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

Optimal. Leaf size=165 \[ \frac{x \left (x^2 (-(d-2 f))+d+f\right )}{6 \left (x^4+x^2+1\right )}-\frac{1}{8} (2 d-f) \log \left (x^2-x+1\right )+\frac{1}{8} (2 d-f) \log \left (x^2+x+1\right )-\frac{(4 d+f) \tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{12 \sqrt{3}}+\frac{(4 d+f) \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{12 \sqrt{3}}+\frac{e \left (2 x^2+1\right )}{6 \left (x^4+x^2+1\right )}+\frac{2 e \tan ^{-1}\left (\frac{2 x^2+1}{\sqrt{3}}\right )}{3 \sqrt{3}} \]

[Out]

(e*(1 + 2*x^2))/(6*(1 + x^2 + x^4)) + (x*(d + f - (d - 2*f)*x^2))/(6*(1 + x^2 + x^4)) - ((4*d + f)*ArcTan[(1 -
2*x)/Sqrt[3]])/(12*Sqrt[3]) + ((4*d + f)*ArcTan[(1 + 2*x)/Sqrt[3]])/(12*Sqrt[3]) + (2*e*ArcTan[(1 + 2*x^2)/Sq
rt[3]])/(3*Sqrt[3]) - ((2*d - f)*Log[1 - x + x^2])/8 + ((2*d - f)*Log[1 + x + x^2])/8

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Rubi [A]  time = 0.129192, antiderivative size = 165, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 10, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.476, Rules used = {1673, 1178, 1169, 634, 618, 204, 628, 12, 1107, 614} \[ \frac{x \left (x^2 (-(d-2 f))+d+f\right )}{6 \left (x^4+x^2+1\right )}-\frac{1}{8} (2 d-f) \log \left (x^2-x+1\right )+\frac{1}{8} (2 d-f) \log \left (x^2+x+1\right )-\frac{(4 d+f) \tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{12 \sqrt{3}}+\frac{(4 d+f) \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{12 \sqrt{3}}+\frac{e \left (2 x^2+1\right )}{6 \left (x^4+x^2+1\right )}+\frac{2 e \tan ^{-1}\left (\frac{2 x^2+1}{\sqrt{3}}\right )}{3 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(e*(1 + 2*x^2))/(6*(1 + x^2 + x^4)) + (x*(d + f - (d - 2*f)*x^2))/(6*(1 + x^2 + x^4)) - ((4*d + f)*ArcTan[(1 -
2*x)/Sqrt[3]])/(12*Sqrt[3]) + ((4*d + f)*ArcTan[(1 + 2*x)/Sqrt[3]])/(12*Sqrt[3]) + (2*e*ArcTan[(1 + 2*x^2)/Sq
rt[3]])/(3*Sqrt[3]) - ((2*d - f)*Log[1 - x + x^2])/8 + ((2*d - f)*Log[1 + x + x^2])/8

Rule 1673

Int[(Pq_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Module[{q = Expon[Pq, x], k}, Int[Sum[Coeff[
Pq, x, 2*k]*x^(2*k), {k, 0, q/2}]*(a + b*x^2 + c*x^4)^p, x] + Int[x*Sum[Coeff[Pq, x, 2*k + 1]*x^(2*k), {k, 0,
(q - 1)/2}]*(a + b*x^2 + c*x^4)^p, x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] &&  !PolyQ[Pq, x^2]

Rule 1178

Int[((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> 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] + Dist[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 1169

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]}, Dist[1/(2*c*q*r), Int[(d*r - (d - e*q)*x)/(q - r*x + x^2), x], x] + Dist[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] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2
- b*d*e + a*e^2, 0] && NegQ[b^2 - 4*a*c]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(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 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 1107

Int[(x_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[(a + b*x + c*x^2)^p, x],
x, x^2], x] /; FreeQ[{a, b, c, p}, x]

Rule 614

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] - Dist[(2*c*(2*p + 3))/((p + 1)*(b^2 - 4*a*c)), Int[(a + b*x + c*x^2)^(p + 1), x], x] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[p, -3/2] && IntegerQ[4*p]

Rubi steps

\begin{align*} \int \frac{d+e x+f x^2}{\left (1+x^2+x^4\right )^2} \, dx &=\int \frac{e x}{\left (1+x^2+x^4\right )^2} \, dx+\int \frac{d+f x^2}{\left (1+x^2+x^4\right )^2} \, dx\\ &=\frac{x \left (d+f-(d-2 f) x^2\right )}{6 \left (1+x^2+x^4\right )}+\frac{1}{6} \int \frac{5 d-f+(-d+2 f) x^2}{1+x^2+x^4} \, dx+e \int \frac{x}{\left (1+x^2+x^4\right )^2} \, dx\\ &=\frac{x \left (d+f-(d-2 f) x^2\right )}{6 \left (1+x^2+x^4\right )}+\frac{1}{12} \int \frac{5 d-f-(6 d-3 f) x}{1-x+x^2} \, dx+\frac{1}{12} \int \frac{5 d-f+(6 d-3 f) x}{1+x+x^2} \, dx+\frac{1}{2} e \operatorname{Subst}\left (\int \frac{1}{\left (1+x+x^2\right )^2} \, dx,x,x^2\right )\\ &=\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{1}{3} e \operatorname{Subst}\left (\int \frac{1}{1+x+x^2} \, dx,x,x^2\right )+\frac{1}{8} (2 d-f) \int \frac{1+2 x}{1+x+x^2} \, dx+\frac{1}{8} (-2 d+f) \int \frac{-1+2 x}{1-x+x^2} \, dx+\frac{1}{24} (4 d+f) \int \frac{1}{1-x+x^2} \, dx+\frac{1}{24} (4 d+f) \int \frac{1}{1+x+x^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{1}{8} (2 d-f) \log \left (1-x+x^2\right )+\frac{1}{8} (2 d-f) \log \left (1+x+x^2\right )-\frac{1}{3} (2 e) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+2 x^2\right )+\frac{1}{12} (-4 d-f) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,-1+2 x\right )+\frac{1}{12} (-4 d-f) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+2 x\right )\\ &=\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) \tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{12 \sqrt{3}}+\frac{(4 d+f) \tan ^{-1}\left (\frac{1+2 x}{\sqrt{3}}\right )}{12 \sqrt{3}}+\frac{2 e \tan ^{-1}\left (\frac{1+2 x^2}{\sqrt{3}}\right )}{3 \sqrt{3}}-\frac{1}{8} (2 d-f) \log \left (1-x+x^2\right )+\frac{1}{8} (2 d-f) \log \left (1+x+x^2\right )\\ \end{align*}

Mathematica [C]  time = 0.42446, size = 186, normalized size = 1.13 \[ \frac{1}{36} \left (\frac{6 \left (x \left (-d x^2+d+2 f x^2+f\right )+2 e x^2+e\right )}{x^4+x^2+1}-\frac{\left (\left (\sqrt{3}-11 i\right ) d-2 \left (\sqrt{3}-2 i\right ) f\right ) \tan ^{-1}\left (\frac{1}{2} \left (\sqrt{3}-i\right ) x\right )}{\sqrt{\frac{1}{6} \left (1+i \sqrt{3}\right )}}-\frac{\left (\left (\sqrt{3}+11 i\right ) d-2 \left (\sqrt{3}+2 i\right ) f\right ) \tan ^{-1}\left (\frac{1}{2} \left (\sqrt{3}+i\right ) x\right )}{\sqrt{\frac{1}{6} \left (1-i \sqrt{3}\right )}}-8 \sqrt{3} e \tan ^{-1}\left (\frac{\sqrt{3}}{2 x^2+1}\right )\right ) \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((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

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Maple [A]  time = 0.014, size = 214, normalized size = 1.3 \begin{align*}{\frac{1}{4\,{x}^{2}+4\,x+4} \left ( \left ( -{\frac{d}{3}}-{\frac{e}{3}}+{\frac{2\,f}{3}} \right ) x-{\frac{2\,d}{3}}+{\frac{e}{3}}+{\frac{f}{3}} \right ) }+{\frac{d\ln \left ({x}^{2}+x+1 \right ) }{4}}-{\frac{\ln \left ({x}^{2}+x+1 \right ) f}{8}}+{\frac{d\sqrt{3}}{9}\arctan \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{3}}{3}} \right ) }-{\frac{2\,\sqrt{3}e}{9}\arctan \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{3}}{3}} \right ) }+{\frac{\sqrt{3}f}{36}\arctan \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{3}}{3}} \right ) }-{\frac{1}{4\,{x}^{2}-4\,x+4} \left ( \left ({\frac{d}{3}}-{\frac{e}{3}}-{\frac{2\,f}{3}} \right ) x-{\frac{2\,d}{3}}-{\frac{e}{3}}+{\frac{f}{3}} \right ) }-{\frac{d\ln \left ({x}^{2}-x+1 \right ) }{4}}+{\frac{\ln \left ({x}^{2}-x+1 \right ) f}{8}}+{\frac{d\sqrt{3}}{9}\arctan \left ({\frac{ \left ( 2\,x-1 \right ) \sqrt{3}}{3}} \right ) }+{\frac{2\,\sqrt{3}e}{9}\arctan \left ({\frac{ \left ( 2\,x-1 \right ) \sqrt{3}}{3}} \right ) }+{\frac{\sqrt{3}f}{36}\arctan \left ({\frac{ \left ( 2\,x-1 \right ) \sqrt{3}}{3}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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)+1/4*d*ln(x^2+x+1)-1/8*ln(x^2+x+1)*f+1/9*d*arctan(1/3*
(1+2*x)*3^(1/2))*3^(1/2)-2/9*3^(1/2)*arctan(1/3*(1+2*x)*3^(1/2))*e+1/36*3^(1/2)*arctan(1/3*(1+2*x)*3^(1/2))*f-
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)-1/4*d*ln(x^2-x+1)+1/8*ln(x^2-x+1)*f+1/9*3^(1/2)*arctan
(1/3*(2*x-1)*3^(1/2))*d+2/9*3^(1/2)*arctan(1/3*(2*x-1)*3^(1/2))*e+1/36*3^(1/2)*arctan(1/3*(2*x-1)*3^(1/2))*f

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Maxima [A]  time = 1.42986, size = 162, normalized size = 0.98 \begin{align*} \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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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)

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Fricas [A]  time = 1.90264, size = 547, normalized size = 3.32 \begin{align*} -\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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-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)*arc
tan(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*
sqrt(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)

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Sympy [C]  time = 33.2861, size = 4107, normalized size = 24.89 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-d/4 + f/8 - sqrt(3)*I*(4*d + 8*e + f)/72)*log(x + (-164944*d**5*e + 16416*d**5*(-d/4 + f/8 - sqrt(3)*I*(4*d
+ 8*e + f)/72) + 336520*d**4*e*f + 200664*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) + 26127
36*d**3*(-d/4 + f/8 - sqrt(3)*I*(4*d + 8*e + f)/72)**3 + 51840*d**2*e**3*f + 881280*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 + 184320*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 + 12
70080*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 - sqrt(3)*I*(4*d + 8*e + f)
/72)**2 + 70848*e**2*f**3*(-d/4 + f/8 - sqrt(3)*I*(4*d + 8*e + f)/72) - 995328*e**2*f*(-d/4 + f/8 - sqrt(3)*I*
(4*d + 8*e + f)/72)**3 + 3956*e*f**5 - 209088*e*f**3*(-d/4 + f/8 - sqrt(3)*I*(4*d + 8*e + f)/72)**2 - 3996*f**
5*(-d/4 + f/8 - sqrt(3)*I*(4*d + 8*e + f)/72) + 233280*f**3*(-d/4 + f/8 - sqrt(3)*I*(4*d + 8*e + f)/72)**3)/(5
3568*d**6 - 69984*d**5*f - 182528*d**4*e**2 + 23652*d**4*f**2 + 377344*d**3*e**2*f + 5400*d**3*f**3 - 126976*d
**2*e**4 - 278400*d**2*e**2*f**2 - 4131*d**2*f**4 + 102400*d*e**4*f + 93568*d*e**2*f**3 + 81*d*f**5 - 28672*e*
*4*f**2 - 11648*e**2*f**4 + 189*f**6)) + (-d/4 + f/8 + sqrt(3)*I*(4*d + 8*e + f)/72)*log(x + (-164944*d**5*e +
16416*d**5*(-d/4 + f/8 + sqrt(3)*I*(4*d + 8*e + f)/72) + 336520*d**4*e*f + 200664*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) - 27
2380*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 + 881280*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 + 184320*d*e**4*(-d/4 + f/8 + s
qrt(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 + sqrt(3)*I*(4*d + 8*e + f)/72)**2 + 70848*e**2*f**3*(-d/4 + f/8 + sqrt(3)*I*(4*d + 8*e + f)/72)
- 995328*e**2*f*(-d/4 + f/8 + sqrt(3)*I*(4*d + 8*e + f)/72)**3 + 3956*e*f**5 - 209088*e*f**3*(-d/4 + f/8 + sq
rt(3)*I*(4*d + 8*e + f)/72)**2 - 3996*f**5*(-d/4 + f/8 + sqrt(3)*I*(4*d + 8*e + f)/72) + 233280*f**3*(-d/4 + f
/8 + sqrt(3)*I*(4*d + 8*e + f)/72)**3)/(53568*d**6 - 69984*d**5*f - 182528*d**4*e**2 + 23652*d**4*f**2 + 37734
4*d**3*e**2*f + 5400*d**3*f**3 - 126976*d**2*e**4 - 278400*d**2*e**2*f**2 - 4131*d**2*f**4 + 102400*d*e**4*f +
93568*d*e**2*f**3 + 81*d*f**5 - 28672*e**4*f**2 - 11648*e**2*f**4 + 189*f**6)) + (d/4 - f/8 - sqrt(3)*I*(4*d
- 8*e + f)/72)*log(x + (-164944*d**5*e + 16416*d**5*(d/4 - f/8 - sqrt(3)*I*(4*d - 8*e + f)/72) + 336520*d**4*e
*f + 200664*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 + 881280*d**2*e**2*f*(d/4 - f/8 - sqrt(3)*I*(4*d - 8*e + f)/72) + 1194
20*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 +
184320*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 - sqrt(3)*I*(4*d - 8*e + f)/72)**2 + 70848*e**2*f**3*(d/4 - f/8 - sqrt(3)*I
*(4*d - 8*e + f)/72) - 995328*e**2*f*(d/4 - f/8 - sqrt(3)*I*(4*d - 8*e + f)/72)**3 + 3956*e*f**5 - 209088*e*f*
*3*(d/4 - f/8 - sqrt(3)*I*(4*d - 8*e + f)/72)**2 - 3996*f**5*(d/4 - f/8 - sqrt(3)*I*(4*d - 8*e + f)/72) + 2332
80*f**3*(d/4 - f/8 - sqrt(3)*I*(4*d - 8*e + f)/72)**3)/(53568*d**6 - 69984*d**5*f - 182528*d**4*e**2 + 23652*d
**4*f**2 + 377344*d**3*e**2*f + 5400*d**3*f**3 - 126976*d**2*e**4 - 278400*d**2*e**2*f**2 - 4131*d**2*f**4 + 1
02400*d*e**4*f + 93568*d*e**2*f**3 + 81*d*f**5 - 28672*e**4*f**2 - 11648*e**2*f**4 + 189*f**6)) + (d/4 - f/8 +
sqrt(3)*I*(4*d - 8*e + f)/72)*log(x + (-164944*d**5*e + 16416*d**5*(d/4 - f/8 + sqrt(3)*I*(4*d - 8*e + f)/72)
+ 336520*d**4*e*f + 200664*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 + 881280*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 + 184320*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 + sqrt(3)*I*(4*d - 8*e + f)/72)**2 + 70848*e**2*f**3*(d/4 -
f/8 + sqrt(3)*I*(4*d - 8*e + f)/72) - 995328*e**2*f*(d/4 - f/8 + sqrt(3)*I*(4*d - 8*e + f)/72)**3 + 3956*e*f*
*5 - 209088*e*f**3*(d/4 - f/8 + sqrt(3)*I*(4*d - 8*e + f)/72)**2 - 3996*f**5*(d/4 - f/8 + sqrt(3)*I*(4*d - 8*e
+ f)/72) + 233280*f**3*(d/4 - f/8 + sqrt(3)*I*(4*d - 8*e + f)/72)**3)/(53568*d**6 - 69984*d**5*f - 182528*d**
4*e**2 + 23652*d**4*f**2 + 377344*d**3*e**2*f + 5400*d**3*f**3 - 126976*d**2*e**4 - 278400*d**2*e**2*f**2 - 41
31*d**2*f**4 + 102400*d*e**4*f + 93568*d*e**2*f**3 + 81*d*f**5 - 28672*e**4*f**2 - 11648*e**2*f**4 + 189*f**6)
) - (-2*e*x**2 - e + x**3*(d - 2*f) + x*(-d - f))/(6*x**4 + 6*x**2 + 6)

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Giac [A]  time = 1.09278, size = 173, normalized size = 1.05 \begin{align*} \frac{1}{36} \, \sqrt{3}{\left (4 \, d + f - 8 \, e\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) + \frac{1}{36} \, \sqrt{3}{\left (4 \, d + f + 8 \, e\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 \, x^{2} e - d x - f x - e}{6 \,{\left (x^{4} + x^{2} + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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