3.378 \(\int \frac{x^8}{1+3 x^4+x^8} \, dx\)
Optimal. Leaf size=458 \[ -\frac{\sqrt [4]{984-440 \sqrt{5}} \log \left (\sqrt{2} x^2-2^{3/4} \sqrt [4]{3-\sqrt{5}} x+\sqrt{3-\sqrt{5}}\right )}{8 \sqrt{10}}+\frac{\sqrt [4]{984-440 \sqrt{5}} \log \left (\sqrt{2} x^2+2^{3/4} \sqrt [4]{3-\sqrt{5}} x+\sqrt{3-\sqrt{5}}\right )}{8 \sqrt{10}}+\frac{\sqrt [4]{123+55 \sqrt{5}} \log \left (\sqrt{2} x^2-2^{3/4} \sqrt [4]{3+\sqrt{5}} x+\sqrt{3+\sqrt{5}}\right )}{4\ 2^{3/4} \sqrt{5}}-\frac{\sqrt [4]{123+55 \sqrt{5}} \log \left (\sqrt{2} x^2+2^{3/4} \sqrt [4]{3+\sqrt{5}} x+\sqrt{3+\sqrt{5}}\right )}{4\ 2^{3/4} \sqrt{5}}+x-\frac{\sqrt [4]{984-440 \sqrt{5}} \tan ^{-1}\left (1-\frac{2^{3/4} x}{\sqrt [4]{3-\sqrt{5}}}\right )}{4 \sqrt{10}}+\frac{\sqrt [4]{984-440 \sqrt{5}} \tan ^{-1}\left (\frac{2^{3/4} x}{\sqrt [4]{3-\sqrt{5}}}+1\right )}{4 \sqrt{10}}+\frac{\sqrt [4]{123+55 \sqrt{5}} \tan ^{-1}\left (1-\frac{2^{3/4} x}{\sqrt [4]{3+\sqrt{5}}}\right )}{2\ 2^{3/4} \sqrt{5}}-\frac{\sqrt [4]{123+55 \sqrt{5}} \tan ^{-1}\left (\frac{2^{3/4} x}{\sqrt [4]{3+\sqrt{5}}}+1\right )}{2\ 2^{3/4} \sqrt{5}} \]
[Out]
x - ((984 - 440*Sqrt[5])^(1/4)*ArcTan[1 - (2^(3/4)*x)/(3 - Sqrt[5])^(1/4)])/(4*S
qrt[10]) + ((984 - 440*Sqrt[5])^(1/4)*ArcTan[1 + (2^(3/4)*x)/(3 - Sqrt[5])^(1/4)
])/(4*Sqrt[10]) + ((123 + 55*Sqrt[5])^(1/4)*ArcTan[1 - (2^(3/4)*x)/(3 + Sqrt[5])
^(1/4)])/(2*2^(3/4)*Sqrt[5]) - ((123 + 55*Sqrt[5])^(1/4)*ArcTan[1 + (2^(3/4)*x)/
(3 + Sqrt[5])^(1/4)])/(2*2^(3/4)*Sqrt[5]) - ((984 - 440*Sqrt[5])^(1/4)*Log[Sqrt[
3 - Sqrt[5]] - 2^(3/4)*(3 - Sqrt[5])^(1/4)*x + Sqrt[2]*x^2])/(8*Sqrt[10]) + ((98
4 - 440*Sqrt[5])^(1/4)*Log[Sqrt[3 - Sqrt[5]] + 2^(3/4)*(3 - Sqrt[5])^(1/4)*x + S
qrt[2]*x^2])/(8*Sqrt[10]) + ((123 + 55*Sqrt[5])^(1/4)*Log[Sqrt[3 + Sqrt[5]] - 2^
(3/4)*(3 + Sqrt[5])^(1/4)*x + Sqrt[2]*x^2])/(4*2^(3/4)*Sqrt[5]) - ((123 + 55*Sqr
t[5])^(1/4)*Log[Sqrt[3 + Sqrt[5]] + 2^(3/4)*(3 + Sqrt[5])^(1/4)*x + Sqrt[2]*x^2]
)/(4*2^(3/4)*Sqrt[5])
_______________________________________________________________________________________
Rubi [A] time = 0.873226, antiderivative size = 440, normalized size of antiderivative =
0.96, number of steps used = 20, number of rules used = 8, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5
\[ -\frac{\sqrt [4]{123-55 \sqrt{5}} \log \left (2 x^2-2 \sqrt [4]{2 \left (3-\sqrt{5}\right )} x+\sqrt{2 \left (3-\sqrt{5}\right )}\right )}{4\ 2^{3/4} \sqrt{5}}+\frac{\sqrt [4]{123-55 \sqrt{5}} \log \left (2 x^2+2 \sqrt [4]{2 \left (3-\sqrt{5}\right )} x+\sqrt{2 \left (3-\sqrt{5}\right )}\right )}{4\ 2^{3/4} \sqrt{5}}+\frac{\sqrt [4]{123+55 \sqrt{5}} \log \left (2 x^2-2 \sqrt [4]{2 \left (3+\sqrt{5}\right )} x+\sqrt{2 \left (3+\sqrt{5}\right )}\right )}{4\ 2^{3/4} \sqrt{5}}-\frac{\sqrt [4]{123+55 \sqrt{5}} \log \left (2 x^2+2 \sqrt [4]{2 \left (3+\sqrt{5}\right )} x+\sqrt{2 \left (3+\sqrt{5}\right )}\right )}{4\ 2^{3/4} \sqrt{5}}+x-\frac{\sqrt [4]{123-55 \sqrt{5}} \tan ^{-1}\left (1-\frac{2^{3/4} x}{\sqrt [4]{3-\sqrt{5}}}\right )}{2\ 2^{3/4} \sqrt{5}}+\frac{\sqrt [4]{123-55 \sqrt{5}} \tan ^{-1}\left (\frac{2^{3/4} x}{\sqrt [4]{3-\sqrt{5}}}+1\right )}{2\ 2^{3/4} \sqrt{5}}+\frac{\sqrt [4]{123+55 \sqrt{5}} \tan ^{-1}\left (1-\frac{2^{3/4} x}{\sqrt [4]{3+\sqrt{5}}}\right )}{2\ 2^{3/4} \sqrt{5}}-\frac{\sqrt [4]{123+55 \sqrt{5}} \tan ^{-1}\left (\frac{2^{3/4} x}{\sqrt [4]{3+\sqrt{5}}}+1\right )}{2\ 2^{3/4} \sqrt{5}} \]
Warning: Unable to verify antiderivative.
[In] Int[x^8/(1 + 3*x^4 + x^8),x]
[Out]
x - ((123 - 55*Sqrt[5])^(1/4)*ArcTan[1 - (2^(3/4)*x)/(3 - Sqrt[5])^(1/4)])/(2*2^
(3/4)*Sqrt[5]) + ((123 - 55*Sqrt[5])^(1/4)*ArcTan[1 + (2^(3/4)*x)/(3 - Sqrt[5])^
(1/4)])/(2*2^(3/4)*Sqrt[5]) + ((123 + 55*Sqrt[5])^(1/4)*ArcTan[1 - (2^(3/4)*x)/(
3 + Sqrt[5])^(1/4)])/(2*2^(3/4)*Sqrt[5]) - ((123 + 55*Sqrt[5])^(1/4)*ArcTan[1 +
(2^(3/4)*x)/(3 + Sqrt[5])^(1/4)])/(2*2^(3/4)*Sqrt[5]) - ((123 - 55*Sqrt[5])^(1/4
)*Log[Sqrt[2*(3 - Sqrt[5])] - 2*(2*(3 - Sqrt[5]))^(1/4)*x + 2*x^2])/(4*2^(3/4)*S
qrt[5]) + ((123 - 55*Sqrt[5])^(1/4)*Log[Sqrt[2*(3 - Sqrt[5])] + 2*(2*(3 - Sqrt[5
]))^(1/4)*x + 2*x^2])/(4*2^(3/4)*Sqrt[5]) + ((123 + 55*Sqrt[5])^(1/4)*Log[Sqrt[2
*(3 + Sqrt[5])] - 2*(2*(3 + Sqrt[5]))^(1/4)*x + 2*x^2])/(4*2^(3/4)*Sqrt[5]) - ((
123 + 55*Sqrt[5])^(1/4)*Log[Sqrt[2*(3 + Sqrt[5])] + 2*(2*(3 + Sqrt[5]))^(1/4)*x
+ 2*x^2])/(4*2^(3/4)*Sqrt[5])
_______________________________________________________________________________________
Rubi in Sympy [A] time = 91.0684, size = 605, normalized size = 1.32 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**8/(x**8+3*x**4+1),x)
[Out]
x + 2**(3/4)*sqrt(-2*sqrt(5) + 6)*(-7*sqrt(5)/10 + 3/2)*log(2*x**2 - 2*2**(1/4)*
x*(-sqrt(5) + 3)**(1/4) + sqrt(-2*sqrt(5) + 6))/(8*(-sqrt(5) + 3)**(5/4)) - 2**(
3/4)*sqrt(-2*sqrt(5) + 6)*(-7*sqrt(5)/10 + 3/2)*log(2*x**2 + 2*2**(1/4)*x*(-sqrt
(5) + 3)**(1/4) + sqrt(-2*sqrt(5) + 6))/(8*(-sqrt(5) + 3)**(5/4)) + 2**(3/4)*(3/
2 + 7*sqrt(5)/10)*sqrt(2*sqrt(5) + 6)*log(2*x**2 - 2*2**(1/4)*x*(sqrt(5) + 3)**(
1/4) + sqrt(2*sqrt(5) + 6))/(8*(sqrt(5) + 3)**(5/4)) - 2**(3/4)*(3/2 + 7*sqrt(5)
/10)*sqrt(2*sqrt(5) + 6)*log(2*x**2 + 2*2**(1/4)*x*(sqrt(5) + 3)**(1/4) + sqrt(2
*sqrt(5) + 6))/(8*(sqrt(5) + 3)**(5/4)) - 2**(3/4)*(-7*sqrt(5)/10 + 3/2)*atan(2*
*(3/4)*(x - (-2*sqrt(5) + 6)**(1/4)/2)/(-sqrt(5) + 3)**(1/4))/(2*sqrt(-2*sqrt(5)
+ 6)*(-sqrt(5) + 3)**(1/4)) - 2**(3/4)*(-7*sqrt(5)/10 + 3/2)*atan(2**(3/4)*(x +
(-2*sqrt(5) + 6)**(1/4)/2)/(-sqrt(5) + 3)**(1/4))/(2*sqrt(-2*sqrt(5) + 6)*(-sqr
t(5) + 3)**(1/4)) - 2**(3/4)*(3/2 + 7*sqrt(5)/10)*atan(2**(3/4)*(x - (2*sqrt(5)
+ 6)**(1/4)/2)/(sqrt(5) + 3)**(1/4))/(2*(sqrt(5) + 3)**(1/4)*sqrt(2*sqrt(5) + 6)
) - 2**(3/4)*(3/2 + 7*sqrt(5)/10)*atan(2**(3/4)*(x + (2*sqrt(5) + 6)**(1/4)/2)/(
sqrt(5) + 3)**(1/4))/(2*(sqrt(5) + 3)**(1/4)*sqrt(2*sqrt(5) + 6))
_______________________________________________________________________________________
Mathematica [C] time = 0.0222967, size = 58, normalized size = 0.13 \[ x-\frac{1}{4} \text{RootSum}\left [\text{$\#$1}^8+3 \text{$\#$1}^4+1\&,\frac{3 \text{$\#$1}^4 \log (x-\text{$\#$1})+\log (x-\text{$\#$1})}{2 \text{$\#$1}^7+3 \text{$\#$1}^3}\&\right ] \]
Antiderivative was successfully verified.
[In] Integrate[x^8/(1 + 3*x^4 + x^8),x]
[Out]
x - RootSum[1 + 3*#1^4 + #1^8 & , (Log[x - #1] + 3*Log[x - #1]*#1^4)/(3*#1^3 + 2
*#1^7) & ]/4
_______________________________________________________________________________________
Maple [C] time = 0.011, size = 46, normalized size = 0.1 \[ x+{\frac{1}{4}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{8}+3\,{{\it \_Z}}^{4}+1 \right ) }{\frac{ \left ( -3\,{{\it \_R}}^{4}-1 \right ) \ln \left ( x-{\it \_R} \right ) }{2\,{{\it \_R}}^{7}+3\,{{\it \_R}}^{3}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^8/(x^8+3*x^4+1),x)
[Out]
x+1/4*sum((-3*_R^4-1)/(2*_R^7+3*_R^3)*ln(x-_R),_R=RootOf(_Z^8+3*_Z^4+1))
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ x - \int \frac{3 \, x^{4} + 1}{x^{8} + 3 \, x^{4} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^8/(x^8 + 3*x^4 + 1),x, algorithm="maxima")
[Out]
x - integrate((3*x^4 + 1)/(x^8 + 3*x^4 + 1), x)
_______________________________________________________________________________________
Fricas [A] time = 0.307641, size = 1837, normalized size = 4.01 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^8/(x^8 + 3*x^4 + 1),x, algorithm="fricas")
[Out]
1/40*sqrt(5)*sqrt(2)*(4*sqrt(2)*sqrt(sqrt(5)*(123*sqrt(5) + 275))*sqrt(sqrt(5)*(
123*sqrt(5) - 275))*x - 4*(1/250)^(1/4)*sqrt(sqrt(5)*(123*sqrt(5) + 275))*(sqrt(
5)*(123*sqrt(5) - 275))^(3/4)*arctan(5*sqrt(1/10)*(1/250)^(1/4)*(sqrt(5)*(123*sq
rt(5) - 275))^(3/4)*(sqrt(5) + 3)/(2*sqrt(5)*sqrt(2)*sqrt(1/10)*sqrt(sqrt(5)*(12
3*sqrt(5) - 275))*x + 5*sqrt(1/10)*(1/250)^(1/4)*(sqrt(5)*(123*sqrt(5) - 275))^(
3/4)*(sqrt(5) + 3) + 2*sqrt(5)*sqrt(2)*sqrt(1/10)*sqrt(sqrt(5)*(123*sqrt(5) - 27
5))*sqrt((123*sqrt(5)*x^2 - 275*x^2 + 2*sqrt(1/10)*sqrt(sqrt(5)*(123*sqrt(5) - 2
75))*(9*sqrt(5) - 20) - 5*(1/250)^(1/4)*(21*sqrt(5)*sqrt(2)*x - 47*sqrt(2)*x)*(s
qrt(5)*(123*sqrt(5) - 275))^(1/4))/(123*sqrt(5) - 275)))) - 4*(1/250)^(1/4)*sqrt
(sqrt(5)*(123*sqrt(5) + 275))*(sqrt(5)*(123*sqrt(5) - 275))^(3/4)*arctan(5*sqrt(
1/10)*(1/250)^(1/4)*(sqrt(5)*(123*sqrt(5) - 275))^(3/4)*(sqrt(5) + 3)/(2*sqrt(5)
*sqrt(2)*sqrt(1/10)*sqrt(sqrt(5)*(123*sqrt(5) - 275))*x - 5*sqrt(1/10)*(1/250)^(
1/4)*(sqrt(5)*(123*sqrt(5) - 275))^(3/4)*(sqrt(5) + 3) + 2*sqrt(5)*sqrt(2)*sqrt(
1/10)*sqrt(sqrt(5)*(123*sqrt(5) - 275))*sqrt((123*sqrt(5)*x^2 - 275*x^2 + 2*sqrt
(1/10)*sqrt(sqrt(5)*(123*sqrt(5) - 275))*(9*sqrt(5) - 20) + 5*(1/250)^(1/4)*(21*
sqrt(5)*sqrt(2)*x - 47*sqrt(2)*x)*(sqrt(5)*(123*sqrt(5) - 275))^(1/4))/(123*sqrt
(5) - 275)))) - 4*(1/250)^(1/4)*(sqrt(5)*(123*sqrt(5) + 275))^(3/4)*sqrt(sqrt(5)
*(123*sqrt(5) - 275))*arctan(5*sqrt(1/10)*(1/250)^(1/4)*(sqrt(5)*(123*sqrt(5) +
275))^(3/4)*(sqrt(5) - 3)/(2*sqrt(5)*sqrt(2)*sqrt(1/10)*sqrt(sqrt(5)*(123*sqrt(5
) + 275))*x + 5*sqrt(1/10)*(1/250)^(1/4)*(sqrt(5)*(123*sqrt(5) + 275))^(3/4)*(sq
rt(5) - 3) + 2*sqrt(5)*sqrt(2)*sqrt(1/10)*sqrt(sqrt(5)*(123*sqrt(5) + 275))*sqrt
((123*sqrt(5)*x^2 + 275*x^2 + 2*sqrt(1/10)*sqrt(sqrt(5)*(123*sqrt(5) + 275))*(9*
sqrt(5) + 20) - 5*(1/250)^(1/4)*(21*sqrt(5)*sqrt(2)*x + 47*sqrt(2)*x)*(sqrt(5)*(
123*sqrt(5) + 275))^(1/4))/(123*sqrt(5) + 275)))) - 4*(1/250)^(1/4)*(sqrt(5)*(12
3*sqrt(5) + 275))^(3/4)*sqrt(sqrt(5)*(123*sqrt(5) - 275))*arctan(5*sqrt(1/10)*(1
/250)^(1/4)*(sqrt(5)*(123*sqrt(5) + 275))^(3/4)*(sqrt(5) - 3)/(2*sqrt(5)*sqrt(2)
*sqrt(1/10)*sqrt(sqrt(5)*(123*sqrt(5) + 275))*x - 5*sqrt(1/10)*(1/250)^(1/4)*(sq
rt(5)*(123*sqrt(5) + 275))^(3/4)*(sqrt(5) - 3) + 2*sqrt(5)*sqrt(2)*sqrt(1/10)*sq
rt(sqrt(5)*(123*sqrt(5) + 275))*sqrt((123*sqrt(5)*x^2 + 275*x^2 + 2*sqrt(1/10)*s
qrt(sqrt(5)*(123*sqrt(5) + 275))*(9*sqrt(5) + 20) + 5*(1/250)^(1/4)*(21*sqrt(5)*
sqrt(2)*x + 47*sqrt(2)*x)*(sqrt(5)*(123*sqrt(5) + 275))^(1/4))/(123*sqrt(5) + 27
5)))) - (1/250)^(1/4)*(sqrt(5)*(123*sqrt(5) + 275))^(3/4)*sqrt(sqrt(5)*(123*sqrt
(5) - 275))*log(123*sqrt(5)*x^2 + 275*x^2 + 2*sqrt(1/10)*sqrt(sqrt(5)*(123*sqrt(
5) + 275))*(9*sqrt(5) + 20) + 5*(1/250)^(1/4)*(21*sqrt(5)*sqrt(2)*x + 47*sqrt(2)
*x)*(sqrt(5)*(123*sqrt(5) + 275))^(1/4)) + (1/250)^(1/4)*(sqrt(5)*(123*sqrt(5) +
275))^(3/4)*sqrt(sqrt(5)*(123*sqrt(5) - 275))*log(123*sqrt(5)*x^2 + 275*x^2 + 2
*sqrt(1/10)*sqrt(sqrt(5)*(123*sqrt(5) + 275))*(9*sqrt(5) + 20) - 5*(1/250)^(1/4)
*(21*sqrt(5)*sqrt(2)*x + 47*sqrt(2)*x)*(sqrt(5)*(123*sqrt(5) + 275))^(1/4)) - (1
/250)^(1/4)*sqrt(sqrt(5)*(123*sqrt(5) + 275))*(sqrt(5)*(123*sqrt(5) - 275))^(3/4
)*log(123*sqrt(5)*x^2 - 275*x^2 + 2*sqrt(1/10)*sqrt(sqrt(5)*(123*sqrt(5) - 275))
*(9*sqrt(5) - 20) + 5*(1/250)^(1/4)*(21*sqrt(5)*sqrt(2)*x - 47*sqrt(2)*x)*(sqrt(
5)*(123*sqrt(5) - 275))^(1/4)) + (1/250)^(1/4)*sqrt(sqrt(5)*(123*sqrt(5) + 275))
*(sqrt(5)*(123*sqrt(5) - 275))^(3/4)*log(123*sqrt(5)*x^2 - 275*x^2 + 2*sqrt(1/10
)*sqrt(sqrt(5)*(123*sqrt(5) - 275))*(9*sqrt(5) - 20) - 5*(1/250)^(1/4)*(21*sqrt(
5)*sqrt(2)*x - 47*sqrt(2)*x)*(sqrt(5)*(123*sqrt(5) - 275))^(1/4)))/(sqrt(123*sqr
t(5) + 275)*sqrt(123*sqrt(5) - 275))
_______________________________________________________________________________________
Sympy [A] time = 3.86152, size = 29, normalized size = 0.06 \[ x + \operatorname{RootSum}{\left (40960000 t^{8} + 787200 t^{4} + 1, \left ( t \mapsto t \log{\left (\frac{15360 t^{5}}{11} + \frac{1288 t}{55} + x \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**8/(x**8+3*x**4+1),x)
[Out]
x + RootSum(40960000*_t**8 + 787200*_t**4 + 1, Lambda(_t, _t*log(15360*_t**5/11
+ 1288*_t/55 + x)))
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{8}}{x^{8} + 3 \, x^{4} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^8/(x^8 + 3*x^4 + 1),x, algorithm="giac")
[Out]
integrate(x^8/(x^8 + 3*x^4 + 1), x)