3.48 \(\int \frac{d+e x+f x^2}{(1+x^2+x^4)^3} \, dx\)

Optimal. Leaf size=223 \[ \frac{x \left (-7 x^2 (d-f)+2 d+3 f\right )}{24 \left (x^4+x^2+1\right )}+\frac{x \left (x^2 (-(d-2 f))+d+f\right )}{12 \left (x^4+x^2+1\right )^2}-\frac{1}{32} (9 d-4 f) \log \left (x^2-x+1\right )+\frac{1}{32} (9 d-4 f) \log \left (x^2+x+1\right )-\frac{(13 d+2 f) \tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{48 \sqrt{3}}+\frac{(13 d+2 f) \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{48 \sqrt{3}}+\frac{e \left (2 x^2+1\right )}{6 \left (x^4+x^2+1\right )}+\frac{e \left (2 x^2+1\right )}{12 \left (x^4+x^2+1\right )^2}+\frac{2 e \tan ^{-1}\left (\frac{2 x^2+1}{\sqrt{3}}\right )}{3 \sqrt{3}} \]

[Out]

(e*(1 + 2*x^2))/(12*(1 + x^2 + x^4)^2) + (x*(d + f - (d - 2*f)*x^2))/(12*(1 + x^2 + x^4)^2) + (e*(1 + 2*x^2))/
(6*(1 + x^2 + x^4)) + (x*(2*d + 3*f - 7*(d - f)*x^2))/(24*(1 + x^2 + x^4)) - ((13*d + 2*f)*ArcTan[(1 - 2*x)/Sq
rt[3]])/(48*Sqrt[3]) + ((13*d + 2*f)*ArcTan[(1 + 2*x)/Sqrt[3]])/(48*Sqrt[3]) + (2*e*ArcTan[(1 + 2*x^2)/Sqrt[3]
])/(3*Sqrt[3]) - ((9*d - 4*f)*Log[1 - x + x^2])/32 + ((9*d - 4*f)*Log[1 + x + x^2])/32

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Rubi [A]  time = 0.214941, antiderivative size = 223, normalized size of antiderivative = 1., number of steps used = 18, 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} \[ \frac{x \left (-7 x^2 (d-f)+2 d+3 f\right )}{24 \left (x^4+x^2+1\right )}+\frac{x \left (x^2 (-(d-2 f))+d+f\right )}{12 \left (x^4+x^2+1\right )^2}-\frac{1}{32} (9 d-4 f) \log \left (x^2-x+1\right )+\frac{1}{32} (9 d-4 f) \log \left (x^2+x+1\right )-\frac{(13 d+2 f) \tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{48 \sqrt{3}}+\frac{(13 d+2 f) \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{48 \sqrt{3}}+\frac{e \left (2 x^2+1\right )}{6 \left (x^4+x^2+1\right )}+\frac{e \left (2 x^2+1\right )}{12 \left (x^4+x^2+1\right )^2}+\frac{2 e \tan ^{-1}\left (\frac{2 x^2+1}{\sqrt{3}}\right )}{3 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x + f*x^2)/(1 + x^2 + x^4)^3,x]

[Out]

(e*(1 + 2*x^2))/(12*(1 + x^2 + x^4)^2) + (x*(d + f - (d - 2*f)*x^2))/(12*(1 + x^2 + x^4)^2) + (e*(1 + 2*x^2))/
(6*(1 + x^2 + x^4)) + (x*(2*d + 3*f - 7*(d - f)*x^2))/(24*(1 + x^2 + x^4)) - ((13*d + 2*f)*ArcTan[(1 - 2*x)/Sq
rt[3]])/(48*Sqrt[3]) + ((13*d + 2*f)*ArcTan[(1 + 2*x)/Sqrt[3]])/(48*Sqrt[3]) + (2*e*ArcTan[(1 + 2*x^2)/Sqrt[3]
])/(3*Sqrt[3]) - ((9*d - 4*f)*Log[1 - x + x^2])/32 + ((9*d - 4*f)*Log[1 + x + x^2])/32

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 )^3} \, dx &=\int \frac{e x}{\left (1+x^2+x^4\right )^3} \, dx+\int \frac{d+f x^2}{\left (1+x^2+x^4\right )^3} \, dx\\ &=\frac{x \left (d+f-(d-2 f) x^2\right )}{12 \left (1+x^2+x^4\right )^2}+\frac{1}{12} \int \frac{11 d-f-5 (d-2 f) x^2}{\left (1+x^2+x^4\right )^2} \, dx+e \int \frac{x}{\left (1+x^2+x^4\right )^3} \, dx\\ &=\frac{x \left (d+f-(d-2 f) x^2\right )}{12 \left (1+x^2+x^4\right )^2}+\frac{x \left (2 d+3 f-7 (d-f) x^2\right )}{24 \left (1+x^2+x^4\right )}+\frac{1}{72} \int \frac{15 (4 d-f)-21 (d-f) x^2}{1+x^2+x^4} \, dx+\frac{1}{2} e \operatorname{Subst}\left (\int \frac{1}{\left (1+x+x^2\right )^3} \, dx,x,x^2\right )\\ &=\frac{e \left (1+2 x^2\right )}{12 \left (1+x^2+x^4\right )^2}+\frac{x \left (d+f-(d-2 f) x^2\right )}{12 \left (1+x^2+x^4\right )^2}+\frac{x \left (2 d+3 f-7 (d-f) x^2\right )}{24 \left (1+x^2+x^4\right )}+\frac{1}{144} \int \frac{15 (4 d-f)-(21 (d-f)+15 (4 d-f)) x}{1-x+x^2} \, dx+\frac{1}{144} \int \frac{15 (4 d-f)+(21 (d-f)+15 (4 d-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 )}{12 \left (1+x^2+x^4\right )^2}+\frac{x \left (d+f-(d-2 f) x^2\right )}{12 \left (1+x^2+x^4\right )^2}+\frac{e \left (1+2 x^2\right )}{6 \left (1+x^2+x^4\right )}+\frac{x \left (2 d+3 f-7 (d-f) x^2\right )}{24 \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}{32} (9 d-4 f) \int \frac{1+2 x}{1+x+x^2} \, dx+\frac{1}{96} (13 d+2 f) \int \frac{1}{1-x+x^2} \, dx+\frac{1}{96} (13 d+2 f) \int \frac{1}{1+x+x^2} \, dx+\frac{1}{32} (-9 d+4 f) \int \frac{-1+2 x}{1-x+x^2} \, dx\\ &=\frac{e \left (1+2 x^2\right )}{12 \left (1+x^2+x^4\right )^2}+\frac{x \left (d+f-(d-2 f) x^2\right )}{12 \left (1+x^2+x^4\right )^2}+\frac{e \left (1+2 x^2\right )}{6 \left (1+x^2+x^4\right )}+\frac{x \left (2 d+3 f-7 (d-f) x^2\right )}{24 \left (1+x^2+x^4\right )}-\frac{1}{32} (9 d-4 f) \log \left (1-x+x^2\right )+\frac{1}{32} (9 d-4 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}{48} (-13 d-2 f) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,-1+2 x\right )+\frac{1}{48} (-13 d-2 f) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+2 x\right )\\ &=\frac{e \left (1+2 x^2\right )}{12 \left (1+x^2+x^4\right )^2}+\frac{x \left (d+f-(d-2 f) x^2\right )}{12 \left (1+x^2+x^4\right )^2}+\frac{e \left (1+2 x^2\right )}{6 \left (1+x^2+x^4\right )}+\frac{x \left (2 d+3 f-7 (d-f) x^2\right )}{24 \left (1+x^2+x^4\right )}-\frac{(13 d+2 f) \tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{48 \sqrt{3}}+\frac{(13 d+2 f) \tan ^{-1}\left (\frac{1+2 x}{\sqrt{3}}\right )}{48 \sqrt{3}}+\frac{2 e \tan ^{-1}\left (\frac{1+2 x^2}{\sqrt{3}}\right )}{3 \sqrt{3}}-\frac{1}{32} (9 d-4 f) \log \left (1-x+x^2\right )+\frac{1}{32} (9 d-4 f) \log \left (1+x+x^2\right )\\ \end{align*}

Mathematica [C]  time = 0.589907, size = 235, normalized size = 1.05 \[ \frac{1}{144} \left (\frac{6 \left (-7 d x^3+2 d x+e \left (8 x^2+4\right )+7 f x^3+3 f x\right )}{x^4+x^2+1}+\frac{12 \left (x \left (-d x^2+d+2 f x^2+f\right )+2 e x^2+e\right )}{\left (x^4+x^2+1\right )^2}-\frac{\left (\left (7 \sqrt{3}-47 i\right ) d+\left (-7 \sqrt{3}+17 i\right ) f\right ) \tan ^{-1}\left (\frac{1}{2} \left (\sqrt{3}-i\right ) x\right )}{\sqrt{\frac{1}{6} \left (1+i \sqrt{3}\right )}}-\frac{\left (\left (7 \sqrt{3}+47 i\right ) d-\left (7 \sqrt{3}+17 i\right ) f\right ) \tan ^{-1}\left (\frac{1}{2} \left (\sqrt{3}+i\right ) x\right )}{\sqrt{\frac{1}{6} \left (1-i \sqrt{3}\right )}}-32 \sqrt{3} e \tan ^{-1}\left (\frac{\sqrt{3}}{2 x^2+1}\right )\right ) \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(d + e*x + f*x^2)/(1 + x^2 + x^4)^3,x]

[Out]

((6*(2*d*x + 3*f*x - 7*d*x^3 + 7*f*x^3 + e*(4 + 8*x^2)))/(1 + x^2 + x^4) + (12*(e + 2*e*x^2 + x*(d + f - d*x^2
+ 2*f*x^2)))/(1 + x^2 + x^4)^2 - (((-47*I + 7*Sqrt[3])*d + (17*I - 7*Sqrt[3])*f)*ArcTan[((-I + Sqrt[3])*x)/2]
)/Sqrt[(1 + I*Sqrt[3])/6] - (((47*I + 7*Sqrt[3])*d - (17*I + 7*Sqrt[3])*f)*ArcTan[((I + Sqrt[3])*x)/2])/Sqrt[(
1 - I*Sqrt[3])/6] - 32*Sqrt[3]*e*ArcTan[Sqrt[3]/(1 + 2*x^2)])/144

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Maple [A]  time = 0.018, size = 264, normalized size = 1.2 \begin{align*}{\frac{1}{16\, \left ({x}^{2}+x+1 \right ) ^{2}} \left ( \left ( -{\frac{7\,d}{3}}+{\frac{7\,f}{3}}-{\frac{4\,e}{3}} \right ){x}^{3}+ \left ( -6\,d+4\,f \right ){x}^{2}+ \left ( -{\frac{20\,d}{3}}+{\frac{13\,f}{3}}+{\frac{e}{3}} \right ) x-4\,d+{\frac{4\,f}{3}}+2\,e \right ) }+{\frac{9\,d\ln \left ({x}^{2}+x+1 \right ) }{32}}-{\frac{\ln \left ({x}^{2}+x+1 \right ) f}{8}}+{\frac{13\,d\sqrt{3}}{144}\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}{72}\arctan \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{3}}{3}} \right ) }-{\frac{1}{16\, \left ({x}^{2}-x+1 \right ) ^{2}} \left ( \left ({\frac{7\,d}{3}}-{\frac{7\,f}{3}}-{\frac{4\,e}{3}} \right ){x}^{3}+ \left ( -6\,d+4\,f \right ){x}^{2}+ \left ({\frac{20\,d}{3}}-{\frac{13\,f}{3}}+{\frac{e}{3}} \right ) x-4\,d+{\frac{4\,f}{3}}-2\,e \right ) }-{\frac{9\,d\ln \left ({x}^{2}-x+1 \right ) }{32}}+{\frac{\ln \left ({x}^{2}-x+1 \right ) f}{8}}+{\frac{13\,d\sqrt{3}}{144}\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}{72}\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)^3,x)

[Out]

1/16*((-7/3*d+7/3*f-4/3*e)*x^3+(-6*d+4*f)*x^2+(-20/3*d+13/3*f+1/3*e)*x-4*d+4/3*f+2*e)/(x^2+x+1)^2+9/32*d*ln(x^
2+x+1)-1/8*ln(x^2+x+1)*f+13/144*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+1/72*3^(1/2)*arctan(1/3*(1+2*x)*3^(1/2))*f-1/16*((7/3*d-7/3*f-4/3*e)*x^3+(-6*d+4*f)*x^2+(20/3*d-13/3*f+1/3*e
)*x-4*d+4/3*f-2*e)/(x^2-x+1)^2-9/32*d*ln(x^2-x+1)+1/8*ln(x^2-x+1)*f+13/144*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+1/72*3^(1/2)*arctan(1/3*(2*x-1)*3^(1/2))*f

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Maxima [A]  time = 1.43095, size = 234, normalized size = 1.05 \begin{align*} \frac{1}{144} \, \sqrt{3}{\left (13 \, d - 32 \, e + 2 \, f\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) + \frac{1}{144} \, \sqrt{3}{\left (13 \, d + 32 \, e + 2 \, f\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) + \frac{1}{32} \,{\left (9 \, d - 4 \, f\right )} \log \left (x^{2} + x + 1\right ) - \frac{1}{32} \,{\left (9 \, d - 4 \, f\right )} \log \left (x^{2} - x + 1\right ) - \frac{7 \,{\left (d - f\right )} x^{7} - 8 \, e x^{6} + 5 \,{\left (d - 2 \, f\right )} x^{5} - 12 \, e x^{4} + 7 \,{\left (d - 2 \, f\right )} x^{3} - 16 \, e x^{2} -{\left (4 \, d + 5 \, f\right )} x - 6 \, e}{24 \,{\left (x^{8} + 2 \, x^{6} + 3 \, x^{4} + 2 \, 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)^3,x, algorithm="maxima")

[Out]

1/144*sqrt(3)*(13*d - 32*e + 2*f)*arctan(1/3*sqrt(3)*(2*x + 1)) + 1/144*sqrt(3)*(13*d + 32*e + 2*f)*arctan(1/3
*sqrt(3)*(2*x - 1)) + 1/32*(9*d - 4*f)*log(x^2 + x + 1) - 1/32*(9*d - 4*f)*log(x^2 - x + 1) - 1/24*(7*(d - f)*
x^7 - 8*e*x^6 + 5*(d - 2*f)*x^5 - 12*e*x^4 + 7*(d - 2*f)*x^3 - 16*e*x^2 - (4*d + 5*f)*x - 6*e)/(x^8 + 2*x^6 +
3*x^4 + 2*x^2 + 1)

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Fricas [A]  time = 2.21264, size = 977, normalized size = 4.38 \begin{align*} -\frac{84 \,{\left (d - f\right )} x^{7} - 96 \, e x^{6} + 60 \,{\left (d - 2 \, f\right )} x^{5} - 144 \, e x^{4} + 84 \,{\left (d - 2 \, f\right )} x^{3} - 192 \, e x^{2} - 2 \, \sqrt{3}{\left ({\left (13 \, d - 32 \, e + 2 \, f\right )} x^{8} + 2 \,{\left (13 \, d - 32 \, e + 2 \, f\right )} x^{6} + 3 \,{\left (13 \, d - 32 \, e + 2 \, f\right )} x^{4} + 2 \,{\left (13 \, d - 32 \, e + 2 \, f\right )} x^{2} + 13 \, d - 32 \, e + 2 \, f\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) - 2 \, \sqrt{3}{\left ({\left (13 \, d + 32 \, e + 2 \, f\right )} x^{8} + 2 \,{\left (13 \, d + 32 \, e + 2 \, f\right )} x^{6} + 3 \,{\left (13 \, d + 32 \, e + 2 \, f\right )} x^{4} + 2 \,{\left (13 \, d + 32 \, e + 2 \, f\right )} x^{2} + 13 \, d + 32 \, e + 2 \, f\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) - 12 \,{\left (4 \, d + 5 \, f\right )} x - 9 \,{\left ({\left (9 \, d - 4 \, f\right )} x^{8} + 2 \,{\left (9 \, d - 4 \, f\right )} x^{6} + 3 \,{\left (9 \, d - 4 \, f\right )} x^{4} + 2 \,{\left (9 \, d - 4 \, f\right )} x^{2} + 9 \, d - 4 \, f\right )} \log \left (x^{2} + x + 1\right ) + 9 \,{\left ({\left (9 \, d - 4 \, f\right )} x^{8} + 2 \,{\left (9 \, d - 4 \, f\right )} x^{6} + 3 \,{\left (9 \, d - 4 \, f\right )} x^{4} + 2 \,{\left (9 \, d - 4 \, f\right )} x^{2} + 9 \, d - 4 \, f\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 )}} \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)^3,x, algorithm="fricas")

[Out]

-1/288*(84*(d - f)*x^7 - 96*e*x^6 + 60*(d - 2*f)*x^5 - 144*e*x^4 + 84*(d - 2*f)*x^3 - 192*e*x^2 - 2*sqrt(3)*((
13*d - 32*e + 2*f)*x^8 + 2*(13*d - 32*e + 2*f)*x^6 + 3*(13*d - 32*e + 2*f)*x^4 + 2*(13*d - 32*e + 2*f)*x^2 + 1
3*d - 32*e + 2*f)*arctan(1/3*sqrt(3)*(2*x + 1)) - 2*sqrt(3)*((13*d + 32*e + 2*f)*x^8 + 2*(13*d + 32*e + 2*f)*x
^6 + 3*(13*d + 32*e + 2*f)*x^4 + 2*(13*d + 32*e + 2*f)*x^2 + 13*d + 32*e + 2*f)*arctan(1/3*sqrt(3)*(2*x - 1))
- 12*(4*d + 5*f)*x - 9*((9*d - 4*f)*x^8 + 2*(9*d - 4*f)*x^6 + 3*(9*d - 4*f)*x^4 + 2*(9*d - 4*f)*x^2 + 9*d - 4*
f)*log(x^2 + x + 1) + 9*((9*d - 4*f)*x^8 + 2*(9*d - 4*f)*x^6 + 3*(9*d - 4*f)*x^4 + 2*(9*d - 4*f)*x^2 + 9*d - 4
*f)*log(x^2 - x + 1) - 72*e)/(x^8 + 2*x^6 + 3*x^4 + 2*x^2 + 1)

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Sympy [C]  time = 36.7534, size = 4498, normalized size = 20.17 \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)**3,x)

[Out]

(-9*d/32 + f/8 - sqrt(3)*I*(13*d + 32*e + 2*f)/288)*log(x + (-1025428432*d**5*e - 334752912*d**5*(-9*d/32 + f/
8 - sqrt(3)*I*(13*d + 32*e + 2*f)/288) + 2008961360*d**4*e*f + 1151575920*d**4*f*(-9*d/32 + f/8 - sqrt(3)*I*(1
3*d + 32*e + 2*f)/288) - 431308800*d**3*e**3 - 3143688192*d**3*e**2*(-9*d/32 + f/8 - sqrt(3)*I*(13*d + 32*e +
2*f)/288) - 1598857120*d**3*e*f**2 + 9917005824*d**3*e*(-9*d/32 + f/8 - sqrt(3)*I*(13*d + 32*e + 2*f)/288)**2
- 944300160*d**3*f**2*(-9*d/32 + f/8 - sqrt(3)*I*(13*d + 32*e + 2*f)/288) + 11878244352*d**3*(-9*d/32 + f/8 -
sqrt(3)*I*(13*d + 32*e + 2*f)/288)**3 + 233164800*d**2*e**3*f + 4409634816*d**2*e**2*f*(-9*d/32 + f/8 - sqrt(3
)*I*(13*d + 32*e + 2*f)/288) + 662937520*d**2*e*f**3 - 13004623872*d**2*e*f*(-9*d/32 + f/8 - sqrt(3)*I*(13*d +
32*e + 2*f)/288)**2 + 231796080*d**2*f**3*(-9*d/32 + f/8 - sqrt(3)*I*(13*d + 32*e + 2*f)/288) - 10089639936*d
**2*f*(-9*d/32 + f/8 - sqrt(3)*I*(13*d + 32*e + 2*f)/288)**3 + 142606336*d*e**5 + 754974720*d*e**4*(-9*d/32 +
f/8 - sqrt(3)*I*(13*d + 32*e + 2*f)/288) - 1843200*d*e**3*f**2 + 3850371072*d*e**3*(-9*d/32 + f/8 - sqrt(3)*I*
(13*d + 32*e + 2*f)/288)**2 - 1926291456*d*e**2*f**2*(-9*d/32 + f/8 - sqrt(3)*I*(13*d + 32*e + 2*f)/288) + 203
84317440*d*e**2*(-9*d/32 + f/8 - sqrt(3)*I*(13*d + 32*e + 2*f)/288)**3 - 146756960*d*e*f**4 + 5813379072*d*e*f
**2*(-9*d/32 + f/8 - sqrt(3)*I*(13*d + 32*e + 2*f)/288)**2 + 12679200*d*f**4*(-9*d/32 + f/8 - sqrt(3)*I*(13*d
+ 32*e + 2*f)/288) + 1116758016*d*f**2*(-9*d/32 + f/8 - sqrt(3)*I*(13*d + 32*e + 2*f)/288)**3 - 79691776*e**5*
f - 188743680*e**4*f*(-9*d/32 + f/8 - sqrt(3)*I*(13*d + 32*e + 2*f)/288) - 7372800*e**3*f**3 - 2151677952*e**3
*f*(-9*d/32 + f/8 - sqrt(3)*I*(13*d + 32*e + 2*f)/288)**2 + 287096832*e**2*f**3*(-9*d/32 + f/8 - sqrt(3)*I*(13
*d + 32*e + 2*f)/288) - 5096079360*e**2*f*(-9*d/32 + f/8 - sqrt(3)*I*(13*d + 32*e + 2*f)/288)**3 + 14093632*e*
f**5 - 859521024*e*f**3*(-9*d/32 + f/8 - sqrt(3)*I*(13*d + 32*e + 2*f)/288)**2 - 7648128*f**5*(-9*d/32 + f/8 -
sqrt(3)*I*(13*d + 32*e + 2*f)/288) + 453869568*f**3*(-9*d/32 + f/8 - sqrt(3)*I*(13*d + 32*e + 2*f)/288)**3)/(
217696167*d**6 - 301346487*d**5*f - 1217128448*d**4*e**2 + 130506255*d**4*f**2 + 2181281792*d**3*e**2*f - 5619
240*d**3*f**3 - 617611264*d**2*e**4 - 1450149888*d**2*e**2*f**2 - 8036820*d**2*f**4 + 495976448*d*e**4*f + 430
088192*d*e**2*f**3 + 783648*d*f**5 - 114294784*e**4*f**2 - 47771648*e**2*f**4 + 188352*f**6)) + (-9*d/32 + f/8
+ sqrt(3)*I*(13*d + 32*e + 2*f)/288)*log(x + (-1025428432*d**5*e - 334752912*d**5*(-9*d/32 + f/8 + sqrt(3)*I*
(13*d + 32*e + 2*f)/288) + 2008961360*d**4*e*f + 1151575920*d**4*f*(-9*d/32 + f/8 + sqrt(3)*I*(13*d + 32*e + 2
*f)/288) - 431308800*d**3*e**3 - 3143688192*d**3*e**2*(-9*d/32 + f/8 + sqrt(3)*I*(13*d + 32*e + 2*f)/288) - 15
98857120*d**3*e*f**2 + 9917005824*d**3*e*(-9*d/32 + f/8 + sqrt(3)*I*(13*d + 32*e + 2*f)/288)**2 - 944300160*d*
*3*f**2*(-9*d/32 + f/8 + sqrt(3)*I*(13*d + 32*e + 2*f)/288) + 11878244352*d**3*(-9*d/32 + f/8 + sqrt(3)*I*(13*
d + 32*e + 2*f)/288)**3 + 233164800*d**2*e**3*f + 4409634816*d**2*e**2*f*(-9*d/32 + f/8 + sqrt(3)*I*(13*d + 32
*e + 2*f)/288) + 662937520*d**2*e*f**3 - 13004623872*d**2*e*f*(-9*d/32 + f/8 + sqrt(3)*I*(13*d + 32*e + 2*f)/2
88)**2 + 231796080*d**2*f**3*(-9*d/32 + f/8 + sqrt(3)*I*(13*d + 32*e + 2*f)/288) - 10089639936*d**2*f*(-9*d/32
+ f/8 + sqrt(3)*I*(13*d + 32*e + 2*f)/288)**3 + 142606336*d*e**5 + 754974720*d*e**4*(-9*d/32 + f/8 + sqrt(3)*
I*(13*d + 32*e + 2*f)/288) - 1843200*d*e**3*f**2 + 3850371072*d*e**3*(-9*d/32 + f/8 + sqrt(3)*I*(13*d + 32*e +
2*f)/288)**2 - 1926291456*d*e**2*f**2*(-9*d/32 + f/8 + sqrt(3)*I*(13*d + 32*e + 2*f)/288) + 20384317440*d*e**
2*(-9*d/32 + f/8 + sqrt(3)*I*(13*d + 32*e + 2*f)/288)**3 - 146756960*d*e*f**4 + 5813379072*d*e*f**2*(-9*d/32 +
f/8 + sqrt(3)*I*(13*d + 32*e + 2*f)/288)**2 + 12679200*d*f**4*(-9*d/32 + f/8 + sqrt(3)*I*(13*d + 32*e + 2*f)/
288) + 1116758016*d*f**2*(-9*d/32 + f/8 + sqrt(3)*I*(13*d + 32*e + 2*f)/288)**3 - 79691776*e**5*f - 188743680*
e**4*f*(-9*d/32 + f/8 + sqrt(3)*I*(13*d + 32*e + 2*f)/288) - 7372800*e**3*f**3 - 2151677952*e**3*f*(-9*d/32 +
f/8 + sqrt(3)*I*(13*d + 32*e + 2*f)/288)**2 + 287096832*e**2*f**3*(-9*d/32 + f/8 + sqrt(3)*I*(13*d + 32*e + 2*
f)/288) - 5096079360*e**2*f*(-9*d/32 + f/8 + sqrt(3)*I*(13*d + 32*e + 2*f)/288)**3 + 14093632*e*f**5 - 8595210
24*e*f**3*(-9*d/32 + f/8 + sqrt(3)*I*(13*d + 32*e + 2*f)/288)**2 - 7648128*f**5*(-9*d/32 + f/8 + sqrt(3)*I*(13
*d + 32*e + 2*f)/288) + 453869568*f**3*(-9*d/32 + f/8 + sqrt(3)*I*(13*d + 32*e + 2*f)/288)**3)/(217696167*d**6
- 301346487*d**5*f - 1217128448*d**4*e**2 + 130506255*d**4*f**2 + 2181281792*d**3*e**2*f - 5619240*d**3*f**3
- 617611264*d**2*e**4 - 1450149888*d**2*e**2*f**2 - 8036820*d**2*f**4 + 495976448*d*e**4*f + 430088192*d*e**2*
f**3 + 783648*d*f**5 - 114294784*e**4*f**2 - 47771648*e**2*f**4 + 188352*f**6)) + (9*d/32 - f/8 - sqrt(3)*I*(1
3*d - 32*e + 2*f)/288)*log(x + (-1025428432*d**5*e - 334752912*d**5*(9*d/32 - f/8 - sqrt(3)*I*(13*d - 32*e + 2
*f)/288) + 2008961360*d**4*e*f + 1151575920*d**4*f*(9*d/32 - f/8 - sqrt(3)*I*(13*d - 32*e + 2*f)/288) - 431308
800*d**3*e**3 - 3143688192*d**3*e**2*(9*d/32 - f/8 - sqrt(3)*I*(13*d - 32*e + 2*f)/288) - 1598857120*d**3*e*f*
*2 + 9917005824*d**3*e*(9*d/32 - f/8 - sqrt(3)*I*(13*d - 32*e + 2*f)/288)**2 - 944300160*d**3*f**2*(9*d/32 - f
/8 - sqrt(3)*I*(13*d - 32*e + 2*f)/288) + 11878244352*d**3*(9*d/32 - f/8 - sqrt(3)*I*(13*d - 32*e + 2*f)/288)*
*3 + 233164800*d**2*e**3*f + 4409634816*d**2*e**2*f*(9*d/32 - f/8 - sqrt(3)*I*(13*d - 32*e + 2*f)/288) + 66293
7520*d**2*e*f**3 - 13004623872*d**2*e*f*(9*d/32 - f/8 - sqrt(3)*I*(13*d - 32*e + 2*f)/288)**2 + 231796080*d**2
*f**3*(9*d/32 - f/8 - sqrt(3)*I*(13*d - 32*e + 2*f)/288) - 10089639936*d**2*f*(9*d/32 - f/8 - sqrt(3)*I*(13*d
- 32*e + 2*f)/288)**3 + 142606336*d*e**5 + 754974720*d*e**4*(9*d/32 - f/8 - sqrt(3)*I*(13*d - 32*e + 2*f)/288)
- 1843200*d*e**3*f**2 + 3850371072*d*e**3*(9*d/32 - f/8 - sqrt(3)*I*(13*d - 32*e + 2*f)/288)**2 - 1926291456*
d*e**2*f**2*(9*d/32 - f/8 - sqrt(3)*I*(13*d - 32*e + 2*f)/288) + 20384317440*d*e**2*(9*d/32 - f/8 - sqrt(3)*I*
(13*d - 32*e + 2*f)/288)**3 - 146756960*d*e*f**4 + 5813379072*d*e*f**2*(9*d/32 - f/8 - sqrt(3)*I*(13*d - 32*e
+ 2*f)/288)**2 + 12679200*d*f**4*(9*d/32 - f/8 - sqrt(3)*I*(13*d - 32*e + 2*f)/288) + 1116758016*d*f**2*(9*d/3
2 - f/8 - sqrt(3)*I*(13*d - 32*e + 2*f)/288)**3 - 79691776*e**5*f - 188743680*e**4*f*(9*d/32 - f/8 - sqrt(3)*I
*(13*d - 32*e + 2*f)/288) - 7372800*e**3*f**3 - 2151677952*e**3*f*(9*d/32 - f/8 - sqrt(3)*I*(13*d - 32*e + 2*f
)/288)**2 + 287096832*e**2*f**3*(9*d/32 - f/8 - sqrt(3)*I*(13*d - 32*e + 2*f)/288) - 5096079360*e**2*f*(9*d/32
- f/8 - sqrt(3)*I*(13*d - 32*e + 2*f)/288)**3 + 14093632*e*f**5 - 859521024*e*f**3*(9*d/32 - f/8 - sqrt(3)*I*
(13*d - 32*e + 2*f)/288)**2 - 7648128*f**5*(9*d/32 - f/8 - sqrt(3)*I*(13*d - 32*e + 2*f)/288) + 453869568*f**3
*(9*d/32 - f/8 - sqrt(3)*I*(13*d - 32*e + 2*f)/288)**3)/(217696167*d**6 - 301346487*d**5*f - 1217128448*d**4*e
**2 + 130506255*d**4*f**2 + 2181281792*d**3*e**2*f - 5619240*d**3*f**3 - 617611264*d**2*e**4 - 1450149888*d**2
*e**2*f**2 - 8036820*d**2*f**4 + 495976448*d*e**4*f + 430088192*d*e**2*f**3 + 783648*d*f**5 - 114294784*e**4*f
**2 - 47771648*e**2*f**4 + 188352*f**6)) + (9*d/32 - f/8 + sqrt(3)*I*(13*d - 32*e + 2*f)/288)*log(x + (-102542
8432*d**5*e - 334752912*d**5*(9*d/32 - f/8 + sqrt(3)*I*(13*d - 32*e + 2*f)/288) + 2008961360*d**4*e*f + 115157
5920*d**4*f*(9*d/32 - f/8 + sqrt(3)*I*(13*d - 32*e + 2*f)/288) - 431308800*d**3*e**3 - 3143688192*d**3*e**2*(9
*d/32 - f/8 + sqrt(3)*I*(13*d - 32*e + 2*f)/288) - 1598857120*d**3*e*f**2 + 9917005824*d**3*e*(9*d/32 - f/8 +
sqrt(3)*I*(13*d - 32*e + 2*f)/288)**2 - 944300160*d**3*f**2*(9*d/32 - f/8 + sqrt(3)*I*(13*d - 32*e + 2*f)/288)
+ 11878244352*d**3*(9*d/32 - f/8 + sqrt(3)*I*(13*d - 32*e + 2*f)/288)**3 + 233164800*d**2*e**3*f + 4409634816
*d**2*e**2*f*(9*d/32 - f/8 + sqrt(3)*I*(13*d - 32*e + 2*f)/288) + 662937520*d**2*e*f**3 - 13004623872*d**2*e*f
*(9*d/32 - f/8 + sqrt(3)*I*(13*d - 32*e + 2*f)/288)**2 + 231796080*d**2*f**3*(9*d/32 - f/8 + sqrt(3)*I*(13*d -
32*e + 2*f)/288) - 10089639936*d**2*f*(9*d/32 - f/8 + sqrt(3)*I*(13*d - 32*e + 2*f)/288)**3 + 142606336*d*e**
5 + 754974720*d*e**4*(9*d/32 - f/8 + sqrt(3)*I*(13*d - 32*e + 2*f)/288) - 1843200*d*e**3*f**2 + 3850371072*d*e
**3*(9*d/32 - f/8 + sqrt(3)*I*(13*d - 32*e + 2*f)/288)**2 - 1926291456*d*e**2*f**2*(9*d/32 - f/8 + sqrt(3)*I*(
13*d - 32*e + 2*f)/288) + 20384317440*d*e**2*(9*d/32 - f/8 + sqrt(3)*I*(13*d - 32*e + 2*f)/288)**3 - 146756960
*d*e*f**4 + 5813379072*d*e*f**2*(9*d/32 - f/8 + sqrt(3)*I*(13*d - 32*e + 2*f)/288)**2 + 12679200*d*f**4*(9*d/3
2 - f/8 + sqrt(3)*I*(13*d - 32*e + 2*f)/288) + 1116758016*d*f**2*(9*d/32 - f/8 + sqrt(3)*I*(13*d - 32*e + 2*f)
/288)**3 - 79691776*e**5*f - 188743680*e**4*f*(9*d/32 - f/8 + sqrt(3)*I*(13*d - 32*e + 2*f)/288) - 7372800*e**
3*f**3 - 2151677952*e**3*f*(9*d/32 - f/8 + sqrt(3)*I*(13*d - 32*e + 2*f)/288)**2 + 287096832*e**2*f**3*(9*d/32
- f/8 + sqrt(3)*I*(13*d - 32*e + 2*f)/288) - 5096079360*e**2*f*(9*d/32 - f/8 + sqrt(3)*I*(13*d - 32*e + 2*f)/
288)**3 + 14093632*e*f**5 - 859521024*e*f**3*(9*d/32 - f/8 + sqrt(3)*I*(13*d - 32*e + 2*f)/288)**2 - 7648128*f
**5*(9*d/32 - f/8 + sqrt(3)*I*(13*d - 32*e + 2*f)/288) + 453869568*f**3*(9*d/32 - f/8 + sqrt(3)*I*(13*d - 32*e
+ 2*f)/288)**3)/(217696167*d**6 - 301346487*d**5*f - 1217128448*d**4*e**2 + 130506255*d**4*f**2 + 2181281792*
d**3*e**2*f - 5619240*d**3*f**3 - 617611264*d**2*e**4 - 1450149888*d**2*e**2*f**2 - 8036820*d**2*f**4 + 495976
448*d*e**4*f + 430088192*d*e**2*f**3 + 783648*d*f**5 - 114294784*e**4*f**2 - 47771648*e**2*f**4 + 188352*f**6)
) - (-8*e*x**6 - 12*e*x**4 - 16*e*x**2 - 6*e + x**7*(7*d - 7*f) + x**5*(5*d - 10*f) + x**3*(7*d - 14*f) + x*(-
4*d - 5*f))/(24*x**8 + 48*x**6 + 72*x**4 + 48*x**2 + 24)

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Giac [A]  time = 1.11395, size = 231, normalized size = 1.04 \begin{align*} \frac{1}{144} \, \sqrt{3}{\left (13 \, d + 2 \, f - 32 \, e\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) + \frac{1}{144} \, \sqrt{3}{\left (13 \, d + 2 \, f + 32 \, e\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) + \frac{1}{32} \,{\left (9 \, d - 4 \, f\right )} \log \left (x^{2} + x + 1\right ) - \frac{1}{32} \,{\left (9 \, d - 4 \, f\right )} \log \left (x^{2} - x + 1\right ) - \frac{7 \, d x^{7} - 7 \, f x^{7} - 8 \, x^{6} e + 5 \, d x^{5} - 10 \, f x^{5} - 12 \, x^{4} e + 7 \, d x^{3} - 14 \, f x^{3} - 16 \, x^{2} e - 4 \, d x - 5 \, f x - 6 \, e}{24 \,{\left (x^{4} + x^{2} + 1\right )}^{2}} \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)^3,x, algorithm="giac")

[Out]

1/144*sqrt(3)*(13*d + 2*f - 32*e)*arctan(1/3*sqrt(3)*(2*x + 1)) + 1/144*sqrt(3)*(13*d + 2*f + 32*e)*arctan(1/3
*sqrt(3)*(2*x - 1)) + 1/32*(9*d - 4*f)*log(x^2 + x + 1) - 1/32*(9*d - 4*f)*log(x^2 - x + 1) - 1/24*(7*d*x^7 -
7*f*x^7 - 8*x^6*e + 5*d*x^5 - 10*f*x^5 - 12*x^4*e + 7*d*x^3 - 14*f*x^3 - 16*x^2*e - 4*d*x - 5*f*x - 6*e)/(x^4
+ x^2 + 1)^2