3.380 \(\int \frac{x^4}{1+3 x^4+x^8} \, dx\)
Optimal. Leaf size=469 \[ \frac{\sqrt [4]{3-\sqrt{5}} \log \left (\sqrt{2} x^2-2^{3/4} \sqrt [4]{3-\sqrt{5}} x+\sqrt{3-\sqrt{5}}\right )}{4\ 2^{3/4} \sqrt{5}}-\frac{\sqrt [4]{3-\sqrt{5}} \log \left (\sqrt{2} x^2+2^{3/4} \sqrt [4]{3-\sqrt{5}} x+\sqrt{3-\sqrt{5}}\right )}{4\ 2^{3/4} \sqrt{5}}-\frac{\sqrt [4]{3+\sqrt{5}} \log \left (\sqrt{2} x^2-2^{3/4} \sqrt [4]{3+\sqrt{5}} x+\sqrt{3+\sqrt{5}}\right )}{4\ 2^{3/4} \sqrt{5}}+\frac{\sqrt [4]{3+\sqrt{5}} \log \left (\sqrt{2} x^2+2^{3/4} \sqrt [4]{3+\sqrt{5}} x+\sqrt{3+\sqrt{5}}\right )}{4\ 2^{3/4} \sqrt{5}}+\frac{\sqrt [4]{3-\sqrt{5}} \tan ^{-1}\left (1-\frac{2^{3/4} x}{\sqrt [4]{3-\sqrt{5}}}\right )}{2\ 2^{3/4} \sqrt{5}}-\frac{\sqrt [4]{3-\sqrt{5}} \tan ^{-1}\left (\frac{2^{3/4} x}{\sqrt [4]{3-\sqrt{5}}}+1\right )}{2\ 2^{3/4} \sqrt{5}}-\frac{\sqrt [4]{3+\sqrt{5}} \tan ^{-1}\left (1-\frac{2^{3/4} x}{\sqrt [4]{3+\sqrt{5}}}\right )}{2\ 2^{3/4} \sqrt{5}}+\frac{\sqrt [4]{3+\sqrt{5}} \tan ^{-1}\left (\frac{2^{3/4} x}{\sqrt [4]{3+\sqrt{5}}}+1\right )}{2\ 2^{3/4} \sqrt{5}} \]
[Out]
((3 - Sqrt[5])^(1/4)*ArcTan[1 - (2^(3/4)*x)/(3 - Sqrt[5])^(1/4)])/(2*2^(3/4)*Sqr
t[5]) - ((3 - Sqrt[5])^(1/4)*ArcTan[1 + (2^(3/4)*x)/(3 - Sqrt[5])^(1/4)])/(2*2^(
3/4)*Sqrt[5]) - ((3 + Sqrt[5])^(1/4)*ArcTan[1 - (2^(3/4)*x)/(3 + Sqrt[5])^(1/4)]
)/(2*2^(3/4)*Sqrt[5]) + ((3 + Sqrt[5])^(1/4)*ArcTan[1 + (2^(3/4)*x)/(3 + Sqrt[5]
)^(1/4)])/(2*2^(3/4)*Sqrt[5]) + ((3 - 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]) - ((3 - 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]) - ((3 + 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]) + ((3 + 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]
)
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Rubi [A] time = 0.661987, antiderivative size = 451, normalized size of antiderivative =
0.96, number of steps used = 19, number of rules used = 7, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.438
\[ \frac{\sqrt [4]{3-\sqrt{5}} \log \left (2 x^2-2 \sqrt [4]{2 \left (3-\sqrt{5}\right )} x+\sqrt{2 \left (3-\sqrt{5}\right )}\right )}{4\ 2^{3/4} \sqrt{5}}-\frac{\sqrt [4]{3-\sqrt{5}} \log \left (2 x^2+2 \sqrt [4]{2 \left (3-\sqrt{5}\right )} x+\sqrt{2 \left (3-\sqrt{5}\right )}\right )}{4\ 2^{3/4} \sqrt{5}}-\frac{\sqrt [4]{3+\sqrt{5}} \log \left (2 x^2-2 \sqrt [4]{2 \left (3+\sqrt{5}\right )} x+\sqrt{2 \left (3+\sqrt{5}\right )}\right )}{4\ 2^{3/4} \sqrt{5}}+\frac{\sqrt [4]{3+\sqrt{5}} \log \left (2 x^2+2 \sqrt [4]{2 \left (3+\sqrt{5}\right )} x+\sqrt{2 \left (3+\sqrt{5}\right )}\right )}{4\ 2^{3/4} \sqrt{5}}+\frac{\sqrt [4]{3-\sqrt{5}} \tan ^{-1}\left (1-\frac{2^{3/4} x}{\sqrt [4]{3-\sqrt{5}}}\right )}{2\ 2^{3/4} \sqrt{5}}-\frac{\sqrt [4]{3-\sqrt{5}} \tan ^{-1}\left (\frac{2^{3/4} x}{\sqrt [4]{3-\sqrt{5}}}+1\right )}{2\ 2^{3/4} \sqrt{5}}-\frac{\sqrt [4]{3+\sqrt{5}} \tan ^{-1}\left (1-\frac{2^{3/4} x}{\sqrt [4]{3+\sqrt{5}}}\right )}{2\ 2^{3/4} \sqrt{5}}+\frac{\sqrt [4]{3+\sqrt{5}} \tan ^{-1}\left (\frac{2^{3/4} x}{\sqrt [4]{3+\sqrt{5}}}+1\right )}{2\ 2^{3/4} \sqrt{5}} \]
Antiderivative was successfully verified.
[In] Int[x^4/(1 + 3*x^4 + x^8),x]
[Out]
((3 - Sqrt[5])^(1/4)*ArcTan[1 - (2^(3/4)*x)/(3 - Sqrt[5])^(1/4)])/(2*2^(3/4)*Sqr
t[5]) - ((3 - Sqrt[5])^(1/4)*ArcTan[1 + (2^(3/4)*x)/(3 - Sqrt[5])^(1/4)])/(2*2^(
3/4)*Sqrt[5]) - ((3 + Sqrt[5])^(1/4)*ArcTan[1 - (2^(3/4)*x)/(3 + Sqrt[5])^(1/4)]
)/(2*2^(3/4)*Sqrt[5]) + ((3 + Sqrt[5])^(1/4)*ArcTan[1 + (2^(3/4)*x)/(3 + Sqrt[5]
)^(1/4)])/(2*2^(3/4)*Sqrt[5]) + ((3 - 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]) - ((3 - 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]) - ((3 + 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]) + ((3 + 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])
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Rubi in Sympy [A] time = 84.7237, size = 604, normalized size = 1.29 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**4/(x**8+3*x**4+1),x)
[Out]
-2**(3/4)*sqrt(-2*sqrt(5) + 6)*(-3*sqrt(5)/10 + 1/2)*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)*(-3*sqrt(5)/10 + 1/2)*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)*(1/2 +
3*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)*(1/2 + 3*sqrt(5)/10
)*sqrt(2*sqrt(5) + 6)*log(2*x**2 + 2*2**(1/4)*x*(sqrt(5) + 3)**(1/4) + sqrt(2*sq
rt(5) + 6))/(8*(sqrt(5) + 3)**(5/4)) + 2**(3/4)*(-3*sqrt(5)/10 + 1/2)*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)*(-3*sqrt(5)/10 + 1/2)*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)*(1/2 + 3*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)*(1/2 + 3*sqrt(5)/10)*atan(2**(3/4)*(x + (2*sqrt(5) + 6)**(1/4)/2)/(sqr
t(5) + 3)**(1/4))/(2*(sqrt(5) + 3)**(1/4)*sqrt(2*sqrt(5) + 6))
_______________________________________________________________________________________
Mathematica [C] time = 0.0155784, size = 39, normalized size = 0.08 \[ \frac{1}{4} \text{RootSum}\left [\text{$\#$1}^8+3 \text{$\#$1}^4+1\&,\frac{\text{$\#$1} \log (x-\text{$\#$1})}{2 \text{$\#$1}^4+3}\&\right ] \]
Antiderivative was successfully verified.
[In] Integrate[x^4/(1 + 3*x^4 + x^8),x]
[Out]
RootSum[1 + 3*#1^4 + #1^8 & , (Log[x - #1]*#1)/(3 + 2*#1^4) & ]/4
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Maple [C] time = 0.01, size = 40, normalized size = 0.1 \[{\frac{1}{4}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{8}+3\,{{\it \_Z}}^{4}+1 \right ) }{\frac{{{\it \_R}}^{4}\ln \left ( x-{\it \_R} \right ) }{2\,{{\it \_R}}^{7}+3\,{{\it \_R}}^{3}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^4/(x^8+3*x^4+1),x)
[Out]
1/4*sum(_R^4/(2*_R^7+3*_R^3)*ln(x-_R),_R=RootOf(_Z^8+3*_Z^4+1))
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4}}{x^{8} + 3 \, x^{4} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^4/(x^8 + 3*x^4 + 1),x, algorithm="maxima")
[Out]
integrate(x^4/(x^8 + 3*x^4 + 1), x)
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Fricas [A] time = 0.310072, size = 1378, normalized size = 2.94 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^4/(x^8 + 3*x^4 + 1),x, algorithm="fricas")
[Out]
-1/40*sqrt(5)*sqrt(2)*(4*(1/250)^(1/4)*(sqrt(5)*(3*sqrt(5) + 5))^(3/4)*sqrt(sqrt
(5)*(3*sqrt(5) - 5))*arctan(5*sqrt(1/10)*(1/250)^(1/4)*(sqrt(5)*(3*sqrt(5) + 5))
^(3/4)/(sqrt(5)*sqrt(2)*sqrt(1/10)*sqrt(sqrt(5)*(3*sqrt(5) + 5))*x + sqrt(5)*sqr
t(2)*sqrt(1/10)*sqrt(sqrt(5)*sqrt(2)*(1/250)^(1/4)*(sqrt(5)*(3*sqrt(5) + 5))^(1/
4)*x + x^2 + sqrt(1/10)*sqrt(sqrt(5)*(3*sqrt(5) + 5)))*sqrt(sqrt(5)*(3*sqrt(5) +
5)) + 5*sqrt(1/10)*(1/250)^(1/4)*(sqrt(5)*(3*sqrt(5) + 5))^(3/4))) + 4*(1/250)^
(1/4)*(sqrt(5)*(3*sqrt(5) + 5))^(3/4)*sqrt(sqrt(5)*(3*sqrt(5) - 5))*arctan(5*sqr
t(1/10)*(1/250)^(1/4)*(sqrt(5)*(3*sqrt(5) + 5))^(3/4)/(sqrt(5)*sqrt(2)*sqrt(1/10
)*sqrt(sqrt(5)*(3*sqrt(5) + 5))*x + sqrt(5)*sqrt(2)*sqrt(1/10)*sqrt(-sqrt(5)*sqr
t(2)*(1/250)^(1/4)*(sqrt(5)*(3*sqrt(5) + 5))^(1/4)*x + x^2 + sqrt(1/10)*sqrt(sqr
t(5)*(3*sqrt(5) + 5)))*sqrt(sqrt(5)*(3*sqrt(5) + 5)) - 5*sqrt(1/10)*(1/250)^(1/4
)*(sqrt(5)*(3*sqrt(5) + 5))^(3/4))) - 4*(1/250)^(1/4)*sqrt(sqrt(5)*(3*sqrt(5) +
5))*(sqrt(5)*(3*sqrt(5) - 5))^(3/4)*arctan(5*sqrt(1/10)*(1/250)^(1/4)*(sqrt(5)*(
3*sqrt(5) - 5))^(3/4)/(sqrt(5)*sqrt(2)*sqrt(1/10)*sqrt(sqrt(5)*(3*sqrt(5) - 5))*
x + sqrt(5)*sqrt(2)*sqrt(1/10)*sqrt(sqrt(5)*sqrt(2)*(1/250)^(1/4)*(sqrt(5)*(3*sq
rt(5) - 5))^(1/4)*x + x^2 + sqrt(1/10)*sqrt(sqrt(5)*(3*sqrt(5) - 5)))*sqrt(sqrt(
5)*(3*sqrt(5) - 5)) + 5*sqrt(1/10)*(1/250)^(1/4)*(sqrt(5)*(3*sqrt(5) - 5))^(3/4)
)) - 4*(1/250)^(1/4)*sqrt(sqrt(5)*(3*sqrt(5) + 5))*(sqrt(5)*(3*sqrt(5) - 5))^(3/
4)*arctan(5*sqrt(1/10)*(1/250)^(1/4)*(sqrt(5)*(3*sqrt(5) - 5))^(3/4)/(sqrt(5)*sq
rt(2)*sqrt(1/10)*sqrt(sqrt(5)*(3*sqrt(5) - 5))*x + sqrt(5)*sqrt(2)*sqrt(1/10)*sq
rt(-sqrt(5)*sqrt(2)*(1/250)^(1/4)*(sqrt(5)*(3*sqrt(5) - 5))^(1/4)*x + x^2 + sqrt
(1/10)*sqrt(sqrt(5)*(3*sqrt(5) - 5)))*sqrt(sqrt(5)*(3*sqrt(5) - 5)) - 5*sqrt(1/1
0)*(1/250)^(1/4)*(sqrt(5)*(3*sqrt(5) - 5))^(3/4))) - (1/250)^(1/4)*(sqrt(5)*(3*s
qrt(5) + 5))^(3/4)*sqrt(sqrt(5)*(3*sqrt(5) - 5))*log(sqrt(5)*sqrt(2)*(1/250)^(1/
4)*(sqrt(5)*(3*sqrt(5) + 5))^(1/4)*x + x^2 + sqrt(1/10)*sqrt(sqrt(5)*(3*sqrt(5)
+ 5))) + (1/250)^(1/4)*(sqrt(5)*(3*sqrt(5) + 5))^(3/4)*sqrt(sqrt(5)*(3*sqrt(5) -
5))*log(-sqrt(5)*sqrt(2)*(1/250)^(1/4)*(sqrt(5)*(3*sqrt(5) + 5))^(1/4)*x + x^2
+ sqrt(1/10)*sqrt(sqrt(5)*(3*sqrt(5) + 5))) + (1/250)^(1/4)*sqrt(sqrt(5)*(3*sqrt
(5) + 5))*(sqrt(5)*(3*sqrt(5) - 5))^(3/4)*log(sqrt(5)*sqrt(2)*(1/250)^(1/4)*(sqr
t(5)*(3*sqrt(5) - 5))^(1/4)*x + x^2 + sqrt(1/10)*sqrt(sqrt(5)*(3*sqrt(5) - 5)))
- (1/250)^(1/4)*sqrt(sqrt(5)*(3*sqrt(5) + 5))*(sqrt(5)*(3*sqrt(5) - 5))^(3/4)*lo
g(-sqrt(5)*sqrt(2)*(1/250)^(1/4)*(sqrt(5)*(3*sqrt(5) - 5))^(1/4)*x + x^2 + sqrt(
1/10)*sqrt(sqrt(5)*(3*sqrt(5) - 5))))/(sqrt(3*sqrt(5) + 5)*sqrt(3*sqrt(5) - 5))
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Sympy [A] time = 3.82122, size = 24, normalized size = 0.05 \[ \operatorname{RootSum}{\left (40960000 t^{8} + 19200 t^{4} + 1, \left ( t \mapsto t \log{\left (- 51200 t^{5} - 12 t + x \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**4/(x**8+3*x**4+1),x)
[Out]
RootSum(40960000*_t**8 + 19200*_t**4 + 1, Lambda(_t, _t*log(-51200*_t**5 - 12*_t
+ x)))
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4}}{x^{8} + 3 \, x^{4} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^4/(x^8 + 3*x^4 + 1),x, algorithm="giac")
[Out]
integrate(x^4/(x^8 + 3*x^4 + 1), x)