {\left (d + f\right )} x - 9 \, {\left ({\left (2 \, d - f\right )} x^{4} + {\left (2 \, d - f\right )} x^{2} + 2 \, d - f\right )} \log \left (x^{2} + x + 1\right ) + 9 \, {\left ({\left (2 \, d - f\right )} x^{4} + {\left (2 \, d - f\right )} x^{2} + 2 \, d - f\right )} \log \left (x^{2} - x + 1\right ) - 12 \, e}{72 \, {\left (x^{4} + x^{2} + 1\right )}} \] Input:

integrate((f*x^2+e*x+d)/(x^4+x^2+1)^2,x, algorithm="fricas")
 

Output:

-1/72*(12*(d - 2*f)*x^3 - 24*e*x^2 - 2*sqrt(3)*((4*d - 8*e + f)*x^4 + (4*d 
 - 8*e + f)*x^2 + 4*d - 8*e + f)*arctan(1/3*sqrt(3)*(2*x + 1)) - 2*sqrt(3) 
*((4*d + 8*e + f)*x^4 + (4*d + 8*e + f)*x^2 + 4*d + 8*e + f)*arctan(1/3*sq 
rt(3)*(2*x - 1)) - 12*(d + f)*x - 9*((2*d - f)*x^4 + (2*d - f)*x^2 + 2*d - 
 f)*log(x^2 + x + 1) + 9*((2*d - f)*x^4 + (2*d - f)*x^2 + 2*d - f)*log(x^2 
 - x + 1) - 12*e)/(x^4 + x^2 + 1)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 62.68 (sec) , antiderivative size = 4106, normalized size of antiderivative = 27.74 \[ \int \frac {d+e x+f x^2}{\left (1+x^2+x^4\right )^2} \, dx=\text {Too large to display} \] Input:

integrate((f*x**2+e*x+d)/(x**4+x**2+1)**2,x)
 

Output:

(-d/4 + f/8 - sqrt(3)*I*(4*d + 8*e + f)/72)*log(x + (-164944*d**5*e + 1641 
6*d**5*(-d/4 + f/8 - sqrt(3)*I*(4*d + 8*e + f)/72) + 336520*d**4*e*f + 200 
664*d**4*f*(-d/4 + f/8 - sqrt(3)*I*(4*d + 8*e + f)/72) - 115200*d**3*e**3 
- 504576*d**3*e**2*(-d/4 + f/8 - sqrt(3)*I*(4*d + 8*e + f)/72) - 272380*d* 
*3*e*f**2 + 1734912*d**3*e*(-d/4 + f/8 - sqrt(3)*I*(4*d + 8*e + f)/72)**2 
- 229500*d**3*f**2*(-d/4 + f/8 - sqrt(3)*I*(4*d + 8*e + f)/72) + 2612736*d 
**3*(-d/4 + f/8 - sqrt(3)*I*(4*d + 8*e + f)/72)**3 + 51840*d**2*e**3*f + 8 
81280*d**2*e**2*f*(-d/4 + f/8 - sqrt(3)*I*(4*d + 8*e + f)/72) + 119420*d** 
2*e*f**3 - 2477952*d**2*e*f*(-d/4 + f/8 - sqrt(3)*I*(4*d + 8*e + f)/72)**2 
 + 50436*d**2*f**3*(-d/4 + f/8 - sqrt(3)*I*(4*d + 8*e + f)/72) - 2519424*d 
**2*f*(-d/4 + f/8 - sqrt(3)*I*(4*d + 8*e + f)/72)**3 + 28672*d*e**5 + 1843 
20*d*e**4*(-d/4 + f/8 - sqrt(3)*I*(4*d + 8*e + f)/72) + 8640*d*e**3*f**2 + 
 774144*d*e**3*(-d/4 + f/8 - sqrt(3)*I*(4*d + 8*e + f)/72)**2 - 409536*d*e 
**2*f**2*(-d/4 + f/8 - sqrt(3)*I*(4*d + 8*e + f)/72) + 4976640*d*e**2*(-d/ 
4 + f/8 - sqrt(3)*I*(4*d + 8*e + f)/72)**3 - 31040*d*e*f**4 + 1270080*d*e* 
f**2*(-d/4 + f/8 - sqrt(3)*I*(4*d + 8*e + f)/72)**2 + 14040*d*f**4*(-d/4 + 
 f/8 - sqrt(3)*I*(4*d + 8*e + f)/72) + 139968*d*f**2*(-d/4 + f/8 - sqrt(3) 
*I*(4*d + 8*e + f)/72)**3 - 20480*e**5*f - 36864*e**4*f*(-d/4 + f/8 - sqrt 
(3)*I*(4*d + 8*e + f)/72) - 2880*e**3*f**3 - 552960*e**3*f*(-d/4 + f/8 - s 
qrt(3)*I*(4*d + 8*e + f)/72)**2 + 70848*e**2*f**3*(-d/4 + f/8 - sqrt(3)...
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.81 \[ \int \frac {d+e x+f x^2}{\left (1+x^2+x^4\right )^2} \, dx=\frac {1}{36} \, \sqrt {3} {\left (4 \, d - 8 \, e + f\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) + \frac {1}{36} \, \sqrt {3} {\left (4 \, d + 8 \, e + f\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + \frac {1}{8} \, {\left (2 \, d - f\right )} \log \left (x^{2} + x + 1\right ) - \frac {1}{8} \, {\left (2 \, d - f\right )} \log \left (x^{2} - x + 1\right ) - \frac {{\left (d - 2 \, f\right )} x^{3} - 2 \, e x^{2} - {\left (d + f\right )} x - e}{6 \, {\left (x^{4} + x^{2} + 1\right )}} \] Input:

integrate((f*x^2+e*x+d)/(x^4+x^2+1)^2,x, algorithm="maxima")
 

Output:

1/36*sqrt(3)*(4*d - 8*e + f)*arctan(1/3*sqrt(3)*(2*x + 1)) + 1/36*sqrt(3)* 
(4*d + 8*e + f)*arctan(1/3*sqrt(3)*(2*x - 1)) + 1/8*(2*d - f)*log(x^2 + x 
+ 1) - 1/8*(2*d - f)*log(x^2 - x + 1) - 1/6*((d - 2*f)*x^3 - 2*e*x^2 - (d 
+ f)*x - e)/(x^4 + x^2 + 1)
 

Giac [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.84 \[ \int \frac {d+e x+f x^2}{\left (1+x^2+x^4\right )^2} \, dx=\frac {1}{36} \, \sqrt {3} {\left (4 \, d - 8 \, e + f\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) + \frac {1}{36} \, \sqrt {3} {\left (4 \, d + 8 \, e + f\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + \frac {1}{8} \, {\left (2 \, d - f\right )} \log \left (x^{2} + x + 1\right ) - \frac {1}{8} \, {\left (2 \, d - f\right )} \log \left (x^{2} - x + 1\right ) - \frac {d x^{3} - 2 \, f x^{3} - 2 \, e x^{2} - d x - f x - e}{6 \, {\left (x^{4} + x^{2} + 1\right )}} \] Input:

integrate((f*x^2+e*x+d)/(x^4+x^2+1)^2,x, algorithm="giac")
                                                                                    
                                                                                    
 

Output:

1/36*sqrt(3)*(4*d - 8*e + f)*arctan(1/3*sqrt(3)*(2*x + 1)) + 1/36*sqrt(3)* 
(4*d + 8*e + f)*arctan(1/3*sqrt(3)*(2*x - 1)) + 1/8*(2*d - f)*log(x^2 + x 
+ 1) - 1/8*(2*d - f)*log(x^2 - x + 1) - 1/6*(d*x^3 - 2*f*x^3 - 2*e*x^2 - d 
*x - f*x - e)/(x^4 + x^2 + 1)
 

Mupad [B] (verification not implemented)

Time = 0.34 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.36 \[ \int \frac {d+e x+f x^2}{\left (1+x^2+x^4\right )^2} \, dx=\frac {\left (\frac {f}{3}-\frac {d}{6}\right )\,x^3+\frac {e\,x^2}{3}+\left (\frac {d}{6}+\frac {f}{6}\right )\,x+\frac {e}{6}}{x^4+x^2+1}-\ln \left (x-\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {d}{4}-\frac {f}{8}+\frac {\sqrt {3}\,d\,1{}\mathrm {i}}{18}+\frac {\sqrt {3}\,e\,1{}\mathrm {i}}{9}+\frac {\sqrt {3}\,f\,1{}\mathrm {i}}{72}\right )-\ln \left (x+\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {f}{8}-\frac {d}{4}+\frac {\sqrt {3}\,d\,1{}\mathrm {i}}{18}-\frac {\sqrt {3}\,e\,1{}\mathrm {i}}{9}+\frac {\sqrt {3}\,f\,1{}\mathrm {i}}{72}\right )+\ln \left (x-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {f}{8}-\frac {d}{4}+\frac {\sqrt {3}\,d\,1{}\mathrm {i}}{18}+\frac {\sqrt {3}\,e\,1{}\mathrm {i}}{9}+\frac {\sqrt {3}\,f\,1{}\mathrm {i}}{72}\right )+\ln \left (x+\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {d}{4}-\frac {f}{8}+\frac {\sqrt {3}\,d\,1{}\mathrm {i}}{18}-\frac {\sqrt {3}\,e\,1{}\mathrm {i}}{9}+\frac {\sqrt {3}\,f\,1{}\mathrm {i}}{72}\right ) \] Input:

int((d + e*x + f*x^2)/(x^2 + x^4 + 1)^2,x)
 

Output:

(e/6 - x^3*(d/6 - f/3) + (e*x^2)/3 + x*(d/6 + f/6))/(x^2 + x^4 + 1) - log( 
x - (3^(1/2)*1i)/2 - 1/2)*(d/4 - f/8 + (3^(1/2)*d*1i)/18 + (3^(1/2)*e*1i)/ 
9 + (3^(1/2)*f*1i)/72) - log(x - (3^(1/2)*1i)/2 + 1/2)*(f/8 - d/4 + (3^(1/ 
2)*d*1i)/18 - (3^(1/2)*e*1i)/9 + (3^(1/2)*f*1i)/72) + log(x + (3^(1/2)*1i) 
/2 - 1/2)*(f/8 - d/4 + (3^(1/2)*d*1i)/18 + (3^(1/2)*e*1i)/9 + (3^(1/2)*f*1 
i)/72) + log(x + (3^(1/2)*1i)/2 + 1/2)*(d/4 - f/8 + (3^(1/2)*d*1i)/18 - (3 
^(1/2)*e*1i)/9 + (3^(1/2)*f*1i)/72)
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 525, normalized size of antiderivative = 3.55 \[ \int \frac {d+e x+f x^2}{\left (1+x^2+x^4\right )^2} \, dx=\frac {-18 \,\mathrm {log}\left (x^{2}-x +1\right ) d \,x^{4}-18 \,\mathrm {log}\left (x^{2}-x +1\right ) d \,x^{2}+9 \,\mathrm {log}\left (x^{2}-x +1\right ) f \,x^{4}+9 \,\mathrm {log}\left (x^{2}-x +1\right ) f \,x^{2}+18 \,\mathrm {log}\left (x^{2}+x +1\right ) d \,x^{4}+18 \,\mathrm {log}\left (x^{2}+x +1\right ) d \,x^{2}-9 \,\mathrm {log}\left (x^{2}+x +1\right ) f \,x^{4}-9 \,\mathrm {log}\left (x^{2}+x +1\right ) f \,x^{2}+8 \sqrt {3}\, \mathit {atan} \left (\frac {2 x -1}{\sqrt {3}}\right ) d +16 \sqrt {3}\, \mathit {atan} \left (\frac {2 x -1}{\sqrt {3}}\right ) e +2 \sqrt {3}\, \mathit {atan} \left (\frac {2 x -1}{\sqrt {3}}\right ) f +8 \sqrt {3}\, \mathit {atan} \left (\frac {2 x +1}{\sqrt {3}}\right ) d -16 \sqrt {3}\, \mathit {atan} \left (\frac {2 x +1}{\sqrt {3}}\right ) e +2 \sqrt {3}\, \mathit {atan} \left (\frac {2 x +1}{\sqrt {3}}\right ) f -12 e -12 d \,x^{3}+24 f \,x^{3}+12 f x -18 \,\mathrm {log}\left (x^{2}-x +1\right ) d +9 \,\mathrm {log}\left (x^{2}-x +1\right ) f +18 \,\mathrm {log}\left (x^{2}+x +1\right ) d -9 \,\mathrm {log}\left (x^{2}+x +1\right ) f +8 \sqrt {3}\, \mathit {atan} \left (\frac {2 x -1}{\sqrt {3}}\right ) d \,x^{4}+8 \sqrt {3}\, \mathit {atan} \left (\frac {2 x -1}{\sqrt {3}}\right ) d \,x^{2}+16 \sqrt {3}\, \mathit {atan} \left (\frac {2 x -1}{\sqrt {3}}\right ) e \,x^{4}+16 \sqrt {3}\, \mathit {atan} \left (\frac {2 x -1}{\sqrt {3}}\right ) e \,x^{2}+2 \sqrt {3}\, \mathit {atan} \left (\frac {2 x -1}{\sqrt {3}}\right ) f \,x^{4}+2 \sqrt {3}\, \mathit {atan} \left (\frac {2 x -1}{\sqrt {3}}\right ) f \,x^{2}+8 \sqrt {3}\, \mathit {atan} \left (\frac {2 x +1}{\sqrt {3}}\right ) d \,x^{4}+8 \sqrt {3}\, \mathit {atan} \left (\frac {2 x +1}{\sqrt {3}}\right ) d \,x^{2}-16 \sqrt {3}\, \mathit {atan} \left (\frac {2 x +1}{\sqrt {3}}\right ) e \,x^{4}-16 \sqrt {3}\, \mathit {atan} \left (\frac {2 x +1}{\sqrt {3}}\right ) e \,x^{2}+2 \sqrt {3}\, \mathit {atan} \left (\frac {2 x +1}{\sqrt {3}}\right ) f \,x^{4}+2 \sqrt {3}\, \mathit {atan} \left (\frac {2 x +1}{\sqrt {3}}\right ) f \,x^{2}-24 e \,x^{4}+12 d x}{72 x^{4}+72 x^{2}+72} \] Input:

int((f*x^2+e*x+d)/(x^4+x^2+1)^2,x)
 

Output:

(8*sqrt(3)*atan((2*x - 1)/sqrt(3))*d*x**4 + 8*sqrt(3)*atan((2*x - 1)/sqrt( 
3))*d*x**2 + 8*sqrt(3)*atan((2*x - 1)/sqrt(3))*d + 16*sqrt(3)*atan((2*x - 
1)/sqrt(3))*e*x**4 + 16*sqrt(3)*atan((2*x - 1)/sqrt(3))*e*x**2 + 16*sqrt(3 
)*atan((2*x - 1)/sqrt(3))*e + 2*sqrt(3)*atan((2*x - 1)/sqrt(3))*f*x**4 + 2 
*sqrt(3)*atan((2*x - 1)/sqrt(3))*f*x**2 + 2*sqrt(3)*atan((2*x - 1)/sqrt(3) 
)*f + 8*sqrt(3)*atan((2*x + 1)/sqrt(3))*d*x**4 + 8*sqrt(3)*atan((2*x + 1)/ 
sqrt(3))*d*x**2 + 8*sqrt(3)*atan((2*x + 1)/sqrt(3))*d - 16*sqrt(3)*atan((2 
*x + 1)/sqrt(3))*e*x**4 - 16*sqrt(3)*atan((2*x + 1)/sqrt(3))*e*x**2 - 16*s 
qrt(3)*atan((2*x + 1)/sqrt(3))*e + 2*sqrt(3)*atan((2*x + 1)/sqrt(3))*f*x** 
4 + 2*sqrt(3)*atan((2*x + 1)/sqrt(3))*f*x**2 + 2*sqrt(3)*atan((2*x + 1)/sq 
rt(3))*f - 18*log(x**2 - x + 1)*d*x**4 - 18*log(x**2 - x + 1)*d*x**2 - 18* 
log(x**2 - x + 1)*d + 9*log(x**2 - x + 1)*f*x**4 + 9*log(x**2 - x + 1)*f*x 
**2 + 9*log(x**2 - x + 1)*f + 18*log(x**2 + x + 1)*d*x**4 + 18*log(x**2 + 
x + 1)*d*x**2 + 18*log(x**2 + x + 1)*d - 9*log(x**2 + x + 1)*f*x**4 - 9*lo 
g(x**2 + x + 1)*f*x**2 - 9*log(x**2 + x + 1)*f - 12*d*x**3 + 12*d*x - 24*e 
*x**4 - 12*e + 24*f*x**3 + 12*f*x)/(72*(x**4 + x**2 + 1))