3.50 \(\int \frac {1}{(8+8 x-x^3+8 x^4)^2} \, dx\)
Optimal. Leaf size=357 \[ -\frac {29 \left (\frac {4}{x}+1\right )^2+207}{336 \left (\left (\frac {4}{x}+1\right )^4-6 \left (\frac {4}{x}+1\right )^2+261\right )}+\frac {5 \left (199 \left (\frac {4}{x}+1\right )^2+5157\right ) \left (\frac {4}{x}+1\right )}{87696 \left (\left (\frac {4}{x}+1\right )^4-6 \left (\frac {4}{x}+1\right )^2+261\right )}-\frac {\sqrt {\frac {45923327 \sqrt {29}-180983329}{1218}} \log \left (\left (\frac {4}{x}+1\right )^2-\sqrt {6 \left (1+\sqrt {29}\right )} \left (\frac {4}{x}+1\right )+3 \sqrt {29}\right )}{175392}+\frac {\sqrt {\frac {45923327 \sqrt {29}-180983329}{1218}} \log \left (\left (\frac {4}{x}+1\right )^2+\sqrt {6 \left (1+\sqrt {29}\right )} \left (\frac {4}{x}+1\right )+3 \sqrt {29}\right )}{175392}-\frac {17 \tan ^{-1}\left (\frac {3-\left (\frac {4}{x}+1\right )^2}{6 \sqrt {7}}\right )}{1008 \sqrt {7}}-\frac {\sqrt {\frac {180983329+45923327 \sqrt {29}}{1218}} \tan ^{-1}\left (\frac {\frac {8}{x}-\sqrt {6 \left (1+\sqrt {29}\right )}+2}{\sqrt {6 \left (\sqrt {29}-1\right )}}\right )}{87696}-\frac {\sqrt {\frac {180983329+45923327 \sqrt {29}}{1218}} \tan ^{-1}\left (\frac {\frac {8}{x}+\sqrt {6 \left (1+\sqrt {29}\right )}+2}{\sqrt {6 \left (\sqrt {29}-1\right )}}\right )}{87696} \]
[Out]
1/336*(-207-29*(1+4/x)^2)/(261-6*(1+4/x)^2+(1+4/x)^4)+5/87696*(5157+199*(1+4/x)^2)*(1+4/x)/(261-6*(1+4/x)^2+(1
+4/x)^4)-17/7056*arctan(1/42*(3-(1+4/x)^2)*7^(1/2))*7^(1/2)-1/213627456*ln((1+4/x)^2+3*29^(1/2)-(1+4/x)*(6+6*2
9^(1/2))^(1/2))*(-220437694722+55934612286*29^(1/2))^(1/2)+1/213627456*ln((1+4/x)^2+3*29^(1/2)+(1+4/x)*(6+6*29
^(1/2))^(1/2))*(-220437694722+55934612286*29^(1/2))^(1/2)-1/106813728*arctan((2+8/x-(6+6*29^(1/2))^(1/2))/(-6+
6*29^(1/2))^(1/2))*(220437694722+55934612286*29^(1/2))^(1/2)-1/106813728*arctan((2+8/x+(6+6*29^(1/2))^(1/2))/(
-6+6*29^(1/2))^(1/2))*(220437694722+55934612286*29^(1/2))^(1/2)
________________________________________________________________________________________
Rubi [A] time = 0.40, antiderivative size = 357, normalized size of antiderivative =
1.00, number of steps used = 18, number of rules used = 11, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\)
= 0.647, Rules used = {2069, 12, 1673, 1678, 1169, 634, 618, 204, 628, 1663, 1660}
\[ -\frac {29 \left (\frac {4}{x}+1\right )^2+207}{336 \left (\left (\frac {4}{x}+1\right )^4-6 \left (\frac {4}{x}+1\right )^2+261\right )}+\frac {5 \left (199 \left (\frac {4}{x}+1\right )^2+5157\right ) \left (\frac {4}{x}+1\right )}{87696 \left (\left (\frac {4}{x}+1\right )^4-6 \left (\frac {4}{x}+1\right )^2+261\right )}-\frac {\sqrt {\frac {45923327 \sqrt {29}-180983329}{1218}} \log \left (\left (\frac {4}{x}+1\right )^2-\sqrt {6 \left (1+\sqrt {29}\right )} \left (\frac {4}{x}+1\right )+3 \sqrt {29}\right )}{175392}+\frac {\sqrt {\frac {45923327 \sqrt {29}-180983329}{1218}} \log \left (\left (\frac {4}{x}+1\right )^2+\sqrt {6 \left (1+\sqrt {29}\right )} \left (\frac {4}{x}+1\right )+3 \sqrt {29}\right )}{175392}-\frac {17 \tan ^{-1}\left (\frac {3-\left (\frac {4}{x}+1\right )^2}{6 \sqrt {7}}\right )}{1008 \sqrt {7}}-\frac {\sqrt {\frac {180983329+45923327 \sqrt {29}}{1218}} \tan ^{-1}\left (\frac {\frac {8}{x}-\sqrt {6 \left (1+\sqrt {29}\right )}+2}{\sqrt {6 \left (\sqrt {29}-1\right )}}\right )}{87696}-\frac {\sqrt {\frac {180983329+45923327 \sqrt {29}}{1218}} \tan ^{-1}\left (\frac {\frac {8}{x}+\sqrt {6 \left (1+\sqrt {29}\right )}+2}{\sqrt {6 \left (\sqrt {29}-1\right )}}\right )}{87696} \]
Antiderivative was successfully verified.
[In]
Int[(8 + 8*x - x^3 + 8*x^4)^(-2),x]
[Out]
-(207 + 29*(1 + 4/x)^2)/(336*(261 - 6*(1 + 4/x)^2 + (1 + 4/x)^4)) + (5*(5157 + 199*(1 + 4/x)^2)*(1 + 4/x))/(87
696*(261 - 6*(1 + 4/x)^2 + (1 + 4/x)^4)) - (17*ArcTan[(3 - (1 + 4/x)^2)/(6*Sqrt[7])])/(1008*Sqrt[7]) - (Sqrt[(
180983329 + 45923327*Sqrt[29])/1218]*ArcTan[(2 - Sqrt[6*(1 + Sqrt[29])] + 8/x)/Sqrt[6*(-1 + Sqrt[29])]])/87696
- (Sqrt[(180983329 + 45923327*Sqrt[29])/1218]*ArcTan[(2 + Sqrt[6*(1 + Sqrt[29])] + 8/x)/Sqrt[6*(-1 + Sqrt[29]
)]])/87696 - (Sqrt[(-180983329 + 45923327*Sqrt[29])/1218]*Log[3*Sqrt[29] - Sqrt[6*(1 + Sqrt[29])]*(1 + 4/x) +
(1 + 4/x)^2])/175392 + (Sqrt[(-180983329 + 45923327*Sqrt[29])/1218]*Log[3*Sqrt[29] + Sqrt[6*(1 + Sqrt[29])]*(1
+ 4/x) + (1 + 4/x)^2])/175392
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 (8+8 x-x^3+8 x^4\right )^2} \, dx &=-\left (1024 \operatorname {Subst}\left (\int \frac {(8-32 x)^6}{64 \left (1069056-393216 x^2+1048576 x^4\right )^2} \, dx,x,\frac {1}{4}+\frac {1}{x}\right )\right )\\ &=-\left (16 \operatorname {Subst}\left (\int \frac {(8-32 x)^6}{\left (1069056-393216 x^2+1048576 x^4\right )^2} \, dx,x,\frac {1}{4}+\frac {1}{x}\right )\right )\\ &=-\left (16 \operatorname {Subst}\left (\int \frac {x \left (-6291456-335544320 x^2-1610612736 x^4\right )}{\left (1069056-393216 x^2+1048576 x^4\right )^2} \, dx,x,\frac {1}{4}+\frac {1}{x}\right )\right )-16 \operatorname {Subst}\left (\int \frac {262144+62914560 x^2+1006632960 x^4+1073741824 x^6}{\left (1069056-393216 x^2+1048576 x^4\right )^2} \, dx,x,\frac {1}{4}+\frac {1}{x}\right )\\ &=\frac {5 \left (5157+199 \left (1+\frac {4}{x}\right )^2\right ) \left (1+\frac {4}{x}\right )}{87696 \left (261-6 \left (1+\frac {4}{x}\right )^2+\left (1+\frac {4}{x}\right )^4\right )}-\frac {\operatorname {Subst}\left (\int \frac {2789277407614152474624+7758008804499473301504 x^2}{1069056-393216 x^2+1048576 x^4} \, dx,x,\frac {1}{4}+\frac {1}{x}\right )}{578536630256664576}-8 \operatorname {Subst}\left (\int \frac {-6291456-335544320 x-1610612736 x^2}{\left (1069056-393216 x+1048576 x^2\right )^2} \, dx,x,\left (\frac {1}{4}+\frac {1}{x}\right )^2\right )\\ &=-\frac {207+29 \left (1+\frac {4}{x}\right )^2}{336 \left (261-6 \left (1+\frac {4}{x}\right )^2+\left (1+\frac {4}{x}\right )^4\right )}+\frac {5 \left (5157+199 \left (1+\frac {4}{x}\right )^2\right ) \left (1+\frac {4}{x}\right )}{87696 \left (261-6 \left (1+\frac {4}{x}\right )^2+\left (1+\frac {4}{x}\right )^4\right )}-\frac {\operatorname {Subst}\left (\int -\frac {3588805953060864}{1069056-393216 x+1048576 x^2} \, dx,x,\left (\frac {1}{4}+\frac {1}{x}\right )^2\right )}{541165879296}-\frac {\operatorname {Subst}\left (\int \frac {697319351903538118656 \sqrt {6 \left (1+\sqrt {29}\right )}-\left (2789277407614152474624-1454626650843651244032 \sqrt {29}\right ) x}{\frac {3 \sqrt {29}}{16}-\frac {1}{2} \sqrt {\frac {3}{2} \left (1+\sqrt {29}\right )} x+x^2} \, dx,x,\frac {1}{4}+\frac {1}{x}\right )}{56872464900751154479104 \sqrt {174 \left (1+\sqrt {29}\right )}}-\frac {\operatorname {Subst}\left (\int \frac {697319351903538118656 \sqrt {6 \left (1+\sqrt {29}\right )}+\left (2789277407614152474624-1454626650843651244032 \sqrt {29}\right ) x}{\frac {3 \sqrt {29}}{16}+\frac {1}{2} \sqrt {\frac {3}{2} \left (1+\sqrt {29}\right )} x+x^2} \, dx,x,\frac {1}{4}+\frac {1}{x}\right )}{56872464900751154479104 \sqrt {174 \left (1+\sqrt {29}\right )}}\\ &=-\frac {207+29 \left (1+\frac {4}{x}\right )^2}{336 \left (261-6 \left (1+\frac {4}{x}\right )^2+\left (1+\frac {4}{x}\right )^4\right )}+\frac {5 \left (5157+199 \left (1+\frac {4}{x}\right )^2\right ) \left (1+\frac {4}{x}\right )}{87696 \left (261-6 \left (1+\frac {4}{x}\right )^2+\left (1+\frac {4}{x}\right )^4\right )}+\frac {139264}{21} \operatorname {Subst}\left (\int \frac {1}{1069056-393216 x+1048576 x^2} \, dx,x,\left (\frac {1}{4}+\frac {1}{x}\right )^2\right )-\frac {\sqrt {\frac {1}{58} \left (82199511+9647143 \sqrt {29}\right )} \operatorname {Subst}\left (\int \frac {1}{\frac {3 \sqrt {29}}{16}-\frac {1}{2} \sqrt {\frac {3}{2} \left (1+\sqrt {29}\right )} x+x^2} \, dx,x,\frac {1}{4}+\frac {1}{x}\right )}{350784}-\frac {\sqrt {\frac {1}{58} \left (82199511+9647143 \sqrt {29}\right )} \operatorname {Subst}\left (\int \frac {1}{\frac {3 \sqrt {29}}{16}+\frac {1}{2} \sqrt {\frac {3}{2} \left (1+\sqrt {29}\right )} x+x^2} \, dx,x,\frac {1}{4}+\frac {1}{x}\right )}{350784}-\frac {\sqrt {\frac {-180983329+45923327 \sqrt {29}}{1218}} \operatorname {Subst}\left (\int \frac {-\frac {1}{2} \sqrt {\frac {3}{2} \left (1+\sqrt {29}\right )}+2 x}{\frac {3 \sqrt {29}}{16}-\frac {1}{2} \sqrt {\frac {3}{2} \left (1+\sqrt {29}\right )} x+x^2} \, dx,x,\frac {1}{4}+\frac {1}{x}\right )}{175392}+\frac {\sqrt {\frac {-180983329+45923327 \sqrt {29}}{1218}} \operatorname {Subst}\left (\int \frac {\frac {1}{2} \sqrt {\frac {3}{2} \left (1+\sqrt {29}\right )}+2 x}{\frac {3 \sqrt {29}}{16}+\frac {1}{2} \sqrt {\frac {3}{2} \left (1+\sqrt {29}\right )} x+x^2} \, dx,x,\frac {1}{4}+\frac {1}{x}\right )}{175392}\\ &=-\frac {207+29 \left (1+\frac {4}{x}\right )^2}{336 \left (261-6 \left (1+\frac {4}{x}\right )^2+\left (1+\frac {4}{x}\right )^4\right )}+\frac {5 \left (5157+199 \left (1+\frac {4}{x}\right )^2\right ) \left (1+\frac {4}{x}\right )}{87696 \left (261-6 \left (1+\frac {4}{x}\right )^2+\left (1+\frac {4}{x}\right )^4\right )}-\frac {\sqrt {\frac {-180983329+45923327 \sqrt {29}}{1218}} \log \left (3 \sqrt {29}-\sqrt {6 \left (1+\sqrt {29}\right )} \left (1+\frac {4}{x}\right )+\left (1+\frac {4}{x}\right )^2\right )}{175392}+\frac {\sqrt {\frac {-180983329+45923327 \sqrt {29}}{1218}} \log \left (3 \sqrt {29}+\sqrt {6 \left (1+\sqrt {29}\right )} \left (1+\frac {4}{x}\right )+\left (1+\frac {4}{x}\right )^2\right )}{175392}-\frac {278528}{21} \operatorname {Subst}\left (\int \frac {1}{-4329327034368-x^2} \, dx,x,-393216+2097152 \left (\frac {1}{4}+\frac {1}{x}\right )^2\right )+\frac {\sqrt {\frac {1}{58} \left (82199511+9647143 \sqrt {29}\right )} \operatorname {Subst}\left (\int \frac {1}{\frac {3}{8} \left (1-\sqrt {29}\right )-x^2} \, dx,x,-\frac {1}{2} \sqrt {\frac {3}{2} \left (1+\sqrt {29}\right )}+2 \left (\frac {1}{4}+\frac {1}{x}\right )\right )}{175392}+\frac {\sqrt {\frac {1}{58} \left (82199511+9647143 \sqrt {29}\right )} \operatorname {Subst}\left (\int \frac {1}{\frac {3}{8} \left (1-\sqrt {29}\right )-x^2} \, dx,x,\frac {1}{4} \left (2+\sqrt {6 \left (1+\sqrt {29}\right )}+\frac {8}{x}\right )\right )}{175392}\\ &=-\frac {207+29 \left (1+\frac {4}{x}\right )^2}{336 \left (261-6 \left (1+\frac {4}{x}\right )^2+\left (1+\frac {4}{x}\right )^4\right )}+\frac {5 \left (5157+199 \left (1+\frac {4}{x}\right )^2\right ) \left (1+\frac {4}{x}\right )}{87696 \left (261-6 \left (1+\frac {4}{x}\right )^2+\left (1+\frac {4}{x}\right )^4\right )}-\frac {17 \tan ^{-1}\left (\frac {3-\left (1+\frac {4}{x}\right )^2}{6 \sqrt {7}}\right )}{1008 \sqrt {7}}-\frac {\sqrt {\frac {180983329+45923327 \sqrt {29}}{1218}} \tan ^{-1}\left (\frac {2+\sqrt {6 \left (1+\sqrt {29}\right )}+\frac {8}{x}}{\sqrt {6 \left (-1+\sqrt {29}\right )}}\right )}{87696}-\frac {\sqrt {\frac {180983329+45923327 \sqrt {29}}{1218}} \tan ^{-1}\left (\frac {8+\left (2-\sqrt {6 \left (1+\sqrt {29}\right )}\right ) x}{\sqrt {6 \left (-1+\sqrt {29}\right )} x}\right )}{87696}-\frac {\sqrt {\frac {-180983329+45923327 \sqrt {29}}{1218}} \log \left (3 \sqrt {29}-\sqrt {6 \left (1+\sqrt {29}\right )} \left (1+\frac {4}{x}\right )+\left (1+\frac {4}{x}\right )^2\right )}{175392}+\frac {\sqrt {\frac {-180983329+45923327 \sqrt {29}}{1218}} \log \left (3 \sqrt {29}+\sqrt {6 \left (1+\sqrt {29}\right )} \left (1+\frac {4}{x}\right )+\left (1+\frac {4}{x}\right )^2\right )}{175392}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 113, normalized size = 0.32 \[ \frac {\text {RootSum}\left [8 \text {$\#$1}^4-\text {$\#$1}^3+8 \text {$\#$1}+8\& ,\frac {392 \text {$\#$1}^2 \log (x-\text {$\#$1})-1097 \text {$\#$1} \log (x-\text {$\#$1})+2243 \log (x-\text {$\#$1})}{32 \text {$\#$1}^3-3 \text {$\#$1}^2+8}\& \right ]}{21924}+\frac {784 x^3-1146 x^2+1539 x+544}{43848 \left (8 x^4-x^3+8 x+8\right )} \]
Antiderivative was successfully verified.
[In]
Integrate[(8 + 8*x - x^3 + 8*x^4)^(-2),x]
[Out]
(544 + 1539*x - 1146*x^2 + 784*x^3)/(43848*(8 + 8*x - x^3 + 8*x^4)) + RootSum[8 + 8*#1 - #1^3 + 8*#1^4 & , (22
43*Log[x - #1] - 1097*Log[x - #1]*#1 + 392*Log[x - #1]*#1^2)/(8 - 3*#1^2 + 32*#1^3) & ]/21924
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fricas [C] time = 2.23, size = 1201, normalized size = 3.36 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(8*x^4-x^3+8*x+8)^2,x, algorithm="fricas")
[Out]
1/213627456*(3819648*x^3 - 15138*(8*x^4 - x^3 + 8*x + 8)*(-17*I*sqrt(7) + 7056*sqrt(4550065/334540596096*I*sqr
t(7) - 180983329/4683568345344))*log(6217850567873065654359973859328*(17/14112*I*sqrt(7) - 1/2*sqrt(4550065/33
4540596096*I*sqrt(7) - 180983329/4683568345344))^3 - 10028767243179717478632775680*(17/14112*I*sqrt(7) - 1/2*s
qrt(4550065/334540596096*I*sqrt(7) - 180983329/4683568345344))^2 + 67481665655469287031416*x + 320944207138750
561964778*I*sqrt(7) - 133210725033589645013145504*sqrt(4550065/334540596096*I*sqrt(7) - 180983329/468356834534
4) + 333979081113202533090737) - 15138*(8*x^4 - x^3 + 8*x + 8)*(17*I*sqrt(7) + 7056*sqrt(-4550065/334540596096
*I*sqrt(7) - 180983329/4683568345344))*log(-777231320984133206794996732416*(17/14112*I*sqrt(7) - 1/2*sqrt(4550
065/334540596096*I*sqrt(7) - 180983329/4683568345344))^3 + 878169064752*(-17/14112*I*sqrt(7) - 1/2*sqrt(-45500
65/334540596096*I*sqrt(7) - 180983329/4683568345344))^2*(-1066184864424603*I*sqrt(7) + 442529435492941104*sqrt
(4550065/334540596096*I*sqrt(7) - 180983329/4683568345344) - 1427510892508480) + 7569*(7276511507810430573072*
(17/14112*I*sqrt(7) - 1/2*sqrt(4550065/334540596096*I*sqrt(7) - 180983329/4683568345344))^2 - 2335942355437154
3)*(17*I*sqrt(7) + 7056*sqrt(-4550065/334540596096*I*sqrt(7) - 180983329/4683568345344)) + 8435208206933660878
927*x - 148449195141328682772633/4*I*sqrt(7) + 15403787072311988024172036*sqrt(4550065/334540596096*I*sqrt(7)
- 180983329/4683568345344) - 47393606606696595067616) - 5583312*x^2 + (56*sqrt(87)*(8*x^4 - x^3 + 8*x + 8)*sqr
t(-125452723536*(17/14112*I*sqrt(7) - 1/2*sqrt(4550065/334540596096*I*sqrt(7) - 180983329/4683568345344))^2 -
125452723536*(-17/14112*I*sqrt(7) - 1/2*sqrt(-4550065/334540596096*I*sqrt(7) - 180983329/4683568345344))^2 - 6
58503/1568*(17*I*sqrt(7) + 7056*sqrt(-4550065/334540596096*I*sqrt(7) - 180983329/4683568345344))*(-17*I*sqrt(7
) + 7056*sqrt(4550065/334540596096*I*sqrt(7) - 180983329/4683568345344)) - 6630191) + 7569*(8*x^4 - x^3 + 8*x
+ 8)*(17*I*sqrt(7) + 7056*sqrt(-4550065/334540596096*I*sqrt(7) - 180983329/4683568345344)) + 7569*(8*x^4 - x^3
+ 8*x + 8)*(-17*I*sqrt(7) + 7056*sqrt(4550065/334540596096*I*sqrt(7) - 180983329/4683568345344)))*log(-439084
532376*(-17/14112*I*sqrt(7) - 1/2*sqrt(-4550065/334540596096*I*sqrt(7) - 180983329/4683568345344))^2*(-1066184
864424603*I*sqrt(7) + 442529435492941104*sqrt(4550065/334540596096*I*sqrt(7) - 180983329/4683568345344) - 1427
510892508480) - 7569/2*(7276511507810430573072*(17/14112*I*sqrt(7) - 1/2*sqrt(4550065/334540596096*I*sqrt(7) -
180983329/4683568345344))^2 - 23359423554371543)*(17*I*sqrt(7) + 7056*sqrt(-4550065/334540596096*I*sqrt(7) -
180983329/4683568345344)) + 626797952698732342414548480*(17/14112*I*sqrt(7) - 1/2*sqrt(4550065/334540596096*I*
sqrt(7) - 180983329/4683568345344))^2 + 1/16*(261*(62716756730859*sqrt(87)*(-17*I*sqrt(7) + 7056*sqrt(4550065/
334540596096*I*sqrt(7) - 180983329/4683568345344)) - 1427510892508480*sqrt(87))*(17*I*sqrt(7) + 7056*sqrt(-455
0065/334540596096*I*sqrt(7) - 180983329/4683568345344)) - 372580342944713280*sqrt(87)*(-17*I*sqrt(7) + 7056*sq
rt(4550065/334540596096*I*sqrt(7) - 180983329/4683568345344)) + 10465021752358451264*sqrt(87))*sqrt(-125452723
536*(17/14112*I*sqrt(7) - 1/2*sqrt(4550065/334540596096*I*sqrt(7) - 180983329/4683568345344))^2 - 125452723536
*(-17/14112*I*sqrt(7) - 1/2*sqrt(-4550065/334540596096*I*sqrt(7) - 180983329/4683568345344))^2 - 658503/1568*(
17*I*sqrt(7) + 7056*sqrt(-4550065/334540596096*I*sqrt(7) - 180983329/4683568345344))*(-17*I*sqrt(7) + 7056*sqr
t(4550065/334540596096*I*sqrt(7) - 180983329/4683568345344)) - 6630191) + 8435208206933660878927*x - 300572710
7011649552439/2*I*sqrt(7) + 623776778443358801235576*sqrt(4550065/334540596096*I*sqrt(7) - 180983329/468356834
5344) + 2295910220839785410704) - (56*sqrt(87)*(8*x^4 - x^3 + 8*x + 8)*sqrt(-125452723536*(17/14112*I*sqrt(7)
- 1/2*sqrt(4550065/334540596096*I*sqrt(7) - 180983329/4683568345344))^2 - 125452723536*(-17/14112*I*sqrt(7) -
1/2*sqrt(-4550065/334540596096*I*sqrt(7) - 180983329/4683568345344))^2 - 658503/1568*(17*I*sqrt(7) + 7056*sqrt
(-4550065/334540596096*I*sqrt(7) - 180983329/4683568345344))*(-17*I*sqrt(7) + 7056*sqrt(4550065/334540596096*I
*sqrt(7) - 180983329/4683568345344)) - 6630191) - 7569*(8*x^4 - x^3 + 8*x + 8)*(17*I*sqrt(7) + 7056*sqrt(-4550
065/334540596096*I*sqrt(7) - 180983329/4683568345344)) - 7569*(8*x^4 - x^3 + 8*x + 8)*(-17*I*sqrt(7) + 7056*sq
rt(4550065/334540596096*I*sqrt(7) - 180983329/4683568345344)))*log(-439084532376*(-17/14112*I*sqrt(7) - 1/2*sq
rt(-4550065/334540596096*I*sqrt(7) - 180983329/4683568345344))^2*(-1066184864424603*I*sqrt(7) + 44252943549294
1104*sqrt(4550065/334540596096*I*sqrt(7) - 180983329/4683568345344) - 1427510892508480) - 7569/2*(727651150781
0430573072*(17/14112*I*sqrt(7) - 1/2*sqrt(4550065/334540596096*I*sqrt(7) - 180983329/4683568345344))^2 - 23359
423554371543)*(17*I*sqrt(7) + 7056*sqrt(-4550065/334540596096*I*sqrt(7) - 180983329/4683568345344)) + 62679795
2698732342414548480*(17/14112*I*sqrt(7) - 1/2*sqrt(4550065/334540596096*I*sqrt(7) - 180983329/4683568345344))^
2 - 1/16*(261*(62716756730859*sqrt(87)*(-17*I*sqrt(7) + 7056*sqrt(4550065/334540596096*I*sqrt(7) - 180983329/4
683568345344)) - 1427510892508480*sqrt(87))*(17*I*sqrt(7) + 7056*sqrt(-4550065/334540596096*I*sqrt(7) - 180983
329/4683568345344)) - 372580342944713280*sqrt(87)*(-17*I*sqrt(7) + 7056*sqrt(4550065/334540596096*I*sqrt(7) -
180983329/4683568345344)) + 10465021752358451264*sqrt(87))*sqrt(-125452723536*(17/14112*I*sqrt(7) - 1/2*sqrt(4
550065/334540596096*I*sqrt(7) - 180983329/4683568345344))^2 - 125452723536*(-17/14112*I*sqrt(7) - 1/2*sqrt(-45
50065/334540596096*I*sqrt(7) - 180983329/4683568345344))^2 - 658503/1568*(17*I*sqrt(7) + 7056*sqrt(-4550065/33
4540596096*I*sqrt(7) - 180983329/4683568345344))*(-17*I*sqrt(7) + 7056*sqrt(4550065/334540596096*I*sqrt(7) - 1
80983329/4683568345344)) - 6630191) + 8435208206933660878927*x - 3005727107011649552439/2*I*sqrt(7) + 62377677
8443358801235576*sqrt(4550065/334540596096*I*sqrt(7) - 180983329/4683568345344) + 2295910220839785410704) + 74
98008*x + 2650368)/(8*x^4 - x^3 + 8*x + 8)
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (8 \, x^{4} - x^{3} + 8 \, x + 8\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(8*x^4-x^3+8*x+8)^2,x, algorithm="giac")
[Out]
integrate((8*x^4 - x^3 + 8*x + 8)^(-2), x)
________________________________________________________________________________________
maple [C] time = 0.01, size = 83, normalized size = 0.23 \[ \frac {\left (392 \RootOf \left (8 \textit {\_Z}^{4}-\textit {\_Z}^{3}+8 \textit {\_Z} +8\right )^{2}-1097 \RootOf \left (8 \textit {\_Z}^{4}-\textit {\_Z}^{3}+8 \textit {\_Z} +8\right )+2243\right ) \ln \left (-\RootOf \left (8 \textit {\_Z}^{4}-\textit {\_Z}^{3}+8 \textit {\_Z} +8\right )+x \right )}{701568 \RootOf \left (8 \textit {\_Z}^{4}-\textit {\_Z}^{3}+8 \textit {\_Z} +8\right )^{3}-65772 \RootOf \left (8 \textit {\_Z}^{4}-\textit {\_Z}^{3}+8 \textit {\_Z} +8\right )^{2}+175392}+\frac {\frac {7}{3132} x^{3}-\frac {191}{58464} x^{2}+\frac {57}{12992} x +\frac {17}{10962}}{x^{4}-\frac {1}{8} x^{3}+x +1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(8*x^4-x^3+8*x+8)^2,x)
[Out]
(7/3132*x^3-191/58464*x^2+57/12992*x+17/10962)/(x^4-1/8*x^3+x+1)+1/21924*sum((392*_R^2-1097*_R+2243)/(32*_R^3-
3*_R^2+8)*ln(-_R+x),_R=RootOf(8*_Z^4-_Z^3+8*_Z+8))
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {784 \, x^{3} - 1146 \, x^{2} + 1539 \, x + 544}{43848 \, {\left (8 \, x^{4} - x^{3} + 8 \, x + 8\right )}} + \frac {1}{21924} \, \int \frac {392 \, x^{2} - 1097 \, x + 2243}{8 \, x^{4} - x^{3} + 8 \, x + 8}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(8*x^4-x^3+8*x+8)^2,x, algorithm="maxima")
[Out]
1/43848*(784*x^3 - 1146*x^2 + 1539*x + 544)/(8*x^4 - x^3 + 8*x + 8) + 1/21924*integrate((392*x^2 - 1097*x + 22
43)/(8*x^4 - x^3 + 8*x + 8), x)
________________________________________________________________________________________
mupad [B] time = 0.21, size = 176, normalized size = 0.49 \[ \left (\sum _{k=1}^4\ln \left (\frac {2615257\,\mathrm {root}\left (z^4+\frac {6630191\,z^2}{167270298048}+\frac {77351105\,z}{674433841729536}+\frac {1114096}{13723971258377709},z,k\right )}{72918171648}+\frac {4225\,x}{40375589184}-\frac {\mathrm {root}\left (z^4+\frac {6630191\,z^2}{167270298048}+\frac {77351105\,z}{674433841729536}+\frac {1114096}{13723971258377709},z,k\right )\,x\,34885379}{72918171648}-\frac {{\mathrm {root}\left (z^4+\frac {6630191\,z^2}{167270298048}+\frac {77351105\,z}{674433841729536}+\frac {1114096}{13723971258377709},z,k\right )}^2\,x\,191555}{475136}-\frac {{\mathrm {root}\left (z^4+\frac {6630191\,z^2}{167270298048}+\frac {77351105\,z}{674433841729536}+\frac {1114096}{13723971258377709},z,k\right )}^3\,x\,9261}{256}-\frac {11205\,{\mathrm {root}\left (z^4+\frac {6630191\,z^2}{167270298048}+\frac {77351105\,z}{674433841729536}+\frac {1114096}{13723971258377709},z,k\right )}^2}{59392}-\frac {24759\,{\mathrm {root}\left (z^4+\frac {6630191\,z^2}{167270298048}+\frac {77351105\,z}{674433841729536}+\frac {1114096}{13723971258377709},z,k\right )}^3}{512}+\frac {10901}{107668237824}\right )\,\mathrm {root}\left (z^4+\frac {6630191\,z^2}{167270298048}+\frac {77351105\,z}{674433841729536}+\frac {1114096}{13723971258377709},z,k\right )\right )+\frac {\frac {7\,x^3}{3132}-\frac {191\,x^2}{58464}+\frac {57\,x}{12992}+\frac {17}{10962}}{x^4-\frac {x^3}{8}+x+1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(8*x - x^3 + 8*x^4 + 8)^2,x)
[Out]
symsum(log((2615257*root(z^4 + (6630191*z^2)/167270298048 + (77351105*z)/674433841729536 + 1114096/13723971258
377709, z, k))/72918171648 + (4225*x)/40375589184 - (34885379*root(z^4 + (6630191*z^2)/167270298048 + (7735110
5*z)/674433841729536 + 1114096/13723971258377709, z, k)*x)/72918171648 - (191555*root(z^4 + (6630191*z^2)/1672
70298048 + (77351105*z)/674433841729536 + 1114096/13723971258377709, z, k)^2*x)/475136 - (9261*root(z^4 + (663
0191*z^2)/167270298048 + (77351105*z)/674433841729536 + 1114096/13723971258377709, z, k)^3*x)/256 - (11205*roo
t(z^4 + (6630191*z^2)/167270298048 + (77351105*z)/674433841729536 + 1114096/13723971258377709, z, k)^2)/59392
- (24759*root(z^4 + (6630191*z^2)/167270298048 + (77351105*z)/674433841729536 + 1114096/13723971258377709, z,
k)^3)/512 + 10901/107668237824)*root(z^4 + (6630191*z^2)/167270298048 + (77351105*z)/674433841729536 + 1114096
/13723971258377709, z, k), k, 1, 4) + ((57*x)/12992 - (191*x^2)/58464 + (7*x^3)/3132 + 17/10962)/(x - x^3/8 +
x^4 + 1)
________________________________________________________________________________________
sympy [B] time = 3.22, size = 3834, normalized size = 10.74 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(8*x**4-x**3+8*x+8)**2,x)
[Out]
(784*x**3 - 1146*x**2 + 1539*x + 544)/(350784*x**4 - 43848*x**3 + 350784*x + 350784) - sqrt(-180983329/3746854
6762752 + 1583563*sqrt(29)/1292018853888)*log(x**2 + x*(-62716756730859*sqrt(1218)*sqrt(-180983329 + 45923327*
sqrt(29))*sqrt(214095423017213*sqrt(29) + 47106822945*sqrt(1218)*sqrt(-180983329 + 45923327*sqrt(29)) + 406998
73480352667)/227008323264998681573683424 - 267658292345340*sqrt(214095423017213*sqrt(29) + 47106822945*sqrt(12
18)*sqrt(-180983329 + 45923327*sqrt(29)) + 40699873480352667)/8435208206933660878927 - 2157374520970352866823*
sqrt(1218)*sqrt(-180983329 + 45923327*sqrt(29))/113504161632499340786841712 + 3881045239007430*sqrt(29)/532672
7264361229 + 435853770857118353330297/33740832827734643515708 + 20905585576953*sqrt(42)*sqrt(-180983329 + 4592
3327*sqrt(29))/85227636229779664) - 2942814074101429415084030510182204250067556953*sqrt(214095423017213*sqrt(2
9) + 47106822945*sqrt(1218)*sqrt(-180983329 + 45923327*sqrt(29)) + 40699873480352667)/888496186751485201253966
401139075287452416534006272 - 14257625632856314835831142972765102609010539559351093/27765505835983912539186450
035596102732888016687696 - 75184631502818837388875900060881355871*sqrt(1218)*sqrt(-180983329 + 45923327*sqrt(2
9))*sqrt(214095423017213*sqrt(29) + 47106822945*sqrt(1218)*sqrt(-180983329 + 45923327*sqrt(29)) + 406998734803
52667)/30637799543154662112205737970312940946635052896768 - 9633141817961412597488587661065704878094062299*sqr
t(1218)*sqrt(-180983329 + 45923327*sqrt(29))/30637799543154662112205737970312940946635052896768 - 139888833400
1652366855237255*sqrt(42)*sqrt(-180983329 + 45923327*sqrt(29))*sqrt(214095423017213*sqrt(29) + 47106822945*sqr
t(1218)*sqrt(-180983329 + 45923327*sqrt(29)) + 40699873480352667)/359456428291497016547944746810895370264 + 91
245981690030498967778233214015591679*sqrt(42)*sqrt(-180983329 + 45923327*sqrt(29))/230052114106558090590684637
95897303696896 + 10304175351841941260676745569701505519*sqrt(29)*sqrt(214095423017213*sqrt(29) + 47106822945*s
qrt(1218)*sqrt(-180983329 + 45923327*sqrt(29)) + 40699873480352667)/193473827963615354186765780523496324090895
36 + 639111088489748962499984017403917984374085485*sqrt(29)/4836845699090383854669144513087408102272384) + sqr
t(-180983329/37468546762752 + 1583563*sqrt(29)/1292018853888)*log(x**2 + x*(-62716756730859*sqrt(1218)*sqrt(-1
80983329 + 45923327*sqrt(29))*sqrt(-47106822945*sqrt(1218)*sqrt(-180983329 + 45923327*sqrt(29)) + 214095423017
213*sqrt(29) + 40699873480352667)/227008323264998681573683424 - 20905585576953*sqrt(42)*sqrt(-180983329 + 4592
3327*sqrt(29))/85227636229779664 + 3881045239007430*sqrt(29)/5326727264361229 + 267658292345340*sqrt(-47106822
945*sqrt(1218)*sqrt(-180983329 + 45923327*sqrt(29)) + 214095423017213*sqrt(29) + 40699873480352667)/8435208206
933660878927 + 2157374520970352866823*sqrt(1218)*sqrt(-180983329 + 45923327*sqrt(29))/113504161632499340786841
712 + 435853770857118353330297/33740832827734643515708) - 1425762563285631483583114297276510260901053955935109
3/27765505835983912539186450035596102732888016687696 - 10304175351841941260676745569701505519*sqrt(29)*sqrt(-4
7106822945*sqrt(1218)*sqrt(-180983329 + 45923327*sqrt(29)) + 214095423017213*sqrt(29) + 40699873480352667)/193
47382796361535418676578052349632409089536 - 91245981690030498967778233214015591679*sqrt(42)*sqrt(-180983329 +
45923327*sqrt(29))/23005211410655809059068463795897303696896 - 75184631502818837388875900060881355871*sqrt(121
8)*sqrt(-180983329 + 45923327*sqrt(29))*sqrt(-47106822945*sqrt(1218)*sqrt(-180983329 + 45923327*sqrt(29)) + 21
4095423017213*sqrt(29) + 40699873480352667)/30637799543154662112205737970312940946635052896768 - 1398888334001
652366855237255*sqrt(42)*sqrt(-180983329 + 45923327*sqrt(29))*sqrt(-47106822945*sqrt(1218)*sqrt(-180983329 + 4
5923327*sqrt(29)) + 214095423017213*sqrt(29) + 40699873480352667)/359456428291497016547944746810895370264 + 96
33141817961412597488587661065704878094062299*sqrt(1218)*sqrt(-180983329 + 45923327*sqrt(29))/30637799543154662
112205737970312940946635052896768 + 2942814074101429415084030510182204250067556953*sqrt(-47106822945*sqrt(1218
)*sqrt(-180983329 + 45923327*sqrt(29)) + 214095423017213*sqrt(29) + 40699873480352667)/88849618675148520125396
6401139075287452416534006272 + 639111088489748962499984017403917984374085485*sqrt(29)/483684569909038385466914
4513087408102272384) - 2*sqrt(199631405/37468546762752 + sqrt(-47106822945*sqrt(1218)*sqrt(-180983329 + 459233
27*sqrt(29)) + 214095423017213*sqrt(29) + 40699873480352667)/9367136690688 + 1583563*sqrt(29)/430672951296)*at
an(454016646529997363147366848*x/(-4509673516272467429860*sqrt(1218)*sqrt(199631405 + 4*sqrt(-47106822945*sqrt
(1218)*sqrt(-180983329 + 45923327*sqrt(29)) + 214095423017213*sqrt(29) + 40699873480352667) + 137769981*sqrt(2
9)) + 3601609981798895040*sqrt(-180983329 + 45923327*sqrt(29))*sqrt(199631405 + 4*sqrt(-47106822945*sqrt(1218)
*sqrt(-180983329 + 45923327*sqrt(29)) + 214095423017213*sqrt(29) + 40699873480352667) + 137769981*sqrt(29)) +
20905585576953*sqrt(1218)*sqrt(199631405 + 4*sqrt(-47106822945*sqrt(1218)*sqrt(-180983329 + 45923327*sqrt(29))
+ 214095423017213*sqrt(29) + 40699873480352667) + 137769981*sqrt(29))*sqrt(-47106822945*sqrt(1218)*sqrt(-1809
83329 + 45923327*sqrt(29)) + 214095423017213*sqrt(29) + 40699873480352667)) + 2932424170326692281206238216/(-4
509673516272467429860*sqrt(1218)*sqrt(199631405 + 4*sqrt(-47106822945*sqrt(1218)*sqrt(-180983329 + 45923327*sq
rt(29)) + 214095423017213*sqrt(29) + 40699873480352667) + 137769981*sqrt(29)) + 3601609981798895040*sqrt(-1809
83329 + 45923327*sqrt(29))*sqrt(199631405 + 4*sqrt(-47106822945*sqrt(1218)*sqrt(-180983329 + 45923327*sqrt(29)
) + 214095423017213*sqrt(29) + 40699873480352667) + 137769981*sqrt(29)) + 20905585576953*sqrt(1218)*sqrt(19963
1405 + 4*sqrt(-47106822945*sqrt(1218)*sqrt(-180983329 + 45923327*sqrt(29)) + 214095423017213*sqrt(29) + 406998
73480352667) + 137769981*sqrt(29))*sqrt(-47106822945*sqrt(1218)*sqrt(-180983329 + 45923327*sqrt(29)) + 2140954
23017213*sqrt(29) + 40699873480352667)) + 4314749041940705733646*sqrt(1218)*sqrt(-180983329 + 45923327*sqrt(29
))/(-4509673516272467429860*sqrt(1218)*sqrt(199631405 + 4*sqrt(-47106822945*sqrt(1218)*sqrt(-180983329 + 45923
327*sqrt(29)) + 214095423017213*sqrt(29) + 40699873480352667) + 137769981*sqrt(29)) + 3601609981798895040*sqrt
(-180983329 + 45923327*sqrt(29))*sqrt(199631405 + 4*sqrt(-47106822945*sqrt(1218)*sqrt(-180983329 + 45923327*sq
rt(29)) + 214095423017213*sqrt(29) + 40699873480352667) + 137769981*sqrt(29)) + 20905585576953*sqrt(1218)*sqrt
(199631405 + 4*sqrt(-47106822945*sqrt(1218)*sqrt(-180983329 + 45923327*sqrt(29)) + 214095423017213*sqrt(29) +
40699873480352667) + 137769981*sqrt(29))*sqrt(-47106822945*sqrt(1218)*sqrt(-180983329 + 45923327*sqrt(29)) + 2
14095423017213*sqrt(29) + 40699873480352667)) + 7203219963597790080*sqrt(-47106822945*sqrt(1218)*sqrt(-1809833
29 + 45923327*sqrt(29)) + 214095423017213*sqrt(29) + 40699873480352667)/(-4509673516272467429860*sqrt(1218)*sq
rt(199631405 + 4*sqrt(-47106822945*sqrt(1218)*sqrt(-180983329 + 45923327*sqrt(29)) + 214095423017213*sqrt(29)
+ 40699873480352667) + 137769981*sqrt(29)) + 3601609981798895040*sqrt(-180983329 + 45923327*sqrt(29))*sqrt(199
631405 + 4*sqrt(-47106822945*sqrt(1218)*sqrt(-180983329 + 45923327*sqrt(29)) + 214095423017213*sqrt(29) + 4069
9873480352667) + 137769981*sqrt(29)) + 20905585576953*sqrt(1218)*sqrt(199631405 + 4*sqrt(-47106822945*sqrt(121
8)*sqrt(-180983329 + 45923327*sqrt(29)) + 214095423017213*sqrt(29) + 40699873480352667) + 137769981*sqrt(29))*
sqrt(-47106822945*sqrt(1218)*sqrt(-180983329 + 45923327*sqrt(29)) + 214095423017213*sqrt(29) + 406998734803526
67)) + 165397912920614705160598080*sqrt(29)/(-4509673516272467429860*sqrt(1218)*sqrt(199631405 + 4*sqrt(-47106
822945*sqrt(1218)*sqrt(-180983329 + 45923327*sqrt(29)) + 214095423017213*sqrt(29) + 40699873480352667) + 13776
9981*sqrt(29)) + 3601609981798895040*sqrt(-180983329 + 45923327*sqrt(29))*sqrt(199631405 + 4*sqrt(-47106822945
*sqrt(1218)*sqrt(-180983329 + 45923327*sqrt(29)) + 214095423017213*sqrt(29) + 40699873480352667) + 137769981*s
qrt(29)) + 20905585576953*sqrt(1218)*sqrt(199631405 + 4*sqrt(-47106822945*sqrt(1218)*sqrt(-180983329 + 4592332
7*sqrt(29)) + 214095423017213*sqrt(29) + 40699873480352667) + 137769981*sqrt(29))*sqrt(-47106822945*sqrt(1218)
*sqrt(-180983329 + 45923327*sqrt(29)) + 214095423017213*sqrt(29) + 40699873480352667)) - 556831344694599843925
98*sqrt(42)*sqrt(-180983329 + 45923327*sqrt(29))/(-4509673516272467429860*sqrt(1218)*sqrt(199631405 + 4*sqrt(-
47106822945*sqrt(1218)*sqrt(-180983329 + 45923327*sqrt(29)) + 214095423017213*sqrt(29) + 40699873480352667) +
137769981*sqrt(29)) + 3601609981798895040*sqrt(-180983329 + 45923327*sqrt(29))*sqrt(199631405 + 4*sqrt(-471068
22945*sqrt(1218)*sqrt(-180983329 + 45923327*sqrt(29)) + 214095423017213*sqrt(29) + 40699873480352667) + 137769
981*sqrt(29)) + 20905585576953*sqrt(1218)*sqrt(199631405 + 4*sqrt(-47106822945*sqrt(1218)*sqrt(-180983329 + 45
923327*sqrt(29)) + 214095423017213*sqrt(29) + 40699873480352667) + 137769981*sqrt(29))*sqrt(-47106822945*sqrt(
1218)*sqrt(-180983329 + 45923327*sqrt(29)) + 214095423017213*sqrt(29) + 40699873480352667)) - 62716756730859*s
qrt(1218)*sqrt(-180983329 + 45923327*sqrt(29))*sqrt(-47106822945*sqrt(1218)*sqrt(-180983329 + 45923327*sqrt(29
)) + 214095423017213*sqrt(29) + 40699873480352667)/(-4509673516272467429860*sqrt(1218)*sqrt(199631405 + 4*sqrt
(-47106822945*sqrt(1218)*sqrt(-180983329 + 45923327*sqrt(29)) + 214095423017213*sqrt(29) + 40699873480352667)
+ 137769981*sqrt(29)) + 3601609981798895040*sqrt(-180983329 + 45923327*sqrt(29))*sqrt(199631405 + 4*sqrt(-4710
6822945*sqrt(1218)*sqrt(-180983329 + 45923327*sqrt(29)) + 214095423017213*sqrt(29) + 40699873480352667) + 1377
69981*sqrt(29)) + 20905585576953*sqrt(1218)*sqrt(199631405 + 4*sqrt(-47106822945*sqrt(1218)*sqrt(-180983329 +
45923327*sqrt(29)) + 214095423017213*sqrt(29) + 40699873480352667) + 137769981*sqrt(29))*sqrt(-47106822945*sqr
t(1218)*sqrt(-180983329 + 45923327*sqrt(29)) + 214095423017213*sqrt(29) + 40699873480352667))) + 2*sqrt(-sqrt(
214095423017213*sqrt(29) + 47106822945*sqrt(1218)*sqrt(-180983329 + 45923327*sqrt(29)) + 40699873480352667)/93
67136690688 + 199631405/37468546762752 + 1583563*sqrt(29)/430672951296)*atan(454016646529997363147366848*x/(36
01609981798895040*sqrt(-180983329 + 45923327*sqrt(29))*sqrt(-4*sqrt(214095423017213*sqrt(29) + 47106822945*sqr
t(1218)*sqrt(-180983329 + 45923327*sqrt(29)) + 40699873480352667) + 199631405 + 137769981*sqrt(29)) + 45096735
16272467429860*sqrt(1218)*sqrt(-4*sqrt(214095423017213*sqrt(29) + 47106822945*sqrt(1218)*sqrt(-180983329 + 459
23327*sqrt(29)) + 40699873480352667) + 199631405 + 137769981*sqrt(29)) + 20905585576953*sqrt(1218)*sqrt(214095
423017213*sqrt(29) + 47106822945*sqrt(1218)*sqrt(-180983329 + 45923327*sqrt(29)) + 40699873480352667)*sqrt(-4*
sqrt(214095423017213*sqrt(29) + 47106822945*sqrt(1218)*sqrt(-180983329 + 45923327*sqrt(29)) + 4069987348035266
7) + 199631405 + 137769981*sqrt(29))) - 62716756730859*sqrt(1218)*sqrt(-180983329 + 45923327*sqrt(29))*sqrt(21
4095423017213*sqrt(29) + 47106822945*sqrt(1218)*sqrt(-180983329 + 45923327*sqrt(29)) + 40699873480352667)/(360
1609981798895040*sqrt(-180983329 + 45923327*sqrt(29))*sqrt(-4*sqrt(214095423017213*sqrt(29) + 47106822945*sqrt
(1218)*sqrt(-180983329 + 45923327*sqrt(29)) + 40699873480352667) + 199631405 + 137769981*sqrt(29)) + 450967351
6272467429860*sqrt(1218)*sqrt(-4*sqrt(214095423017213*sqrt(29) + 47106822945*sqrt(1218)*sqrt(-180983329 + 4592
3327*sqrt(29)) + 40699873480352667) + 199631405 + 137769981*sqrt(29)) + 20905585576953*sqrt(1218)*sqrt(2140954
23017213*sqrt(29) + 47106822945*sqrt(1218)*sqrt(-180983329 + 45923327*sqrt(29)) + 40699873480352667)*sqrt(-4*s
qrt(214095423017213*sqrt(29) + 47106822945*sqrt(1218)*sqrt(-180983329 + 45923327*sqrt(29)) + 40699873480352667
) + 199631405 + 137769981*sqrt(29))) - 7203219963597790080*sqrt(214095423017213*sqrt(29) + 47106822945*sqrt(12
18)*sqrt(-180983329 + 45923327*sqrt(29)) + 40699873480352667)/(3601609981798895040*sqrt(-180983329 + 45923327*
sqrt(29))*sqrt(-4*sqrt(214095423017213*sqrt(29) + 47106822945*sqrt(1218)*sqrt(-180983329 + 45923327*sqrt(29))
+ 40699873480352667) + 199631405 + 137769981*sqrt(29)) + 4509673516272467429860*sqrt(1218)*sqrt(-4*sqrt(214095
423017213*sqrt(29) + 47106822945*sqrt(1218)*sqrt(-180983329 + 45923327*sqrt(29)) + 40699873480352667) + 199631
405 + 137769981*sqrt(29)) + 20905585576953*sqrt(1218)*sqrt(214095423017213*sqrt(29) + 47106822945*sqrt(1218)*s
qrt(-180983329 + 45923327*sqrt(29)) + 40699873480352667)*sqrt(-4*sqrt(214095423017213*sqrt(29) + 47106822945*s
qrt(1218)*sqrt(-180983329 + 45923327*sqrt(29)) + 40699873480352667) + 199631405 + 137769981*sqrt(29))) - 43147
49041940705733646*sqrt(1218)*sqrt(-180983329 + 45923327*sqrt(29))/(3601609981798895040*sqrt(-180983329 + 45923
327*sqrt(29))*sqrt(-4*sqrt(214095423017213*sqrt(29) + 47106822945*sqrt(1218)*sqrt(-180983329 + 45923327*sqrt(2
9)) + 40699873480352667) + 199631405 + 137769981*sqrt(29)) + 4509673516272467429860*sqrt(1218)*sqrt(-4*sqrt(21
4095423017213*sqrt(29) + 47106822945*sqrt(1218)*sqrt(-180983329 + 45923327*sqrt(29)) + 40699873480352667) + 19
9631405 + 137769981*sqrt(29)) + 20905585576953*sqrt(1218)*sqrt(214095423017213*sqrt(29) + 47106822945*sqrt(121
8)*sqrt(-180983329 + 45923327*sqrt(29)) + 40699873480352667)*sqrt(-4*sqrt(214095423017213*sqrt(29) + 471068229
45*sqrt(1218)*sqrt(-180983329 + 45923327*sqrt(29)) + 40699873480352667) + 199631405 + 137769981*sqrt(29))) + 1
65397912920614705160598080*sqrt(29)/(3601609981798895040*sqrt(-180983329 + 45923327*sqrt(29))*sqrt(-4*sqrt(214
095423017213*sqrt(29) + 47106822945*sqrt(1218)*sqrt(-180983329 + 45923327*sqrt(29)) + 40699873480352667) + 199
631405 + 137769981*sqrt(29)) + 4509673516272467429860*sqrt(1218)*sqrt(-4*sqrt(214095423017213*sqrt(29) + 47106
822945*sqrt(1218)*sqrt(-180983329 + 45923327*sqrt(29)) + 40699873480352667) + 199631405 + 137769981*sqrt(29))
+ 20905585576953*sqrt(1218)*sqrt(214095423017213*sqrt(29) + 47106822945*sqrt(1218)*sqrt(-180983329 + 45923327*
sqrt(29)) + 40699873480352667)*sqrt(-4*sqrt(214095423017213*sqrt(29) + 47106822945*sqrt(1218)*sqrt(-180983329
+ 45923327*sqrt(29)) + 40699873480352667) + 199631405 + 137769981*sqrt(29))) + 2932424170326692281206238216/(3
601609981798895040*sqrt(-180983329 + 45923327*sqrt(29))*sqrt(-4*sqrt(214095423017213*sqrt(29) + 47106822945*sq
rt(1218)*sqrt(-180983329 + 45923327*sqrt(29)) + 40699873480352667) + 199631405 + 137769981*sqrt(29)) + 4509673
516272467429860*sqrt(1218)*sqrt(-4*sqrt(214095423017213*sqrt(29) + 47106822945*sqrt(1218)*sqrt(-180983329 + 45
923327*sqrt(29)) + 40699873480352667) + 199631405 + 137769981*sqrt(29)) + 20905585576953*sqrt(1218)*sqrt(21409
5423017213*sqrt(29) + 47106822945*sqrt(1218)*sqrt(-180983329 + 45923327*sqrt(29)) + 40699873480352667)*sqrt(-4
*sqrt(214095423017213*sqrt(29) + 47106822945*sqrt(1218)*sqrt(-180983329 + 45923327*sqrt(29)) + 406998734803526
67) + 199631405 + 137769981*sqrt(29))) + 55683134469459984392598*sqrt(42)*sqrt(-180983329 + 45923327*sqrt(29))
/(3601609981798895040*sqrt(-180983329 + 45923327*sqrt(29))*sqrt(-4*sqrt(214095423017213*sqrt(29) + 47106822945
*sqrt(1218)*sqrt(-180983329 + 45923327*sqrt(29)) + 40699873480352667) + 199631405 + 137769981*sqrt(29)) + 4509
673516272467429860*sqrt(1218)*sqrt(-4*sqrt(214095423017213*sqrt(29) + 47106822945*sqrt(1218)*sqrt(-180983329 +
45923327*sqrt(29)) + 40699873480352667) + 199631405 + 137769981*sqrt(29)) + 20905585576953*sqrt(1218)*sqrt(21
4095423017213*sqrt(29) + 47106822945*sqrt(1218)*sqrt(-180983329 + 45923327*sqrt(29)) + 40699873480352667)*sqrt
(-4*sqrt(214095423017213*sqrt(29) + 47106822945*sqrt(1218)*sqrt(-180983329 + 45923327*sqrt(29)) + 406998734803
52667) + 199631405 + 137769981*sqrt(29))))
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