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]
1/2*(-17+(1+1/x)^2)/(5-2*(1+1/x)^2+(1+1/x)^4)+1/10*(59-17*(1+1/x)^2)*(1+1/x)/(5-2*(1+1/x)^2+(1+1/x)^4)+7/4*arc
tan(-1/2+1/2*(1+1/x)^2)+1/400*ln((1+1/x)^2+5^(1/2)-(1+1/x)*(2+2*5^(1/2))^(1/2))*(-59590+26650*5^(1/2))^(1/2)-1
/400*ln((1+1/x)^2+5^(1/2)+(1+1/x)*(2+2*5^(1/2))^(1/2))*(-59590+26650*5^(1/2))^(1/2)-1/200*arctan((2+2/x-(2+2*5
^(1/2))^(1/2))/(-2+2*5^(1/2))^(1/2))*(59590+26650*5^(1/2))^(1/2)-1/200*arctan((2+2/x+(2+2*5^(1/2))^(1/2))/(-2+
2*5^(1/2))^(1/2))*(59590+26650*5^(1/2))^(1/2)
________________________________________________________________________________________
Rubi [A] time = 0.33, antiderivative size = 317, normalized size of antiderivative =
1.00, 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 12
Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] && !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]
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 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 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 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 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 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 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 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 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]
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 = 0.02, 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
________________________________________________________________________________________
fricas [C] time = 1.34, size = 704, normalized size = 2.22 \[ \text {result too large to display} \]
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)
________________________________________________________________________________________
giac [C] time = 0.72, size = 315, normalized size = 0.99 \[ -\frac {1}{400} \, {\left (-\left (i + 3\right ) \, \sqrt {2665 \, \sqrt {5} - 4790} {\left (\frac {709 i}{533 \, \sqrt {5} - 958} + 1\right )} - 350 i\right )} \log \left (\left (2534636224790 i + 16853816172010\right ) \, \sqrt {5} x - \left (3913528401620 i + 26022625108780\right ) \, x + 5049076145 \, \sqrt {5} \sqrt {1424281 \, \sqrt {5} - 2199118} - \left (8426908086005 i - 1267318112395\right ) \, \sqrt {5} + \left (8166407345 i - 7795873310\right ) \, \sqrt {1424281 \, \sqrt {5} - 2199118} + 13011312554390 i - 1956764200810\right ) - \frac {1}{400} \, {\left (\left (i + 3\right ) \, \sqrt {2665 \, \sqrt {5} - 4790} {\left (\frac {709 i}{533 \, \sqrt {5} - 958} + 1\right )} - 350 i\right )} \log \left (\left (2534636224790 i + 16853816172010\right ) \, \sqrt {5} x - \left (3913528401620 i + 26022625108780\right ) \, x - 5049076145 \, \sqrt {5} \sqrt {1424281 \, \sqrt {5} - 2199118} - \left (8426908086005 i - 1267318112395\right ) \, \sqrt {5} - \left (8166407345 i - 7795873310\right ) \, \sqrt {1424281 \, \sqrt {5} - 2199118} + 13011312554390 i - 1956764200810\right ) - \frac {1}{400} \, {\left (\left (3 i + 1\right ) \, \sqrt {2665 \, \sqrt {5} + 4790} {\left (\frac {709 i}{533 \, \sqrt {5} + 958} + 1\right )} + 350 i\right )} \log \left (\left (16722951192450 i + 2480822188910\right ) \, \sqrt {5} x + \left (25712356272300 i + 3814385585140\right ) \, x + 5021907265 \, \sqrt {5} \sqrt {1416617 \, \sqrt {5} + 2178118} + \left (1240411094455 i - 8361475596225\right ) \, \sqrt {5} + \left (8153361745 i + 7721428310\right ) \, \sqrt {1416617 \, \sqrt {5} + 2178118} + 1907192792570 i - 12856178136150\right ) - \frac {1}{400} \, {\left (-\left (3 i + 1\right ) \, \sqrt {2665 \, \sqrt {5} + 4790} {\left (\frac {709 i}{533 \, \sqrt {5} + 958} + 1\right )} + 350 i\right )} \log \left (\left (16722951192450 i + 2480822188910\right ) \, \sqrt {5} x + \left (25712356272300 i + 3814385585140\right ) \, x - 5021907265 \, \sqrt {5} \sqrt {1416617 \, \sqrt {5} + 2178118} + \left (1240411094455 i - 8361475596225\right ) \, \sqrt {5} - \left (8153361745 i + 7721428310\right ) \, \sqrt {1416617 \, \sqrt {5} + 2178118} + 1907192792570 i - 12856178136150\right ) + \frac {36 \, x^{3} - 16 \, x^{2} + 42 \, x + 19}{20 \, {\left (4 \, x^{4} + 4 \, x^{2} + 4 \, x + 1\right )}} \]
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]
-1/400*(-(I + 3)*sqrt(2665*sqrt(5) - 4790)*(709*I/(533*sqrt(5) - 958) + 1) - 350*I)*log((2534636224790*I + 168
53816172010)*sqrt(5)*x - (3913528401620*I + 26022625108780)*x + 5049076145*sqrt(5)*sqrt(1424281*sqrt(5) - 2199
118) - (8426908086005*I - 1267318112395)*sqrt(5) + (8166407345*I - 7795873310)*sqrt(1424281*sqrt(5) - 2199118)
+ 13011312554390*I - 1956764200810) - 1/400*((I + 3)*sqrt(2665*sqrt(5) - 4790)*(709*I/(533*sqrt(5) - 958) + 1
) - 350*I)*log((2534636224790*I + 16853816172010)*sqrt(5)*x - (3913528401620*I + 26022625108780)*x - 504907614
5*sqrt(5)*sqrt(1424281*sqrt(5) - 2199118) - (8426908086005*I - 1267318112395)*sqrt(5) - (8166407345*I - 779587
3310)*sqrt(1424281*sqrt(5) - 2199118) + 13011312554390*I - 1956764200810) - 1/400*((3*I + 1)*sqrt(2665*sqrt(5)
+ 4790)*(709*I/(533*sqrt(5) + 958) + 1) + 350*I)*log((16722951192450*I + 2480822188910)*sqrt(5)*x + (25712356
272300*I + 3814385585140)*x + 5021907265*sqrt(5)*sqrt(1416617*sqrt(5) + 2178118) + (1240411094455*I - 83614755
96225)*sqrt(5) + (8153361745*I + 7721428310)*sqrt(1416617*sqrt(5) + 2178118) + 1907192792570*I - 1285617813615
0) - 1/400*(-(3*I + 1)*sqrt(2665*sqrt(5) + 4790)*(709*I/(533*sqrt(5) + 958) + 1) + 350*I)*log((16722951192450*
I + 2480822188910)*sqrt(5)*x + (25712356272300*I + 3814385585140)*x - 5021907265*sqrt(5)*sqrt(1416617*sqrt(5)
+ 2178118) + (1240411094455*I - 8361475596225)*sqrt(5) - (8153361745*I + 7721428310)*sqrt(1416617*sqrt(5) + 21
78118) + 1907192792570*I - 12856178136150) + 1/20*(36*x^3 - 16*x^2 + 42*x + 19)/(4*x^4 + 4*x^2 + 4*x + 1)
________________________________________________________________________________________
maple [C] time = 0.01, size = 79, normalized size = 0.25 \[ \frac {\left (18 \RootOf \left (4 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+4 \textit {\_Z} +1\right )^{2}-16 \RootOf \left (4 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+4 \textit {\_Z} +1\right )+27\right ) \ln \left (-\RootOf \left (4 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+4 \textit {\_Z} +1\right )+x \right )}{160 \RootOf \left (4 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+4 \textit {\_Z} +1\right )^{3}+80 \RootOf \left (4 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+4 \textit {\_Z} +1\right )+40}+\frac {\frac {9}{20} x^{3}-\frac {1}{5} x^{2}+\frac {21}{40} x +\frac {19}{80}}{x^{4}+x^{2}+x +\frac {1}{4}} \]
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(-_R+x),_R=Root
Of(4*_Z^4+4*_Z^2+4*_Z+1))
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {36 \, x^{3} - 16 \, x^{2} + 42 \, x + 19}{20 \, {\left (4 \, x^{4} + 4 \, x^{2} + 4 \, x + 1\right )}} + \frac {1}{10} \, \int \frac {18 \, x^{2} - 16 \, x + 27}{4 \, x^{4} + 4 \, x^{2} + 4 \, x + 1}\,{d x} \]
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)
________________________________________________________________________________________
mupad [B] time = 2.21, size = 174, normalized size = 0.55 \[ \left (\sum _{k=1}^4\ln \left (-\frac {169\,\mathrm {root}\left (z^4+\frac {3021\,z^2}{1000}-\frac {133\,z}{8000}+\frac {29}{64000},z,k\right )}{100}+\frac {11\,x}{1600}+\frac {\mathrm {root}\left (z^4+\frac {3021\,z^2}{1000}-\frac {133\,z}{8000}+\frac {29}{64000},z,k\right )\,x\,131}{100}-\frac {{\mathrm {root}\left (z^4+\frac {3021\,z^2}{1000}-\frac {133\,z}{8000}+\frac {29}{64000},z,k\right )}^2\,x\,72}{5}-{\mathrm {root}\left (z^4+\frac {3021\,z^2}{1000}-\frac {133\,z}{8000}+\frac {29}{64000},z,k\right )}^3\,x\,36+\frac {59\,{\mathrm {root}\left (z^4+\frac {3021\,z^2}{1000}-\frac {133\,z}{8000}+\frac {29}{64000},z,k\right )}^2}{20}-16\,{\mathrm {root}\left (z^4+\frac {3021\,z^2}{1000}-\frac {133\,z}{8000}+\frac {29}{64000},z,k\right )}^3+\frac {27}{1600}\right )\,\mathrm {root}\left (z^4+\frac {3021\,z^2}{1000}-\frac {133\,z}{8000}+\frac {29}{64000},z,k\right )\right )+\frac {\frac {9\,x^3}{20}-\frac {x^2}{5}+\frac {21\,x}{40}+\frac {19}{80}}{x^4+x^2+x+\frac {1}{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(4*x + 4*x^2 + 4*x^4 + 1)^2,x)
[Out]
symsum(log((11*x)/1600 - (169*root(z^4 + (3021*z^2)/1000 - (133*z)/8000 + 29/64000, z, k))/100 + (131*root(z^4
+ (3021*z^2)/1000 - (133*z)/8000 + 29/64000, z, k)*x)/100 - (72*root(z^4 + (3021*z^2)/1000 - (133*z)/8000 + 2
9/64000, z, k)^2*x)/5 - 36*root(z^4 + (3021*z^2)/1000 - (133*z)/8000 + 29/64000, z, k)^3*x + (59*root(z^4 + (3
021*z^2)/1000 - (133*z)/8000 + 29/64000, z, k)^2)/20 - 16*root(z^4 + (3021*z^2)/1000 - (133*z)/8000 + 29/64000
, z, k)^3 + 27/1600)*root(z^4 + (3021*z^2)/1000 - (133*z)/8000 + 29/64000, z, k), k, 1, 4) + ((21*x)/40 - x^2/
5 + (9*x^3)/20 + 19/80)/(x + x^2 + x^4 + 1/4)
________________________________________________________________________________________
sympy [B] time = 3.66, size = 3834, normalized size = 12.09 \[ \text {result too large to display} \]
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) - sqrt(-5959/16000 + 533*sqrt(5)/3200)*log(x**
2 + x*(-1601676*sqrt(10)*sqrt(-5959 + 2665*sqrt(5))*sqrt(-665*sqrt(10)*sqrt(-5959 + 2665*sqrt(5)) + 221195*sqr
t(5) + 36004639)/13543383425 - 1067784*sqrt(2)*sqrt(-5959 + 2665*sqrt(5))/1016389 + 3131659367*sqrt(10)*sqrt(-
5959 + 2665*sqrt(5))/13543383425 + 291689395/1083470674 + 470215*sqrt(5)/2032778 + 94043*sqrt(-665*sqrt(10)*sq
rt(-5959 + 2665*sqrt(5)) + 221195*sqrt(5) + 36004639)/541735337) - 40634464149111451*sqrt(5)*sqrt(-665*sqrt(10
)*sqrt(-5959 + 2665*sqrt(5)) + 221195*sqrt(5) + 36004639)/27530691871904650 - 2885835544225227917282997/146738
587677251784500 - 83803227754187*sqrt(2)*sqrt(-5959 + 2665*sqrt(5))/100111606806926 - 50208805356*sqrt(2)*sqrt
(-5959 + 2665*sqrt(5))*sqrt(-665*sqrt(10)*sqrt(-5959 + 2665*sqrt(5)) + 221195*sqrt(5) + 36004639)/550613837438
093 - 538485754891933*sqrt(10)*sqrt(-5959 + 2665*sqrt(5))*sqrt(-665*sqrt(10)*sqrt(-5959 + 2665*sqrt(5)) + 2211
95*sqrt(5) + 36004639)/14673858767725178450 - 925321955096901411*sqrt(10)*sqrt(-5959 + 2665*sqrt(5))/293477175
35450356900 + 484304611938766076267*sqrt(5)/55061383743809300 + 22013036087014785403*sqrt(-665*sqrt(10)*sqrt(-
5959 + 2665*sqrt(5)) + 221195*sqrt(5) + 36004639)/6669935803511444750) + sqrt(-5959/16000 + 533*sqrt(5)/3200)*
log(x**2 + x*(-94043*sqrt(665*sqrt(10)*sqrt(-5959 + 2665*sqrt(5)) + 221195*sqrt(5) + 36004639)/541735337 - 160
1676*sqrt(10)*sqrt(-5959 + 2665*sqrt(5))*sqrt(665*sqrt(10)*sqrt(-5959 + 2665*sqrt(5)) + 221195*sqrt(5) + 36004
639)/13543383425 - 3131659367*sqrt(10)*sqrt(-5959 + 2665*sqrt(5))/13543383425 + 291689395/1083470674 + 1067784
*sqrt(2)*sqrt(-5959 + 2665*sqrt(5))/1016389 + 470215*sqrt(5)/2032778) - 22013036087014785403*sqrt(665*sqrt(10)
*sqrt(-5959 + 2665*sqrt(5)) + 221195*sqrt(5) + 36004639)/6669935803511444750 - 2885835544225227917282997/14673
8587677251784500 - 50208805356*sqrt(2)*sqrt(-5959 + 2665*sqrt(5))*sqrt(665*sqrt(10)*sqrt(-5959 + 2665*sqrt(5))
+ 221195*sqrt(5) + 36004639)/550613837438093 - 538485754891933*sqrt(10)*sqrt(-5959 + 2665*sqrt(5))*sqrt(665*s
qrt(10)*sqrt(-5959 + 2665*sqrt(5)) + 221195*sqrt(5) + 36004639)/14673858767725178450 + 925321955096901411*sqrt
(10)*sqrt(-5959 + 2665*sqrt(5))/29347717535450356900 + 83803227754187*sqrt(2)*sqrt(-5959 + 2665*sqrt(5))/10011
1606806926 + 484304611938766076267*sqrt(5)/55061383743809300 + 40634464149111451*sqrt(5)*sqrt(665*sqrt(10)*sqr
t(-5959 + 2665*sqrt(5)) + 221195*sqrt(5) + 36004639)/27530691871904650) + 2*sqrt(6291/16000 + 1599*sqrt(5)/320
0 + sqrt(-665*sqrt(10)*sqrt(-5959 + 2665*sqrt(5)) + 221195*sqrt(5) + 36004639)/4000)*atan(54173533700*x/(-6440
570878*sqrt(10)*sqrt(6291 + 7995*sqrt(5) + 4*sqrt(-665*sqrt(10)*sqrt(-5959 + 2665*sqrt(5)) + 221195*sqrt(5) +
36004639)) + 2351075*sqrt(-5959 + 2665*sqrt(5))*sqrt(6291 + 7995*sqrt(5) + 4*sqrt(-665*sqrt(10)*sqrt(-5959 + 2
665*sqrt(5)) + 221195*sqrt(5) + 36004639)) + 1067784*sqrt(10)*sqrt(6291 + 7995*sqrt(5) + 4*sqrt(-665*sqrt(10)*
sqrt(-5959 + 2665*sqrt(5)) + 221195*sqrt(5) + 36004639))*sqrt(-665*sqrt(10)*sqrt(-5959 + 2665*sqrt(5)) + 22119
5*sqrt(5) + 36004639)) - 3203352*sqrt(10)*sqrt(-5959 + 2665*sqrt(5))*sqrt(-665*sqrt(10)*sqrt(-5959 + 2665*sqrt
(5)) + 221195*sqrt(5) + 36004639)/(-6440570878*sqrt(10)*sqrt(6291 + 7995*sqrt(5) + 4*sqrt(-665*sqrt(10)*sqrt(-
5959 + 2665*sqrt(5)) + 221195*sqrt(5) + 36004639)) + 2351075*sqrt(-5959 + 2665*sqrt(5))*sqrt(6291 + 7995*sqrt(
5) + 4*sqrt(-665*sqrt(10)*sqrt(-5959 + 2665*sqrt(5)) + 221195*sqrt(5) + 36004639)) + 1067784*sqrt(10)*sqrt(629
1 + 7995*sqrt(5) + 4*sqrt(-665*sqrt(10)*sqrt(-5959 + 2665*sqrt(5)) + 221195*sqrt(5) + 36004639))*sqrt(-665*sqr
t(10)*sqrt(-5959 + 2665*sqrt(5)) + 221195*sqrt(5) + 36004639)) - 28456443600*sqrt(2)*sqrt(-5959 + 2665*sqrt(5)
)/(-6440570878*sqrt(10)*sqrt(6291 + 7995*sqrt(5) + 4*sqrt(-665*sqrt(10)*sqrt(-5959 + 2665*sqrt(5)) + 221195*sq
rt(5) + 36004639)) + 2351075*sqrt(-5959 + 2665*sqrt(5))*sqrt(6291 + 7995*sqrt(5) + 4*sqrt(-665*sqrt(10)*sqrt(-
5959 + 2665*sqrt(5)) + 221195*sqrt(5) + 36004639)) + 1067784*sqrt(10)*sqrt(6291 + 7995*sqrt(5) + 4*sqrt(-665*s
qrt(10)*sqrt(-5959 + 2665*sqrt(5)) + 221195*sqrt(5) + 36004639))*sqrt(-665*sqrt(10)*sqrt(-5959 + 2665*sqrt(5))
+ 221195*sqrt(5) + 36004639)) + 6263318734*sqrt(10)*sqrt(-5959 + 2665*sqrt(5))/(-6440570878*sqrt(10)*sqrt(629
1 + 7995*sqrt(5) + 4*sqrt(-665*sqrt(10)*sqrt(-5959 + 2665*sqrt(5)) + 221195*sqrt(5) + 36004639)) + 2351075*sqr
t(-5959 + 2665*sqrt(5))*sqrt(6291 + 7995*sqrt(5) + 4*sqrt(-665*sqrt(10)*sqrt(-5959 + 2665*sqrt(5)) + 221195*sq
rt(5) + 36004639)) + 1067784*sqrt(10)*sqrt(6291 + 7995*sqrt(5) + 4*sqrt(-665*sqrt(10)*sqrt(-5959 + 2665*sqrt(5
)) + 221195*sqrt(5) + 36004639))*sqrt(-665*sqrt(10)*sqrt(-5959 + 2665*sqrt(5)) + 221195*sqrt(5) + 36004639)) +
7292234875/(-6440570878*sqrt(10)*sqrt(6291 + 7995*sqrt(5) + 4*sqrt(-665*sqrt(10)*sqrt(-5959 + 2665*sqrt(5)) +
221195*sqrt(5) + 36004639)) + 2351075*sqrt(-5959 + 2665*sqrt(5))*sqrt(6291 + 7995*sqrt(5) + 4*sqrt(-665*sqrt(
10)*sqrt(-5959 + 2665*sqrt(5)) + 221195*sqrt(5) + 36004639)) + 1067784*sqrt(10)*sqrt(6291 + 7995*sqrt(5) + 4*s
qrt(-665*sqrt(10)*sqrt(-5959 + 2665*sqrt(5)) + 221195*sqrt(5) + 36004639))*sqrt(-665*sqrt(10)*sqrt(-5959 + 266
5*sqrt(5)) + 221195*sqrt(5) + 36004639)) + 6265614875*sqrt(5)/(-6440570878*sqrt(10)*sqrt(6291 + 7995*sqrt(5) +
4*sqrt(-665*sqrt(10)*sqrt(-5959 + 2665*sqrt(5)) + 221195*sqrt(5) + 36004639)) + 2351075*sqrt(-5959 + 2665*sqr
t(5))*sqrt(6291 + 7995*sqrt(5) + 4*sqrt(-665*sqrt(10)*sqrt(-5959 + 2665*sqrt(5)) + 221195*sqrt(5) + 36004639))
+ 1067784*sqrt(10)*sqrt(6291 + 7995*sqrt(5) + 4*sqrt(-665*sqrt(10)*sqrt(-5959 + 2665*sqrt(5)) + 221195*sqrt(5
) + 36004639))*sqrt(-665*sqrt(10)*sqrt(-5959 + 2665*sqrt(5)) + 221195*sqrt(5) + 36004639)) + 4702150*sqrt(-665
*sqrt(10)*sqrt(-5959 + 2665*sqrt(5)) + 221195*sqrt(5) + 36004639)/(-6440570878*sqrt(10)*sqrt(6291 + 7995*sqrt(
5) + 4*sqrt(-665*sqrt(10)*sqrt(-5959 + 2665*sqrt(5)) + 221195*sqrt(5) + 36004639)) + 2351075*sqrt(-5959 + 2665
*sqrt(5))*sqrt(6291 + 7995*sqrt(5) + 4*sqrt(-665*sqrt(10)*sqrt(-5959 + 2665*sqrt(5)) + 221195*sqrt(5) + 360046
39)) + 1067784*sqrt(10)*sqrt(6291 + 7995*sqrt(5) + 4*sqrt(-665*sqrt(10)*sqrt(-5959 + 2665*sqrt(5)) + 221195*sq
rt(5) + 36004639))*sqrt(-665*sqrt(10)*sqrt(-5959 + 2665*sqrt(5)) + 221195*sqrt(5) + 36004639))) - 2*sqrt(-sqrt
(665*sqrt(10)*sqrt(-5959 + 2665*sqrt(5)) + 221195*sqrt(5) + 36004639)/4000 + 6291/16000 + 1599*sqrt(5)/3200)*a
tan(54173533700*x/(2351075*sqrt(-5959 + 2665*sqrt(5))*sqrt(-4*sqrt(665*sqrt(10)*sqrt(-5959 + 2665*sqrt(5)) + 2
21195*sqrt(5) + 36004639) + 6291 + 7995*sqrt(5)) + 6440570878*sqrt(10)*sqrt(-4*sqrt(665*sqrt(10)*sqrt(-5959 +
2665*sqrt(5)) + 221195*sqrt(5) + 36004639) + 6291 + 7995*sqrt(5)) + 1067784*sqrt(10)*sqrt(665*sqrt(10)*sqrt(-5
959 + 2665*sqrt(5)) + 221195*sqrt(5) + 36004639)*sqrt(-4*sqrt(665*sqrt(10)*sqrt(-5959 + 2665*sqrt(5)) + 221195
*sqrt(5) + 36004639) + 6291 + 7995*sqrt(5))) - 4702150*sqrt(665*sqrt(10)*sqrt(-5959 + 2665*sqrt(5)) + 221195*s
qrt(5) + 36004639)/(2351075*sqrt(-5959 + 2665*sqrt(5))*sqrt(-4*sqrt(665*sqrt(10)*sqrt(-5959 + 2665*sqrt(5)) +
221195*sqrt(5) + 36004639) + 6291 + 7995*sqrt(5)) + 6440570878*sqrt(10)*sqrt(-4*sqrt(665*sqrt(10)*sqrt(-5959 +
2665*sqrt(5)) + 221195*sqrt(5) + 36004639) + 6291 + 7995*sqrt(5)) + 1067784*sqrt(10)*sqrt(665*sqrt(10)*sqrt(-
5959 + 2665*sqrt(5)) + 221195*sqrt(5) + 36004639)*sqrt(-4*sqrt(665*sqrt(10)*sqrt(-5959 + 2665*sqrt(5)) + 22119
5*sqrt(5) + 36004639) + 6291 + 7995*sqrt(5))) - 3203352*sqrt(10)*sqrt(-5959 + 2665*sqrt(5))*sqrt(665*sqrt(10)*
sqrt(-5959 + 2665*sqrt(5)) + 221195*sqrt(5) + 36004639)/(2351075*sqrt(-5959 + 2665*sqrt(5))*sqrt(-4*sqrt(665*s
qrt(10)*sqrt(-5959 + 2665*sqrt(5)) + 221195*sqrt(5) + 36004639) + 6291 + 7995*sqrt(5)) + 6440570878*sqrt(10)*s
qrt(-4*sqrt(665*sqrt(10)*sqrt(-5959 + 2665*sqrt(5)) + 221195*sqrt(5) + 36004639) + 6291 + 7995*sqrt(5)) + 1067
784*sqrt(10)*sqrt(665*sqrt(10)*sqrt(-5959 + 2665*sqrt(5)) + 221195*sqrt(5) + 36004639)*sqrt(-4*sqrt(665*sqrt(1
0)*sqrt(-5959 + 2665*sqrt(5)) + 221195*sqrt(5) + 36004639) + 6291 + 7995*sqrt(5))) - 6263318734*sqrt(10)*sqrt(
-5959 + 2665*sqrt(5))/(2351075*sqrt(-5959 + 2665*sqrt(5))*sqrt(-4*sqrt(665*sqrt(10)*sqrt(-5959 + 2665*sqrt(5))
+ 221195*sqrt(5) + 36004639) + 6291 + 7995*sqrt(5)) + 6440570878*sqrt(10)*sqrt(-4*sqrt(665*sqrt(10)*sqrt(-595
9 + 2665*sqrt(5)) + 221195*sqrt(5) + 36004639) + 6291 + 7995*sqrt(5)) + 1067784*sqrt(10)*sqrt(665*sqrt(10)*sqr
t(-5959 + 2665*sqrt(5)) + 221195*sqrt(5) + 36004639)*sqrt(-4*sqrt(665*sqrt(10)*sqrt(-5959 + 2665*sqrt(5)) + 22
1195*sqrt(5) + 36004639) + 6291 + 7995*sqrt(5))) + 7292234875/(2351075*sqrt(-5959 + 2665*sqrt(5))*sqrt(-4*sqrt
(665*sqrt(10)*sqrt(-5959 + 2665*sqrt(5)) + 221195*sqrt(5) + 36004639) + 6291 + 7995*sqrt(5)) + 6440570878*sqrt
(10)*sqrt(-4*sqrt(665*sqrt(10)*sqrt(-5959 + 2665*sqrt(5)) + 221195*sqrt(5) + 36004639) + 6291 + 7995*sqrt(5))
+ 1067784*sqrt(10)*sqrt(665*sqrt(10)*sqrt(-5959 + 2665*sqrt(5)) + 221195*sqrt(5) + 36004639)*sqrt(-4*sqrt(665*
sqrt(10)*sqrt(-5959 + 2665*sqrt(5)) + 221195*sqrt(5) + 36004639) + 6291 + 7995*sqrt(5))) + 28456443600*sqrt(2)
*sqrt(-5959 + 2665*sqrt(5))/(2351075*sqrt(-5959 + 2665*sqrt(5))*sqrt(-4*sqrt(665*sqrt(10)*sqrt(-5959 + 2665*sq
rt(5)) + 221195*sqrt(5) + 36004639) + 6291 + 7995*sqrt(5)) + 6440570878*sqrt(10)*sqrt(-4*sqrt(665*sqrt(10)*sqr
t(-5959 + 2665*sqrt(5)) + 221195*sqrt(5) + 36004639) + 6291 + 7995*sqrt(5)) + 1067784*sqrt(10)*sqrt(665*sqrt(1
0)*sqrt(-5959 + 2665*sqrt(5)) + 221195*sqrt(5) + 36004639)*sqrt(-4*sqrt(665*sqrt(10)*sqrt(-5959 + 2665*sqrt(5)
) + 221195*sqrt(5) + 36004639) + 6291 + 7995*sqrt(5))) + 6265614875*sqrt(5)/(2351075*sqrt(-5959 + 2665*sqrt(5)
)*sqrt(-4*sqrt(665*sqrt(10)*sqrt(-5959 + 2665*sqrt(5)) + 221195*sqrt(5) + 36004639) + 6291 + 7995*sqrt(5)) + 6
440570878*sqrt(10)*sqrt(-4*sqrt(665*sqrt(10)*sqrt(-5959 + 2665*sqrt(5)) + 221195*sqrt(5) + 36004639) + 6291 +
7995*sqrt(5)) + 1067784*sqrt(10)*sqrt(665*sqrt(10)*sqrt(-5959 + 2665*sqrt(5)) + 221195*sqrt(5) + 36004639)*sqr
t(-4*sqrt(665*sqrt(10)*sqrt(-5959 + 2665*sqrt(5)) + 221195*sqrt(5) + 36004639) + 6291 + 7995*sqrt(5))))
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