3.56 \(\int \frac{1}{(1+4 x+4 x^2+4 x^4)^2} \, dx\)

Optimal. Leaf size=317 \[ -\frac{17-\left (\frac{1}{x}+1\right )^2}{2 \left (\left (\frac{1}{x}+1\right )^4-2 \left (\frac{1}{x}+1\right )^2+5\right )}+\frac{\left (59-17 \left (\frac{1}{x}+1\right )^2\right ) \left (\frac{1}{x}+1\right )}{10 \left (\left (\frac{1}{x}+1\right )^4-2 \left (\frac{1}{x}+1\right )^2+5\right )}+\frac{1}{40} \sqrt{\frac{1}{10} \left (2665 \sqrt{5}-5959\right )} \log \left (\left (\frac{1}{x}+1\right )^2-\sqrt{2 \left (1+\sqrt{5}\right )} \left (\frac{1}{x}+1\right )+\sqrt{5}\right )-\frac{1}{40} \sqrt{\frac{1}{10} \left (2665 \sqrt{5}-5959\right )} \log \left (\left (\frac{1}{x}+1\right )^2+\sqrt{2 \left (1+\sqrt{5}\right )} \left (\frac{1}{x}+1\right )+\sqrt{5}\right )+\frac{7}{4} \tan ^{-1}\left (\frac{1}{2} \left (\left (\frac{1}{x}+1\right )^2-1\right )\right )-\frac{1}{20} \sqrt{\frac{1}{10} \left (5959+2665 \sqrt{5}\right )} \tan ^{-1}\left (\frac{\frac{2}{x}-\sqrt{2 \left (1+\sqrt{5}\right )}+2}{\sqrt{2 \left (\sqrt{5}-1\right )}}\right )-\frac{1}{20} \sqrt{\frac{1}{10} \left (5959+2665 \sqrt{5}\right )} \tan ^{-1}\left (\frac{\frac{2}{x}+\sqrt{2 \left (1+\sqrt{5}\right )}+2}{\sqrt{2 \left (\sqrt{5}-1\right )}}\right ) \]

[Out]

-(17 - (1 + x^(-1))^2)/(2*(5 - 2*(1 + x^(-1))^2 + (1 + x^(-1))^4)) + ((59 - 17*(1 + x^(-1))^2)*(1 + x^(-1)))/(
10*(5 - 2*(1 + x^(-1))^2 + (1 + x^(-1))^4)) + (7*ArcTan[(-1 + (1 + x^(-1))^2)/2])/4 - (Sqrt[(5959 + 2665*Sqrt[
5])/10]*ArcTan[(2 - Sqrt[2*(1 + Sqrt[5])] + 2/x)/Sqrt[2*(-1 + Sqrt[5])]])/20 - (Sqrt[(5959 + 2665*Sqrt[5])/10]
*ArcTan[(2 + Sqrt[2*(1 + Sqrt[5])] + 2/x)/Sqrt[2*(-1 + Sqrt[5])]])/20 + (Sqrt[(-5959 + 2665*Sqrt[5])/10]*Log[S
qrt[5] - Sqrt[2*(1 + Sqrt[5])]*(1 + x^(-1)) + (1 + x^(-1))^2])/40 - (Sqrt[(-5959 + 2665*Sqrt[5])/10]*Log[Sqrt[
5] + Sqrt[2*(1 + Sqrt[5])]*(1 + x^(-1)) + (1 + x^(-1))^2])/40

________________________________________________________________________________________

Rubi [A]  time = 0.334724, antiderivative size = 317, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 11, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.647, Rules used = {2069, 1673, 1678, 1169, 634, 618, 204, 628, 1663, 1660, 12} \[ -\frac{17-\left (\frac{1}{x}+1\right )^2}{2 \left (\left (\frac{1}{x}+1\right )^4-2 \left (\frac{1}{x}+1\right )^2+5\right )}+\frac{\left (59-17 \left (\frac{1}{x}+1\right )^2\right ) \left (\frac{1}{x}+1\right )}{10 \left (\left (\frac{1}{x}+1\right )^4-2 \left (\frac{1}{x}+1\right )^2+5\right )}+\frac{1}{40} \sqrt{\frac{1}{10} \left (2665 \sqrt{5}-5959\right )} \log \left (\left (\frac{1}{x}+1\right )^2-\sqrt{2 \left (1+\sqrt{5}\right )} \left (\frac{1}{x}+1\right )+\sqrt{5}\right )-\frac{1}{40} \sqrt{\frac{1}{10} \left (2665 \sqrt{5}-5959\right )} \log \left (\left (\frac{1}{x}+1\right )^2+\sqrt{2 \left (1+\sqrt{5}\right )} \left (\frac{1}{x}+1\right )+\sqrt{5}\right )+\frac{7}{4} \tan ^{-1}\left (\frac{1}{2} \left (\left (\frac{1}{x}+1\right )^2-1\right )\right )-\frac{1}{20} \sqrt{\frac{1}{10} \left (5959+2665 \sqrt{5}\right )} \tan ^{-1}\left (\frac{\frac{2}{x}-\sqrt{2 \left (1+\sqrt{5}\right )}+2}{\sqrt{2 \left (\sqrt{5}-1\right )}}\right )-\frac{1}{20} \sqrt{\frac{1}{10} \left (5959+2665 \sqrt{5}\right )} \tan ^{-1}\left (\frac{\frac{2}{x}+\sqrt{2 \left (1+\sqrt{5}\right )}+2}{\sqrt{2 \left (\sqrt{5}-1\right )}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[(1 + 4*x + 4*x^2 + 4*x^4)^(-2),x]

[Out]

-(17 - (1 + x^(-1))^2)/(2*(5 - 2*(1 + x^(-1))^2 + (1 + x^(-1))^4)) + ((59 - 17*(1 + x^(-1))^2)*(1 + x^(-1)))/(
10*(5 - 2*(1 + x^(-1))^2 + (1 + x^(-1))^4)) + (7*ArcTan[(-1 + (1 + x^(-1))^2)/2])/4 - (Sqrt[(5959 + 2665*Sqrt[
5])/10]*ArcTan[(2 - Sqrt[2*(1 + Sqrt[5])] + 2/x)/Sqrt[2*(-1 + Sqrt[5])]])/20 - (Sqrt[(5959 + 2665*Sqrt[5])/10]
*ArcTan[(2 + Sqrt[2*(1 + Sqrt[5])] + 2/x)/Sqrt[2*(-1 + Sqrt[5])]])/20 + (Sqrt[(-5959 + 2665*Sqrt[5])/10]*Log[S
qrt[5] - Sqrt[2*(1 + Sqrt[5])]*(1 + x^(-1)) + (1 + x^(-1))^2])/40 - (Sqrt[(-5959 + 2665*Sqrt[5])/10]*Log[Sqrt[
5] + Sqrt[2*(1 + Sqrt[5])]*(1 + x^(-1)) + (1 + x^(-1))^2])/40

Rule 2069

Int[(P4_)^(p_), x_Symbol] :> With[{a = Coeff[P4, x, 0], b = Coeff[P4, x, 1], c = Coeff[P4, x, 2], d = Coeff[P4
, x, 3], e = Coeff[P4, x, 4]}, Dist[-16*a^2, Subst[Int[(1*((a*(-3*b^4 + 16*a*b^2*c - 64*a^2*b*d + 256*a^3*e -
32*a^2*(3*b^2 - 8*a*c)*x^2 + 256*a^4*x^4))/(b - 4*a*x)^4)^p)/(b - 4*a*x)^2, x], x, b/(4*a) + 1/x], x] /; NeQ[a
, 0] && NeQ[b, 0] && EqQ[b^3 - 4*a*b*c + 8*a^2*d, 0]] /; FreeQ[p, x] && PolyQ[P4, x, 4] && IntegerQ[2*p] &&  !
IGtQ[p, 0]

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 1678

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
]

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 1663

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 1660

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 12

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

Rubi steps

\begin{align*} \int \frac{1}{\left (1+4 x+4 x^2+4 x^4\right )^2} \, dx &=-\left (16 \operatorname{Subst}\left (\int \frac{(4-4 x)^6}{\left (1280-512 x^2+256 x^4\right )^2} \, dx,x,1+\frac{1}{x}\right )\right )\\ &=-\left (16 \operatorname{Subst}\left (\int \frac{x \left (-24576-81920 x^2-24576 x^4\right )}{\left (1280-512 x^2+256 x^4\right )^2} \, dx,x,1+\frac{1}{x}\right )\right )-16 \operatorname{Subst}\left (\int \frac{4096+61440 x^2+61440 x^4+4096 x^6}{\left (1280-512 x^2+256 x^4\right )^2} \, dx,x,1+\frac{1}{x}\right )\\ &=\frac{\left (59-17 \left (1+\frac{1}{x}\right )^2\right ) \left (1+\frac{1}{x}\right )}{10 \left (5-2 \left (1+\frac{1}{x}\right )^2+\left (1+\frac{1}{x}\right )^4\right )}-\frac{\operatorname{Subst}\left (\int \frac{261993005056+115964116992 x^2}{1280-512 x^2+256 x^4} \, dx,x,1+\frac{1}{x}\right )}{167772160}-8 \operatorname{Subst}\left (\int \frac{-24576-81920 x-24576 x^2}{\left (1280-512 x+256 x^2\right )^2} \, dx,x,\left (1+\frac{1}{x}\right )^2\right )\\ &=-\frac{17-\left (1+\frac{1}{x}\right )^2}{2 \left (5-2 \left (1+\frac{1}{x}\right )^2+\left (1+\frac{1}{x}\right )^4\right )}+\frac{\left (59-17 \left (1+\frac{1}{x}\right )^2\right ) \left (1+\frac{1}{x}\right )}{10 \left (5-2 \left (1+\frac{1}{x}\right )^2+\left (1+\frac{1}{x}\right )^4\right )}-\frac{\operatorname{Subst}\left (\int -\frac{117440512}{1280-512 x+256 x^2} \, dx,x,\left (1+\frac{1}{x}\right )^2\right )}{131072}-\frac{\operatorname{Subst}\left (\int \frac{261993005056 \sqrt{2 \left (1+\sqrt{5}\right )}-\left (261993005056-115964116992 \sqrt{5}\right ) x}{\sqrt{5}-\sqrt{2 \left (1+\sqrt{5}\right )} x+x^2} \, dx,x,1+\frac{1}{x}\right )}{85899345920 \sqrt{10 \left (1+\sqrt{5}\right )}}-\frac{\operatorname{Subst}\left (\int \frac{261993005056 \sqrt{2 \left (1+\sqrt{5}\right )}+\left (261993005056-115964116992 \sqrt{5}\right ) x}{\sqrt{5}+\sqrt{2 \left (1+\sqrt{5}\right )} x+x^2} \, dx,x,1+\frac{1}{x}\right )}{85899345920 \sqrt{10 \left (1+\sqrt{5}\right )}}\\ &=-\frac{17-\left (1+\frac{1}{x}\right )^2}{2 \left (5-2 \left (1+\frac{1}{x}\right )^2+\left (1+\frac{1}{x}\right )^4\right )}+\frac{\left (59-17 \left (1+\frac{1}{x}\right )^2\right ) \left (1+\frac{1}{x}\right )}{10 \left (5-2 \left (1+\frac{1}{x}\right )^2+\left (1+\frac{1}{x}\right )^4\right )}+896 \operatorname{Subst}\left (\int \frac{1}{1280-512 x+256 x^2} \, dx,x,\left (1+\frac{1}{x}\right )^2\right )+\frac{\left (61-27 \sqrt{5}\right ) \operatorname{Subst}\left (\int \frac{-\sqrt{2 \left (1+\sqrt{5}\right )}+2 x}{\sqrt{5}-\sqrt{2 \left (1+\sqrt{5}\right )} x+x^2} \, dx,x,1+\frac{1}{x}\right )}{40 \sqrt{10 \left (1+\sqrt{5}\right )}}-\frac{\left (61-27 \sqrt{5}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{2 \left (1+\sqrt{5}\right )}+2 x}{\sqrt{5}+\sqrt{2 \left (1+\sqrt{5}\right )} x+x^2} \, dx,x,1+\frac{1}{x}\right )}{40 \sqrt{10 \left (1+\sqrt{5}\right )}}-\frac{1}{20} \sqrt{\frac{1}{10} \left (3683+1647 \sqrt{5}\right )} \operatorname{Subst}\left (\int \frac{1}{\sqrt{5}-\sqrt{2 \left (1+\sqrt{5}\right )} x+x^2} \, dx,x,1+\frac{1}{x}\right )-\frac{1}{20} \sqrt{\frac{1}{10} \left (3683+1647 \sqrt{5}\right )} \operatorname{Subst}\left (\int \frac{1}{\sqrt{5}+\sqrt{2 \left (1+\sqrt{5}\right )} x+x^2} \, dx,x,1+\frac{1}{x}\right )\\ &=-\frac{17-\left (1+\frac{1}{x}\right )^2}{2 \left (5-2 \left (1+\frac{1}{x}\right )^2+\left (1+\frac{1}{x}\right )^4\right )}+\frac{\left (59-17 \left (1+\frac{1}{x}\right )^2\right ) \left (1+\frac{1}{x}\right )}{10 \left (5-2 \left (1+\frac{1}{x}\right )^2+\left (1+\frac{1}{x}\right )^4\right )}+\frac{1}{40} \sqrt{-\frac{5959}{10}+\frac{533 \sqrt{5}}{2}} \log \left (\sqrt{5}-\sqrt{2 \left (1+\sqrt{5}\right )} \left (1+\frac{1}{x}\right )+\left (1+\frac{1}{x}\right )^2\right )-\frac{1}{40} \sqrt{-\frac{5959}{10}+\frac{533 \sqrt{5}}{2}} \log \left (\sqrt{5}+\sqrt{2 \left (1+\sqrt{5}\right )} \left (1+\frac{1}{x}\right )+\left (1+\frac{1}{x}\right )^2\right )-1792 \operatorname{Subst}\left (\int \frac{1}{-1048576-x^2} \, dx,x,-512+512 \left (1+\frac{1}{x}\right )^2\right )+\frac{1}{10} \sqrt{\frac{1}{10} \left (3683+1647 \sqrt{5}\right )} \operatorname{Subst}\left (\int \frac{1}{2 \left (1-\sqrt{5}\right )-x^2} \, dx,x,-\sqrt{2 \left (1+\sqrt{5}\right )}+2 \left (1+\frac{1}{x}\right )\right )+\frac{1}{10} \sqrt{\frac{1}{10} \left (3683+1647 \sqrt{5}\right )} \operatorname{Subst}\left (\int \frac{1}{2 \left (1-\sqrt{5}\right )-x^2} \, dx,x,\sqrt{2 \left (1+\sqrt{5}\right )}+2 \left (1+\frac{1}{x}\right )\right )\\ &=-\frac{17-\left (1+\frac{1}{x}\right )^2}{2 \left (5-2 \left (1+\frac{1}{x}\right )^2+\left (1+\frac{1}{x}\right )^4\right )}+\frac{\left (59-17 \left (1+\frac{1}{x}\right )^2\right ) \left (1+\frac{1}{x}\right )}{10 \left (5-2 \left (1+\frac{1}{x}\right )^2+\left (1+\frac{1}{x}\right )^4\right )}+\frac{7}{4} \tan ^{-1}\left (\frac{1}{2} \left (-1+\left (1+\frac{1}{x}\right )^2\right )\right )-\frac{1}{20} \sqrt{\frac{1}{10} \left (5959+2665 \sqrt{5}\right )} \tan ^{-1}\left (\frac{2-\sqrt{2 \left (1+\sqrt{5}\right )}+\frac{2}{x}}{\sqrt{2 \left (-1+\sqrt{5}\right )}}\right )-\frac{1}{20} \sqrt{\frac{1}{10} \left (5959+2665 \sqrt{5}\right )} \tan ^{-1}\left (\frac{2+\sqrt{2 \left (1+\sqrt{5}\right )}+\frac{2}{x}}{\sqrt{2 \left (-1+\sqrt{5}\right )}}\right )+\frac{1}{40} \sqrt{-\frac{5959}{10}+\frac{533 \sqrt{5}}{2}} \log \left (\sqrt{5}-\sqrt{2 \left (1+\sqrt{5}\right )} \left (1+\frac{1}{x}\right )+\left (1+\frac{1}{x}\right )^2\right )-\frac{1}{40} \sqrt{-\frac{5959}{10}+\frac{533 \sqrt{5}}{2}} \log \left (\sqrt{5}+\sqrt{2 \left (1+\sqrt{5}\right )} \left (1+\frac{1}{x}\right )+\left (1+\frac{1}{x}\right )^2\right )\\ \end{align*}

Mathematica [C]  time = 0.0233888, size = 108, normalized size = 0.34 \[ \frac{1}{40} \left (\text{RootSum}\left [4 \text{$\#$1}^4+4 \text{$\#$1}^2+4 \text{$\#$1}+1\& ,\frac{18 \text{$\#$1}^2 \log (x-\text{$\#$1})-16 \text{$\#$1} \log (x-\text{$\#$1})+27 \log (x-\text{$\#$1})}{4 \text{$\#$1}^3+2 \text{$\#$1}+1}\& \right ]+\frac{72 x^3-32 x^2+84 x+38}{4 x^4+4 x^2+4 x+1}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + 4*x + 4*x^2 + 4*x^4)^(-2),x]

[Out]

((38 + 84*x - 32*x^2 + 72*x^3)/(1 + 4*x + 4*x^2 + 4*x^4) + RootSum[1 + 4*#1 + 4*#1^2 + 4*#1^4 & , (27*Log[x -
#1] - 16*Log[x - #1]*#1 + 18*Log[x - #1]*#1^2)/(1 + 2*#1 + 4*#1^3) & ])/40

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Maple [C]  time = 0.008, size = 79, normalized size = 0.3 \begin{align*}{ \left ({\frac{9\,{x}^{3}}{20}}-{\frac{{x}^{2}}{5}}+{\frac{21\,x}{40}}+{\frac{19}{80}} \right ) \left ({x}^{4}+{x}^{2}+x+{\frac{1}{4}} \right ) ^{-1}}+{\frac{1}{40}\sum _{{\it \_R}={\it RootOf} \left ( 4\,{{\it \_Z}}^{4}+4\,{{\it \_Z}}^{2}+4\,{\it \_Z}+1 \right ) }{\frac{ \left ( 18\,{{\it \_R}}^{2}-16\,{\it \_R}+27 \right ) \ln \left ( x-{\it \_R} \right ) }{4\,{{\it \_R}}^{3}+2\,{\it \_R}+1}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(9/20*x^3-1/5*x^2+21/40*x+19/80)/(x^4+x^2+x+1/4)+1/40*sum((18*_R^2-16*_R+27)/(4*_R^3+2*_R+1)*ln(x-_R),_R=RootO
f(4*_Z^4+4*_Z^2+4*_Z+1))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{36 \, x^{3} - 16 \, x^{2} + 42 \, x + 19}{20 \,{\left (4 \, x^{4} + 4 \, x^{2} + 4 \, x + 1\right )}} + \frac{1}{10} \, \int \frac{18 \, x^{2} - 16 \, x + 27}{4 \, x^{4} + 4 \, x^{2} + 4 \, x + 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/20*(36*x^3 - 16*x^2 + 42*x + 19)/(4*x^4 + 4*x^2 + 4*x + 1) + 1/10*integrate((18*x^2 - 16*x + 27)/(4*x^4 + 4*
x^2 + 4*x + 1), x)

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Fricas [C]  time = 9.9968, size = 4027, normalized size = 12.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/400*(720*x^3 - 50*(4*x^4 + 4*x^2 + 4*x + 1)*(4*sqrt(19/1000*I - 5959/2000) + 7*I)*log(33368250*(4*sqrt(19/10
00*I - 5959/2000) + 7*I)^3 - 11755375/4*(4*sqrt(19/1000*I - 5959/2000) + 7*I)^2 + 541735337*x + 25784243612*sq
rt(19/1000*I - 5959/2000) + 45122426321*I - 71080995) - 50*(4*x^4 + 4*x^2 + 4*x + 1)*(4*sqrt(-19/1000*I - 5959
/2000) - 7*I)*log(-33368250*(4*sqrt(19/1000*I - 5959/2000) + 7*I)^3 - 125/4*(4271136*sqrt(19/1000*I - 5959/200
0) + 7474488*I + 94043)*(4*sqrt(-19/1000*I - 5959/2000) - 7*I)^2 - 25*(1334730*(4*sqrt(19/1000*I - 5959/2000)
+ 7*I)^2 + 219601)*(4*sqrt(-19/1000*I - 5959/2000) - 7*I) + 541735337*x - 25806203712*sqrt(19/1000*I - 5959/20
00) - 45160856496*I - 355111539) - 320*x^2 - (4*sqrt(10)*(4*x^4 + 4*x^2 + 4*x + 1)*sqrt(-375/32*(4*sqrt(19/100
0*I - 5959/2000) + 7*I)^2 - 125/16*(4*sqrt(19/1000*I - 5959/2000) + 7*I)*(4*sqrt(-19/1000*I - 5959/2000) - 7*I
) - 375/32*(4*sqrt(-19/1000*I - 5959/2000) - 7*I)^2 - 3021) - 25*(4*x^4 + 4*x^2 + 4*x + 1)*(4*sqrt(19/1000*I -
5959/2000) + 7*I) - 25*(4*x^4 + 4*x^2 + 4*x + 1)*(4*sqrt(-19/1000*I - 5959/2000) - 7*I))*log(125/8*(4271136*s
qrt(19/1000*I - 5959/2000) + 7474488*I + 94043)*(4*sqrt(-19/1000*I - 5959/2000) - 7*I)^2 + 11755375/8*(4*sqrt(
19/1000*I - 5959/2000) + 7*I)^2 + 25/2*(1334730*(4*sqrt(19/1000*I - 5959/2000) + 7*I)^2 + 219601)*(4*sqrt(-19/
1000*I - 5959/2000) - 7*I) + 1/2*sqrt(-375/32*(4*sqrt(19/1000*I - 5959/2000) + 7*I)^2 - 125/16*(4*sqrt(19/1000
*I - 5959/2000) + 7*I)*(4*sqrt(-19/1000*I - 5959/2000) - 7*I) - 375/32*(4*sqrt(-19/1000*I - 5959/2000) - 7*I)^
2 - 3021)*(5*(1067784*sqrt(10)*(4*sqrt(19/1000*I - 5959/2000) + 7*I) + 94043*sqrt(10))*(4*sqrt(-19/1000*I - 59
59/2000) - 7*I) + 470215*sqrt(10)*(4*sqrt(19/1000*I - 5959/2000) + 7*I) - 878404*sqrt(10)) + 541735337*x + 109
80050*sqrt(19/1000*I - 5959/2000) + 38430175/2*I + 213096267) + (4*sqrt(10)*(4*x^4 + 4*x^2 + 4*x + 1)*sqrt(-37
5/32*(4*sqrt(19/1000*I - 5959/2000) + 7*I)^2 - 125/16*(4*sqrt(19/1000*I - 5959/2000) + 7*I)*(4*sqrt(-19/1000*I
- 5959/2000) - 7*I) - 375/32*(4*sqrt(-19/1000*I - 5959/2000) - 7*I)^2 - 3021) + 25*(4*x^4 + 4*x^2 + 4*x + 1)*
(4*sqrt(19/1000*I - 5959/2000) + 7*I) + 25*(4*x^4 + 4*x^2 + 4*x + 1)*(4*sqrt(-19/1000*I - 5959/2000) - 7*I))*l
og(125/8*(4271136*sqrt(19/1000*I - 5959/2000) + 7474488*I + 94043)*(4*sqrt(-19/1000*I - 5959/2000) - 7*I)^2 +
11755375/8*(4*sqrt(19/1000*I - 5959/2000) + 7*I)^2 + 25/2*(1334730*(4*sqrt(19/1000*I - 5959/2000) + 7*I)^2 + 2
19601)*(4*sqrt(-19/1000*I - 5959/2000) - 7*I) - 1/2*sqrt(-375/32*(4*sqrt(19/1000*I - 5959/2000) + 7*I)^2 - 125
/16*(4*sqrt(19/1000*I - 5959/2000) + 7*I)*(4*sqrt(-19/1000*I - 5959/2000) - 7*I) - 375/32*(4*sqrt(-19/1000*I -
5959/2000) - 7*I)^2 - 3021)*(5*(1067784*sqrt(10)*(4*sqrt(19/1000*I - 5959/2000) + 7*I) + 94043*sqrt(10))*(4*s
qrt(-19/1000*I - 5959/2000) - 7*I) + 470215*sqrt(10)*(4*sqrt(19/1000*I - 5959/2000) + 7*I) - 878404*sqrt(10))
+ 541735337*x + 10980050*sqrt(19/1000*I - 5959/2000) + 38430175/2*I + 213096267) + 840*x + 380)/(4*x^4 + 4*x^2
+ 4*x + 1)

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Sympy [A]  time = 0.875286, size = 71, normalized size = 0.22 \begin{align*} \frac{36 x^{3} - 16 x^{2} + 42 x + 19}{80 x^{4} + 80 x^{2} + 80 x + 20} + \operatorname{RootSum}{\left (64000 t^{4} + 193344 t^{2} - 1064 t + 29, \left ( t \mapsto t \log{\left (- \frac{17084544000 t^{3}}{541735337} - \frac{188086000 t^{2}}{541735337} - \frac{51568487224 t}{541735337} + x - \frac{71080995}{541735337} \right )} \right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(4*x**4+4*x**2+4*x+1)**2,x)

[Out]

(36*x**3 - 16*x**2 + 42*x + 19)/(80*x**4 + 80*x**2 + 80*x + 20) + RootSum(64000*_t**4 + 193344*_t**2 - 1064*_t
+ 29, Lambda(_t, _t*log(-17084544000*_t**3/541735337 - 188086000*_t**2/541735337 - 51568487224*_t/541735337 +
x - 71080995/541735337)))

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Giac [A]  time = 1.2926, size = 463, normalized size = 1.46 \begin{align*} -\frac{{\left (\sqrt{2665 \, \sqrt{5} + 2062}{\left (\frac{5591 \, i}{2665 \, \sqrt{5} + 2062} + 1\right )} + 63 \, i + 91\right )} \log \left (11182 \,{\left (3765 \, i + 113\right )} x + 5591 \, \sqrt{7093997 \, \sqrt{5} + 6230338}{\left (\frac{14587901 \, i}{7093997 \, \sqrt{5} + 6230338} - 1\right )} - 631783 \, i + 21050115\right )}{8 \,{\left (13 \, i - 9\right )}} + \frac{{\left (\sqrt{2665 \, \sqrt{5} + 2062}{\left (\frac{5591 \, i}{2665 \, \sqrt{5} + 2062} + 1\right )} - 63 \, i - 91\right )} \log \left (11182 \,{\left (3765 \, i + 113\right )} x - 5591 \, \sqrt{7093997 \, \sqrt{5} + 6230338}{\left (\frac{14587901 \, i}{7093997 \, \sqrt{5} + 6230338} - 1\right )} - 631783 \, i + 21050115\right )}{8 \,{\left (13 \, i - 9\right )}} - \frac{{\left (\sqrt{2665 \, \sqrt{5} - 2062}{\left (\frac{5591 \, i}{2665 \, \sqrt{5} - 2062} + 1\right )} - 91 \, i - 63\right )} \log \left (11182 \,{\left (125 \, i + 3769\right )} x + 5591 \, \sqrt{7110493 \, \sqrt{5} - 6152618}{\left (\frac{14660861 \, i}{7110493 \, \sqrt{5} - 6152618} - 1\right )} + 21072479 \, i - 698875\right )}{8 \,{\left (9 \, i - 13\right )}} + \frac{{\left (\sqrt{2665 \, \sqrt{5} - 2062}{\left (\frac{5591 \, i}{2665 \, \sqrt{5} - 2062} + 1\right )} + 91 \, i + 63\right )} \log \left (11182 \,{\left (125 \, i + 3769\right )} x - 5591 \, \sqrt{7110493 \, \sqrt{5} - 6152618}{\left (\frac{14660861 \, i}{7110493 \, \sqrt{5} - 6152618} - 1\right )} + 21072479 \, i - 698875\right )}{8 \,{\left (9 \, i - 13\right )}} + \frac{36 \, x^{3} - 16 \, x^{2} + 42 \, x + 19}{20 \,{\left (4 \, x^{4} + 4 \, x^{2} + 4 \, x + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(sqrt(2665*sqrt(5) + 2062)*(5591*i/(2665*sqrt(5) + 2062) + 1) + 63*i + 91)*log(11182*(3765*i + 113)*x + 5
591*sqrt(7093997*sqrt(5) + 6230338)*(14587901*i/(7093997*sqrt(5) + 6230338) - 1) - 631783*i + 21050115)/(13*i
- 9) + 1/8*(sqrt(2665*sqrt(5) + 2062)*(5591*i/(2665*sqrt(5) + 2062) + 1) - 63*i - 91)*log(11182*(3765*i + 113)
*x - 5591*sqrt(7093997*sqrt(5) + 6230338)*(14587901*i/(7093997*sqrt(5) + 6230338) - 1) - 631783*i + 21050115)/
(13*i - 9) - 1/8*(sqrt(2665*sqrt(5) - 2062)*(5591*i/(2665*sqrt(5) - 2062) + 1) - 91*i - 63)*log(11182*(125*i +
3769)*x + 5591*sqrt(7110493*sqrt(5) - 6152618)*(14660861*i/(7110493*sqrt(5) - 6152618) - 1) + 21072479*i - 69
8875)/(9*i - 13) + 1/8*(sqrt(2665*sqrt(5) - 2062)*(5591*i/(2665*sqrt(5) - 2062) + 1) + 91*i + 63)*log(11182*(1
25*i + 3769)*x - 5591*sqrt(7110493*sqrt(5) - 6152618)*(14660861*i/(7110493*sqrt(5) - 6152618) - 1) + 21072479*
i - 698875)/(9*i - 13) + 1/20*(36*x^3 - 16*x^2 + 42*x + 19)/(4*x^4 + 4*x^2 + 4*x + 1)