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

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

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 1.1466, antiderivative size = 263, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 9, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.409 \[ \frac{1}{4} \sqrt{\frac{3}{13}} \tan ^{-1}\left (\frac{-5 x^2+12 x+8}{\sqrt{39} x^2}\right )-\frac{1}{8} \sqrt{\frac{235 \sqrt{517}-5167}{40326}} \log \left (\left (\frac{4}{x}+3\right )^2-\sqrt{2 \left (19+\sqrt{517}\right )} \left (\frac{4}{x}+3\right )+\sqrt{517}\right )+\frac{1}{8} \sqrt{\frac{235 \sqrt{517}-5167}{40326}} \log \left (\left (\frac{4}{x}+3\right )^2+\sqrt{2 \left (19+\sqrt{517}\right )} \left (\frac{4}{x}+3\right )+\sqrt{517}\right )-\frac{1}{4} \sqrt{\frac{5167+235 \sqrt{517}}{40326}} \tan ^{-1}\left (\frac{\frac{8}{x}-\sqrt{2 \left (19+\sqrt{517}\right )}+6}{\sqrt{2 \left (\sqrt{517}-19\right )}}\right )-\frac{1}{4} \sqrt{\frac{5167+235 \sqrt{517}}{40326}} \tan ^{-1}\left (\frac{\frac{8}{x}+\sqrt{2 \left (19+\sqrt{517}\right )}+6}{\sqrt{2 \left (\sqrt{517}-19\right )}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 89.2643, size = 372, normalized size = 1.41 \[ \frac{\sqrt{1034} \left (- 32 \sqrt{517} + 288\right ) \log{\left (\sqrt{2} \left (- \frac{3}{16} - \frac{1}{4 x}\right ) \sqrt{19 + \sqrt{517}} + \left (\frac{3}{4} + \frac{1}{x}\right )^{2} + \frac{\sqrt{517}}{16} \right )}}{264704 \sqrt{19 + \sqrt{517}}} - \frac{\sqrt{1034} \left (- 32 \sqrt{517} + 288\right ) \log{\left (\sqrt{2} \left (\frac{3}{16} + \frac{1}{4 x}\right ) \sqrt{19 + \sqrt{517}} + \left (\frac{3}{4} + \frac{1}{x}\right )^{2} + \frac{\sqrt{517}}{16} \right )}}{264704 \sqrt{19 + \sqrt{517}}} + \frac{\sqrt{39} \operatorname{atan}{\left (\sqrt{39} \left (\frac{8 \left (\frac{3}{4} + \frac{1}{x}\right )^{2}}{39} - \frac{19}{78}\right ) \right )}}{52} - \frac{\sqrt{517} \left (- \frac{\sqrt{2} \sqrt{19 + \sqrt{517}} \left (- 64 \sqrt{517} + 576\right )}{8} + 144 \sqrt{2} \sqrt{19 + \sqrt{517}}\right ) \operatorname{atan}{\left (\frac{\sqrt{2} \left (3 + \frac{\sqrt{38 + 2 \sqrt{517}}}{2} + \frac{4}{x}\right )}{\sqrt{-19 + \sqrt{517}}} \right )}}{33088 \sqrt{-19 + \sqrt{517}} \sqrt{19 + \sqrt{517}}} - \frac{\sqrt{517} \left (- \frac{\sqrt{2} \sqrt{19 + \sqrt{517}} \left (- 64 \sqrt{517} + 576\right )}{8} + 144 \sqrt{2} \sqrt{19 + \sqrt{517}}\right ) \operatorname{atan}{\left (\frac{\sqrt{2} \left (- \frac{\sqrt{38 + 2 \sqrt{517}}}{2} + 3 + \frac{4}{x}\right )}{\sqrt{-19 + \sqrt{517}}} \right )}}{33088 \sqrt{-19 + \sqrt{517}} \sqrt{19 + \sqrt{517}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(8*x**4-15*x**3+8*x**2+24*x+8),x)

[Out]

sqrt(1034)*(-32*sqrt(517) + 288)*log(sqrt(2)*(-3/16 - 1/(4*x))*sqrt(19 + sqrt(51
7)) + (3/4 + 1/x)**2 + sqrt(517)/16)/(264704*sqrt(19 + sqrt(517))) - sqrt(1034)*
(-32*sqrt(517) + 288)*log(sqrt(2)*(3/16 + 1/(4*x))*sqrt(19 + sqrt(517)) + (3/4 +
1/x)**2 + sqrt(517)/16)/(264704*sqrt(19 + sqrt(517))) + sqrt(39)*atan(sqrt(39)*
(8*(3/4 + 1/x)**2/39 - 19/78))/52 - sqrt(517)*(-sqrt(2)*sqrt(19 + sqrt(517))*(-6
4*sqrt(517) + 576)/8 + 144*sqrt(2)*sqrt(19 + sqrt(517)))*atan(sqrt(2)*(3 + sqrt(
38 + 2*sqrt(517))/2 + 4/x)/sqrt(-19 + sqrt(517)))/(33088*sqrt(-19 + sqrt(517))*s
qrt(19 + sqrt(517))) - sqrt(517)*(-sqrt(2)*sqrt(19 + sqrt(517))*(-64*sqrt(517) +
576)/8 + 144*sqrt(2)*sqrt(19 + sqrt(517)))*atan(sqrt(2)*(-sqrt(38 + 2*sqrt(517)
)/2 + 3 + 4/x)/sqrt(-19 + sqrt(517)))/(33088*sqrt(-19 + sqrt(517))*sqrt(19 + sqr
t(517)))

_______________________________________________________________________________________

Mathematica [C]  time = 0.0158968, size = 55, normalized size = 0.21 \[ \text{RootSum}\left [8 \text{$\#$1}^4-15 \text{$\#$1}^3+8 \text{$\#$1}^2+24 \text{$\#$1}+8\&,\frac{\log (x-\text{$\#$1})}{32 \text{$\#$1}^3-45 \text{$\#$1}^2+16 \text{$\#$1}+24}\&\right ] \]

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) & ]

_______________________________________________________________________________________

Maple [C]  time = 0.007, size = 49, normalized size = 0.2 \[ \sum _{{\it \_R}={\it RootOf} \left ( 8\,{{\it \_Z}}^{4}-15\,{{\it \_Z}}^{3}+8\,{{\it \_Z}}^{2}+24\,{\it \_Z}+8 \right ) }{\frac{\ln \left ( x-{\it \_R} \right ) }{32\,{{\it \_R}}^{3}-45\,{{\it \_R}}^{2}+16\,{\it \_R}+24}} \]

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)

[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))

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{8 \, x^{4} - 15 \, x^{3} + 8 \, x^{2} + 24 \, x + 8}\,{d x} \]

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),x, algorithm="maxima")

[Out]

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

_______________________________________________________________________________________

Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

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]

Exception raised: NotImplementedError

_______________________________________________________________________________________

Sympy [A]  time = 3.04575, size = 41, normalized size = 0.16 \[ \operatorname{RootSum}{\left (50326848 t^{4} + 765960 t^{2} + 12753 t + 64, \left ( t \mapsto t \log{\left (\frac{100785893208 t^{3}}{4758335} - \frac{1430512512 t^{2}}{4758335} + \frac{72982352521 t}{223641745} + x + \frac{2270349121}{1789133960} \right )} \right )\right )} \]

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(1007858
93208*_t**3/4758335 - 1430512512*_t**2/4758335 + 72982352521*_t/223641745 + x +
2270349121/1789133960)))

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{8 \, x^{4} - 15 \, x^{3} + 8 \, x^{2} + 24 \, x + 8}\,{d x} \]

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),x, algorithm="giac")

[Out]

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