\(\int \frac {d+e x+f x^2+g x^3+h x^4+i x^5}{(1+x^2+x^4)^2} \, dx\) [35]

Optimal result

Integrand size = 36, antiderivative size = 194 \[ \int \frac {d+e x+f x^2+g x^3+h x^4+i x^5}{\left (1+x^2+x^4\right )^2} \, dx=\frac {x \left (d+f-2 h-(d-2 f+h) x^2\right )}{6 \left (1+x^2+x^4\right )}+\frac {e-2 g+i+(2 e-g-i) x^2}{6 \left (1+x^2+x^4\right )}-\frac {(4 d+f+h) \arctan \left (\frac {1-2 x}{\sqrt {3}}\right )}{12 \sqrt {3}}+\frac {(4 d+f+h) \arctan \left (\frac {1+2 x}{\sqrt {3}}\right )}{12 \sqrt {3}}+\frac {(2 e-g+2 i) \arctan \left (\frac {1+2 x^2}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {1}{8} (2 d-f+h) \log \left (1-x+x^2\right )+\frac {1}{8} (2 d-f+h) \log \left (1+x+x^2\right ) \]

[Out]

1/6*x*(d+f-2*h-(d-2*f+h)*x^2)/(x^4+x^2+1)+1/6*(e-2*g+i+(2*e-g-i)*x^2)/(x^4+x^2+1)-1/8*(2*d-f+h)*ln(x^2-x+1)+1/
8*(2*d-f+h)*ln(x^2+x+1)-1/36*(4*d+f+h)*arctan(1/3*(1-2*x)*3^(1/2))*3^(1/2)+1/36*(4*d+f+h)*arctan(1/3*(1+2*x)*3
^(1/2))*3^(1/2)+1/9*(2*e-g+2*i)*arctan(1/3*(2*x^2+1)*3^(1/2))*3^(1/2)

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {1687, 1692, 1183, 648, 632, 210, 642, 1677, 1674, 12} \[ \int \frac {d+e x+f x^2+g x^3+h x^4+i x^5}{\left (1+x^2+x^4\right )^2} \, dx=-\frac {\arctan \left (\frac {1-2 x}{\sqrt {3}}\right ) (4 d+f+h)}{12 \sqrt {3}}+\frac {\arctan \left (\frac {2 x+1}{\sqrt {3}}\right ) (4 d+f+h)}{12 \sqrt {3}}+\frac {\arctan \left (\frac {2 x^2+1}{\sqrt {3}}\right ) (2 e-g+2 i)}{3 \sqrt {3}}-\frac {1}{8} \log \left (x^2-x+1\right ) (2 d-f+h)+\frac {1}{8} \log \left (x^2+x+1\right ) (2 d-f+h)+\frac {x \left (-\left (x^2 (d-2 f+h)\right )+d+f-2 h\right )}{6 \left (x^4+x^2+1\right )}+\frac {x^2 (2 e-g-i)+e-2 g+i}{6 \left (x^4+x^2+1\right )} \]

[In]

Int[(d + e*x + f*x^2 + g*x^3 + h*x^4 + i*x^5)/(1 + x^2 + x^4)^2,x]

[Out]

(x*(d + f - 2*h - (d - 2*f + h)*x^2))/(6*(1 + x^2 + x^4)) + (e - 2*g + i + (2*e - g - i)*x^2)/(6*(1 + x^2 + x^
4)) - ((4*d + f + h)*ArcTan[(1 - 2*x)/Sqrt[3]])/(12*Sqrt[3]) + ((4*d + f + h)*ArcTan[(1 + 2*x)/Sqrt[3]])/(12*S
qrt[3]) + ((2*e - g + 2*i)*ArcTan[(1 + 2*x^2)/Sqrt[3]])/(3*Sqrt[3]) - ((2*d - f + h)*Log[1 - x + x^2])/8 + ((2
*d - f + h)*Log[1 + x + x^2])/8

Rule 12

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

Rule 210

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 632

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 642

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 648

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 1183

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 1674

Int[(Pq_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, a + b*x + c*
x^2, x], f = Coeff[PolynomialRemainder[Pq, a + b*x + c*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b
*x + c*x^2, x], x, 1]}, Simp[(b*f - 2*a*g + (2*c*f - b*g)*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)
)), x] + Dist[1/((p + 1)*(b^2 - 4*a*c)), Int[(a + b*x + c*x^2)^(p + 1)*ExpandToSum[(p + 1)*(b^2 - 4*a*c)*Q - (
2*p + 3)*(2*c*f - b*g), x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1
]

Rule 1677

Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Dist[1/2, Subst[Int[x^((m - 1)/2)
*SubstFor[x^2, Pq, x]*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x^2] && Inte
gerQ[(m - 1)/2]

Rule 1687

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 1692

Int[(Pq_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{d = Coeff[PolynomialRemainder[Pq, a +
b*x^2 + c*x^4, x], x, 0], e = Coeff[PolynomialRemainder[Pq, a + b*x^2 + c*x^4, x], x, 2]}, Simp[x*(a + b*x^2 +
c*x^4)^(p + 1)*((a*b*e - d*(b^2 - 2*a*c) - c*(b*d - 2*a*e)*x^2)/(2*a*(p + 1)*(b^2 - 4*a*c))), x] + Dist[1/(2*
a*(p + 1)*(b^2 - 4*a*c)), Int[(a + b*x^2 + c*x^4)^(p + 1)*ExpandToSum[2*a*(p + 1)*(b^2 - 4*a*c)*PolynomialQuot
ient[Pq, a + b*x^2 + c*x^4, x] + b^2*d*(2*p + 3) - 2*a*c*d*(4*p + 5) - a*b*e + c*(4*p + 7)*(b*d - 2*a*e)*x^2,
x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x^2] && Expon[Pq, x^2] > 1 && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1
]

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.39 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.25 \[ \int \frac {d+e x+f x^2+g x^3+h x^4+i x^5}{\left (1+x^2+x^4\right )^2} \, dx=\frac {1}{36} \left (\frac {6 \left (e+i+d x+f x-2 h x+2 e x^2-i x^2-d x^3+2 f x^3-h x^3-g \left (2+x^2\right )\right )}{1+x^2+x^4}-\frac {\left (\left (-11 i+\sqrt {3}\right ) d-2 \left (-2 i+\sqrt {3}\right ) f+\left (-5 i+\sqrt {3}\right ) h\right ) \arctan \left (\frac {1}{2} \left (-i+\sqrt {3}\right ) x\right )}{\sqrt {\frac {1}{6} \left (1+i \sqrt {3}\right )}}-\frac {\left (\left (11 i+\sqrt {3}\right ) d-2 \left (2 i+\sqrt {3}\right ) f+\left (5 i+\sqrt {3}\right ) h\right ) \arctan \left (\frac {1}{2} \left (i+\sqrt {3}\right ) x\right )}{\sqrt {\frac {1}{6} \left (1-i \sqrt {3}\right )}}-4 \sqrt {3} (2 e-g+2 i) \arctan \left (\frac {\sqrt {3}}{1+2 x^2}\right )\right ) \]

[In]

Integrate[(d + e*x + f*x^2 + g*x^3 + h*x^4 + i*x^5)/(1 + x^2 + x^4)^2,x]

[Out]

((6*(e + i + d*x + f*x - 2*h*x + 2*e*x^2 - i*x^2 - d*x^3 + 2*f*x^3 - h*x^3 - g*(2 + x^2)))/(1 + x^2 + x^4) - (
((-11*I + Sqrt[3])*d - 2*(-2*I + Sqrt[3])*f + (-5*I + Sqrt[3])*h)*ArcTan[((-I + Sqrt[3])*x)/2])/Sqrt[(1 + I*Sq
rt[3])/6] - (((11*I + Sqrt[3])*d - 2*(2*I + Sqrt[3])*f + (5*I + Sqrt[3])*h)*ArcTan[((I + Sqrt[3])*x)/2])/Sqrt[
(1 - I*Sqrt[3])/6] - 4*Sqrt[3]*(2*e - g + 2*i)*ArcTan[Sqrt[3]/(1 + 2*x^2)])/36

Maple [A] (verified)

Time = 0.51 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.10

method	result	size
default	\(-\frac {\left (\frac {d}{3}-\frac {e}{3}-\frac {g}{3}+\frac {h}{3}-\frac {2 f}{3}+\frac {2 i}{3}\right ) x -\frac {2 d}{3}-\frac {e}{3}+\frac {2 g}{3}+\frac {h}{3}+\frac {f}{3}-\frac {i}{3}}{4 \left (x^{2}-x +1\right )}-\frac {\left (6 d -3 f +3 h \right ) \ln \left (x^{2}-x +1\right )}{24}-\frac {\left (-2 d -4 e -\frac {f}{2}+2 g -\frac {h}{2}-4 i \right ) \sqrt {3}\, \arctan \left (\frac {\left (2 x -1\right ) \sqrt {3}}{3}\right )}{18}+\frac {\left (-\frac {d}{3}-\frac {e}{3}-\frac {g}{3}-\frac {h}{3}+\frac {2 f}{3}+\frac {2 i}{3}\right ) x -\frac {2 d}{3}+\frac {e}{3}-\frac {2 g}{3}+\frac {h}{3}+\frac {f}{3}+\frac {i}{3}}{4 x^{2}+4 x +4}+\frac {\left (6 d -3 f +3 h \right ) \ln \left (x^{2}+x +1\right )}{24}+\frac {\left (2 d -4 e +\frac {f}{2}+2 g +\frac {h}{2}-4 i \right ) \arctan \left (\frac {\left (1+2 x \right ) \sqrt {3}}{3}\right ) \sqrt {3}}{18}\)	\(214\)
risch	\(\text {Expression too large to display}\)	\(463209\)

[In]

int((i*x^5+h*x^4+g*x^3+f*x^2+e*x+d)/(x^4+x^2+1)^2,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 4.87 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.44 \[ \int \frac {d+e x+f x^2+g x^3+h x^4+i x^5}{\left (1+x^2+x^4\right )^2} \, dx=-\frac {12 \, {\left (d - 2 \, f + h\right )} x^{3} - 12 \, {\left (2 \, e - g - i\right )} x^{2} - 2 \, \sqrt {3} {\left ({\left (4 \, d - 8 \, e + f + 4 \, g + h - 8 \, i\right )} x^{4} + {\left (4 \, d - 8 \, e + f + 4 \, g + h - 8 \, i\right )} x^{2} + 4 \, d - 8 \, e + f + 4 \, g + h - 8 \, i\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) - 2 \, \sqrt {3} {\left ({\left (4 \, d + 8 \, e + f - 4 \, g + h + 8 \, i\right )} x^{4} + {\left (4 \, d + 8 \, e + f - 4 \, g + h + 8 \, i\right )} x^{2} + 4 \, d + 8 \, e + f - 4 \, g + h + 8 \, i\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) - 12 \, {\left (d + f - 2 \, h\right )} x - 9 \, {\left ({\left (2 \, d - f + h\right )} x^{4} + {\left (2 \, d - f + h\right )} x^{2} + 2 \, d - f + h\right )} \log \left (x^{2} + x + 1\right ) + 9 \, {\left ({\left (2 \, d - f + h\right )} x^{4} + {\left (2 \, d - f + h\right )} x^{2} + 2 \, d - f + h\right )} \log \left (x^{2} - x + 1\right ) - 12 \, e + 24 \, g - 12 \, i}{72 \, {\left (x^{4} + x^{2} + 1\right )}} \]

[In]

integrate((i*x^5+h*x^4+g*x^3+f*x^2+e*x+d)/(x^4+x^2+1)^2,x, algorithm="fricas")

[Out]

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

Sympy [F(-1)]

Timed out. \[ \int \frac {d+e x+f x^2+g x^3+h x^4+i x^5}{\left (1+x^2+x^4\right )^2} \, dx=\text {Timed out} \]

[In]

integrate((i*x**5+h*x**4+g*x**3+f*x**2+e*x+d)/(x**4+x**2+1)**2,x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.80 \[ \int \frac {d+e x+f x^2+g x^3+h x^4+i x^5}{\left (1+x^2+x^4\right )^2} \, dx=\frac {1}{36} \, \sqrt {3} {\left (4 \, d - 8 \, e + f + 4 \, g + h - 8 \, i\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) + \frac {1}{36} \, \sqrt {3} {\left (4 \, d + 8 \, e + f - 4 \, g + h + 8 \, i\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + \frac {1}{8} \, {\left (2 \, d - f + h\right )} \log \left (x^{2} + x + 1\right ) - \frac {1}{8} \, {\left (2 \, d - f + h\right )} \log \left (x^{2} - x + 1\right ) - \frac {{\left (d - 2 \, f + h\right )} x^{3} - {\left (2 \, e - g - i\right )} x^{2} - {\left (d + f - 2 \, h\right )} x - e + 2 \, g - i}{6 \, {\left (x^{4} + x^{2} + 1\right )}} \]

[In]

integrate((i*x^5+h*x^4+g*x^3+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 + 4*g + h - 8*i)*arctan(1/3*sqrt(3)*(2*x + 1)) + 1/36*sqrt(3)*(4*d + 8*e + f - 4*g
+ h + 8*i)*arctan(1/3*sqrt(3)*(2*x - 1)) + 1/8*(2*d - f + h)*log(x^2 + x + 1) - 1/8*(2*d - f + h)*log(x^2 - x
+ 1) - 1/6*((d - 2*f + h)*x^3 - (2*e - g - i)*x^2 - (d + f - 2*h)*x - e + 2*g - i)/(x^4 + x^2 + 1)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.85 \[ \int \frac {d+e x+f x^2+g x^3+h x^4+i x^5}{\left (1+x^2+x^4\right )^2} \, dx=\frac {1}{36} \, \sqrt {3} {\left (4 \, d - 8 \, e + f + 4 \, g + h - 8 \, i\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) + \frac {1}{36} \, \sqrt {3} {\left (4 \, d + 8 \, e + f - 4 \, g + h + 8 \, i\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + \frac {1}{8} \, {\left (2 \, d - f + h\right )} \log \left (x^{2} + x + 1\right ) - \frac {1}{8} \, {\left (2 \, d - f + h\right )} \log \left (x^{2} - x + 1\right ) - \frac {d x^{3} - 2 \, f x^{3} + h x^{3} - 2 \, e x^{2} + g x^{2} + i x^{2} - d x - f x + 2 \, h x - e + 2 \, g - i}{6 \, {\left (x^{4} + x^{2} + 1\right )}} \]

[In]

integrate((i*x^5+h*x^4+g*x^3+f*x^2+e*x+d)/(x^4+x^2+1)^2,x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 13.24 (sec) , antiderivative size = 1894, normalized size of antiderivative = 9.76 \[ \int \frac {d+e x+f x^2+g x^3+h x^4+i x^5}{\left (1+x^2+x^4\right )^2} \, dx=\text {Too large to display} \]

[In]

int((d + e*x + f*x^2 + g*x^3 + h*x^4 + i*x^5)/(x^2 + x^4 + 1)^2,x)

[Out]

(e/6 - g/3 + i/6 + x*(d/6 + f/6 - h/3) - x^3*(d/6 - f/3 + h/6) - x^2*(g/6 - e/3 + i/6))/(x^2 + x^4 + 1) - log(
60*d*g - 153*d*f - 120*d*e + 24*e*f + 135*d*h - 120*d*i - 48*e*h - 12*f*g - 81*f*h + 24*f*i + 24*g*h - 48*h*i
+ 3^(1/2)*d^2*90i + 3^(1/2)*f^2*9i + 3^(1/2)*h^2*18i - 198*d^2*x - 36*f^2*x - 45*h^2*x + 126*d^2 + 45*f^2 + 36
*h^2 + 3^(1/2)*d*e*56i - 3^(1/2)*d*f*63i - 3^(1/2)*d*g*28i - 3^(1/2)*e*f*40i + 3^(1/2)*d*h*81i + 3^(1/2)*d*i*5
6i + 3^(1/2)*e*h*32i + 3^(1/2)*f*g*20i - 3^(1/2)*f*h*27i - 3^(1/2)*f*i*40i - 3^(1/2)*g*h*16i + 3^(1/2)*h*i*32i
- 24*d*e*x + 171*d*f*x + 12*d*g*x + 48*e*f*x - 189*d*h*x - 24*d*i*x - 24*e*h*x - 24*f*g*x + 81*f*h*x + 48*f*i
*x + 12*g*h*x - 24*h*i*x + 3^(1/2)*d^2*x*18i + 3^(1/2)*f^2*x*18i + 3^(1/2)*h^2*x*9i - 3^(1/2)*d*f*x*45i + 3^(1
/2)*d*g*x*44i + 3^(1/2)*e*f*x*32i + 3^(1/2)*d*h*x*27i - 3^(1/2)*d*i*x*88i - 3^(1/2)*e*h*x*40i - 3^(1/2)*f*g*x*
16i - 3^(1/2)*f*h*x*27i + 3^(1/2)*f*i*x*32i + 3^(1/2)*g*h*x*20i - 3^(1/2)*h*i*x*40i - 3^(1/2)*d*e*x*88i)*(d/4
- f/8 + h/8 + (3^(1/2)*d*1i)/18 + (3^(1/2)*e*1i)/9 + (3^(1/2)*f*1i)/72 - (3^(1/2)*g*1i)/18 + (3^(1/2)*h*1i)/72
+ (3^(1/2)*i*1i)/9) - log(120*d*e - 153*d*f - 60*d*g - 24*e*f + 135*d*h + 120*d*i + 48*e*h + 12*f*g - 81*f*h
- 24*f*i - 24*g*h + 48*h*i - 3^(1/2)*d^2*90i - 3^(1/2)*f^2*9i - 3^(1/2)*h^2*18i + 198*d^2*x + 36*f^2*x + 45*h^
2*x + 126*d^2 + 45*f^2 + 36*h^2 + 3^(1/2)*d*e*56i + 3^(1/2)*d*f*63i - 3^(1/2)*d*g*28i - 3^(1/2)*e*f*40i - 3^(1
/2)*d*h*81i + 3^(1/2)*d*i*56i + 3^(1/2)*e*h*32i + 3^(1/2)*f*g*20i + 3^(1/2)*f*h*27i - 3^(1/2)*f*i*40i - 3^(1/2
)*g*h*16i + 3^(1/2)*h*i*32i - 24*d*e*x - 171*d*f*x + 12*d*g*x + 48*e*f*x + 189*d*h*x - 24*d*i*x - 24*e*h*x - 2
4*f*g*x - 81*f*h*x + 48*f*i*x + 12*g*h*x - 24*h*i*x + 3^(1/2)*d^2*x*18i + 3^(1/2)*f^2*x*18i + 3^(1/2)*h^2*x*9i
- 3^(1/2)*d*f*x*45i - 3^(1/2)*d*g*x*44i - 3^(1/2)*e*f*x*32i + 3^(1/2)*d*h*x*27i + 3^(1/2)*d*i*x*88i + 3^(1/2)
*e*h*x*40i + 3^(1/2)*f*g*x*16i - 3^(1/2)*f*h*x*27i - 3^(1/2)*f*i*x*32i - 3^(1/2)*g*h*x*20i + 3^(1/2)*h*i*x*40i
+ 3^(1/2)*d*e*x*88i)*(f/8 - d/4 - h/8 + (3^(1/2)*d*1i)/18 - (3^(1/2)*e*1i)/9 + (3^(1/2)*f*1i)/72 + (3^(1/2)*g
*1i)/18 + (3^(1/2)*h*1i)/72 - (3^(1/2)*i*1i)/9) + log(120*d*e - 153*d*f - 60*d*g - 24*e*f + 135*d*h + 120*d*i
+ 48*e*h + 12*f*g - 81*f*h - 24*f*i - 24*g*h + 48*h*i + 3^(1/2)*d^2*90i + 3^(1/2)*f^2*9i + 3^(1/2)*h^2*18i + 1
98*d^2*x + 36*f^2*x + 45*h^2*x + 126*d^2 + 45*f^2 + 36*h^2 - 3^(1/2)*d*e*56i - 3^(1/2)*d*f*63i + 3^(1/2)*d*g*2
8i + 3^(1/2)*e*f*40i + 3^(1/2)*d*h*81i - 3^(1/2)*d*i*56i - 3^(1/2)*e*h*32i - 3^(1/2)*f*g*20i - 3^(1/2)*f*h*27i
+ 3^(1/2)*f*i*40i + 3^(1/2)*g*h*16i - 3^(1/2)*h*i*32i - 24*d*e*x - 171*d*f*x + 12*d*g*x + 48*e*f*x + 189*d*h*
x - 24*d*i*x - 24*e*h*x - 24*f*g*x - 81*f*h*x + 48*f*i*x + 12*g*h*x - 24*h*i*x - 3^(1/2)*d^2*x*18i - 3^(1/2)*f
^2*x*18i - 3^(1/2)*h^2*x*9i + 3^(1/2)*d*f*x*45i + 3^(1/2)*d*g*x*44i + 3^(1/2)*e*f*x*32i - 3^(1/2)*d*h*x*27i -
3^(1/2)*d*i*x*88i - 3^(1/2)*e*h*x*40i - 3^(1/2)*f*g*x*16i + 3^(1/2)*f*h*x*27i + 3^(1/2)*f*i*x*32i + 3^(1/2)*g*
h*x*20i - 3^(1/2)*h*i*x*40i - 3^(1/2)*d*e*x*88i)*(d/4 - f/8 + h/8 + (3^(1/2)*d*1i)/18 - (3^(1/2)*e*1i)/9 + (3^
(1/2)*f*1i)/72 + (3^(1/2)*g*1i)/18 + (3^(1/2)*h*1i)/72 - (3^(1/2)*i*1i)/9) + log(120*d*e + 153*d*f - 60*d*g -
24*e*f - 135*d*h + 120*d*i + 48*e*h + 12*f*g + 81*f*h - 24*f*i - 24*g*h + 48*h*i + 3^(1/2)*d^2*90i + 3^(1/2)*f
^2*9i + 3^(1/2)*h^2*18i + 198*d^2*x + 36*f^2*x + 45*h^2*x - 126*d^2 - 45*f^2 - 36*h^2 + 3^(1/2)*d*e*56i - 3^(1
/2)*d*f*63i - 3^(1/2)*d*g*28i - 3^(1/2)*e*f*40i + 3^(1/2)*d*h*81i + 3^(1/2)*d*i*56i + 3^(1/2)*e*h*32i + 3^(1/2
)*f*g*20i - 3^(1/2)*f*h*27i - 3^(1/2)*f*i*40i - 3^(1/2)*g*h*16i + 3^(1/2)*h*i*32i + 24*d*e*x - 171*d*f*x - 12*
d*g*x - 48*e*f*x + 189*d*h*x + 24*d*i*x + 24*e*h*x + 24*f*g*x - 81*f*h*x - 48*f*i*x - 12*g*h*x + 24*h*i*x + 3^
(1/2)*d^2*x*18i + 3^(1/2)*f^2*x*18i + 3^(1/2)*h^2*x*9i - 3^(1/2)*d*f*x*45i + 3^(1/2)*d*g*x*44i + 3^(1/2)*e*f*x
*32i + 3^(1/2)*d*h*x*27i - 3^(1/2)*d*i*x*88i - 3^(1/2)*e*h*x*40i - 3^(1/2)*f*g*x*16i - 3^(1/2)*f*h*x*27i + 3^(
1/2)*f*i*x*32i + 3^(1/2)*g*h*x*20i - 3^(1/2)*h*i*x*40i - 3^(1/2)*d*e*x*88i)*(f/8 - d/4 - h/8 + (3^(1/2)*d*1i)/
18 + (3^(1/2)*e*1i)/9 + (3^(1/2)*f*1i)/72 - (3^(1/2)*g*1i)/18 + (3^(1/2)*h*1i)/72 + (3^(1/2)*i*1i)/9)