3.49 \(\int \frac{1}{8+8 x-x^3+8 x^4} \, dx\)

Optimal. Leaf size=268 \[ -\frac{1}{24} \sqrt{\frac{67 \sqrt{29}-109}{1218}} \log \left (\left (\frac{4}{x}+1\right )^2-\sqrt{6 \left (1+\sqrt{29}\right )} \left (\frac{4}{x}+1\right )+3 \sqrt{29}\right )+\frac{1}{24} \sqrt{\frac{67 \sqrt{29}-109}{1218}} \log \left (\left (\frac{4}{x}+1\right )^2+\sqrt{6 \left (1+\sqrt{29}\right )} \left (\frac{4}{x}+1\right )+3 \sqrt{29}\right )-\frac{\tan ^{-1}\left (\frac{3-\left (\frac{4}{x}+1\right )^2}{6 \sqrt{7}}\right )}{12 \sqrt{7}}-\frac{1}{12} \sqrt{\frac{109+67 \sqrt{29}}{1218}} \tan ^{-1}\left (\frac{\frac{8}{x}-\sqrt{6 \left (1+\sqrt{29}\right )}+2}{\sqrt{6 \left (\sqrt{29}-1\right )}}\right )-\frac{1}{12} \sqrt{\frac{109+67 \sqrt{29}}{1218}} \tan ^{-1}\left (\frac{\frac{8}{x}+\sqrt{6 \left (1+\sqrt{29}\right )}+2}{\sqrt{6 \left (\sqrt{29}-1\right )}}\right ) \]

[Out]

-ArcTan[(3 - (1 + 4/x)^2)/(6*Sqrt[7])]/(12*Sqrt[7]) - (Sqrt[(109 + 67*Sqrt[29])/1218]*ArcTan[(2 - Sqrt[6*(1 +
Sqrt[29])] + 8/x)/Sqrt[6*(-1 + Sqrt[29])]])/12 - (Sqrt[(109 + 67*Sqrt[29])/1218]*ArcTan[(2 + Sqrt[6*(1 + Sqrt[
29])] + 8/x)/Sqrt[6*(-1 + Sqrt[29])]])/12 - (Sqrt[(-109 + 67*Sqrt[29])/1218]*Log[3*Sqrt[29] - Sqrt[6*(1 + Sqrt
[29])]*(1 + 4/x) + (1 + 4/x)^2])/24 + (Sqrt[(-109 + 67*Sqrt[29])/1218]*Log[3*Sqrt[29] + Sqrt[6*(1 + Sqrt[29])]
*(1 + 4/x) + (1 + 4/x)^2])/24

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Rubi [A]  time = 0.39625, antiderivative size = 268, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 9, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.529, Rules used = {2069, 12, 1673, 1169, 634, 618, 204, 628, 1107} \[ -\frac{1}{24} \sqrt{\frac{67 \sqrt{29}-109}{1218}} \log \left (\left (\frac{4}{x}+1\right )^2-\sqrt{6 \left (1+\sqrt{29}\right )} \left (\frac{4}{x}+1\right )+3 \sqrt{29}\right )+\frac{1}{24} \sqrt{\frac{67 \sqrt{29}-109}{1218}} \log \left (\left (\frac{4}{x}+1\right )^2+\sqrt{6 \left (1+\sqrt{29}\right )} \left (\frac{4}{x}+1\right )+3 \sqrt{29}\right )-\frac{\tan ^{-1}\left (\frac{3-\left (\frac{4}{x}+1\right )^2}{6 \sqrt{7}}\right )}{12 \sqrt{7}}-\frac{1}{12} \sqrt{\frac{109+67 \sqrt{29}}{1218}} \tan ^{-1}\left (\frac{\frac{8}{x}-\sqrt{6 \left (1+\sqrt{29}\right )}+2}{\sqrt{6 \left (\sqrt{29}-1\right )}}\right )-\frac{1}{12} \sqrt{\frac{109+67 \sqrt{29}}{1218}} \tan ^{-1}\left (\frac{\frac{8}{x}+\sqrt{6 \left (1+\sqrt{29}\right )}+2}{\sqrt{6 \left (\sqrt{29}-1\right )}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[(8 + 8*x - x^3 + 8*x^4)^(-1),x]

[Out]

-ArcTan[(3 - (1 + 4/x)^2)/(6*Sqrt[7])]/(12*Sqrt[7]) - (Sqrt[(109 + 67*Sqrt[29])/1218]*ArcTan[(2 - Sqrt[6*(1 +
Sqrt[29])] + 8/x)/Sqrt[6*(-1 + Sqrt[29])]])/12 - (Sqrt[(109 + 67*Sqrt[29])/1218]*ArcTan[(2 + Sqrt[6*(1 + Sqrt[
29])] + 8/x)/Sqrt[6*(-1 + Sqrt[29])]])/12 - (Sqrt[(-109 + 67*Sqrt[29])/1218]*Log[3*Sqrt[29] - Sqrt[6*(1 + Sqrt
[29])]*(1 + 4/x) + (1 + 4/x)^2])/24 + (Sqrt[(-109 + 67*Sqrt[29])/1218]*Log[3*Sqrt[29] + Sqrt[6*(1 + Sqrt[29])]
*(1 + 4/x) + (1 + 4/x)^2])/24

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 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 1107

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

Rubi steps

\begin{align*} \int \frac{1}{8+8 x-x^3+8 x^4} \, dx &=-\left (1024 \operatorname{Subst}\left (\int \frac{(8-32 x)^2}{8 \left (1069056-393216 x^2+1048576 x^4\right )} \, dx,x,\frac{1}{4}+\frac{1}{x}\right )\right )\\ &=-\left (128 \operatorname{Subst}\left (\int \frac{(8-32 x)^2}{1069056-393216 x^2+1048576 x^4} \, dx,x,\frac{1}{4}+\frac{1}{x}\right )\right )\\ &=-\left (128 \operatorname{Subst}\left (\int -\frac{512 x}{1069056-393216 x^2+1048576 x^4} \, dx,x,\frac{1}{4}+\frac{1}{x}\right )\right )-128 \operatorname{Subst}\left (\int \frac{64+1024 x^2}{1069056-393216 x^2+1048576 x^4} \, dx,x,\frac{1}{4}+\frac{1}{x}\right )\\ &=65536 \operatorname{Subst}\left (\int \frac{x}{1069056-393216 x^2+1048576 x^4} \, dx,x,\frac{1}{4}+\frac{1}{x}\right )-\frac{\operatorname{Subst}\left (\int \frac{16 \sqrt{6 \left (1+\sqrt{29}\right )}-\left (64-192 \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 )}{768 \sqrt{174 \left (1+\sqrt{29}\right )}}-\frac{\operatorname{Subst}\left (\int \frac{16 \sqrt{6 \left (1+\sqrt{29}\right )}+\left (64-192 \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 )}{768 \sqrt{174 \left (1+\sqrt{29}\right )}}\\ &=32768 \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{\left (87+\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 )}{2784}-\frac{\left (87+\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 )}{2784}-\frac{1}{24} \sqrt{\frac{-109+67 \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 )+\frac{1}{24} \sqrt{\frac{-109+67 \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 )\\ &=-\frac{1}{24} \sqrt{\frac{-109+67 \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 )+\frac{1}{24} \sqrt{\frac{-109+67 \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 )-65536 \operatorname{Subst}\left (\int \frac{1}{-4329327034368-x^2} \, dx,x,-393216+2097152 \left (\frac{1}{4}+\frac{1}{x}\right )^2\right )+\frac{\left (87+\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 )}{1392}+\frac{\left (87+\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 )}{1392}\\ &=-\frac{\tan ^{-1}\left (\frac{3-\left (1+\frac{4}{x}\right )^2}{6 \sqrt{7}}\right )}{12 \sqrt{7}}-\frac{1}{12} \sqrt{\frac{109+67 \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 )-\frac{1}{12} \sqrt{\frac{109+67 \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 )-\frac{1}{24} \sqrt{\frac{-109+67 \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 )+\frac{1}{24} \sqrt{\frac{-109+67 \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 )\\ \end{align*}

Mathematica [C]  time = 0.0084298, size = 45, normalized size = 0.17 \[ \text{RootSum}\left [8 \text{$\#$1}^4-\text{$\#$1}^3+8 \text{$\#$1}+8\& ,\frac{\log (x-\text{$\#$1})}{32 \text{$\#$1}^3-3 \text{$\#$1}^2+8}\& \right ] \]

Antiderivative was successfully verified.

[In]

Integrate[(8 + 8*x - x^3 + 8*x^4)^(-1),x]

[Out]

RootSum[8 + 8*#1 - #1^3 + 8*#1^4 & , Log[x - #1]/(8 - 3*#1^2 + 32*#1^3) & ]

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Maple [C]  time = 0.003, size = 41, normalized size = 0.2 \begin{align*} \sum _{{\it \_R}={\it RootOf} \left ( 8\,{{\it \_Z}}^{4}-{{\it \_Z}}^{3}+8\,{\it \_Z}+8 \right ) }{\frac{\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),x)

[Out]

sum(1/(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*} \int \frac{1}{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),x, algorithm="maxima")

[Out]

integrate(1/(8*x^4 - x^3 + 8*x + 8), x)

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Fricas [C]  time = 10.6498, size = 4797, normalized size = 17.9 \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),x, algorithm="fricas")

[Out]

-1/168*(-I*sqrt(7) + 84*sqrt(65/43848*I*sqrt(7) - 109/87696))*log(287314195392*(1/168*I*sqrt(7) - 1/2*sqrt(65/
43848*I*sqrt(7) - 109/87696))^3 - 12038906880*(1/168*I*sqrt(7) - 1/2*sqrt(65/43848*I*sqrt(7) - 109/87696))^2 +
16878104*x + 4897683*I*sqrt(7) - 411405372*sqrt(65/43848*I*sqrt(7) - 109/87696) + 6055613) - 1/168*(I*sqrt(7)
+ 84*sqrt(-65/43848*I*sqrt(7) - 109/87696))*log(-35914274424*(1/168*I*sqrt(7) - 1/2*sqrt(65/43848*I*sqrt(7) -
109/87696))^3 + 16443*(-1/168*I*sqrt(7) - 1/2*sqrt(-65/43848*I*sqrt(7) - 109/87696))^2*(-13001*I*sqrt(7) + 10
92084*sqrt(65/43848*I*sqrt(7) - 109/87696) - 91520) + 609*(351027*(1/168*I*sqrt(7) - 1/2*sqrt(65/43848*I*sqrt(
7) - 109/87696))^2 - 613)*(I*sqrt(7) + 84*sqrt(-65/43848*I*sqrt(7) - 109/87696)) + 2109763*x - 1911147/8*I*sqr
t(7) + 40134087/2*sqrt(65/43848*I*sqrt(7) - 109/87696) - 1461344) + 1/1044*(sqrt(174)*sqrt(-4698*(1/168*I*sqrt
(7) - 1/2*sqrt(65/43848*I*sqrt(7) - 109/87696))^2 - 4698*(-1/168*I*sqrt(7) - 1/2*sqrt(-65/43848*I*sqrt(7) - 10
9/87696))^2 - 87/784*(I*sqrt(7) + 84*sqrt(-65/43848*I*sqrt(7) - 109/87696))*(-I*sqrt(7) + 84*sqrt(65/43848*I*s
qrt(7) - 109/87696)) - 7) + 261*sqrt(65/43848*I*sqrt(7) - 109/87696) + 261*sqrt(-65/43848*I*sqrt(7) - 109/8769
6))*log(-16443/2*(-1/168*I*sqrt(7) - 1/2*sqrt(-65/43848*I*sqrt(7) - 109/87696))^2*(-13001*I*sqrt(7) + 1092084*
sqrt(65/43848*I*sqrt(7) - 109/87696) - 91520) - 609/2*(351027*(1/168*I*sqrt(7) - 1/2*sqrt(65/43848*I*sqrt(7) -
109/87696))^2 - 613)*(I*sqrt(7) + 84*sqrt(-65/43848*I*sqrt(7) - 109/87696)) + 752431680*(1/168*I*sqrt(7) - 1/
2*sqrt(65/43848*I*sqrt(7) - 109/87696))^2 + 1/32*(3*(13001*sqrt(174)*(-I*sqrt(7) + 84*sqrt(65/43848*I*sqrt(7)
- 109/87696)) - 91520*sqrt(174))*(I*sqrt(7) + 84*sqrt(-65/43848*I*sqrt(7) - 109/87696)) - 274560*sqrt(174)*(-I
*sqrt(7) + 84*sqrt(65/43848*I*sqrt(7) - 109/87696)) + 1922368*sqrt(174))*sqrt(-4698*(1/168*I*sqrt(7) - 1/2*sqr
t(65/43848*I*sqrt(7) - 109/87696))^2 - 4698*(-1/168*I*sqrt(7) - 1/2*sqrt(-65/43848*I*sqrt(7) - 109/87696))^2 -
87/784*(I*sqrt(7) + 84*sqrt(-65/43848*I*sqrt(7) - 109/87696))*(-I*sqrt(7) + 84*sqrt(65/43848*I*sqrt(7) - 109/
87696)) - 7) + 2109763*x - 373317/2*I*sqrt(7) + 15679314*sqrt(65/43848*I*sqrt(7) - 109/87696) + 220336) - 1/10
44*(sqrt(174)*sqrt(-4698*(1/168*I*sqrt(7) - 1/2*sqrt(65/43848*I*sqrt(7) - 109/87696))^2 - 4698*(-1/168*I*sqrt(
7) - 1/2*sqrt(-65/43848*I*sqrt(7) - 109/87696))^2 - 87/784*(I*sqrt(7) + 84*sqrt(-65/43848*I*sqrt(7) - 109/8769
6))*(-I*sqrt(7) + 84*sqrt(65/43848*I*sqrt(7) - 109/87696)) - 7) - 261*sqrt(65/43848*I*sqrt(7) - 109/87696) - 2
61*sqrt(-65/43848*I*sqrt(7) - 109/87696))*log(-16443/2*(-1/168*I*sqrt(7) - 1/2*sqrt(-65/43848*I*sqrt(7) - 109/
87696))^2*(-13001*I*sqrt(7) + 1092084*sqrt(65/43848*I*sqrt(7) - 109/87696) - 91520) - 609/2*(351027*(1/168*I*s
qrt(7) - 1/2*sqrt(65/43848*I*sqrt(7) - 109/87696))^2 - 613)*(I*sqrt(7) + 84*sqrt(-65/43848*I*sqrt(7) - 109/876
96)) + 752431680*(1/168*I*sqrt(7) - 1/2*sqrt(65/43848*I*sqrt(7) - 109/87696))^2 - 1/32*(3*(13001*sqrt(174)*(-I
*sqrt(7) + 84*sqrt(65/43848*I*sqrt(7) - 109/87696)) - 91520*sqrt(174))*(I*sqrt(7) + 84*sqrt(-65/43848*I*sqrt(7
) - 109/87696)) - 274560*sqrt(174)*(-I*sqrt(7) + 84*sqrt(65/43848*I*sqrt(7) - 109/87696)) + 1922368*sqrt(174))
*sqrt(-4698*(1/168*I*sqrt(7) - 1/2*sqrt(65/43848*I*sqrt(7) - 109/87696))^2 - 4698*(-1/168*I*sqrt(7) - 1/2*sqrt
(-65/43848*I*sqrt(7) - 109/87696))^2 - 87/784*(I*sqrt(7) + 84*sqrt(-65/43848*I*sqrt(7) - 109/87696))*(-I*sqrt(
7) + 84*sqrt(65/43848*I*sqrt(7) - 109/87696)) - 7) + 2109763*x - 373317/2*I*sqrt(7) + 15679314*sqrt(65/43848*I
*sqrt(7) - 109/87696) + 220336)

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Sympy [A]  time = 0.799543, size = 41, normalized size = 0.15 \begin{align*} \operatorname{RootSum}{\left (66298176 t^{4} + 74088 t^{2} + 4095 t + 64, \left ( t \mapsto t \log{\left (\frac{35914274424 t^{3}}{2109763} - \frac{1504863360 t^{2}}{2109763} + \frac{102851343 t}{2109763} + x + \frac{6055613}{16878104} \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),x)

[Out]

RootSum(66298176*_t**4 + 74088*_t**2 + 4095*_t + 64, Lambda(_t, _t*log(35914274424*_t**3/2109763 - 1504863360*
_t**2/2109763 + 102851343*_t/2109763 + x + 6055613/16878104)))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{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),x, algorithm="giac")

[Out]

integrate(1/(8*x^4 - x^3 + 8*x + 8), x)