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]

-(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

________________________________________________________________________________________

Rubi [A]  time = 0.396672, antiderivative size = 357, normalized size of antiderivative = 1., 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 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 12

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

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
]

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*}

Mathematica [C]  time = 0.0157122, 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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Maple [C]  time = 0.007, size = 83, normalized size = 0.2 \begin{align*}{ \left ({\frac{7\,{x}^{3}}{3132}}-{\frac{191\,{x}^{2}}{58464}}+{\frac{57\,x}{12992}}+{\frac{17}{10962}} \right ) \left ({x}^{4}-{\frac{{x}^{3}}{8}}+x+1 \right ) ^{-1}}+{\frac{1}{21924}\sum _{{\it \_R}={\it RootOf} \left ( 8\,{{\it \_Z}}^{4}-{{\it \_Z}}^{3}+8\,{\it \_Z}+8 \right ) }{\frac{ \left ( 392\,{{\it \_R}}^{2}-1097\,{\it \_R}+2243 \right ) \ln \left ( x-{\it \_R} \right ) }{32\,{{\it \_R}}^{3}-3\,{{\it \_R}}^{2}+8}}} \end{align*}

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(x-_R),_R=RootOf(8*_Z^4-_Z^3+8*_Z+8))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

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)

________________________________________________________________________________________

Fricas [C]  time = 12.6073, size = 8227, normalized size = 23.04 \begin{align*} \text{result too large to display} \end{align*}

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)

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Sympy [A]  time = 0.916419, size = 71, normalized size = 0.2 \begin{align*} \frac{784 x^{3} - 1146 x^{2} + 1539 x + 544}{350784 x^{4} - 43848 x^{3} + 350784 x + 350784} + \operatorname{RootSum}{\left (56213386274315096064 t^{4} + 2228162991905088 t^{2} + 6447137250645 t + 4563337216, \left ( t \mapsto t \log{\left (\frac{777231320984133206794996732416 t^{3}}{8435208206933660878927} - \frac{1253595905397464684829096960 t^{2}}{8435208206933660878927} + \frac{900072466443173277115848 t}{227978600187396239971} + x + \frac{333979081113202533090737}{67481665655469287031416} \right )} \right )\right )} \end{align*}

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) + RootSum(562133862743150
96064*_t**4 + 2228162991905088*_t**2 + 6447137250645*_t + 4563337216, Lambda(_t, _t*log(7772313209841332067949
96732416*_t**3/8435208206933660878927 - 1253595905397464684829096960*_t**2/8435208206933660878927 + 9000724664
43173277115848*_t/227978600187396239971 + x + 333979081113202533090737/67481665655469287031416)))

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (8 \, x^{4} - x^{3} + 8 \, x + 8\right )}^{2}}\,{d x} \end{align*}

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)