∫ 1_________ (8+24x+8x2−15x3+8x4)2 dx

3.62 $\int \frac {1}{(8+24 x+8 x^2-15 x^3+8 x^4)^2} \, dx$

Optimal. Leaf size=366 \[ \frac {73}{208} \sqrt {\frac {3}{13}} \tan ^{-1}\left (\frac {-5 x^2+12 x+8}{\sqrt {39} x^2}\right )-\frac {3 \left (3359-107 \left (\frac {4}{x}+3\right )^2\right )}{208 \left (\left (\frac {4}{x}+3\right )^4-38 \left (\frac {4}{x}+3\right )^2+517\right )}+\frac {\left (3327931-129631 \left (\frac {4}{x}+3\right )^2\right ) \left (\frac {4}{x}+3\right )}{322608 \left (\left (\frac {4}{x}+3\right )^4-38 \left (\frac {4}{x}+3\right )^2+517\right )}-\frac {\sqrt {\frac {2623170438295 \sqrt {517}-59644114671451}{40326}} \log \left (\left (\frac {4}{x}+3\right )^2-\sqrt {2 \left (19+\sqrt {517}\right )} \left (\frac {4}{x}+3\right )+\sqrt {517}\right )}{645216}+\frac {\sqrt {\frac {2623170438295 \sqrt {517}-59644114671451}{40326}} \log \left (\left (\frac {4}{x}+3\right )^2+\sqrt {2 \left (19+\sqrt {517}\right )} \left (\frac {4}{x}+3\right )+\sqrt {517}\right )}{645216}-\frac {\sqrt {\frac {19+\sqrt {517}}{40326}} \left (1678181+74897 \sqrt {517}\right ) \tan ^{-1}\left (\frac {\frac {8}{x}-\sqrt {2 \left (19+\sqrt {517}\right )}+6}{\sqrt {2 \left (\sqrt {517}-19\right )}}\right )}{645216}-\frac {\sqrt {\frac {19+\sqrt {517}}{40326}} \left (1678181+74897 \sqrt {517}\right ) \tan ^{-1}\left (\frac {\frac {8}{x}+\sqrt {2 \left (19+\sqrt {517}\right )}+6}{\sqrt {2 \left (\sqrt {517}-19\right )}}\right )}{645216} \]

[Out]

-3/208*(3359-107*(3+4/x)^2)/(517-38*(3+4/x)^2+(3+4/x)^4)+1/322608*(3327931-129631*(3+4/x)^2)*(3+4/x)/(517-38*(

3+4/x)^2+(3+4/x)^4)+73/2704*arctan(1/39*(-5*x^2+12*x+8)/x^2*39^(1/2))*39^(1/2)-1/26018980416*arctan((6+8/x-(38

+2*517^(1/2))^(1/2))/(-38+2*517^(1/2))^(1/2))*(1678181+74897*517^(1/2))*(766194+40326*517^(1/2))^(1/2)-1/26018

980416*arctan((6+8/x+(38+2*517^(1/2))^(1/2))/(-38+2*517^(1/2))^(1/2))*(1678181+74897*517^(1/2))*(766194+40326*

517^(1/2))^(1/2)-1/26018980416*ln((3+4/x)^2+517^(1/2)-(3+4/x)*(38+2*517^(1/2))^(1/2))*(-2405208568240933026+10

5781971094684170*517^(1/2))^(1/2)+1/26018980416*ln((3+4/x)^2+517^(1/2)+(3+4/x)*(38+2*517^(1/2))^(1/2))*(-24052

08568240933026+105781971094684170*517^(1/2))^(1/2)

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Rubi [A] time = 0.51, antiderivative size = 366, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 11, integrand size = 22, $\frac {\text {number of rules}}{\text {integrand size}}$ = 0.500, Rules used = {2069, 12, 1673, 1678, 1169, 634, 618, 204, 628, 1663, 1660} \[ \frac {73}{208} \sqrt {\frac {3}{13}} \tan ^{-1}\left (\frac {-5 x^2+12 x+8}{\sqrt {39} x^2}\right )-\frac {3 \left (3359-107 \left (\frac {4}{x}+3\right )^2\right )}{208 \left (\left (\frac {4}{x}+3\right )^4-38 \left (\frac {4}{x}+3\right )^2+517\right )}+\frac {\left (3327931-129631 \left (\frac {4}{x}+3\right )^2\right ) \left (\frac {4}{x}+3\right )}{322608 \left (\left (\frac {4}{x}+3\right )^4-38 \left (\frac {4}{x}+3\right )^2+517\right )}-\frac {\sqrt {\frac {2623170438295 \sqrt {517}-59644114671451}{40326}} \log \left (\left (\frac {4}{x}+3\right )^2-\sqrt {2 \left (19+\sqrt {517}\right )} \left (\frac {4}{x}+3\right )+\sqrt {517}\right )}{645216}+\frac {\sqrt {\frac {2623170438295 \sqrt {517}-59644114671451}{40326}} \log \left (\left (\frac {4}{x}+3\right )^2+\sqrt {2 \left (19+\sqrt {517}\right )} \left (\frac {4}{x}+3\right )+\sqrt {517}\right )}{645216}-\frac {\sqrt {\frac {19+\sqrt {517}}{40326}} \left (1678181+74897 \sqrt {517}\right ) \tan ^{-1}\left (\frac {\frac {8}{x}-\sqrt {2 \left (19+\sqrt {517}\right )}+6}{\sqrt {2 \left (\sqrt {517}-19\right )}}\right )}{645216}-\frac {\sqrt {\frac {19+\sqrt {517}}{40326}} \left (1678181+74897 \sqrt {517}\right ) \tan ^{-1}\left (\frac {\frac {8}{x}+\sqrt {2 \left (19+\sqrt {517}\right )}+6}{\sqrt {2 \left (\sqrt {517}-19\right )}}\right )}{645216} \]

Antiderivative was successfully veriﬁed.

[In]

Int[(8 + 24*x + 8*x^2 - 15*x^3 + 8*x^4)^(-2),x]

[Out]

(-3*(3359 - 107*(3 + 4/x)^2))/(208*(517 - 38*(3 + 4/x)^2 + (3 + 4/x)^4)) + ((3327931 - 129631*(3 + 4/x)^2)*(3

+ 4/x))/(322608*(517 - 38*(3 + 4/x)^2 + (3 + 4/x)^4)) - (Sqrt[(19 + Sqrt[517])/40326]*(1678181 + 74897*Sqrt[51

7])*ArcTan[(6 - Sqrt[2*(19 + Sqrt[517])] + 8/x)/Sqrt[2*(-19 + Sqrt[517])]])/645216 - (Sqrt[(19 + Sqrt[517])/40

326]*(1678181 + 74897*Sqrt[517])*ArcTan[(6 + Sqrt[2*(19 + Sqrt[517])] + 8/x)/Sqrt[2*(-19 + Sqrt[517])]])/64521

6 + (73*Sqrt[3/13]*ArcTan[(8 + 12*x - 5*x^2)/(Sqrt[39]*x^2)])/208 - (Sqrt[(-59644114671451 + 2623170438295*Sqr

t[517])/40326]*Log[Sqrt[517] - Sqrt[2*(19 + Sqrt[517])]*(3 + 4/x) + (3 + 4/x)^2])/645216 + (Sqrt[(-59644114671

451 + 2623170438295*Sqrt[517])/40326]*Log[Sqrt[517] + Sqrt[2*(19 + Sqrt[517])]*(3 + 4/x) + (3 + 4/x)^2])/64521

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+24 x+8 x^2-15 x^3+8 x^4\right )^2} \, dx &=-\left (1024 \operatorname {Subst}\left (\int \frac {(24-32 x)^6}{64 \left (2117632-2490368 x^2+1048576 x^4\right )^2} \, dx,x,\frac {3}{4}+\frac {1}{x}\right )\right )\\ &=-\left (16 \operatorname {Subst}\left (\int \frac {(24-32 x)^6}{\left (2117632-2490368 x^2+1048576 x^4\right )^2} \, dx,x,\frac {3}{4}+\frac {1}{x}\right )\right )\\ &=-\left (16 \operatorname {Subst}\left (\int \frac {x \left (-1528823808-9059696640 x^2-4831838208 x^4\right )}{\left (2117632-2490368 x^2+1048576 x^4\right )^2} \, dx,x,\frac {3}{4}+\frac {1}{x}\right )\right )-16 \operatorname {Subst}\left (\int \frac {191102976+5096079360 x^2+9059696640 x^4+1073741824 x^6}{\left (2117632-2490368 x^2+1048576 x^4\right )^2} \, dx,x,\frac {3}{4}+\frac {1}{x}\right )\\ &=\frac {\left (3327931-129631 \left (3+\frac {4}{x}\right )^2\right ) \left (3+\frac {4}{x}\right )}{322608 \left (517-38 \left (3+\frac {4}{x}\right )^2+\left (3+\frac {4}{x}\right )^4\right )}-\frac {\operatorname {Subst}\left (\int \frac {120925685220163941564416+86350361930539017961472 x^2}{2117632-2490368 x^2+1048576 x^4} \, dx,x,\frac {3}{4}+\frac {1}{x}\right )}{709422494427119616}-8 \operatorname {Subst}\left (\int \frac {-1528823808-9059696640 x-4831838208 x^2}{\left (2117632-2490368 x+1048576 x^2\right )^2} \, dx,x,\left (\frac {3}{4}+\frac {1}{x}\right )^2\right )\\ &=-\frac {3 \left (3359-107 \left (3+\frac {4}{x}\right )^2\right )}{208 \left (517-38 \left (3+\frac {4}{x}\right )^2+\left (3+\frac {4}{x}\right )^4\right )}+\frac {\left (3327931-129631 \left (3+\frac {4}{x}\right )^2\right ) \left (3+\frac {4}{x}\right )}{322608 \left (517-38 \left (3+\frac {4}{x}\right )^2+\left (3+\frac {4}{x}\right )^4\right )}-\frac {\operatorname {Subst}\left (\int -\frac {46232264924725248}{2117632-2490368 x+1048576 x^2} \, dx,x,\left (\frac {3}{4}+\frac {1}{x}\right )^2\right )}{335007449088}-\frac {\operatorname {Subst}\left (\int \frac {30231421305040985391104 \sqrt {2 \left (19+\sqrt {517}\right )}-\left (120925685220163941564416-5396897620658688622592 \sqrt {517}\right ) x}{\frac {\sqrt {517}}{16}-\frac {1}{2} \sqrt {\frac {1}{2} \left (19+\sqrt {517}\right )} x+x^2} \, dx,x,\frac {3}{4}+\frac {1}{x}\right )}{23246356297387855577088 \sqrt {1034 \left (19+\sqrt {517}\right )}}-\frac {\operatorname {Subst}\left (\int \frac {30231421305040985391104 \sqrt {2 \left (19+\sqrt {517}\right )}+\left (120925685220163941564416-5396897620658688622592 \sqrt {517}\right ) x}{\frac {\sqrt {517}}{16}+\frac {1}{2} \sqrt {\frac {1}{2} \left (19+\sqrt {517}\right )} x+x^2} \, dx,x,\frac {3}{4}+\frac {1}{x}\right )}{23246356297387855577088 \sqrt {1034 \left (19+\sqrt {517}\right )}}\\ &=-\frac {3 \left (3359-107 \left (3+\frac {4}{x}\right )^2\right )}{208 \left (517-38 \left (3+\frac {4}{x}\right )^2+\left (3+\frac {4}{x}\right )^4\right )}+\frac {\left (3327931-129631 \left (3+\frac {4}{x}\right )^2\right ) \left (3+\frac {4}{x}\right )}{322608 \left (517-38 \left (3+\frac {4}{x}\right )^2+\left (3+\frac {4}{x}\right )^4\right )}+\frac {1794048}{13} \operatorname {Subst}\left (\int \frac {1}{2117632-2490368 x+1048576 x^2} \, dx,x,\left (\frac {3}{4}+\frac {1}{x}\right )^2\right )+\frac {\left (1678181-74897 \sqrt {517}\right ) \operatorname {Subst}\left (\int \frac {-\frac {1}{2} \sqrt {\frac {1}{2} \left (19+\sqrt {517}\right )}+2 x}{\frac {\sqrt {517}}{16}-\frac {1}{2} \sqrt {\frac {1}{2} \left (19+\sqrt {517}\right )} x+x^2} \, dx,x,\frac {3}{4}+\frac {1}{x}\right )}{645216 \sqrt {1034 \left (19+\sqrt {517}\right )}}-\frac {\left (1678181-74897 \sqrt {517}\right ) \operatorname {Subst}\left (\int \frac {\frac {1}{2} \sqrt {\frac {1}{2} \left (19+\sqrt {517}\right )}+2 x}{\frac {\sqrt {517}}{16}+\frac {1}{2} \sqrt {\frac {1}{2} \left (19+\sqrt {517}\right )} x+x^2} \, dx,x,\frac {3}{4}+\frac {1}{x}\right )}{645216 \sqrt {1034 \left (19+\sqrt {517}\right )}}-\frac {\left (38721749+1678181 \sqrt {517}\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt {517}}{16}-\frac {1}{2} \sqrt {\frac {1}{2} \left (19+\sqrt {517}\right )} x+x^2} \, dx,x,\frac {3}{4}+\frac {1}{x}\right )}{1334306688}-\frac {\left (38721749+1678181 \sqrt {517}\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt {517}}{16}+\frac {1}{2} \sqrt {\frac {1}{2} \left (19+\sqrt {517}\right )} x+x^2} \, dx,x,\frac {3}{4}+\frac {1}{x}\right )}{1334306688}\\ &=-\frac {3 \left (3359-107 \left (3+\frac {4}{x}\right )^2\right )}{208 \left (517-38 \left (3+\frac {4}{x}\right )^2+\left (3+\frac {4}{x}\right )^4\right )}+\frac {\left (3327931-129631 \left (3+\frac {4}{x}\right )^2\right ) \left (3+\frac {4}{x}\right )}{322608 \left (517-38 \left (3+\frac {4}{x}\right )^2+\left (3+\frac {4}{x}\right )^4\right )}-\frac {\sqrt {-\frac {59644114671451}{40326}+\frac {5073830635 \sqrt {517}}{78}} \log \left (\sqrt {517}-\sqrt {2 \left (19+\sqrt {517}\right )} \left (3+\frac {4}{x}\right )+\left (3+\frac {4}{x}\right )^2\right )}{645216}+\frac {\sqrt {-\frac {59644114671451}{40326}+\frac {5073830635 \sqrt {517}}{78}} \log \left (\sqrt {517}+\sqrt {2 \left (19+\sqrt {517}\right )} \left (3+\frac {4}{x}\right )+\left (3+\frac {4}{x}\right )^2\right )}{645216}-\frac {3588096}{13} \operatorname {Subst}\left (\int \frac {1}{-2680059592704-x^2} \, dx,x,-2490368+2097152 \left (\frac {3}{4}+\frac {1}{x}\right )^2\right )+\frac {\left (38721749+1678181 \sqrt {517}\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {1}{8} \left (19-\sqrt {517}\right )-x^2} \, dx,x,-\frac {1}{2} \sqrt {\frac {1}{2} \left (19+\sqrt {517}\right )}+2 \left (\frac {3}{4}+\frac {1}{x}\right )\right )}{667153344}+\frac {\left (38721749+1678181 \sqrt {517}\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {1}{8} \left (19-\sqrt {517}\right )-x^2} \, dx,x,\frac {1}{4} \left (6+\sqrt {2 \left (19+\sqrt {517}\right )}+\frac {8}{x}\right )\right )}{667153344}\\ &=-\frac {3 \left (3359-107 \left (3+\frac {4}{x}\right )^2\right )}{208 \left (517-38 \left (3+\frac {4}{x}\right )^2+\left (3+\frac {4}{x}\right )^4\right )}+\frac {\left (3327931-129631 \left (3+\frac {4}{x}\right )^2\right ) \left (3+\frac {4}{x}\right )}{322608 \left (517-38 \left (3+\frac {4}{x}\right )^2+\left (3+\frac {4}{x}\right )^4\right )}-\frac {73}{208} \sqrt {\frac {3}{13}} \tan ^{-1}\left (\frac {19-\left (3+\frac {4}{x}\right )^2}{2 \sqrt {39}}\right )-\frac {\left (1678181+74897 \sqrt {517}\right ) \tan ^{-1}\left (\frac {6+\sqrt {2 \left (19+\sqrt {517}\right )}+\frac {8}{x}}{\sqrt {2 \left (-19+\sqrt {517}\right )}}\right )}{322608 \sqrt {1034 \left (-19+\sqrt {517}\right )}}-\frac {\left (1678181+74897 \sqrt {517}\right ) \tan ^{-1}\left (\frac {8+\left (6-\sqrt {2 \left (19+\sqrt {517}\right )}\right ) x}{\sqrt {2 \left (-19+\sqrt {517}\right )} x}\right )}{322608 \sqrt {1034 \left (-19+\sqrt {517}\right )}}-\frac {\sqrt {-\frac {59644114671451}{40326}+\frac {5073830635 \sqrt {517}}{78}} \log \left (\sqrt {517}-\sqrt {2 \left (19+\sqrt {517}\right )} \left (3+\frac {4}{x}\right )+\left (3+\frac {4}{x}\right )^2\right )}{645216}+\frac {\sqrt {-\frac {59644114671451}{40326}+\frac {5073830635 \sqrt {517}}{78}} \log \left (\sqrt {517}+\sqrt {2 \left (19+\sqrt {517}\right )} \left (3+\frac {4}{x}\right )+\left (3+\frac {4}{x}\right )^2\right )}{645216}\\ \end {align*}

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Mathematica [C] time = 0.03, size = 128, normalized size = 0.35 \[ \frac {\text {RootSum}\left [8 \text {$\#$1}^4-15 \text {$\#$1}^3+8 \text {$\#$1}^2+24 \text {$\#$1}+8\& ,\frac {19640 \text {$\#$1}^2 \log (x-\text {$\#$1})-57489 \text {$\#$1} \log (x-\text {$\#$1})+74897 \log (x-\text {$\#$1})}{32 \text {$\#$1}^3-45 \text {$\#$1}^2+16 \text {$\#$1}+24}\& \right ]}{80652}+\frac {39280 x^3-94314 x^2+89033 x+72888}{161304 \left (8 x^4-15 x^3+8 x^2+24 x+8\right )} \]

Antiderivative was successfully veriﬁed.

[In]

Integrate[(8 + 24*x + 8*x^2 - 15*x^3 + 8*x^4)^(-2),x]

[Out]

(72888 + 89033*x - 94314*x^2 + 39280*x^3)/(161304*(8 + 24*x + 8*x^2 - 15*x^3 + 8*x^4)) + RootSum[8 + 24*#1 + 8

*#1^2 - 15*#1^3 + 8*#1^4 & , (74897*Log[x - #1] - 57489*Log[x - #1]*#1 + 19640*Log[x - #1]*#1^2)/(24 + 16*#1 -

45*#1^2 + 32*#1^3) & ]/80652

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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(8*x^4-15*x^3+8*x^2+24*x+8)^2,x, algorithm="fricas")

[Out]

Timed out

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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (8 \, x^{4} - 15 \, x^{3} + 8 \, x^{2} + 24 \, x + 8\right )}^{2}}\,{d x} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(8*x^4-15*x^3+8*x^2+24*x+8)^2,x, algorithm="giac")

[Out]

integrate((8*x^4 - 15*x^3 + 8*x^2 + 24*x + 8)^(-2), x)

________________________________________________________________________________________

maple [C] time = 0.01, size = 96, normalized size = 0.26 \[ \frac {\left (19640 \RootOf \left (8 \textit {\_Z}^{4}-15 \textit {\_Z}^{3}+8 \textit {\_Z}^{2}+24 \textit {\_Z} +8\right )^{2}-57489 \RootOf \left (8 \textit {\_Z}^{4}-15 \textit {\_Z}^{3}+8 \textit {\_Z}^{2}+24 \textit {\_Z} +8\right )+74897\right ) \ln \left (-\RootOf \left (8 \textit {\_Z}^{4}-15 \textit {\_Z}^{3}+8 \textit {\_Z}^{2}+24 \textit {\_Z} +8\right )+x \right )}{2580864 \RootOf \left (8 \textit {\_Z}^{4}-15 \textit {\_Z}^{3}+8 \textit {\_Z}^{2}+24 \textit {\_Z} +8\right )^{3}-3629340 \RootOf \left (8 \textit {\_Z}^{4}-15 \textit {\_Z}^{3}+8 \textit {\_Z}^{2}+24 \textit {\_Z} +8\right )^{2}+1290432 \RootOf \left (8 \textit {\_Z}^{4}-15 \textit {\_Z}^{3}+8 \textit {\_Z}^{2}+24 \textit {\_Z} +8\right )+1935648}+\frac {\frac {2455}{80652} x^{3}-\frac {1429}{19552} x^{2}+\frac {89033}{1290432} x +\frac {3037}{53768}}{x^{4}-\frac {15}{8} x^{3}+x^{2}+3 x +1} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(1/(8*x^4-15*x^3+8*x^2+24*x+8)^2,x)

[Out]

(2455/80652*x^3-1429/19552*x^2+89033/1290432*x+3037/53768)/(x^4-15/8*x^3+x^2+3*x+1)+1/80652*sum((19640*_R^2-57

489*_R+74897)/(32*_R^3-45*_R^2+16*_R+24)*ln(-_R+x),_R=RootOf(8*_Z^4-15*_Z^3+8*_Z^2+24*_Z+8))

________________________________________________________________________________________

maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {39280 \, x^{3} - 94314 \, x^{2} + 89033 \, x + 72888}{161304 \, {\left (8 \, x^{4} - 15 \, x^{3} + 8 \, x^{2} + 24 \, x + 8\right )}} + \frac {1}{80652} \, \int \frac {19640 \, x^{2} - 57489 \, x + 74897}{8 \, x^{4} - 15 \, x^{3} + 8 \, x^{2} + 24 \, x + 8}\,{d x} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(8*x^4-15*x^3+8*x^2+24*x+8)^2,x, algorithm="maxima")

[Out]

1/161304*(39280*x^3 - 94314*x^2 + 89033*x + 72888)/(8*x^4 - 15*x^3 + 8*x^2 + 24*x + 8) + 1/80652*integrate((19

640*x^2 - 57489*x + 74897)/(8*x^4 - 15*x^3 + 8*x^2 + 24*x + 8), x)

________________________________________________________________________________________

mupad [B] time = 0.21, size = 181, normalized size = 0.49 \[ \frac {\frac {2455\,x^3}{80652}-\frac {1429\,x^2}{19552}+\frac {89033\,x}{1290432}+\frac {3037}{53768}}{x^4-\frac {15\,x^3}{8}+x^2+3\,x+1}+\left (\sum _{k=1}^4\ln \left (\frac {2146659825\,\mathrm {root}\left (z^4+\frac {14911625619311\,z^2}{524620702127808}+\frac {39238139261\,z}{3730636104019968}+\frac {43023440}{44204510553294663},z,k\right )}{2960381771776}+\frac {2222183\,x}{338246745408}+\frac {\mathrm {root}\left (z^4+\frac {14911625619311\,z^2}{524620702127808}+\frac {39238139261\,z}{3730636104019968}+\frac {43023440}{44204510553294663},z,k\right )\,x\,924124364159}{26643435945984}-\frac {{\mathrm {root}\left (z^4+\frac {14911625619311\,z^2}{524620702127808}+\frac {39238139261\,z}{3730636104019968}+\frac {43023440}{44204510553294663},z,k\right )}^2\,x\,72451101}{8470528}-\frac {{\mathrm {root}\left (z^4+\frac {14911625619311\,z^2}{524620702127808}+\frac {39238139261\,z}{3730636104019968}+\frac {43023440}{44204510553294663},z,k\right )}^3\,x\,95745}{256}+\frac {389551\,{\mathrm {root}\left (z^4+\frac {14911625619311\,z^2}{524620702127808}+\frac {39238139261\,z}{3730636104019968}+\frac {43023440}{44204510553294663},z,k\right )}^2}{264704}-\frac {100737\,{\mathrm {root}\left (z^4+\frac {14911625619311\,z^2}{524620702127808}+\frac {39238139261\,z}{3730636104019968}+\frac {43023440}{44204510553294663},z,k\right )}^3}{512}+\frac {271033}{624455529984}\right )\,\mathrm {root}\left (z^4+\frac {14911625619311\,z^2}{524620702127808}+\frac {39238139261\,z}{3730636104019968}+\frac {43023440}{44204510553294663},z,k\right )\right ) \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(1/(24*x + 8*x^2 - 15*x^3 + 8*x^4 + 8)^2,x)

[Out]

((89033*x)/1290432 - (1429*x^2)/19552 + (2455*x^3)/80652 + 3037/53768)/(3*x + x^2 - (15*x^3)/8 + x^4 + 1) + sy

msum(log((2146659825*root(z^4 + (14911625619311*z^2)/524620702127808 + (39238139261*z)/3730636104019968 + 4302

3440/44204510553294663, z, k))/2960381771776 + (2222183*x)/338246745408 + (924124364159*root(z^4 + (1491162561

9311*z^2)/524620702127808 + (39238139261*z)/3730636104019968 + 43023440/44204510553294663, z, k)*x)/2664343594

5984 - (72451101*root(z^4 + (14911625619311*z^2)/524620702127808 + (39238139261*z)/3730636104019968 + 43023440

/44204510553294663, z, k)^2*x)/8470528 - (95745*root(z^4 + (14911625619311*z^2)/524620702127808 + (39238139261

*z)/3730636104019968 + 43023440/44204510553294663, z, k)^3*x)/256 + (389551*root(z^4 + (14911625619311*z^2)/52

4620702127808 + (39238139261*z)/3730636104019968 + 43023440/44204510553294663, z, k)^2)/264704 - (100737*root(

z^4 + (14911625619311*z^2)/524620702127808 + (39238139261*z)/3730636104019968 + 43023440/44204510553294663, z,

k)^3)/512 + 271033/624455529984)*root(z^4 + (14911625619311*z^2)/524620702127808 + (39238139261*z)/3730636104

019968 + 43023440/44204510553294663, z, k), k, 1, 4)

________________________________________________________________________________________

sympy [B] time = 3.95, size = 3839, normalized size = 10.49 \[ \text {result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(8*x**4-15*x**3+8*x**2+24*x+8)**2,x)

[Out]

(39280*x**3 - 94314*x**2 + 89033*x + 72888)/(1290432*x**4 - 2419560*x**3 + 1290432*x**2 + 3871296*x + 1290432)

+ sqrt(-59644114671451/16787862468089856 + 5073830635*sqrt(517)/32471687559168)*log(x**2 + x*(-11239699502046

8503306932567484755463/603722125611976319526135612861060 - 29643869829812833230907750777733957*sqrt(40326)*sqr

t(-59644114671451 + 2623170438295*sqrt(517))/1936419398792394461637855141912238396080 - 181533261043120360732*

sqrt(-7120427417275887*sqrt(40326)*sqrt(-59644114671451 + 2623170438295*sqrt(517)) + 6263621568587150042935*sq

rt(517) + 3557579971691991294769382675)/150930531402994079881533903215265 - 46926347979646613249222*sqrt(517)/

29746860362632912338339 + 994065243322493861977*sqrt(78)*sqrt(-59644114671451 + 2623170438295*sqrt(517))/14278

49297406379792240272 + 994065243322493861977*sqrt(40326)*sqrt(-59644114671451 + 2623170438295*sqrt(517))*sqrt(

-7120427417275887*sqrt(40326)*sqrt(-59644114671451 + 2623170438295*sqrt(517)) + 6263621568587150042935*sqrt(51

7) + 3557579971691991294769382675)/1290946265861596307758570094608158930720) - 4597149706773066968921854791223

560238809189313591735176029*sqrt(517)*sqrt(-7120427417275887*sqrt(40326)*sqrt(-59644114671451 + 2623170438295*

sqrt(517)) + 6263621568587150042935*sqrt(517) + 3557579971691991294769382675)/18432767186998698626450604048374

76389014874805380627572841955840 - 102213276372026717588278042506361313108860193595830387808115871094971545996

7411486447/302201812380681690634631534385892067350866441656553353409624708723614680800 - 106380947173342801261

7611152668277664372838283556533836993*sqrt(78)*sqrt(-59644114671451 + 2623170438295*sqrt(517))/689619370306997

2436744723519626607862706189949382980866560 - 890360389298500646731845595593034670326044595870824169313*sqrt(4

0326)*sqrt(-59644114671451 + 2623170438295*sqrt(517))*sqrt(-7120427417275887*sqrt(40326)*sqrt(-59644114671451

+ 2623170438295*sqrt(517)) + 6263621568587150042935*sqrt(517) + 3557579971691991294769382675)/9352473884121677

079601749613489889898672849258229891014611209554309158400 - 45113976327488809325094501633826014671791*sqrt(78)

*sqrt(-59644114671451 + 2623170438295*sqrt(517))*sqrt(-7120427417275887*sqrt(40326)*sqrt(-59644114671451 + 262

3170438295*sqrt(517)) + 6263621568587150042935*sqrt(517) + 3557579971691991294769382675)/107753026610468319324

136304994165747854784217959109076040 + 42698009636515468718900942734274005212255282299552802371283821308121207

*sqrt(40326)*sqrt(-59644114671451 + 2623170438295*sqrt(517))/9352473884121677079601749613489889898672849258229

891014611209554309158400 + 68548776709669674081892851407413209373218007060934353137152573209405073*sqrt(517)/4

60819179674967465661265101209369097253718701345156893210488960 + 274196431933554153007434570764680602132735098

624644758583157528631167033*sqrt(-7120427417275887*sqrt(40326)*sqrt(-59644114671451 + 2623170438295*sqrt(517))

+ 6263621568587150042935*sqrt(517) + 3557579971691991294769382675)/483522899809090705015410455017427307761386

3066504853654553995339577834892800) - sqrt(-59644114671451/16787862468089856 + 5073830635*sqrt(517)/3247168755

9168)*log(x**2 + x*(-112396995020468503306932567484755463/603722125611976319526135612861060 - 9940652433224938

61977*sqrt(78)*sqrt(-59644114671451 + 2623170438295*sqrt(517))/1427849297406379792240272 - 4692634797964661324

9222*sqrt(517)/29746860362632912338339 + 181533261043120360732*sqrt(7120427417275887*sqrt(40326)*sqrt(-5964411

4671451 + 2623170438295*sqrt(517)) + 6263621568587150042935*sqrt(517) + 3557579971691991294769382675)/15093053

1402994079881533903215265 + 29643869829812833230907750777733957*sqrt(40326)*sqrt(-59644114671451 + 26231704382

95*sqrt(517))/1936419398792394461637855141912238396080 + 994065243322493861977*sqrt(40326)*sqrt(-5964411467145

1 + 2623170438295*sqrt(517))*sqrt(7120427417275887*sqrt(40326)*sqrt(-59644114671451 + 2623170438295*sqrt(517))

+ 6263621568587150042935*sqrt(517) + 3557579971691991294769382675)/1290946265861596307758570094608158930720)

- 274196431933554153007434570764680602132735098624644758583157528631167033*sqrt(7120427417275887*sqrt(40326)*s

qrt(-59644114671451 + 2623170438295*sqrt(517)) + 6263621568587150042935*sqrt(517) + 35575799716919912947693826

75)/4835228998090907050154104550174273077613863066504853654553995339577834892800 - 102213276372026717588278042

5063613131088601935958303878081158710949715459967411486447/302201812380681690634631534385892067350866441656553

353409624708723614680800 - 890360389298500646731845595593034670326044595870824169313*sqrt(40326)*sqrt(-5964411

4671451 + 2623170438295*sqrt(517))*sqrt(7120427417275887*sqrt(40326)*sqrt(-59644114671451 + 2623170438295*sqrt

(517)) + 6263621568587150042935*sqrt(517) + 3557579971691991294769382675)/935247388412167707960174961348988989

8672849258229891014611209554309158400 - 4269800963651546871890094273427400521225528229955280237128382130812120

7*sqrt(40326)*sqrt(-59644114671451 + 2623170438295*sqrt(517))/935247388412167707960174961348988989867284925822

9891014611209554309158400 - 45113976327488809325094501633826014671791*sqrt(78)*sqrt(-59644114671451 + 26231704

38295*sqrt(517))*sqrt(7120427417275887*sqrt(40326)*sqrt(-59644114671451 + 2623170438295*sqrt(517)) + 626362156

8587150042935*sqrt(517) + 3557579971691991294769382675)/107753026610468319324136304994165747854784217959109076

040 + 1063809471733428012617611152668277664372838283556533836993*sqrt(78)*sqrt(-59644114671451 + 2623170438295

*sqrt(517))/6896193703069972436744723519626607862706189949382980866560 + 6854877670966967408189285140741320937

3218007060934353137152573209405073*sqrt(517)/460819179674967465661265101209369097253718701345156893210488960 +

4597149706773066968921854791223560238809189313591735176029*sqrt(517)*sqrt(7120427417275887*sqrt(40326)*sqrt(-

59644114671451 + 2623170438295*sqrt(517)) + 6263621568587150042935*sqrt(517) + 3557579971691991294769382675)/1

843276718699869862645060404837476389014874805380627572841955840) - 2*sqrt(59653665894623/16787862468089856 + 5

073830635*sqrt(517)/10823895853056 + sqrt(-7120427417275887*sqrt(40326)*sqrt(-59644114671451 + 2623170438295*s

qrt(517)) + 6263621568587150042935*sqrt(517) + 3557579971691991294769382675)/4196965617022464)*atan(-774567759

5169577846551420567648953584320*x/(-59292486929118917272637172801533436*sqrt(40326)*sqrt(59653665894623 + 7869

511314885*sqrt(517) + 4*sqrt(-7120427417275887*sqrt(40326)*sqrt(-59644114671451 + 2623170438295*sqrt(517)) + 6

263621568587150042935*sqrt(517) + 3557579971691991294769382675)) + 2329048502925820708785386304*sqrt(-59644114

671451 + 2623170438295*sqrt(517))*sqrt(59653665894623 + 7869511314885*sqrt(517) + 4*sqrt(-7120427417275887*sqr

t(40326)*sqrt(-59644114671451 + 2623170438295*sqrt(517)) + 6263621568587150042935*sqrt(517) + 3557579971691991

294769382675)) + 994065243322493861977*sqrt(40326)*sqrt(59653665894623 + 7869511314885*sqrt(517) + 4*sqrt(-712

0427417275887*sqrt(40326)*sqrt(-59644114671451 + 2623170438295*sqrt(517)) + 6263621568587150042935*sqrt(517) +

3557579971691991294769382675))*sqrt(-7120427417275887*sqrt(40326)*sqrt(-59644114671451 + 2623170438295*sqrt(5

17)) + 6263621568587150042935*sqrt(517) + 3557579971691991294769382675)) - 2982195729967481585931*sqrt(40326)*

sqrt(-59644114671451 + 2623170438295*sqrt(517))*sqrt(-7120427417275887*sqrt(40326)*sqrt(-59644114671451 + 2623

170438295*sqrt(517)) + 6263621568587150042935*sqrt(517) + 3557579971691991294769382675)/(-59292486929118917272

637172801533436*sqrt(40326)*sqrt(59653665894623 + 7869511314885*sqrt(517) + 4*sqrt(-7120427417275887*sqrt(4032

6)*sqrt(-59644114671451 + 2623170438295*sqrt(517)) + 6263621568587150042935*sqrt(517) + 3557579971691991294769

382675)) + 2329048502925820708785386304*sqrt(-59644114671451 + 2623170438295*sqrt(517))*sqrt(59653665894623 +

7869511314885*sqrt(517) + 4*sqrt(-7120427417275887*sqrt(40326)*sqrt(-59644114671451 + 2623170438295*sqrt(517))

+ 6263621568587150042935*sqrt(517) + 3557579971691991294769382675)) + 994065243322493861977*sqrt(40326)*sqrt(

59653665894623 + 7869511314885*sqrt(517) + 4*sqrt(-7120427417275887*sqrt(40326)*sqrt(-59644114671451 + 2623170

438295*sqrt(517)) + 6263621568587150042935*sqrt(517) + 3557579971691991294769382675))*sqrt(-7120427417275887*s

qrt(40326)*sqrt(-59644114671451 + 2623170438295*sqrt(517)) + 6263621568587150042935*sqrt(517) + 35575799716919

91294769382675)) - 2696261047060775175517112572266328310*sqrt(78)*sqrt(-59644114671451 + 2623170438295*sqrt(51

7))/(-59292486929118917272637172801533436*sqrt(40326)*sqrt(59653665894623 + 7869511314885*sqrt(517) + 4*sqrt(-

7120427417275887*sqrt(40326)*sqrt(-59644114671451 + 2623170438295*sqrt(517)) + 6263621568587150042935*sqrt(517

) + 3557579971691991294769382675)) + 2329048502925820708785386304*sqrt(-59644114671451 + 2623170438295*sqrt(51

7))*sqrt(59653665894623 + 7869511314885*sqrt(517) + 4*sqrt(-7120427417275887*sqrt(40326)*sqrt(-59644114671451

+ 2623170438295*sqrt(517)) + 6263621568587150042935*sqrt(517) + 3557579971691991294769382675)) + 9940652433224

93861977*sqrt(40326)*sqrt(59653665894623 + 7869511314885*sqrt(517) + 4*sqrt(-7120427417275887*sqrt(40326)*sqrt

(-59644114671451 + 2623170438295*sqrt(517)) + 6263621568587150042935*sqrt(517) + 3557579971691991294769382675)

)*sqrt(-7120427417275887*sqrt(40326)*sqrt(-59644114671451 + 2623170438295*sqrt(517)) + 6263621568587150042935*

sqrt(517) + 3557579971691991294769382675)) + 6109491182230238698537149348154570111680*sqrt(517)/(-592924869291

18917272637172801533436*sqrt(40326)*sqrt(59653665894623 + 7869511314885*sqrt(517) + 4*sqrt(-7120427417275887*s

qrt(40326)*sqrt(-59644114671451 + 2623170438295*sqrt(517)) + 6263621568587150042935*sqrt(517) + 35575799716919

91294769382675)) + 2329048502925820708785386304*sqrt(-59644114671451 + 2623170438295*sqrt(517))*sqrt(596536658

94623 + 7869511314885*sqrt(517) + 4*sqrt(-7120427417275887*sqrt(40326)*sqrt(-59644114671451 + 2623170438295*sq

rt(517)) + 6263621568587150042935*sqrt(517) + 3557579971691991294769382675)) + 994065243322493861977*sqrt(4032

6)*sqrt(59653665894623 + 7869511314885*sqrt(517) + 4*sqrt(-7120427417275887*sqrt(40326)*sqrt(-59644114671451 +

2623170438295*sqrt(517)) + 6263621568587150042935*sqrt(517) + 3557579971691991294769382675))*sqrt(-7120427417

275887*sqrt(40326)*sqrt(-59644114671451 + 2623170438295*sqrt(517)) + 6263621568587150042935*sqrt(517) + 355757

9971691991294769382675)) + 4658097005851641417570772608*sqrt(-7120427417275887*sqrt(40326)*sqrt(-5964411467145

1 + 2623170438295*sqrt(517)) + 6263621568587150042935*sqrt(517) + 3557579971691991294769382675)/(-592924869291