3.1.61 \(\int \frac {1}{8+24 x+8 x^2-15 x^3+8 x^4} \, dx\) [61]

Optimal. Leaf size=263 \[ -\frac {1}{4} \sqrt {\frac {5167+235 \sqrt {517}}{40326}} \tan ^{-1}\left (\frac {6-\sqrt {2 \left (19+\sqrt {517}\right )}+\frac {8}{x}}{\sqrt {2 \left (-19+\sqrt {517}\right )}}\right )-\frac {1}{4} \sqrt {\frac {5167+235 \sqrt {517}}{40326}} \tan ^{-1}\left (\frac {6+\sqrt {2 \left (19+\sqrt {517}\right )}+\frac {8}{x}}{\sqrt {2 \left (-19+\sqrt {517}\right )}}\right )+\frac {1}{4} \sqrt {\frac {3}{13}} \tan ^{-1}\left (\frac {8+12 x-5 x^2}{\sqrt {39} x^2}\right )-\frac {1}{8} \sqrt {\frac {-5167+235 \sqrt {517}}{40326}} \log \left (\sqrt {517}-\sqrt {2 \left (19+\sqrt {517}\right )} \left (3+\frac {4}{x}\right )+\left (3+\frac {4}{x}\right )^2\right )+\frac {1}{8} \sqrt {\frac {-5167+235 \sqrt {517}}{40326}} \log \left (\sqrt {517}+\sqrt {2 \left (19+\sqrt {517}\right )} \left (3+\frac {4}{x}\right )+\left (3+\frac {4}{x}\right )^2\right ) \]

[Out]

1/52*arctan(1/39*(-5*x^2+12*x+8)/x^2*39^(1/2))*39^(1/2)-1/322608*ln((3+4/x)^2+517^(1/2)-(3+4/x)*(38+2*517^(1/2
))^(1/2))*(-208364442+9476610*517^(1/2))^(1/2)+1/322608*ln((3+4/x)^2+517^(1/2)+(3+4/x)*(38+2*517^(1/2))^(1/2))
*(-208364442+9476610*517^(1/2))^(1/2)-1/161304*arctan((6+8/x-(38+2*517^(1/2))^(1/2))/(-38+2*517^(1/2))^(1/2))*
(208364442+9476610*517^(1/2))^(1/2)-1/161304*arctan((6+8/x+(38+2*517^(1/2))^(1/2))/(-38+2*517^(1/2))^(1/2))*(2
08364442+9476610*517^(1/2))^(1/2)

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Rubi [A]
time = 0.39, antiderivative size = 263, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 9, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {2094, 12, 1687, 1183, 648, 632, 210, 642, 1121} \begin {gather*} \frac {1}{4} \sqrt {\frac {3}{13}} \text {ArcTan}\left (\frac {-5 x^2+12 x+8}{\sqrt {39} x^2}\right )-\frac {1}{4} \sqrt {\frac {5167+235 \sqrt {517}}{40326}} \text {ArcTan}\left (\frac {\frac {8}{x}-\sqrt {2 \left (19+\sqrt {517}\right )}+6}{\sqrt {2 \left (\sqrt {517}-19\right )}}\right )-\frac {1}{4} \sqrt {\frac {5167+235 \sqrt {517}}{40326}} \text {ArcTan}\left (\frac {\frac {8}{x}+\sqrt {2 \left (19+\sqrt {517}\right )}+6}{\sqrt {2 \left (\sqrt {517}-19\right )}}\right )-\frac {1}{8} \sqrt {\frac {235 \sqrt {517}-5167}{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 )+\frac {1}{8} \sqrt {\frac {235 \sqrt {517}-5167}{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 ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(8 + 24*x + 8*x^2 - 15*x^3 + 8*x^4)^(-1),x]

[Out]

-1/4*(Sqrt[(5167 + 235*Sqrt[517])/40326]*ArcTan[(6 - Sqrt[2*(19 + Sqrt[517])] + 8/x)/Sqrt[2*(-19 + Sqrt[517])]
]) - (Sqrt[(5167 + 235*Sqrt[517])/40326]*ArcTan[(6 + Sqrt[2*(19 + Sqrt[517])] + 8/x)/Sqrt[2*(-19 + Sqrt[517])]
])/4 + (Sqrt[3/13]*ArcTan[(8 + 12*x - 5*x^2)/(Sqrt[39]*x^2)])/4 - (Sqrt[(-5167 + 235*Sqrt[517])/40326]*Log[Sqr
t[517] - Sqrt[2*(19 + Sqrt[517])]*(3 + 4/x) + (3 + 4/x)^2])/8 + (Sqrt[(-5167 + 235*Sqrt[517])/40326]*Log[Sqrt[
517] + Sqrt[2*(19 + Sqrt[517])]*(3 + 4/x) + (3 + 4/x)^2])/8

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 1121

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]

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 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 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*} \int \frac {1}{8+24 x+8 x^2-15 x^3+8 x^4} \, dx &=-\left (1024 \text {Subst}\left (\int \frac {(24-32 x)^2}{8 \left (2117632-2490368 x^2+1048576 x^4\right )} \, dx,x,\frac {3}{4}+\frac {1}{x}\right )\right )\\ &=-\left (128 \text {Subst}\left (\int \frac {(24-32 x)^2}{2117632-2490368 x^2+1048576 x^4} \, dx,x,\frac {3}{4}+\frac {1}{x}\right )\right )\\ &=-\left (128 \text {Subst}\left (\int -\frac {1536 x}{2117632-2490368 x^2+1048576 x^4} \, dx,x,\frac {3}{4}+\frac {1}{x}\right )\right )-128 \text {Subst}\left (\int \frac {576+1024 x^2}{2117632-2490368 x^2+1048576 x^4} \, dx,x,\frac {3}{4}+\frac {1}{x}\right )\\ &=196608 \text {Subst}\left (\int \frac {x}{2117632-2490368 x^2+1048576 x^4} \, dx,x,\frac {3}{4}+\frac {1}{x}\right )-\frac {\text {Subst}\left (\int \frac {144 \sqrt {2 \left (19+\sqrt {517}\right )}-\left (576-64 \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 )}{256 \sqrt {1034 \left (19+\sqrt {517}\right )}}-\frac {\text {Subst}\left (\int \frac {144 \sqrt {2 \left (19+\sqrt {517}\right )}+\left (576-64 \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 )}{256 \sqrt {1034 \left (19+\sqrt {517}\right )}}\\ &=98304 \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 (517+9 \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 )}{16544}-\frac {\left (517+9 \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 )}{16544}-\frac {1}{8} \sqrt {\frac {-5167+235 \sqrt {517}}{40326}} \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 )+\frac {1}{8} \sqrt {\frac {-5167+235 \sqrt {517}}{40326}} \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 )\\ &=-\frac {1}{8} \sqrt {\frac {-5167+235 \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 )+\frac {1}{8} \sqrt {\frac {-5167+235 \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 )-196608 \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 (517+9 \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 )}{8272}+\frac {\left (517+9 \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 )}{8272}\\ &=-\frac {1}{4} \sqrt {\frac {3}{13}} \tan ^{-1}\left (\frac {19-\left (3+\frac {4}{x}\right )^2}{2 \sqrt {39}}\right )-\frac {1}{4} \sqrt {\frac {5167+235 \sqrt {517}}{40326}} \tan ^{-1}\left (\frac {6+\sqrt {2 \left (19+\sqrt {517}\right )}+\frac {8}{x}}{\sqrt {2 \left (-19+\sqrt {517}\right )}}\right )-\frac {1}{4} \sqrt {\frac {5167+235 \sqrt {517}}{40326}} \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 )-\frac {1}{8} \sqrt {\frac {-5167+235 \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 )+\frac {1}{8} \sqrt {\frac {-5167+235 \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 )\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 0.01, size = 55, normalized size = 0.21 \begin {gather*} \text {RootSum}\left [8+24 \text {$\#$1}+8 \text {$\#$1}^2-15 \text {$\#$1}^3+8 \text {$\#$1}^4\&,\frac {\log (x-\text {$\#$1})}{24+16 \text {$\#$1}-45 \text {$\#$1}^2+32 \text {$\#$1}^3}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(8 + 24*x + 8*x^2 - 15*x^3 + 8*x^4)^(-1),x]

[Out]

RootSum[8 + 24*#1 + 8*#1^2 - 15*#1^3 + 8*#1^4 & , Log[x - #1]/(24 + 16*#1 - 45*#1^2 + 32*#1^3) & ]

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.02, size = 49, normalized size = 0.19

method	result	size
default	\(\munderset {\textit {\_R} =\RootOf \left (8 \textit {\_Z}^{4}-15 \textit {\_Z}^{3}+8 \textit {\_Z}^{2}+24 \textit {\_Z} +8\right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{32 \textit {\_R}^{3}-45 \textit {\_R}^{2}+16 \textit {\_R} +24}\)	\(49\)
risch	\(\munderset {\textit {\_R} =\RootOf \left (8 \textit {\_Z}^{4}-15 \textit {\_Z}^{3}+8 \textit {\_Z}^{2}+24 \textit {\_Z} +8\right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{32 \textit {\_R}^{3}-45 \textit {\_R}^{2}+16 \textit {\_R} +24}\)	\(49\)

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),x,method=_RETURNVERBOSE)

[Out]

sum(1/(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))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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),x, algorithm="maxima")

[Out]

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

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Fricas [C] Result contains complex when optimal does not.
time = 1.18, size = 1297, normalized size = 4.93 \begin {gather*} \text {Too large to display} \end {gather*}

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),x, algorithm="fricas")

[Out]

-1/104*(-I*sqrt(13)*sqrt(3) + 52*sqrt(109/161304*I*sqrt(13)*sqrt(3) - 5167/322608))*log(37895495846208*(1/104*
I*sqrt(13)*sqrt(3) - 1/2*sqrt(109/161304*I*sqrt(13)*sqrt(3) - 5167/322608))^3 - 537872704512*(1/104*I*sqrt(13)
*sqrt(3) - 1/2*sqrt(109/161304*I*sqrt(13)*sqrt(3) - 5167/322608))^2 + 5614027117*I*sqrt(13)*sqrt(3) + 17891339
60*x - 291929410084*sqrt(109/161304*I*sqrt(13)*sqrt(3) - 5167/322608) + 2270349121) - 1/104*(I*sqrt(13)*sqrt(3
) + 52*sqrt(-109/161304*I*sqrt(13)*sqrt(3) - 5167/322608))*log(-4736936980776*(1/104*I*sqrt(13)*sqrt(3) - 1/2*
sqrt(109/161304*I*sqrt(13)*sqrt(3) - 5167/322608))^3 + 20163*(-1/104*I*sqrt(13)*sqrt(3) - 1/2*sqrt(-109/161304
*I*sqrt(13)*sqrt(3) - 5167/322608))^2*(-2258963*I*sqrt(13)*sqrt(3) + 117466076*sqrt(109/161304*I*sqrt(13)*sqrt
(3) - 5167/322608) - 3334528) + 517*(88099557*(1/104*I*sqrt(13)*sqrt(3) - 1/2*sqrt(109/161304*I*sqrt(13)*sqrt(
3) - 5167/322608))^2 - 16507)*(I*sqrt(13)*sqrt(3) + 52*sqrt(-109/161304*I*sqrt(13)*sqrt(3) - 5167/322608)) - 5
545754165/8*I*sqrt(13)*sqrt(3) + 223641745*x + 72094804145/2*sqrt(109/161304*I*sqrt(13)*sqrt(3) - 5167/322608)
- 916562824) + 1/80652*(sqrt(40326)*sqrt(-120978*(1/104*I*sqrt(13)*sqrt(3) - 1/2*sqrt(109/161304*I*sqrt(13)*s
qrt(3) - 5167/322608))^2 - 120978*(-1/104*I*sqrt(13)*sqrt(3) - 1/2*sqrt(-109/161304*I*sqrt(13)*sqrt(3) - 5167/
322608))^2 - 1551/208*(I*sqrt(13)*sqrt(3) + 52*sqrt(-109/161304*I*sqrt(13)*sqrt(3) - 5167/322608))*(-I*sqrt(13
)*sqrt(3) + 52*sqrt(109/161304*I*sqrt(13)*sqrt(3) - 5167/322608)) - 2455) + 20163*sqrt(109/161304*I*sqrt(13)*s
qrt(3) - 5167/322608) + 20163*sqrt(-109/161304*I*sqrt(13)*sqrt(3) - 5167/322608))*log(-60489/2*(-1/104*I*sqrt(
13)*sqrt(3) - 1/2*sqrt(-109/161304*I*sqrt(13)*sqrt(3) - 5167/322608))^2*(-2258963*I*sqrt(13)*sqrt(3) + 1174660
76*sqrt(109/161304*I*sqrt(13)*sqrt(3) - 5167/322608) - 3334528) - 1551/2*(88099557*(1/104*I*sqrt(13)*sqrt(3) -
1/2*sqrt(109/161304*I*sqrt(13)*sqrt(3) - 5167/322608))^2 - 16507)*(I*sqrt(13)*sqrt(3) + 52*sqrt(-109/161304*I
*sqrt(13)*sqrt(3) - 5167/322608)) + 100851132096*(1/104*I*sqrt(13)*sqrt(3) - 1/2*sqrt(109/161304*I*sqrt(13)*sq
rt(3) - 5167/322608))^2 + 1/416*(3*(2258963*sqrt(40326)*(-I*sqrt(13)*sqrt(3) + 52*sqrt(109/161304*I*sqrt(13)*s
qrt(3) - 5167/322608)) - 3334528*sqrt(40326))*(I*sqrt(13)*sqrt(3) + 52*sqrt(-109/161304*I*sqrt(13)*sqrt(3) - 5
167/322608)) - 10003584*sqrt(40326)*(-I*sqrt(13)*sqrt(3) + 52*sqrt(109/161304*I*sqrt(13)*sqrt(3) - 5167/322608
)) + 13733824*sqrt(40326))*sqrt(-120978*(1/104*I*sqrt(13)*sqrt(3) - 1/2*sqrt(109/161304*I*sqrt(13)*sqrt(3) - 5
167/322608))^2 - 120978*(-1/104*I*sqrt(13)*sqrt(3) - 1/2*sqrt(-109/161304*I*sqrt(13)*sqrt(3) - 5167/322608))^2
- 1551/208*(I*sqrt(13)*sqrt(3) + 52*sqrt(-109/161304*I*sqrt(13)*sqrt(3) - 5167/322608))*(-I*sqrt(13)*sqrt(3)
+ 52*sqrt(109/161304*I*sqrt(13)*sqrt(3) - 5167/322608)) - 2455) - 25602357/2*I*sqrt(13)*sqrt(3) + 670925235*x
+ 665661282*sqrt(109/161304*I*sqrt(13)*sqrt(3) - 5167/322608) + 320161368) - 1/80652*(sqrt(40326)*sqrt(-120978
*(1/104*I*sqrt(13)*sqrt(3) - 1/2*sqrt(109/161304*I*sqrt(13)*sqrt(3) - 5167/322608))^2 - 120978*(-1/104*I*sqrt(
13)*sqrt(3) - 1/2*sqrt(-109/161304*I*sqrt(13)*sqrt(3) - 5167/322608))^2 - 1551/208*(I*sqrt(13)*sqrt(3) + 52*sq
rt(-109/161304*I*sqrt(13)*sqrt(3) - 5167/322608))*(-I*sqrt(13)*sqrt(3) + 52*sqrt(109/161304*I*sqrt(13)*sqrt(3)
- 5167/322608)) - 2455) - 20163*sqrt(109/161304*I*sqrt(13)*sqrt(3) - 5167/322608) - 20163*sqrt(-109/161304*I*
sqrt(13)*sqrt(3) - 5167/322608))*log(-60489/2*(-1/104*I*sqrt(13)*sqrt(3) - 1/2*sqrt(-109/161304*I*sqrt(13)*sqr
t(3) - 5167/322608))^2*(-2258963*I*sqrt(13)*sqrt(3) + 117466076*sqrt(109/161304*I*sqrt(13)*sqrt(3) - 5167/3226
08) - 3334528) - 1551/2*(88099557*(1/104*I*sqrt(13)*sqrt(3) - 1/2*sqrt(109/161304*I*sqrt(13)*sqrt(3) - 5167/32
2608))^2 - 16507)*(I*sqrt(13)*sqrt(3) + 52*sqrt(-109/161304*I*sqrt(13)*sqrt(3) - 5167/322608)) + 100851132096*
(1/104*I*sqrt(13)*sqrt(3) - 1/2*sqrt(109/161304*I*sqrt(13)*sqrt(3) - 5167/322608))^2 - 1/416*(3*(2258963*sqrt(
40326)*(-I*sqrt(13)*sqrt(3) + 52*sqrt(109/161304*I*sqrt(13)*sqrt(3) - 5167/322608)) - 3334528*sqrt(40326))*(I*
sqrt(13)*sqrt(3) + 52*sqrt(-109/161304*I*sqrt(13)*sqrt(3) - 5167/322608)) - 10003584*sqrt(40326)*(-I*sqrt(13)*
sqrt(3) + 52*sqrt(109/161304*I*sqrt(13)*sqrt(3) - 5167/322608)) + 13733824*sqrt(40326))*sqrt(-120978*(1/104*I*
sqrt(13)*sqrt(3) - 1/2*sqrt(109/161304*I*sqrt(13)*sqrt(3) - 5167/322608))^2 - 120978*(-1/104*I*sqrt(13)*sqrt(3
) - 1/2*sqrt(-109/161304*I*sqrt(13)*sqrt(3) - 5167/322608))^2 - 1551/208*(I*sqrt(13)*sqrt(3) + 52*sqrt(-109/16
1304*I*sqrt(13)*sqrt(3) - 5167/322608))*(-I*sqrt(13)*sqrt(3) + 52*sqrt(109/161304*I*sqrt(13)*sqrt(3) - 5167/32
2608)) - 2455) - 25602357/2*I*sqrt(13)*sqrt(3) + 670925235*x + 665661282*sqrt(109/161304*I*sqrt(13)*sqrt(3) -
5167/322608) + 320161368)

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Sympy [A]
time = 1.30, size = 41, normalized size = 0.16 \begin {gather*} \operatorname {RootSum} {\left (50326848 t^{4} + 765960 t^{2} + 12753 t + 64, \left ( t \mapsto t \log {\left (\frac {100785893208 t^{3}}{4758335} - \frac {1430512512 t^{2}}{4758335} + \frac {72982352521 t}{223641745} + x + \frac {2270349121}{1789133960} \right )} \right )\right )} \end {gather*}

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),x)

[Out]

RootSum(50326848*_t**4 + 765960*_t**2 + 12753*_t + 64, Lambda(_t, _t*log(100785893208*_t**3/4758335 - 14305125
12*_t**2/4758335 + 72982352521*_t/223641745 + x + 2270349121/1789133960)))

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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),x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 0.41, size = 123, normalized size = 0.47 \begin {gather*} \sum _{k=1}^4\ln \left (-\frac {\mathrm {root}\left (z^4+\frac {2455\,z^2}{161304}+\frac {109\,z}{430144}+\frac {1}{786357},z,k\right )\,\left (2184\,\mathrm {root}\left (z^4+\frac {2455\,z^2}{161304}+\frac {109\,z}{430144}+\frac {1}{786357},z,k\right )+256\,x+\mathrm {root}\left (z^4+\frac {2455\,z^2}{161304}+\frac {109\,z}{430144}+\frac {1}{786357},z,k\right )\,x\,38259+{\mathrm {root}\left (z^4+\frac {2455\,z^2}{161304}+\frac {109\,z}{430144}+\frac {1}{786357},z,k\right )}^2\,x\,1531920+805896\,{\mathrm {root}\left (z^4+\frac {2455\,z^2}{161304}+\frac {109\,z}{430144}+\frac {1}{786357},z,k\right )}^2-120\right )}{4096}\right )\,\mathrm {root}\left (z^4+\frac {2455\,z^2}{161304}+\frac {109\,z}{430144}+\frac {1}{786357},z,k\right ) \end {gather*}

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),x)

[Out]

symsum(log(-(root(z^4 + (2455*z^2)/161304 + (109*z)/430144 + 1/786357, z, k)*(2184*root(z^4 + (2455*z^2)/16130
4 + (109*z)/430144 + 1/786357, z, k) + 256*x + 38259*root(z^4 + (2455*z^2)/161304 + (109*z)/430144 + 1/786357,
z, k)*x + 1531920*root(z^4 + (2455*z^2)/161304 + (109*z)/430144 + 1/786357, z, k)^2*x + 805896*root(z^4 + (24
55*z^2)/161304 + (109*z)/430144 + 1/786357, z, k)^2 - 120))/4096)*root(z^4 + (2455*z^2)/161304 + (109*z)/43014
4 + 1/786357, z, k), k, 1, 4)

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