3.381 \(\int \frac{x^2}{1+3 x^4+x^8} \, dx\)

Optimal. Leaf size=449 \[ \frac{\sqrt [4]{3+\sqrt{5}} \log \left (\sqrt{2} x^2-2^{3/4} \sqrt [4]{3-\sqrt{5}} x+\sqrt{3-\sqrt{5}}\right )}{4\ 2^{3/4} \sqrt{5}}-\frac{\sqrt [4]{3+\sqrt{5}} \log \left (\sqrt{2} x^2+2^{3/4} \sqrt [4]{3-\sqrt{5}} x+\sqrt{3-\sqrt{5}}\right )}{4\ 2^{3/4} \sqrt{5}}-\frac{\log \left (\sqrt{2} x^2-2^{3/4} \sqrt [4]{3+\sqrt{5}} x+\sqrt{3+\sqrt{5}}\right )}{4 \sqrt{5} \sqrt [4]{2 \left (3+\sqrt{5}\right )}}+\frac{\log \left (\sqrt{2} x^2+2^{3/4} \sqrt [4]{3+\sqrt{5}} x+\sqrt{3+\sqrt{5}}\right )}{4 \sqrt{5} \sqrt [4]{2 \left (3+\sqrt{5}\right )}}-\frac{\sqrt [4]{3+\sqrt{5}} \tan ^{-1}\left (1-\frac{2^{3/4} x}{\sqrt [4]{3-\sqrt{5}}}\right )}{2\ 2^{3/4} \sqrt{5}}+\frac{\sqrt [4]{3+\sqrt{5}} \tan ^{-1}\left (\frac{2^{3/4} x}{\sqrt [4]{3-\sqrt{5}}}+1\right )}{2\ 2^{3/4} \sqrt{5}}+\frac{\tan ^{-1}\left (1-\frac{2^{3/4} x}{\sqrt [4]{3+\sqrt{5}}}\right )}{2 \sqrt{5} \sqrt [4]{2 \left (3+\sqrt{5}\right )}}-\frac{\tan ^{-1}\left (\frac{2^{3/4} x}{\sqrt [4]{3+\sqrt{5}}}+1\right )}{2 \sqrt{5} \sqrt [4]{2 \left (3+\sqrt{5}\right )}} \]

[Out]

-((3 + Sqrt[5])^(1/4)*ArcTan[1 - (2^(3/4)*x)/(3 - Sqrt[5])^(1/4)])/(2*2^(3/4)*Sq
rt[5]) + ((3 + Sqrt[5])^(1/4)*ArcTan[1 + (2^(3/4)*x)/(3 - Sqrt[5])^(1/4)])/(2*2^
(3/4)*Sqrt[5]) + ArcTan[1 - (2^(3/4)*x)/(3 + Sqrt[5])^(1/4)]/(2*Sqrt[5]*(2*(3 +
Sqrt[5]))^(1/4)) - ArcTan[1 + (2^(3/4)*x)/(3 + Sqrt[5])^(1/4)]/(2*Sqrt[5]*(2*(3
+ Sqrt[5]))^(1/4)) + ((3 + Sqrt[5])^(1/4)*Log[Sqrt[3 - Sqrt[5]] - 2^(3/4)*(3 - S
qrt[5])^(1/4)*x + Sqrt[2]*x^2])/(4*2^(3/4)*Sqrt[5]) - ((3 + Sqrt[5])^(1/4)*Log[S
qrt[3 - Sqrt[5]] + 2^(3/4)*(3 - Sqrt[5])^(1/4)*x + Sqrt[2]*x^2])/(4*2^(3/4)*Sqrt
[5]) - Log[Sqrt[3 + Sqrt[5]] - 2^(3/4)*(3 + Sqrt[5])^(1/4)*x + Sqrt[2]*x^2]/(4*S
qrt[5]*(2*(3 + Sqrt[5]))^(1/4)) + Log[Sqrt[3 + Sqrt[5]] + 2^(3/4)*(3 + Sqrt[5])^
(1/4)*x + Sqrt[2]*x^2]/(4*Sqrt[5]*(2*(3 + Sqrt[5]))^(1/4))

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Rubi [A]  time = 0.518147, antiderivative size = 431, 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 \[ \frac{\sqrt [4]{3+\sqrt{5}} \log \left (2 x^2-2 \sqrt [4]{2 \left (3-\sqrt{5}\right )} x+\sqrt{2 \left (3-\sqrt{5}\right )}\right )}{4\ 2^{3/4} \sqrt{5}}-\frac{\sqrt [4]{3+\sqrt{5}} \log \left (2 x^2+2 \sqrt [4]{2 \left (3-\sqrt{5}\right )} x+\sqrt{2 \left (3-\sqrt{5}\right )}\right )}{4\ 2^{3/4} \sqrt{5}}-\frac{\log \left (2 x^2-2 \sqrt [4]{2 \left (3+\sqrt{5}\right )} x+\sqrt{2 \left (3+\sqrt{5}\right )}\right )}{4 \sqrt{5} \sqrt [4]{2 \left (3+\sqrt{5}\right )}}+\frac{\log \left (2 x^2+2 \sqrt [4]{2 \left (3+\sqrt{5}\right )} x+\sqrt{2 \left (3+\sqrt{5}\right )}\right )}{4 \sqrt{5} \sqrt [4]{2 \left (3+\sqrt{5}\right )}}-\frac{\sqrt [4]{3+\sqrt{5}} \tan ^{-1}\left (1-\frac{2^{3/4} x}{\sqrt [4]{3-\sqrt{5}}}\right )}{2\ 2^{3/4} \sqrt{5}}+\frac{\sqrt [4]{3+\sqrt{5}} \tan ^{-1}\left (\frac{2^{3/4} x}{\sqrt [4]{3-\sqrt{5}}}+1\right )}{2\ 2^{3/4} \sqrt{5}}+\frac{\tan ^{-1}\left (1-\frac{2^{3/4} x}{\sqrt [4]{3+\sqrt{5}}}\right )}{2 \sqrt{5} \sqrt [4]{2 \left (3+\sqrt{5}\right )}}-\frac{\tan ^{-1}\left (\frac{2^{3/4} x}{\sqrt [4]{3+\sqrt{5}}}+1\right )}{2 \sqrt{5} \sqrt [4]{2 \left (3+\sqrt{5}\right )}} \]

Antiderivative was successfully verified.

[In]  Int[x^2/(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)*Sq
rt[5]) + ((3 + Sqrt[5])^(1/4)*ArcTan[1 + (2^(3/4)*x)/(3 - Sqrt[5])^(1/4)])/(2*2^
(3/4)*Sqrt[5]) + ArcTan[1 - (2^(3/4)*x)/(3 + Sqrt[5])^(1/4)]/(2*Sqrt[5]*(2*(3 +
Sqrt[5]))^(1/4)) - ArcTan[1 + (2^(3/4)*x)/(3 + Sqrt[5])^(1/4)]/(2*Sqrt[5]*(2*(3
+ Sqrt[5]))^(1/4)) + ((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]) - L
og[Sqrt[2*(3 + Sqrt[5])] - 2*(2*(3 + Sqrt[5]))^(1/4)*x + 2*x^2]/(4*Sqrt[5]*(2*(3
+ Sqrt[5]))^(1/4)) + Log[Sqrt[2*(3 + Sqrt[5])] + 2*(2*(3 + Sqrt[5]))^(1/4)*x +
2*x^2]/(4*Sqrt[5]*(2*(3 + Sqrt[5]))^(1/4))

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Rubi in Sympy [A]  time = 86.542, size = 488, normalized size = 1.09 \[ \frac{2^{\frac{3}{4}} \sqrt{5} \left (- 2 \sqrt{5} + 6\right ) \log{\left (2 x^{2} - 2 \sqrt [4]{2} x \sqrt [4]{- \sqrt{5} + 3} + \sqrt{- 2 \sqrt{5} + 6} \right )}}{80 \left (- \sqrt{5} + 3\right )^{\frac{5}{4}}} - \frac{2^{\frac{3}{4}} \sqrt{5} \left (- 2 \sqrt{5} + 6\right ) \log{\left (2 x^{2} + 2 \sqrt [4]{2} x \sqrt [4]{- \sqrt{5} + 3} + \sqrt{- 2 \sqrt{5} + 6} \right )}}{80 \left (- \sqrt{5} + 3\right )^{\frac{5}{4}}} - \frac{2^{\frac{3}{4}} \sqrt{5} \left (2 \sqrt{5} + 6\right ) \log{\left (2 x^{2} - 2 \sqrt [4]{2} x \sqrt [4]{\sqrt{5} + 3} + \sqrt{2 \sqrt{5} + 6} \right )}}{80 \left (\sqrt{5} + 3\right )^{\frac{5}{4}}} + \frac{2^{\frac{3}{4}} \sqrt{5} \left (2 \sqrt{5} + 6\right ) \log{\left (2 x^{2} + 2 \sqrt [4]{2} x \sqrt [4]{\sqrt{5} + 3} + \sqrt{2 \sqrt{5} + 6} \right )}}{80 \left (\sqrt{5} + 3\right )^{\frac{5}{4}}} + \frac{2^{\frac{3}{4}} \sqrt{5} \operatorname{atan}{\left (\frac{2^{\frac{3}{4}} \left (x - \frac{\sqrt [4]{- 2 \sqrt{5} + 6}}{2}\right )}{\sqrt [4]{- \sqrt{5} + 3}} \right )}}{20 \sqrt [4]{- \sqrt{5} + 3}} + \frac{2^{\frac{3}{4}} \sqrt{5} \operatorname{atan}{\left (\frac{2^{\frac{3}{4}} \left (x + \frac{\sqrt [4]{- 2 \sqrt{5} + 6}}{2}\right )}{\sqrt [4]{- \sqrt{5} + 3}} \right )}}{20 \sqrt [4]{- \sqrt{5} + 3}} - \frac{2^{\frac{3}{4}} \sqrt{5} \operatorname{atan}{\left (\frac{2^{\frac{3}{4}} \left (x - \frac{\sqrt [4]{2 \sqrt{5} + 6}}{2}\right )}{\sqrt [4]{\sqrt{5} + 3}} \right )}}{20 \sqrt [4]{\sqrt{5} + 3}} - \frac{2^{\frac{3}{4}} \sqrt{5} \operatorname{atan}{\left (\frac{2^{\frac{3}{4}} \left (x + \frac{\sqrt [4]{2 \sqrt{5} + 6}}{2}\right )}{\sqrt [4]{\sqrt{5} + 3}} \right )}}{20 \sqrt [4]{\sqrt{5} + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2**(3/4)*sqrt(5)*(-2*sqrt(5) + 6)*log(2*x**2 - 2*2**(1/4)*x*(-sqrt(5) + 3)**(1/4
) + sqrt(-2*sqrt(5) + 6))/(80*(-sqrt(5) + 3)**(5/4)) - 2**(3/4)*sqrt(5)*(-2*sqrt
(5) + 6)*log(2*x**2 + 2*2**(1/4)*x*(-sqrt(5) + 3)**(1/4) + sqrt(-2*sqrt(5) + 6))
/(80*(-sqrt(5) + 3)**(5/4)) - 2**(3/4)*sqrt(5)*(2*sqrt(5) + 6)*log(2*x**2 - 2*2*
*(1/4)*x*(sqrt(5) + 3)**(1/4) + sqrt(2*sqrt(5) + 6))/(80*(sqrt(5) + 3)**(5/4)) +
2**(3/4)*sqrt(5)*(2*sqrt(5) + 6)*log(2*x**2 + 2*2**(1/4)*x*(sqrt(5) + 3)**(1/4)
+ sqrt(2*sqrt(5) + 6))/(80*(sqrt(5) + 3)**(5/4)) + 2**(3/4)*sqrt(5)*atan(2**(3/
4)*(x - (-2*sqrt(5) + 6)**(1/4)/2)/(-sqrt(5) + 3)**(1/4))/(20*(-sqrt(5) + 3)**(1
/4)) + 2**(3/4)*sqrt(5)*atan(2**(3/4)*(x + (-2*sqrt(5) + 6)**(1/4)/2)/(-sqrt(5)
+ 3)**(1/4))/(20*(-sqrt(5) + 3)**(1/4)) - 2**(3/4)*sqrt(5)*atan(2**(3/4)*(x - (2
*sqrt(5) + 6)**(1/4)/2)/(sqrt(5) + 3)**(1/4))/(20*(sqrt(5) + 3)**(1/4)) - 2**(3/
4)*sqrt(5)*atan(2**(3/4)*(x + (2*sqrt(5) + 6)**(1/4)/2)/(sqrt(5) + 3)**(1/4))/(2
0*(sqrt(5) + 3)**(1/4))

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Mathematica [C]  time = 0.0154862, size = 40, normalized size = 0.09 \[ \frac{1}{4} \text{RootSum}\left [\text{$\#$1}^8+3 \text{$\#$1}^4+1\&,\frac{\log (x-\text{$\#$1})}{2 \text{$\#$1}^5+3 \text{$\#$1}}\&\right ] \]

Antiderivative was successfully verified.

[In]  Integrate[x^2/(1 + 3*x^4 + x^8),x]

[Out]

RootSum[1 + 3*#1^4 + #1^8 & , Log[x - #1]/(3*#1 + 2*#1^5) & ]/4

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Maple [C]  time = 0.009, size = 40, normalized size = 0.1 \[{\frac{1}{4}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{8}+3\,{{\it \_Z}}^{4}+1 \right ) }{\frac{{{\it \_R}}^{2}\ln \left ( x-{\it \_R} \right ) }{2\,{{\it \_R}}^{7}+3\,{{\it \_R}}^{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(x^8+3*x^4+1),x)

[Out]

1/4*sum(_R^2/(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. \[ \int \frac{x^{2}}{x^{8} + 3 \, x^{4} + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(x^8 + 3*x^4 + 1),x, algorithm="maxima")

[Out]

integrate(x^2/(x^8 + 3*x^4 + 1), x)

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Fricas [A]  time = 0.311206, size = 1640, normalized size = 3.65 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(x^8 + 3*x^4 + 1),x, algorithm="fricas")

[Out]

1/40*sqrt(5)*sqrt(2)*(4*(1/250)^(1/4)*sqrt(sqrt(5)*(3*sqrt(5) + 5))*(sqrt(5)*(3*
sqrt(5) - 5))^(3/4)*arctan(1/4*sqrt(5)*(1/250)^(1/4)*(sqrt(5)*(3*sqrt(5) - 5))^(
1/4)*(3*sqrt(5) - 5)*(sqrt(5) + 3)/(sqrt(5)*sqrt(2)*sqrt(1/10)*sqrt(sqrt(5)*(3*s
qrt(5) - 5))*x + sqrt(5)*sqrt(2)*sqrt(1/10)*sqrt(sqrt(5)*(3*sqrt(5) - 5))*sqrt((
10*sqrt(2)*sqrt(1/10)*(1/250)^(1/4)*(sqrt(5)*(3*sqrt(5) - 5))^(3/4)*x + 3*sqrt(5
)*x^2 - 5*x^2 + 2*sqrt(5)*sqrt(1/10)*sqrt(sqrt(5)*(3*sqrt(5) - 5)))/(3*sqrt(5) -
5)) + 5*(1/250)^(1/4)*(sqrt(5)*(3*sqrt(5) - 5))^(1/4))) + 4*(1/250)^(1/4)*sqrt(
sqrt(5)*(3*sqrt(5) + 5))*(sqrt(5)*(3*sqrt(5) - 5))^(3/4)*arctan(1/4*sqrt(5)*(1/2
50)^(1/4)*(sqrt(5)*(3*sqrt(5) - 5))^(1/4)*(3*sqrt(5) - 5)*(sqrt(5) + 3)/(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)*(3*sqrt(5) - 5))*sqrt(-(10*sqrt(2)*sqrt(1/10)*(1/250)^(1/4)*(sqrt(
5)*(3*sqrt(5) - 5))^(3/4)*x - 3*sqrt(5)*x^2 + 5*x^2 - 2*sqrt(5)*sqrt(1/10)*sqrt(
sqrt(5)*(3*sqrt(5) - 5)))/(3*sqrt(5) - 5)) - 5*(1/250)^(1/4)*(sqrt(5)*(3*sqrt(5)
- 5))^(1/4))) + 4*(1/250)^(1/4)*(sqrt(5)*(3*sqrt(5) + 5))^(3/4)*sqrt(sqrt(5)*(3
*sqrt(5) - 5))*arctan(1/4*sqrt(5)*(1/250)^(1/4)*(sqrt(5)*(3*sqrt(5) + 5))^(1/4)*
(3*sqrt(5) + 5)*(sqrt(5) - 3)/(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)*(3*sqrt(5) + 5))*sqrt((10*sq
rt(2)*sqrt(1/10)*(1/250)^(1/4)*(sqrt(5)*(3*sqrt(5) + 5))^(3/4)*x + 3*sqrt(5)*x^2
+ 5*x^2 + 2*sqrt(5)*sqrt(1/10)*sqrt(sqrt(5)*(3*sqrt(5) + 5)))/(3*sqrt(5) + 5))
+ 5*(1/250)^(1/4)*(sqrt(5)*(3*sqrt(5) + 5))^(1/4))) + 4*(1/250)^(1/4)*(sqrt(5)*(
3*sqrt(5) + 5))^(3/4)*sqrt(sqrt(5)*(3*sqrt(5) - 5))*arctan(1/4*sqrt(5)*(1/250)^(
1/4)*(sqrt(5)*(3*sqrt(5) + 5))^(1/4)*(3*sqrt(5) + 5)*(sqrt(5) - 3)/(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)*(3*sqrt(5) + 5))*sqrt(-(10*sqrt(2)*sqrt(1/10)*(1/250)^(1/4)*(sqrt(5)*(3
*sqrt(5) + 5))^(3/4)*x - 3*sqrt(5)*x^2 - 5*x^2 - 2*sqrt(5)*sqrt(1/10)*sqrt(sqrt(
5)*(3*sqrt(5) + 5)))/(3*sqrt(5) + 5)) - 5*(1/250)^(1/4)*(sqrt(5)*(3*sqrt(5) + 5)
)^(1/4))) - (1/250)^(1/4)*(sqrt(5)*(3*sqrt(5) + 5))^(3/4)*sqrt(sqrt(5)*(3*sqrt(5
) - 5))*log(10*sqrt(2)*sqrt(1/10)*(1/250)^(1/4)*(sqrt(5)*(3*sqrt(5) + 5))^(3/4)*
x + 3*sqrt(5)*x^2 + 5*x^2 + 2*sqrt(5)*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))*lo
g(-10*sqrt(2)*sqrt(1/10)*(1/250)^(1/4)*(sqrt(5)*(3*sqrt(5) + 5))^(3/4)*x + 3*sqr
t(5)*x^2 + 5*x^2 + 2*sqrt(5)*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(10*sqrt
(2)*sqrt(1/10)*(1/250)^(1/4)*(sqrt(5)*(3*sqrt(5) - 5))^(3/4)*x + 3*sqrt(5)*x^2 -
5*x^2 + 2*sqrt(5)*sqrt(1/10)*sqrt(sqrt(5)*(3*sqrt(5) - 5))) - (1/250)^(1/4)*sqr
t(sqrt(5)*(3*sqrt(5) + 5))*(sqrt(5)*(3*sqrt(5) - 5))^(3/4)*log(-10*sqrt(2)*sqrt(
1/10)*(1/250)^(1/4)*(sqrt(5)*(3*sqrt(5) - 5))^(3/4)*x + 3*sqrt(5)*x^2 - 5*x^2 +
2*sqrt(5)*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 = 3.75336, size = 26, normalized size = 0.06 \[ \operatorname{RootSum}{\left (40960000 t^{8} + 19200 t^{4} + 1, \left ( t \mapsto t \log{\left (- 6144000 t^{7} - 2240 t^{3} + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2/(x**8+3*x**4+1),x)

[Out]

RootSum(40960000*_t**8 + 19200*_t**4 + 1, Lambda(_t, _t*log(-6144000*_t**7 - 224
0*_t**3 + x)))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(x^8 + 3*x^4 + 1),x, algorithm="giac")

[Out]

integrate(x^2/(x^8 + 3*x^4 + 1), x)