\(\int \frac {1}{(8+24 x+8 x^2-15 x^3+8 x^4)^2} \, dx\) [62]
Optimal result
Integrand size = 22, antiderivative size = 366 \[
\int \frac {1}{\left (8+24 x+8 x^2-15 x^3+8 x^4\right )^2} \, dx=-\frac {3 \left (3359-107 \left (3+\frac {4}{x}\right )^2\right )}{208 \left (517-38 \left (3+\frac {4}{x}\right )^2+\left (3+\frac {4}{x}\right )^4\right )}+\frac {\left (3327931-129631 \left (3+\frac {4}{x}\right )^2\right ) \left (3+\frac {4}{x}\right )}{322608 \left (517-38 \left (3+\frac {4}{x}\right )^2+\left (3+\frac {4}{x}\right )^4\right )}-\frac {\sqrt {\frac {19+\sqrt {517}}{40326}} \left (1678181+74897 \sqrt {517}\right ) \arctan \left (\frac {6-\sqrt {2 \left (19+\sqrt {517}\right )}+\frac {8}{x}}{\sqrt {2 \left (-19+\sqrt {517}\right )}}\right )}{645216}-\frac {\sqrt {\frac {19+\sqrt {517}}{40326}} \left (1678181+74897 \sqrt {517}\right ) \arctan \left (\frac {6+\sqrt {2 \left (19+\sqrt {517}\right )}+\frac {8}{x}}{\sqrt {2 \left (-19+\sqrt {517}\right )}}\right )}{645216}+\frac {73}{208} \sqrt {\frac {3}{13}} \arctan \left (\frac {8+12 x-5 x^2}{\sqrt {39} x^2}\right )-\frac {\sqrt {\frac {-59644114671451+2623170438295 \sqrt {517}}{40326}} \log \left (\sqrt {517}-\sqrt {2 \left (19+\sqrt {517}\right )} \left (3+\frac {4}{x}\right )+\left (3+\frac {4}{x}\right )^2\right )}{645216}+\frac {\sqrt {\frac {-59644114671451+2623170438295 \sqrt {517}}{40326}} \log \left (\sqrt {517}+\sqrt {2 \left (19+\sqrt {517}\right )} \left (3+\frac {4}{x}\right )+\left (3+\frac {4}{x}\right )^2\right )}{645216}
\]
[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)
Rubi [A] (verified)
Time = 0.41 (sec) , antiderivative size = 366, normalized size of antiderivative = 1.00, number of
steps used = 18, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2094, 12, 1687, 1692, 1183,
648, 632, 210, 642, 1677, 1674} \[
\int \frac {1}{\left (8+24 x+8 x^2-15 x^3+8 x^4\right )^2} \, dx=\frac {73}{208} \sqrt {\frac {3}{13}} \arctan \left (\frac {-5 x^2+12 x+8}{\sqrt {39} x^2}\right )-\frac {\sqrt {\frac {19+\sqrt {517}}{40326}} \left (1678181+74897 \sqrt {517}\right ) \arctan \left (\frac {\frac {8}{x}-\sqrt {2 \left (19+\sqrt {517}\right )}+6}{\sqrt {2 \left (\sqrt {517}-19\right )}}\right )}{645216}-\frac {\sqrt {\frac {19+\sqrt {517}}{40326}} \left (1678181+74897 \sqrt {517}\right ) \arctan \left (\frac {\frac {8}{x}+\sqrt {2 \left (19+\sqrt {517}\right )}+6}{\sqrt {2 \left (\sqrt {517}-19\right )}}\right )}{645216}-\frac {3 \left (3359-107 \left (\frac {4}{x}+3\right )^2\right )}{208 \left (\left (\frac {4}{x}+3\right )^4-38 \left (\frac {4}{x}+3\right )^2+517\right )}+\frac {\left (3327931-129631 \left (\frac {4}{x}+3\right )^2\right ) \left (\frac {4}{x}+3\right )}{322608 \left (\left (\frac {4}{x}+3\right )^4-38 \left (\frac {4}{x}+3\right )^2+517\right )}-\frac {\sqrt {\frac {2623170438295 \sqrt {517}-59644114671451}{40326}} \log \left (\left (\frac {4}{x}+3\right )^2-\sqrt {2 \left (19+\sqrt {517}\right )} \left (\frac {4}{x}+3\right )+\sqrt {517}\right )}{645216}+\frac {\sqrt {\frac {2623170438295 \sqrt {517}-59644114671451}{40326}} \log \left (\left (\frac {4}{x}+3\right )^2+\sqrt {2 \left (19+\sqrt {517}\right )} \left (\frac {4}{x}+3\right )+\sqrt {517}\right )}{645216}
\]
[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
6
Rule 12
Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] && !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]
Rule 210
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])
Rule 632
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 642
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 648
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 1183
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 1674
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 1677
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 1687
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 1692
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 2094
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/(b - 4*a*x)^2)*(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, 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*}
\text {integral}& = -\left (1024 \text {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 \text {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 \text {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 \text {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 {\text {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 \text {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 {\text {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 {\text {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 {\text {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} \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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} \text {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 ) \text {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 ) \text {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*}
Mathematica [C] (verified)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.01 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.35
\[
\int \frac {1}{\left (8+24 x+8 x^2-15 x^3+8 x^4\right )^2} \, dx=\frac {72888+89033 x-94314 x^2+39280 x^3}{161304 \left (8+24 x+8 x^2-15 x^3+8 x^4\right )}+\frac {\text {RootSum}\left [8+24 \text {$\#$1}+8 \text {$\#$1}^2-15 \text {$\#$1}^3+8 \text {$\#$1}^4\&,\frac {74897 \log (x-\text {$\#$1})-57489 \log (x-\text {$\#$1}) \text {$\#$1}+19640 \log (x-\text {$\#$1}) \text {$\#$1}^2}{24+16 \text {$\#$1}-45 \text {$\#$1}^2+32 \text {$\#$1}^3}\&\right ]}{80652}
\]
[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
Maple [C] (verified)
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.06 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.26
| | |
method | result | size |
| | |
default |
\(\frac {\frac {2455}{80652} x^{3}-\frac {1429}{19552} x^{2}+\frac {89033}{1290432} x +\frac {3037}{53768}}{x^{4}-\frac {15}{8} x^{3}+x^{2}+3 x +1}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (8 \textit {\_Z}^{4}-15 \textit {\_Z}^{3}+8 \textit {\_Z}^{2}+24 \textit {\_Z} +8\right )}{\sum }\frac {\left (19640 \textit {\_R}^{2}-57489 \textit {\_R} +74897\right ) \ln \left (x -\textit {\_R} \right )}{32 \textit {\_R}^{3}-45 \textit {\_R}^{2}+16 \textit {\_R} +24}\right )}{80652}\) |
\(96\) |
risch |
\(\frac {\frac {2455}{80652} x^{3}-\frac {1429}{19552} x^{2}+\frac {89033}{1290432} x +\frac {3037}{53768}}{x^{4}-\frac {15}{8} x^{3}+x^{2}+3 x +1}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (8 \textit {\_Z}^{4}-15 \textit {\_Z}^{3}+8 \textit {\_Z}^{2}+24 \textit {\_Z} +8\right )}{\sum }\frac {\left (19640 \textit {\_R}^{2}-57489 \textit {\_R} +74897\right ) \ln \left (x -\textit {\_R} \right )}{32 \textit {\_R}^{3}-45 \textit {\_R}^{2}+16 \textit {\_R} +24}\right )}{80652}\) |
\(96\) |
| | |
|
|
|
[In]
int(1/(8*x^4-15*x^3+8*x^2+24*x+8)^2,x,method=_RETURNVERBOSE)
[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(x-_R),_R=RootOf(8*_Z^4-15*_Z^3+8*_Z^2+24*_Z+8))
Fricas [C] (verification not implemented)
Result contains complex when optimal does not.
Time = 1.10 (sec) , antiderivative size = 1540, normalized size of antiderivative = 4.21
\[
\int \frac {1}{\left (8+24 x+8 x^2-15 x^3+8 x^4\right )^2} \, dx=\text {Too large to display}
\]
[In]
integrate(1/(8*x^4-15*x^3+8*x^2+24*x+8)^2,x, algorithm="fricas")
[Out]
1/26018980416*(6336021120*x^3 - 4811202*(8*x^4 - 15*x^3 + 8*x^2 + 24*x + 8)*(-73*I*sqrt(13)*sqrt(3) + 2704*sqr
t(537508757/26903625750144*I*sqrt(13)*sqrt(3) - 59644114671451/2098482808511232))*log(-131155175531952*(217700
288287626155772963*I*sqrt(13)*sqrt(3) + 8063857253832070208357424*sqrt(-537508757/26903625750144*I*sqrt(13)*sq
rt(3) - 59644114671451/2098482808511232) - 2904532176689925771712)*(73/5408*I*sqrt(13)*sqrt(3) - 1/2*sqrt(5375
08757/26903625750144*I*sqrt(13)*sqrt(3) - 59644114671451/2098482808511232))^2 + 211523322718189916535976363749
0696823296*(-73/5408*I*sqrt(13)*sqrt(3) - 1/2*sqrt(-537508757/26903625750144*I*sqrt(13)*sqrt(3) - 596441146714
51/2098482808511232))^3 - 801867*(487774661427048094833332220336*(-73/5408*I*sqrt(13)*sqrt(3) - 1/2*sqrt(-5375
08757/26903625750144*I*sqrt(13)*sqrt(3) - 59644114671451/2098482808511232))^2 + 3281707530577268651899)*(-73*I
*sqrt(13)*sqrt(3) + 2704*sqrt(537508757/26903625750144*I*sqrt(13)*sqrt(3) - 59644114671451/2098482808511232))
- 3246265196161156614776051552784488493/4*I*sqrt(13)*sqrt(3) + 150930531402994079881533903215265*x - 300613051
04177285912172751365511153716*sqrt(-537508757/26903625750144*I*sqrt(13)*sqrt(3) - 59644114671451/2098482808511
232) + 10905071149176173110139073138101752) - 4811202*(8*x^4 - 15*x^3 + 8*x^2 + 24*x + 8)*(73*I*sqrt(13)*sqrt(
3) + 2704*sqrt(-537508757/26903625750144*I*sqrt(13)*sqrt(3) - 59644114671451/2098482808511232))*log(-169218658
17455193322878109099925574586368*(-73/5408*I*sqrt(13)*sqrt(3) - 1/2*sqrt(-537508757/26903625750144*I*sqrt(13)*
sqrt(3) - 59644114671451/2098482808511232))^3 + 3047555419775758686243654429709934592*(-73/5408*I*sqrt(13)*sqr
t(3) - 1/2*sqrt(-537508757/26903625750144*I*sqrt(13)*sqrt(3) - 59644114671451/2098482808511232))^2 + 649252885
5530417335452936763668444114*I*sqrt(13)*sqrt(3) + 1207444251223952639052271225722120*x + 240490383908962307877
599191903554423072*sqrt(-537508757/26903625750144*I*sqrt(13)*sqrt(3) - 59644114671451/2098482808511232) - 9074
0479879102500787082477443749295) - 15213225456*x^2 - (8*sqrt(20163)*(8*x^4 - 15*x^3 + 8*x^2 + 24*x + 8)*sqrt(-
393465526595856*(73/5408*I*sqrt(13)*sqrt(3) - 1/2*sqrt(537508757/26903625750144*I*sqrt(13)*sqrt(3) - 596441146
71451/2098482808511232))^2 - 393465526595856*(-73/5408*I*sqrt(13)*sqrt(3) - 1/2*sqrt(-537508757/26903625750144
*I*sqrt(13)*sqrt(3) - 59644114671451/2098482808511232))^2 - 3731087151/416*(73*I*sqrt(13)*sqrt(3) + 2704*sqrt(
-537508757/26903625750144*I*sqrt(13)*sqrt(3) - 59644114671451/2098482808511232))*(-73*I*sqrt(13)*sqrt(3) + 270
4*sqrt(537508757/26903625750144*I*sqrt(13)*sqrt(3) - 59644114671451/2098482808511232)) - 14911625619311) - 240
5601*(8*x^4 - 15*x^3 + 8*x^2 + 24*x + 8)*(73*I*sqrt(13)*sqrt(3) + 2704*sqrt(-537508757/26903625750144*I*sqrt(1
3)*sqrt(3) - 59644114671451/2098482808511232)) - 2405601*(8*x^4 - 15*x^3 + 8*x^2 + 24*x + 8)*(-73*I*sqrt(13)*s
qrt(3) + 2704*sqrt(537508757/26903625750144*I*sqrt(13)*sqrt(3) - 59644114671451/2098482808511232)))*log(196732
763297928*(217700288287626155772963*I*sqrt(13)*sqrt(3) + 8063857253832070208357424*sqrt(-537508757/26903625750
144*I*sqrt(13)*sqrt(3) - 59644114671451/2098482808511232) - 2904532176689925771712)*(73/5408*I*sqrt(13)*sqrt(3
) - 1/2*sqrt(537508757/26903625750144*I*sqrt(13)*sqrt(3) - 59644114671451/2098482808511232))^2 - 5714166412079
54753670685205570612736*(-73/5408*I*sqrt(13)*sqrt(3) - 1/2*sqrt(-537508757/26903625750144*I*sqrt(13)*sqrt(3) -
59644114671451/2098482808511232))^2 + 2405601/2*(487774661427048094833332220336*(-73/5408*I*sqrt(13)*sqrt(3)
- 1/2*sqrt(-537508757/26903625750144*I*sqrt(13)*sqrt(3) - 59644114671451/2098482808511232))^2 + 32817075305772
68651899)*(-73*I*sqrt(13)*sqrt(3) + 2704*sqrt(537508757/26903625750144*I*sqrt(13)*sqrt(3) - 59644114671451/209
8482808511232)) + 1/208*sqrt(-393465526595856*(73/5408*I*sqrt(13)*sqrt(3) - 1/2*sqrt(537508757/26903625750144*
I*sqrt(13)*sqrt(3) - 59644114671451/2098482808511232))^2 - 393465526595856*(-73/5408*I*sqrt(13)*sqrt(3) - 1/2*
sqrt(-537508757/26903625750144*I*sqrt(13)*sqrt(3) - 59644114671451/2098482808511232))^2 - 3731087151/416*(73*I
*sqrt(13)*sqrt(3) + 2704*sqrt(-537508757/26903625750144*I*sqrt(13)*sqrt(3) - 59644114671451/2098482808511232))
*(-73*I*sqrt(13)*sqrt(3) + 2704*sqrt(537508757/26903625750144*I*sqrt(13)*sqrt(3) - 59644114671451/209848280851
1232)) - 14911625619311)*(4653*(2982195729967481585931*sqrt(20163)*(73*I*sqrt(13)*sqrt(3) + 2704*sqrt(-5375087
57/26903625750144*I*sqrt(13)*sqrt(3) - 59644114671451/2098482808511232)) - 2904532176689925771712*sqrt(20163))
*(-73*I*sqrt(13)*sqrt(3) + 2704*sqrt(537508757/26903625750144*I*sqrt(13)*sqrt(3) - 59644114671451/209848280851
1232)) - 13514788218138224615775936*sqrt(20163)*(73*I*sqrt(13)*sqrt(3) + 2704*sqrt(-537508757/26903625750144*I
*sqrt(13)*sqrt(3) - 59644114671451/2098482808511232)) - 2730380665440287518379968*sqrt(20163)) + 5762969609602
87187378212699827/2*I*sqrt(13)*sqrt(3) + 452791594208982239644601709645795*x + 1067333549614120927856635027624
8*sqrt(-537508757/26903625750144*I*sqrt(13)*sqrt(3) - 59644114671451/2098482808511232) + 231741133996538382702
540710757432) + (8*sqrt(20163)*(8*x^4 - 15*x^3 + 8*x^2 + 24*x + 8)*sqrt(-393465526595856*(73/5408*I*sqrt(13)*s
qrt(3) - 1/2*sqrt(537508757/26903625750144*I*sqrt(13)*sqrt(3) - 59644114671451/2098482808511232))^2 - 39346552
6595856*(-73/5408*I*sqrt(13)*sqrt(3) - 1/2*sqrt(-537508757/26903625750144*I*sqrt(13)*sqrt(3) - 59644114671451/
2098482808511232))^2 - 3731087151/416*(73*I*sqrt(13)*sqrt(3) + 2704*sqrt(-537508757/26903625750144*I*sqrt(13)*
sqrt(3) - 59644114671451/2098482808511232))*(-73*I*sqrt(13)*sqrt(3) + 2704*sqrt(537508757/26903625750144*I*sqr
t(13)*sqrt(3) - 59644114671451/2098482808511232)) - 14911625619311) + 2405601*(8*x^4 - 15*x^3 + 8*x^2 + 24*x +
8)*(73*I*sqrt(13)*sqrt(3) + 2704*sqrt(-537508757/26903625750144*I*sqrt(13)*sqrt(3) - 59644114671451/209848280
8511232)) + 2405601*(8*x^4 - 15*x^3 + 8*x^2 + 24*x + 8)*(-73*I*sqrt(13)*sqrt(3) + 2704*sqrt(537508757/26903625
750144*I*sqrt(13)*sqrt(3) - 59644114671451/2098482808511232)))*log(196732763297928*(217700288287626155772963*I
*sqrt(13)*sqrt(3) + 8063857253832070208357424*sqrt(-537508757/26903625750144*I*sqrt(13)*sqrt(3) - 596441146714
51/2098482808511232) - 2904532176689925771712)*(73/5408*I*sqrt(13)*sqrt(3) - 1/2*sqrt(537508757/26903625750144
*I*sqrt(13)*sqrt(3) - 59644114671451/2098482808511232))^2 - 571416641207954753670685205570612736*(-73/5408*I*s
qrt(13)*sqrt(3) - 1/2*sqrt(-537508757/26903625750144*I*sqrt(13)*sqrt(3) - 59644114671451/2098482808511232))^2
+ 2405601/2*(487774661427048094833332220336*(-73/5408*I*sqrt(13)*sqrt(3) - 1/2*sqrt(-537508757/26903625750144*
I*sqrt(13)*sqrt(3) - 59644114671451/2098482808511232))^2 + 3281707530577268651899)*(-73*I*sqrt(13)*sqrt(3) + 2
704*sqrt(537508757/26903625750144*I*sqrt(13)*sqrt(3) - 59644114671451/2098482808511232)) - 1/208*sqrt(-3934655
26595856*(73/5408*I*sqrt(13)*sqrt(3) - 1/2*sqrt(537508757/26903625750144*I*sqrt(13)*sqrt(3) - 59644114671451/2
098482808511232))^2 - 393465526595856*(-73/5408*I*sqrt(13)*sqrt(3) - 1/2*sqrt(-537508757/26903625750144*I*sqrt
(13)*sqrt(3) - 59644114671451/2098482808511232))^2 - 3731087151/416*(73*I*sqrt(13)*sqrt(3) + 2704*sqrt(-537508
757/26903625750144*I*sqrt(13)*sqrt(3) - 59644114671451/2098482808511232))*(-73*I*sqrt(13)*sqrt(3) + 2704*sqrt(
537508757/26903625750144*I*sqrt(13)*sqrt(3) - 59644114671451/2098482808511232)) - 14911625619311)*(4653*(29821
95729967481585931*sqrt(20163)*(73*I*sqrt(13)*sqrt(3) + 2704*sqrt(-537508757/26903625750144*I*sqrt(13)*sqrt(3)
- 59644114671451/2098482808511232)) - 2904532176689925771712*sqrt(20163))*(-73*I*sqrt(13)*sqrt(3) + 2704*sqrt(
537508757/26903625750144*I*sqrt(13)*sqrt(3) - 59644114671451/2098482808511232)) - 13514788218138224615775936*s
qrt(20163)*(73*I*sqrt(13)*sqrt(3) + 2704*sqrt(-537508757/26903625750144*I*sqrt(13)*sqrt(3) - 59644114671451/20
98482808511232)) - 2730380665440287518379968*sqrt(20163)) + 576296960960287187378212699827/2*I*sqrt(13)*sqrt(3
) + 452791594208982239644601709645795*x + 10673335496141209278566350276248*sqrt(-537508757/26903625750144*I*sq
rt(13)*sqrt(3) - 59644114671451/2098482808511232) + 231741133996538382702540710757432) + 14361379032*x + 11757
125952)/(8*x^4 - 15*x^3 + 8*x^2 + 24*x + 8)
Sympy [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 3839 vs. \(2 (292) = 584\).
Time = 2.23 (sec) , antiderivative size = 3839, normalized size of antiderivative = 10.49
\[
\int \frac {1}{\left (8+24 x+8 x^2-15 x^3+8 x^4\right )^2} \, dx=\text {Too large to display}
\]
[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
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)) + 59287739659625666461815501555467914*sqrt(40326)*sqrt(-59644114671451 + 262317043829
5*sqrt(517))/(-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 + 262317043829
5*sqrt(517))*sqrt(59653665894623 + 7869511314885*sqrt(517) + 4*sqrt(-7120427417275887*sqrt(40326)*sqrt(-596441
14671451 + 2623170438295*sqrt(517)) + 6263621568587150042935*sqrt(517) + 3557579971691991294769382675)) + 9940
65243322493861977*sqrt(40326)*sqrt(59653665894623 + 7869511314885*sqrt(517) + 4*sqrt(-7120427417275887*sqrt(40
326)*sqrt(-59644114671451 + 2623170438295*sqrt(517)) + 6263621568587150042935*sqrt(517) + 35575799716919912947
69382675))*sqrt(-7120427417275887*sqrt(40326)*sqrt(-59644114671451 + 2623170438295*sqrt(517)) + 62636215685871
50042935*sqrt(517) + 3557579971691991294769382675)) + 721019529648624138729760776730387270795368/(-59292486929
118917272637172801533436*sqrt(40326)*sqrt(59653665894623 + 7869511314885*sqrt(517) + 4*sqrt(-7120427417275887*
sqrt(40326)*sqrt(-59644114671451 + 2623170438295*sqrt(517)) + 6263621568587150042935*sqrt(517) + 3557579971691
991294769382675)) + 2329048502925820708785386304*sqrt(-59644114671451 + 2623170438295*sqrt(517))*sqrt(59653665
894623 + 7869511314885*sqrt(517) + 4*sqrt(-7120427417275887*sqrt(40326)*sqrt(-59644114671451 + 2623170438295*s
qrt(517)) + 6263621568587150042935*sqrt(517) + 3557579971691991294769382675)) + 994065243322493861977*sqrt(403
26)*sqrt(59653665894623 + 7869511314885*sqrt(517) + 4*sqrt(-7120427417275887*sqrt(40326)*sqrt(-59644114671451
+ 2623170438295*sqrt(517)) + 6263621568587150042935*sqrt(517) + 3557579971691991294769382675))*sqrt(-712042741
7275887*sqrt(40326)*sqrt(-59644114671451 + 2623170438295*sqrt(517)) + 6263621568587150042935*sqrt(517) + 35575
79971691991294769382675))) - 2*sqrt(-sqrt(7120427417275887*sqrt(40326)*sqrt(-59644114671451 + 2623170438295*sq
rt(517)) + 6263621568587150042935*sqrt(517) + 3557579971691991294769382675)/4196965617022464 + 59653665894623/
16787862468089856 + 5073830635*sqrt(517)/10823895853056)*atan(7745677595169577846551420567648953584320*x/(2329
048502925820708785386304*sqrt(-59644114671451 + 2623170438295*sqrt(517))*sqrt(-4*sqrt(7120427417275887*sqrt(40
326)*sqrt(-59644114671451 + 2623170438295*sqrt(517)) + 6263621568587150042935*sqrt(517) + 35575799716919912947
69382675) + 59653665894623 + 7869511314885*sqrt(517)) + 59292486929118917272637172801533436*sqrt(40326)*sqrt(-
4*sqrt(7120427417275887*sqrt(40326)*sqrt(-59644114671451 + 2623170438295*sqrt(517)) + 6263621568587150042935*s
qrt(517) + 3557579971691991294769382675) + 59653665894623 + 7869511314885*sqrt(517)) + 994065243322493861977*s
qrt(40326)*sqrt(7120427417275887*sqrt(40326)*sqrt(-59644114671451 + 2623170438295*sqrt(517)) + 626362156858715
0042935*sqrt(517) + 3557579971691991294769382675)*sqrt(-4*sqrt(7120427417275887*sqrt(40326)*sqrt(-596441146714
51 + 2623170438295*sqrt(517)) + 6263621568587150042935*sqrt(517) + 3557579971691991294769382675) + 59653665894
623 + 7869511314885*sqrt(517))) - 721019529648624138729760776730387270795368/(2329048502925820708785386304*sqr
t(-59644114671451 + 2623170438295*sqrt(517))*sqrt(-4*sqrt(7120427417275887*sqrt(40326)*sqrt(-59644114671451 +
2623170438295*sqrt(517)) + 6263621568587150042935*sqrt(517) + 3557579971691991294769382675) + 59653665894623 +
7869511314885*sqrt(517)) + 59292486929118917272637172801533436*sqrt(40326)*sqrt(-4*sqrt(7120427417275887*sqrt
(40326)*sqrt(-59644114671451 + 2623170438295*sqrt(517)) + 6263621568587150042935*sqrt(517) + 35575799716919912
94769382675) + 59653665894623 + 7869511314885*sqrt(517)) + 994065243322493861977*sqrt(40326)*sqrt(712042741727
5887*sqrt(40326)*sqrt(-59644114671451 + 2623170438295*sqrt(517)) + 6263621568587150042935*sqrt(517) + 35575799
71691991294769382675)*sqrt(-4*sqrt(7120427417275887*sqrt(40326)*sqrt(-59644114671451 + 2623170438295*sqrt(517)
) + 6263621568587150042935*sqrt(517) + 3557579971691991294769382675) + 59653665894623 + 7869511314885*sqrt(517
))) - 2696261047060775175517112572266328310*sqrt(78)*sqrt(-59644114671451 + 2623170438295*sqrt(517))/(23290485
02925820708785386304*sqrt(-59644114671451 + 2623170438295*sqrt(517))*sqrt(-4*sqrt(7120427417275887*sqrt(40326)
*sqrt(-59644114671451 + 2623170438295*sqrt(517)) + 6263621568587150042935*sqrt(517) + 355757997169199129476938
2675) + 59653665894623 + 7869511314885*sqrt(517)) + 59292486929118917272637172801533436*sqrt(40326)*sqrt(-4*sq
rt(7120427417275887*sqrt(40326)*sqrt(-59644114671451 + 2623170438295*sqrt(517)) + 6263621568587150042935*sqrt(
517) + 3557579971691991294769382675) + 59653665894623 + 7869511314885*sqrt(517)) + 994065243322493861977*sqrt(
40326)*sqrt(7120427417275887*sqrt(40326)*sqrt(-59644114671451 + 2623170438295*sqrt(517)) + 6263621568587150042
935*sqrt(517) + 3557579971691991294769382675)*sqrt(-4*sqrt(7120427417275887*sqrt(40326)*sqrt(-59644114671451 +
2623170438295*sqrt(517)) + 6263621568587150042935*sqrt(517) + 3557579971691991294769382675) + 59653665894623
+ 7869511314885*sqrt(517))) - 6109491182230238698537149348154570111680*sqrt(517)/(2329048502925820708785386304
*sqrt(-59644114671451 + 2623170438295*sqrt(517))*sqrt(-4*sqrt(7120427417275887*sqrt(40326)*sqrt(-5964411467145
1 + 2623170438295*sqrt(517)) + 6263621568587150042935*sqrt(517) + 3557579971691991294769382675) + 596536658946
23 + 7869511314885*sqrt(517)) + 59292486929118917272637172801533436*sqrt(40326)*sqrt(-4*sqrt(7120427417275887*
sqrt(40326)*sqrt(-59644114671451 + 2623170438295*sqrt(517)) + 6263621568587150042935*sqrt(517) + 3557579971691
991294769382675) + 59653665894623 + 7869511314885*sqrt(517)) + 994065243322493861977*sqrt(40326)*sqrt(71204274
17275887*sqrt(40326)*sqrt(-59644114671451 + 2623170438295*sqrt(517)) + 6263621568587150042935*sqrt(517) + 3557
579971691991294769382675)*sqrt(-4*sqrt(7120427417275887*sqrt(40326)*sqrt(-59644114671451 + 2623170438295*sqrt(
517)) + 6263621568587150042935*sqrt(517) + 3557579971691991294769382675) + 59653665894623 + 7869511314885*sqrt
(517))) + 4658097005851641417570772608*sqrt(7120427417275887*sqrt(40326)*sqrt(-59644114671451 + 2623170438295*
sqrt(517)) + 6263621568587150042935*sqrt(517) + 3557579971691991294769382675)/(2329048502925820708785386304*sq
rt(-59644114671451 + 2623170438295*sqrt(517))*sqrt(-4*sqrt(7120427417275887*sqrt(40326)*sqrt(-59644114671451 +
2623170438295*sqrt(517)) + 6263621568587150042935*sqrt(517) + 3557579971691991294769382675) + 59653665894623
+ 7869511314885*sqrt(517)) + 59292486929118917272637172801533436*sqrt(40326)*sqrt(-4*sqrt(7120427417275887*sqr
t(40326)*sqrt(-59644114671451 + 2623170438295*sqrt(517)) + 6263621568587150042935*sqrt(517) + 3557579971691991
294769382675) + 59653665894623 + 7869511314885*sqrt(517)) + 994065243322493861977*sqrt(40326)*sqrt(71204274172
75887*sqrt(40326)*sqrt(-59644114671451 + 2623170438295*sqrt(517)) + 6263621568587150042935*sqrt(517) + 3557579
971691991294769382675)*sqrt(-4*sqrt(7120427417275887*sqrt(40326)*sqrt(-59644114671451 + 2623170438295*sqrt(517
)) + 6263621568587150042935*sqrt(517) + 3557579971691991294769382675) + 59653665894623 + 7869511314885*sqrt(51
7))) + 59287739659625666461815501555467914*sqrt(40326)*sqrt(-59644114671451 + 2623170438295*sqrt(517))/(232904
8502925820708785386304*sqrt(-59644114671451 + 2623170438295*sqrt(517))*sqrt(-4*sqrt(7120427417275887*sqrt(4032
6)*sqrt(-59644114671451 + 2623170438295*sqrt(517)) + 6263621568587150042935*sqrt(517) + 3557579971691991294769
382675) + 59653665894623 + 7869511314885*sqrt(517)) + 59292486929118917272637172801533436*sqrt(40326)*sqrt(-4*
sqrt(7120427417275887*sqrt(40326)*sqrt(-59644114671451 + 2623170438295*sqrt(517)) + 6263621568587150042935*sqr
t(517) + 3557579971691991294769382675) + 59653665894623 + 7869511314885*sqrt(517)) + 994065243322493861977*sqr
t(40326)*sqrt(7120427417275887*sqrt(40326)*sqrt(-59644114671451 + 2623170438295*sqrt(517)) + 62636215685871500
42935*sqrt(517) + 3557579971691991294769382675)*sqrt(-4*sqrt(7120427417275887*sqrt(40326)*sqrt(-59644114671451
+ 2623170438295*sqrt(517)) + 6263621568587150042935*sqrt(517) + 3557579971691991294769382675) + 5965366589462
3 + 7869511314885*sqrt(517))) + 2982195729967481585931*sqrt(40326)*sqrt(-59644114671451 + 2623170438295*sqrt(5
17))*sqrt(7120427417275887*sqrt(40326)*sqrt(-59644114671451 + 2623170438295*sqrt(517)) + 626362156858715004293
5*sqrt(517) + 3557579971691991294769382675)/(2329048502925820708785386304*sqrt(-59644114671451 + 2623170438295
*sqrt(517))*sqrt(-4*sqrt(7120427417275887*sqrt(40326)*sqrt(-59644114671451 + 2623170438295*sqrt(517)) + 626362
1568587150042935*sqrt(517) + 3557579971691991294769382675) + 59653665894623 + 7869511314885*sqrt(517)) + 59292
486929118917272637172801533436*sqrt(40326)*sqrt(-4*sqrt(7120427417275887*sqrt(40326)*sqrt(-59644114671451 + 26
23170438295*sqrt(517)) + 6263621568587150042935*sqrt(517) + 3557579971691991294769382675) + 59653665894623 + 7
869511314885*sqrt(517)) + 994065243322493861977*sqrt(40326)*sqrt(7120427417275887*sqrt(40326)*sqrt(-5964411467
1451 + 2623170438295*sqrt(517)) + 6263621568587150042935*sqrt(517) + 3557579971691991294769382675)*sqrt(-4*sqr
t(7120427417275887*sqrt(40326)*sqrt(-59644114671451 + 2623170438295*sqrt(517)) + 6263621568587150042935*sqrt(5
17) + 3557579971691991294769382675) + 59653665894623 + 7869511314885*sqrt(517))))
Maxima [F]
\[
\int \frac {1}{\left (8+24 x+8 x^2-15 x^3+8 x^4\right )^2} \, dx=\int { \frac {1}{{\left (8 \, x^{4} - 15 \, x^{3} + 8 \, x^{2} + 24 \, x + 8\right )}^{2}} \,d x }
\]
[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)
Giac [F]
\[
\int \frac {1}{\left (8+24 x+8 x^2-15 x^3+8 x^4\right )^2} \, dx=\int { \frac {1}{{\left (8 \, x^{4} - 15 \, x^{3} + 8 \, x^{2} + 24 \, x + 8\right )}^{2}} \,d x }
\]
[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)
Mupad [B] (verification not implemented)
Time = 0.15 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.49
\[
\int \frac {1}{\left (8+24 x+8 x^2-15 x^3+8 x^4\right )^2} \, dx=\frac {\frac {2455\,x^3}{80652}-\frac {1429\,x^2}{19552}+\frac {89033\,x}{1290432}+\frac {3037}{53768}}{x^4-\frac {15\,x^3}{8}+x^2+3\,x+1}+\left (\sum _{k=1}^4\ln \left (\frac {2146659825\,\mathrm {root}\left (z^4+\frac {14911625619311\,z^2}{524620702127808}+\frac {39238139261\,z}{3730636104019968}+\frac {43023440}{44204510553294663},z,k\right )}{2960381771776}+\frac {2222183\,x}{338246745408}+\frac {\mathrm {root}\left (z^4+\frac {14911625619311\,z^2}{524620702127808}+\frac {39238139261\,z}{3730636104019968}+\frac {43023440}{44204510553294663},z,k\right )\,x\,924124364159}{26643435945984}-\frac {{\mathrm {root}\left (z^4+\frac {14911625619311\,z^2}{524620702127808}+\frac {39238139261\,z}{3730636104019968}+\frac {43023440}{44204510553294663},z,k\right )}^2\,x\,72451101}{8470528}-\frac {{\mathrm {root}\left (z^4+\frac {14911625619311\,z^2}{524620702127808}+\frac {39238139261\,z}{3730636104019968}+\frac {43023440}{44204510553294663},z,k\right )}^3\,x\,95745}{256}+\frac {389551\,{\mathrm {root}\left (z^4+\frac {14911625619311\,z^2}{524620702127808}+\frac {39238139261\,z}{3730636104019968}+\frac {43023440}{44204510553294663},z,k\right )}^2}{264704}-\frac {100737\,{\mathrm {root}\left (z^4+\frac {14911625619311\,z^2}{524620702127808}+\frac {39238139261\,z}{3730636104019968}+\frac {43023440}{44204510553294663},z,k\right )}^3}{512}+\frac {271033}{624455529984}\right )\,\mathrm {root}\left (z^4+\frac {14911625619311\,z^2}{524620702127808}+\frac {39238139261\,z}{3730636104019968}+\frac {43023440}{44204510553294663},z,k\right )\right )
\]
[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)