\(\int \frac {d+e x+f x^2+g x^3+h x^4}{(1+x^2+x^4)^3} \, dx\) [50]

Optimal result

Integrand size = 31, antiderivative size = 263 \[ \int \frac {d+e x+f x^2+g x^3+h x^4}{\left (1+x^2+x^4\right )^3} \, dx=\frac {e-2 g+(2 e-g) x^2}{12 \left (1+x^2+x^4\right )^2}+\frac {x \left (d+f-2 h-(d-2 f+h) x^2\right )}{12 \left (1+x^2+x^4\right )^2}+\frac {(2 e-g) \left (1+2 x^2\right )}{12 \left (1+x^2+x^4\right )}+\frac {x \left (2 d+3 f-h-(7 d-7 f+4 h) x^2\right )}{24 \left (1+x^2+x^4\right )}-\frac {(13 d+2 f+h) \arctan \left (\frac {1-2 x}{\sqrt {3}}\right )}{48 \sqrt {3}}+\frac {(13 d+2 f+h) \arctan \left (\frac {1+2 x}{\sqrt {3}}\right )}{48 \sqrt {3}}+\frac {(2 e-g) \arctan \left (\frac {1+2 x^2}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {1}{32} (9 d-4 f+3 h) \log \left (1-x+x^2\right )+\frac {1}{32} (9 d-4 f+3 h) \log \left (1+x+x^2\right ) \]

[Out]

1/12*(e-2*g+(2*e-g)*x^2)/(x^4+x^2+1)^2+1/12*x*(d+f-2*h-(d-2*f+h)*x^2)/(x^4+x^2+1)^2+1/12*(2*e-g)*(2*x^2+1)/(x^
4+x^2+1)+1/24*x*(2*d+3*f-h-(7*d-7*f+4*h)*x^2)/(x^4+x^2+1)-1/32*(9*d-4*f+3*h)*ln(x^2-x+1)+1/32*(9*d-4*f+3*h)*ln
(x^2+x+1)-1/144*(13*d+2*f+h)*arctan(1/3*(1-2*x)*3^(1/2))*3^(1/2)+1/144*(13*d+2*f+h)*arctan(1/3*(1+2*x)*3^(1/2)
)*3^(1/2)+1/9*(2*e-g)*arctan(1/3*(2*x^2+1)*3^(1/2))*3^(1/2)

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.355, Rules used = {1687, 1692, 1192, 1183, 648, 632, 210, 642, 1261, 652, 628} \[ \int \frac {d+e x+f x^2+g x^3+h x^4}{\left (1+x^2+x^4\right )^3} \, dx=-\frac {\arctan \left (\frac {1-2 x}{\sqrt {3}}\right ) (13 d+2 f+h)}{48 \sqrt {3}}+\frac {\arctan \left (\frac {2 x+1}{\sqrt {3}}\right ) (13 d+2 f+h)}{48 \sqrt {3}}+\frac {\arctan \left (\frac {2 x^2+1}{\sqrt {3}}\right ) (2 e-g)}{3 \sqrt {3}}-\frac {1}{32} \log \left (x^2-x+1\right ) (9 d-4 f+3 h)+\frac {1}{32} \log \left (x^2+x+1\right ) (9 d-4 f+3 h)+\frac {x \left (-\left (x^2 (7 d-7 f+4 h)\right )+2 d+3 f-h\right )}{24 \left (x^4+x^2+1\right )}+\frac {x \left (-\left (x^2 (d-2 f+h)\right )+d+f-2 h\right )}{12 \left (x^4+x^2+1\right )^2}+\frac {\left (2 x^2+1\right ) (2 e-g)}{12 \left (x^4+x^2+1\right )}+\frac {x^2 (2 e-g)+e-2 g}{12 \left (x^4+x^2+1\right )^2} \]

[In]

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

[Out]

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

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 628

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]

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 652

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b*d - 2*a*e + (2*c*d -
b*e)*x)/((p + 1)*(b^2 - 4*a*c)))*(a + b*x + c*x^2)^(p + 1), x] - Dist[(2*p + 3)*((2*c*d - b*e)/((p + 1)*(b^2 -
4*a*c))), Int[(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && NeQ[b^
2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[p, -3/2]

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 1192

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 1261

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

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.57 (sec) , antiderivative size = 303, normalized size of antiderivative = 1.15 \[ \int \frac {d+e x+f x^2+g x^3+h x^4}{\left (1+x^2+x^4\right )^3} \, dx=\frac {1}{144} \left (-\frac {6 \left (-4 e \left (1+2 x^2\right )+g \left (2+4 x^2\right )+x \left (-2 d-3 f+h+7 d x^2-7 f x^2+4 h x^2\right )\right )}{1+x^2+x^4}+\frac {12 \left (e+2 e x^2-g \left (2+x^2\right )+x \left (d+f-d x^2+2 f x^2-h \left (2+x^2\right )\right )\right )}{\left (1+x^2+x^4\right )^2}-\frac {\left (\left (-47 i+7 \sqrt {3}\right ) d+\left (17 i-7 \sqrt {3}\right ) f+2 \left (-7 i+2 \sqrt {3}\right ) h\right ) \arctan \left (\frac {1}{2} \left (-i+\sqrt {3}\right ) x\right )}{\sqrt {\frac {1}{6} \left (1+i \sqrt {3}\right )}}-\frac {\left (\left (47 i+7 \sqrt {3}\right ) d-\left (17 i+7 \sqrt {3}\right ) f+2 \left (7 i+2 \sqrt {3}\right ) h\right ) \arctan \left (\frac {1}{2} \left (i+\sqrt {3}\right ) x\right )}{\sqrt {\frac {1}{6} \left (1-i \sqrt {3}\right )}}-16 \sqrt {3} (2 e-g) \arctan \left (\frac {\sqrt {3}}{1+2 x^2}\right )\right ) \]

[In]

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

[Out]

((-6*(-4*e*(1 + 2*x^2) + g*(2 + 4*x^2) + x*(-2*d - 3*f + h + 7*d*x^2 - 7*f*x^2 + 4*h*x^2)))/(1 + x^2 + x^4) +
(12*(e + 2*e*x^2 - g*(2 + x^2) + x*(d + f - d*x^2 + 2*f*x^2 - h*(2 + x^2))))/(1 + x^2 + x^4)^2 - (((-47*I + 7*
Sqrt[3])*d + (17*I - 7*Sqrt[3])*f + 2*(-7*I + 2*Sqrt[3])*h)*ArcTan[((-I + Sqrt[3])*x)/2])/Sqrt[(1 + I*Sqrt[3])
/6] - (((47*I + 7*Sqrt[3])*d - (17*I + 7*Sqrt[3])*f + 2*(7*I + 2*Sqrt[3])*h)*ArcTan[((I + Sqrt[3])*x)/2])/Sqrt
[(1 - I*Sqrt[3])/6] - 16*Sqrt[3]*(2*e - g)*ArcTan[Sqrt[3]/(1 + 2*x^2)])/144

Maple [A] (verified)

Time = 0.43 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.00

method	result	size
default	\(-\frac {\left (\frac {7 d}{3}-\frac {7 f}{3}+\frac {4 h}{3}-\frac {4 e}{3}-\frac {g}{3}\right ) x^{3}+\left (-6 d +4 f -2 h +2 g \right ) x^{2}+\left (\frac {20 d}{3}-\frac {13 f}{3}+\frac {5 h}{3}+\frac {e}{3}-\frac {8 g}{3}\right ) x -4 d +\frac {4 f}{3}-2 e +2 g}{16 \left (x^{2}-x +1\right )^{2}}-\frac {\left (27 d -12 f +9 h \right ) \ln \left (x^{2}-x +1\right )}{96}-\frac {\left (-\frac {13 d}{2}-16 e -f +8 g -\frac {h}{2}\right ) \sqrt {3}\, \arctan \left (\frac {\left (2 x -1\right ) \sqrt {3}}{3}\right )}{72}+\frac {\left (-\frac {7 d}{3}+\frac {7 f}{3}-\frac {4 h}{3}-\frac {4 e}{3}-\frac {g}{3}\right ) x^{3}+\left (-6 d +4 f -2 h -2 g \right ) x^{2}+\left (-\frac {20 d}{3}+\frac {13 f}{3}-\frac {5 h}{3}+\frac {e}{3}-\frac {8 g}{3}\right ) x -4 d +\frac {4 f}{3}+2 e -2 g}{16 \left (x^{2}+x +1\right )^{2}}+\frac {\left (27 d -12 f +9 h \right ) \ln \left (x^{2}+x +1\right )}{96}+\frac {\left (\frac {13 d}{2}-16 e +f +8 g +\frac {h}{2}\right ) \arctan \left (\frac {\left (1+2 x \right ) \sqrt {3}}{3}\right ) \sqrt {3}}{72}\)	\(262\)
risch	\(\text {Expression too large to display}\)	\(169779\)

[In]

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

[Out]

-1/16*((7/3*d-7/3*f+4/3*h-4/3*e-1/3*g)*x^3+(-6*d+4*f-2*h+2*g)*x^2+(20/3*d-13/3*f+5/3*h+1/3*e-8/3*g)*x-4*d+4/3*
f-2*e+2*g)/(x^2-x+1)^2-1/96*(27*d-12*f+9*h)*ln(x^2-x+1)-1/72*(-13/2*d-16*e-f+8*g-1/2*h)*3^(1/2)*arctan(1/3*(2*
x-1)*3^(1/2))+1/16*((-7/3*d+7/3*f-4/3*h-4/3*e-1/3*g)*x^3+(-6*d+4*f-2*h-2*g)*x^2+(-20/3*d+13/3*f-5/3*h+1/3*e-8/
3*g)*x-4*d+4/3*f+2*e-2*g)/(x^2+x+1)^2+1/96*(27*d-12*f+9*h)*ln(x^2+x+1)+1/72*(13/2*d-16*e+f+8*g+1/2*h)*arctan(1
/3*(1+2*x)*3^(1/2))*3^(1/2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 485 vs. \(2 (236) = 472\).

Time = 1.22 (sec) , antiderivative size = 485, normalized size of antiderivative = 1.84 \[ \int \frac {d+e x+f x^2+g x^3+h x^4}{\left (1+x^2+x^4\right )^3} \, dx=-\frac {12 \, {\left (7 \, d - 7 \, f + 4 \, h\right )} x^{7} - 48 \, {\left (2 \, e - g\right )} x^{6} + 60 \, {\left (d - 2 \, f + h\right )} x^{5} - 72 \, {\left (2 \, e - g\right )} x^{4} + 84 \, {\left (d - 2 \, f + h\right )} x^{3} - 96 \, {\left (2 \, e - g\right )} x^{2} - 2 \, \sqrt {3} {\left ({\left (13 \, d - 32 \, e + 2 \, f + 16 \, g + h\right )} x^{8} + 2 \, {\left (13 \, d - 32 \, e + 2 \, f + 16 \, g + h\right )} x^{6} + 3 \, {\left (13 \, d - 32 \, e + 2 \, f + 16 \, g + h\right )} x^{4} + 2 \, {\left (13 \, d - 32 \, e + 2 \, f + 16 \, g + h\right )} x^{2} + 13 \, d - 32 \, e + 2 \, f + 16 \, g + h\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) - 2 \, \sqrt {3} {\left ({\left (13 \, d + 32 \, e + 2 \, f - 16 \, g + h\right )} x^{8} + 2 \, {\left (13 \, d + 32 \, e + 2 \, f - 16 \, g + h\right )} x^{6} + 3 \, {\left (13 \, d + 32 \, e + 2 \, f - 16 \, g + h\right )} x^{4} + 2 \, {\left (13 \, d + 32 \, e + 2 \, f - 16 \, g + h\right )} x^{2} + 13 \, d + 32 \, e + 2 \, f - 16 \, g + h\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) - 12 \, {\left (4 \, d + 5 \, f - 5 \, h\right )} x - 9 \, {\left ({\left (9 \, d - 4 \, f + 3 \, h\right )} x^{8} + 2 \, {\left (9 \, d - 4 \, f + 3 \, h\right )} x^{6} + 3 \, {\left (9 \, d - 4 \, f + 3 \, h\right )} x^{4} + 2 \, {\left (9 \, d - 4 \, f + 3 \, h\right )} x^{2} + 9 \, d - 4 \, f + 3 \, h\right )} \log \left (x^{2} + x + 1\right ) + 9 \, {\left ({\left (9 \, d - 4 \, f + 3 \, h\right )} x^{8} + 2 \, {\left (9 \, d - 4 \, f + 3 \, h\right )} x^{6} + 3 \, {\left (9 \, d - 4 \, f + 3 \, h\right )} x^{4} + 2 \, {\left (9 \, d - 4 \, f + 3 \, h\right )} x^{2} + 9 \, d - 4 \, f + 3 \, h\right )} \log \left (x^{2} - x + 1\right ) - 72 \, e + 72 \, g}{288 \, {\left (x^{8} + 2 \, x^{6} + 3 \, x^{4} + 2 \, x^{2} + 1\right )}} \]

[In]

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

[Out]

-1/288*(12*(7*d - 7*f + 4*h)*x^7 - 48*(2*e - g)*x^6 + 60*(d - 2*f + h)*x^5 - 72*(2*e - g)*x^4 + 84*(d - 2*f +
h)*x^3 - 96*(2*e - g)*x^2 - 2*sqrt(3)*((13*d - 32*e + 2*f + 16*g + h)*x^8 + 2*(13*d - 32*e + 2*f + 16*g + h)*x
^6 + 3*(13*d - 32*e + 2*f + 16*g + h)*x^4 + 2*(13*d - 32*e + 2*f + 16*g + h)*x^2 + 13*d - 32*e + 2*f + 16*g +
h)*arctan(1/3*sqrt(3)*(2*x + 1)) - 2*sqrt(3)*((13*d + 32*e + 2*f - 16*g + h)*x^8 + 2*(13*d + 32*e + 2*f - 16*g
+ h)*x^6 + 3*(13*d + 32*e + 2*f - 16*g + h)*x^4 + 2*(13*d + 32*e + 2*f - 16*g + h)*x^2 + 13*d + 32*e + 2*f -
16*g + h)*arctan(1/3*sqrt(3)*(2*x - 1)) - 12*(4*d + 5*f - 5*h)*x - 9*((9*d - 4*f + 3*h)*x^8 + 2*(9*d - 4*f + 3
*h)*x^6 + 3*(9*d - 4*f + 3*h)*x^4 + 2*(9*d - 4*f + 3*h)*x^2 + 9*d - 4*f + 3*h)*log(x^2 + x + 1) + 9*((9*d - 4*
f + 3*h)*x^8 + 2*(9*d - 4*f + 3*h)*x^6 + 3*(9*d - 4*f + 3*h)*x^4 + 2*(9*d - 4*f + 3*h)*x^2 + 9*d - 4*f + 3*h)*
log(x^2 - x + 1) - 72*e + 72*g)/(x^8 + 2*x^6 + 3*x^4 + 2*x^2 + 1)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 217, normalized size of antiderivative = 0.83 \[ \int \frac {d+e x+f x^2+g x^3+h x^4}{\left (1+x^2+x^4\right )^3} \, dx=\frac {1}{144} \, \sqrt {3} {\left (13 \, d - 32 \, e + 2 \, f + 16 \, g + h\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) + \frac {1}{144} \, \sqrt {3} {\left (13 \, d + 32 \, e + 2 \, f - 16 \, g + h\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + \frac {1}{32} \, {\left (9 \, d - 4 \, f + 3 \, h\right )} \log \left (x^{2} + x + 1\right ) - \frac {1}{32} \, {\left (9 \, d - 4 \, f + 3 \, h\right )} \log \left (x^{2} - x + 1\right ) - \frac {{\left (7 \, d - 7 \, f + 4 \, h\right )} x^{7} - 4 \, {\left (2 \, e - g\right )} x^{6} + 5 \, {\left (d - 2 \, f + h\right )} x^{5} - 6 \, {\left (2 \, e - g\right )} x^{4} + 7 \, {\left (d - 2 \, f + h\right )} x^{3} - 8 \, {\left (2 \, e - g\right )} x^{2} - {\left (4 \, d + 5 \, f - 5 \, h\right )} x - 6 \, e + 6 \, g}{24 \, {\left (x^{8} + 2 \, x^{6} + 3 \, x^{4} + 2 \, x^{2} + 1\right )}} \]

[In]

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

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 222, normalized size of antiderivative = 0.84 \[ \int \frac {d+e x+f x^2+g x^3+h x^4}{\left (1+x^2+x^4\right )^3} \, dx=\frac {1}{144} \, \sqrt {3} {\left (13 \, d - 32 \, e + 2 \, f + 16 \, g + h\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) + \frac {1}{144} \, \sqrt {3} {\left (13 \, d + 32 \, e + 2 \, f - 16 \, g + h\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + \frac {1}{32} \, {\left (9 \, d - 4 \, f + 3 \, h\right )} \log \left (x^{2} + x + 1\right ) - \frac {1}{32} \, {\left (9 \, d - 4 \, f + 3 \, h\right )} \log \left (x^{2} - x + 1\right ) - \frac {7 \, d x^{7} - 7 \, f x^{7} + 4 \, h x^{7} - 8 \, e x^{6} + 4 \, g x^{6} + 5 \, d x^{5} - 10 \, f x^{5} + 5 \, h x^{5} - 12 \, e x^{4} + 6 \, g x^{4} + 7 \, d x^{3} - 14 \, f x^{3} + 7 \, h x^{3} - 16 \, e x^{2} + 8 \, g x^{2} - 4 \, d x - 5 \, f x + 5 \, h x - 6 \, e + 6 \, g}{24 \, {\left (x^{4} + x^{2} + 1\right )}^{2}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 3.44 (sec) , antiderivative size = 1611, normalized size of antiderivative = 6.13 \[ \int \frac {d+e x+f x^2+g x^3+h x^4}{\left (1+x^2+x^4\right )^3} \, dx=\text {Too large to display} \]

[In]

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

[Out]

(e/4 - g/4 + x^2*((2*e)/3 - g/3) + x^4*(e/2 - g/4) + x^6*(e/3 - g/6) + x*(d/6 + (5*f)/24 - (5*h)/24) - x^7*((7
*d)/24 - (7*f)/24 + h/6) - x^5*((5*d)/24 - (5*f)/12 + (5*h)/24) - x^3*((7*d)/24 - (7*f)/12 + (7*h)/24))/(2*x^2
+ 3*x^4 + 2*x^6 + x^8 + 1) - log(960*d*g - 2763*d*f - 1920*d*e + 480*e*f + 1971*d*h - 480*e*h - 240*f*g - 981
*f*h + 240*g*h + 3^(1/2)*d^2*1620i + 3^(1/2)*f^2*180i + 3^(1/2)*h^2*135i - 3807*d^2*x - 612*f^2*x - 378*h^2*x
+ 2754*d^2 + 684*f^2 + 351*h^2 + 3^(1/2)*d*e*1088i - 3^(1/2)*d*f*1125i - 3^(1/2)*d*g*544i - 3^(1/2)*e*f*608i +
3^(1/2)*d*h*945i + 3^(1/2)*e*h*416i + 3^(1/2)*f*g*304i - 3^(1/2)*f*h*315i - 3^(1/2)*g*h*208i - 672*d*e*x + 30
69*d*f*x + 336*d*g*x + 672*e*f*x - 2403*d*h*x - 384*e*h*x - 336*f*g*x + 963*f*h*x + 192*g*h*x + 3^(1/2)*d^2*x*
567i + 3^(1/2)*f^2*x*252i + 3^(1/2)*h^2*x*108i - 3^(1/2)*d*f*x*819i + 3^(1/2)*d*g*x*752i + 3^(1/2)*e*f*x*544i
+ 3^(1/2)*d*h*x*513i - 3^(1/2)*e*h*x*448i - 3^(1/2)*f*g*x*272i - 3^(1/2)*f*h*x*333i + 3^(1/2)*g*h*x*224i - 3^(
1/2)*d*e*x*1504i)*((9*d)/32 - f/8 + (3*h)/32 + (3^(1/2)*d*13i)/288 + (3^(1/2)*e*1i)/9 + (3^(1/2)*f*1i)/144 - (
3^(1/2)*g*1i)/18 + (3^(1/2)*h*1i)/288) - log(1920*d*e - 2763*d*f - 960*d*g - 480*e*f + 1971*d*h + 480*e*h + 24
0*f*g - 981*f*h - 240*g*h - 3^(1/2)*d^2*1620i - 3^(1/2)*f^2*180i - 3^(1/2)*h^2*135i + 3807*d^2*x + 612*f^2*x +
378*h^2*x + 2754*d^2 + 684*f^2 + 351*h^2 + 3^(1/2)*d*e*1088i + 3^(1/2)*d*f*1125i - 3^(1/2)*d*g*544i - 3^(1/2)
*e*f*608i - 3^(1/2)*d*h*945i + 3^(1/2)*e*h*416i + 3^(1/2)*f*g*304i + 3^(1/2)*f*h*315i - 3^(1/2)*g*h*208i - 672
*d*e*x - 3069*d*f*x + 336*d*g*x + 672*e*f*x + 2403*d*h*x - 384*e*h*x - 336*f*g*x - 963*f*h*x + 192*g*h*x + 3^(
1/2)*d^2*x*567i + 3^(1/2)*f^2*x*252i + 3^(1/2)*h^2*x*108i - 3^(1/2)*d*f*x*819i - 3^(1/2)*d*g*x*752i - 3^(1/2)*
e*f*x*544i + 3^(1/2)*d*h*x*513i + 3^(1/2)*e*h*x*448i + 3^(1/2)*f*g*x*272i - 3^(1/2)*f*h*x*333i - 3^(1/2)*g*h*x
*224i + 3^(1/2)*d*e*x*1504i)*(f/8 - (9*d)/32 - (3*h)/32 + (3^(1/2)*d*13i)/288 - (3^(1/2)*e*1i)/9 + (3^(1/2)*f*
1i)/144 + (3^(1/2)*g*1i)/18 + (3^(1/2)*h*1i)/288) + log(1920*d*e - 2763*d*f - 960*d*g - 480*e*f + 1971*d*h + 4
80*e*h + 240*f*g - 981*f*h - 240*g*h + 3^(1/2)*d^2*1620i + 3^(1/2)*f^2*180i + 3^(1/2)*h^2*135i + 3807*d^2*x +
612*f^2*x + 378*h^2*x + 2754*d^2 + 684*f^2 + 351*h^2 - 3^(1/2)*d*e*1088i - 3^(1/2)*d*f*1125i + 3^(1/2)*d*g*544
i + 3^(1/2)*e*f*608i + 3^(1/2)*d*h*945i - 3^(1/2)*e*h*416i - 3^(1/2)*f*g*304i - 3^(1/2)*f*h*315i + 3^(1/2)*g*h
*208i - 672*d*e*x - 3069*d*f*x + 336*d*g*x + 672*e*f*x + 2403*d*h*x - 384*e*h*x - 336*f*g*x - 963*f*h*x + 192*
g*h*x - 3^(1/2)*d^2*x*567i - 3^(1/2)*f^2*x*252i - 3^(1/2)*h^2*x*108i + 3^(1/2)*d*f*x*819i + 3^(1/2)*d*g*x*752i
+ 3^(1/2)*e*f*x*544i - 3^(1/2)*d*h*x*513i - 3^(1/2)*e*h*x*448i - 3^(1/2)*f*g*x*272i + 3^(1/2)*f*h*x*333i + 3^
(1/2)*g*h*x*224i - 3^(1/2)*d*e*x*1504i)*((9*d)/32 - f/8 + (3*h)/32 + (3^(1/2)*d*13i)/288 - (3^(1/2)*e*1i)/9 +
(3^(1/2)*f*1i)/144 + (3^(1/2)*g*1i)/18 + (3^(1/2)*h*1i)/288) + log(1920*d*e + 2763*d*f - 960*d*g - 480*e*f - 1
971*d*h + 480*e*h + 240*f*g + 981*f*h - 240*g*h + 3^(1/2)*d^2*1620i + 3^(1/2)*f^2*180i + 3^(1/2)*h^2*135i + 38
07*d^2*x + 612*f^2*x + 378*h^2*x - 2754*d^2 - 684*f^2 - 351*h^2 + 3^(1/2)*d*e*1088i - 3^(1/2)*d*f*1125i - 3^(1
/2)*d*g*544i - 3^(1/2)*e*f*608i + 3^(1/2)*d*h*945i + 3^(1/2)*e*h*416i + 3^(1/2)*f*g*304i - 3^(1/2)*f*h*315i -
3^(1/2)*g*h*208i + 672*d*e*x - 3069*d*f*x - 336*d*g*x - 672*e*f*x + 2403*d*h*x + 384*e*h*x + 336*f*g*x - 963*f
*h*x - 192*g*h*x + 3^(1/2)*d^2*x*567i + 3^(1/2)*f^2*x*252i + 3^(1/2)*h^2*x*108i - 3^(1/2)*d*f*x*819i + 3^(1/2)
*d*g*x*752i + 3^(1/2)*e*f*x*544i + 3^(1/2)*d*h*x*513i - 3^(1/2)*e*h*x*448i - 3^(1/2)*f*g*x*272i - 3^(1/2)*f*h*
x*333i + 3^(1/2)*g*h*x*224i - 3^(1/2)*d*e*x*1504i)*(f/8 - (9*d)/32 - (3*h)/32 + (3^(1/2)*d*13i)/288 + (3^(1/2)
*e*1i)/9 + (3^(1/2)*f*1i)/144 - (3^(1/2)*g*1i)/18 + (3^(1/2)*h*1i)/288)