3.379 \(\int \frac{x^6}{1+3 x^4+x^8} \, dx\)

Optimal. Leaf size=449 \[ -\frac{\sqrt [4]{9-4 \sqrt{5}} \log \left (\sqrt{2} x^2-2^{3/4} \sqrt [4]{3-\sqrt{5}} x+\sqrt{3-\sqrt{5}}\right )}{4 \sqrt{10}}+\frac{\sqrt [4]{9-4 \sqrt{5}} \log \left (\sqrt{2} x^2+2^{3/4} \sqrt [4]{3-\sqrt{5}} x+\sqrt{3-\sqrt{5}}\right )}{4 \sqrt{10}}+\frac{\left (3+\sqrt{5}\right )^{3/4} \log \left (\sqrt{2} x^2-2^{3/4} \sqrt [4]{3+\sqrt{5}} x+\sqrt{3+\sqrt{5}}\right )}{8 \sqrt [4]{2} \sqrt{5}}-\frac{\left (3+\sqrt{5}\right )^{3/4} \log \left (\sqrt{2} x^2+2^{3/4} \sqrt [4]{3+\sqrt{5}} x+\sqrt{3+\sqrt{5}}\right )}{8 \sqrt [4]{2} \sqrt{5}}+\frac{\sqrt [4]{9-4 \sqrt{5}} \tan ^{-1}\left (1-\frac{2^{3/4} x}{\sqrt [4]{3-\sqrt{5}}}\right )}{2 \sqrt{10}}-\frac{\sqrt [4]{9-4 \sqrt{5}} \tan ^{-1}\left (\frac{2^{3/4} x}{\sqrt [4]{3-\sqrt{5}}}+1\right )}{2 \sqrt{10}}-\frac{\left (3+\sqrt{5}\right )^{3/4} \tan ^{-1}\left (1-\frac{2^{3/4} x}{\sqrt [4]{3+\sqrt{5}}}\right )}{4 \sqrt [4]{2} \sqrt{5}}+\frac{\left (3+\sqrt{5}\right )^{3/4} \tan ^{-1}\left (\frac{2^{3/4} x}{\sqrt [4]{3+\sqrt{5}}}+1\right )}{4 \sqrt [4]{2} \sqrt{5}} \]

[Out]

((9 - 4*Sqrt[5])^(1/4)*ArcTan[1 - (2^(3/4)*x)/(3 - Sqrt[5])^(1/4)])/(2*Sqrt[10])
- ((9 - 4*Sqrt[5])^(1/4)*ArcTan[1 + (2^(3/4)*x)/(3 - Sqrt[5])^(1/4)])/(2*Sqrt[1
0]) - ((3 + Sqrt[5])^(3/4)*ArcTan[1 - (2^(3/4)*x)/(3 + Sqrt[5])^(1/4)])/(4*2^(1/
4)*Sqrt[5]) + ((3 + Sqrt[5])^(3/4)*ArcTan[1 + (2^(3/4)*x)/(3 + Sqrt[5])^(1/4)])/
(4*2^(1/4)*Sqrt[5]) - ((9 - 4*Sqrt[5])^(1/4)*Log[Sqrt[3 - Sqrt[5]] - 2^(3/4)*(3
- Sqrt[5])^(1/4)*x + Sqrt[2]*x^2])/(4*Sqrt[10]) + ((9 - 4*Sqrt[5])^(1/4)*Log[Sqr
t[3 - Sqrt[5]] + 2^(3/4)*(3 - Sqrt[5])^(1/4)*x + Sqrt[2]*x^2])/(4*Sqrt[10]) + ((
3 + Sqrt[5])^(3/4)*Log[Sqrt[3 + Sqrt[5]] - 2^(3/4)*(3 + Sqrt[5])^(1/4)*x + Sqrt[
2]*x^2])/(8*2^(1/4)*Sqrt[5]) - ((3 + Sqrt[5])^(3/4)*Log[Sqrt[3 + Sqrt[5]] + 2^(3
/4)*(3 + Sqrt[5])^(1/4)*x + Sqrt[2]*x^2])/(8*2^(1/4)*Sqrt[5])

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Rubi [A]  time = 0.704739, 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 \[ -\frac{\sqrt [4]{9-4 \sqrt{5}} \log \left (2 x^2-2 \sqrt [4]{2 \left (3-\sqrt{5}\right )} x+\sqrt{2 \left (3-\sqrt{5}\right )}\right )}{4 \sqrt{10}}+\frac{\sqrt [4]{9-4 \sqrt{5}} \log \left (2 x^2+2 \sqrt [4]{2 \left (3-\sqrt{5}\right )} x+\sqrt{2 \left (3-\sqrt{5}\right )}\right )}{4 \sqrt{10}}+\frac{\left (3+\sqrt{5}\right )^{3/4} \log \left (2 x^2-2 \sqrt [4]{2 \left (3+\sqrt{5}\right )} x+\sqrt{2 \left (3+\sqrt{5}\right )}\right )}{8 \sqrt [4]{2} \sqrt{5}}-\frac{\left (3+\sqrt{5}\right )^{3/4} \log \left (2 x^2+2 \sqrt [4]{2 \left (3+\sqrt{5}\right )} x+\sqrt{2 \left (3+\sqrt{5}\right )}\right )}{8 \sqrt [4]{2} \sqrt{5}}+\frac{\sqrt [4]{9-4 \sqrt{5}} \tan ^{-1}\left (1-\frac{2^{3/4} x}{\sqrt [4]{3-\sqrt{5}}}\right )}{2 \sqrt{10}}-\frac{\sqrt [4]{9-4 \sqrt{5}} \tan ^{-1}\left (\frac{2^{3/4} x}{\sqrt [4]{3-\sqrt{5}}}+1\right )}{2 \sqrt{10}}-\frac{\left (3+\sqrt{5}\right )^{3/4} \tan ^{-1}\left (1-\frac{2^{3/4} x}{\sqrt [4]{3+\sqrt{5}}}\right )}{4 \sqrt [4]{2} \sqrt{5}}+\frac{\left (3+\sqrt{5}\right )^{3/4} \tan ^{-1}\left (\frac{2^{3/4} x}{\sqrt [4]{3+\sqrt{5}}}+1\right )}{4 \sqrt [4]{2} \sqrt{5}} \]

Warning: Unable to verify antiderivative.

[In]  Int[x^6/(1 + 3*x^4 + x^8),x]

[Out]

((9 - 4*Sqrt[5])^(1/4)*ArcTan[1 - (2^(3/4)*x)/(3 - Sqrt[5])^(1/4)])/(2*Sqrt[10])
- ((9 - 4*Sqrt[5])^(1/4)*ArcTan[1 + (2^(3/4)*x)/(3 - Sqrt[5])^(1/4)])/(2*Sqrt[1
0]) - ((3 + Sqrt[5])^(3/4)*ArcTan[1 - (2^(3/4)*x)/(3 + Sqrt[5])^(1/4)])/(4*2^(1/
4)*Sqrt[5]) + ((3 + Sqrt[5])^(3/4)*ArcTan[1 + (2^(3/4)*x)/(3 + Sqrt[5])^(1/4)])/
(4*2^(1/4)*Sqrt[5]) - ((9 - 4*Sqrt[5])^(1/4)*Log[Sqrt[2*(3 - Sqrt[5])] - 2*(2*(3
- Sqrt[5]))^(1/4)*x + 2*x^2])/(4*Sqrt[10]) + ((9 - 4*Sqrt[5])^(1/4)*Log[Sqrt[2*
(3 - Sqrt[5])] + 2*(2*(3 - Sqrt[5]))^(1/4)*x + 2*x^2])/(4*Sqrt[10]) + ((3 + Sqrt
[5])^(3/4)*Log[Sqrt[2*(3 + Sqrt[5])] - 2*(2*(3 + Sqrt[5]))^(1/4)*x + 2*x^2])/(8*
2^(1/4)*Sqrt[5]) - ((3 + Sqrt[5])^(3/4)*Log[Sqrt[2*(3 + Sqrt[5])] + 2*(2*(3 + Sq
rt[5]))^(1/4)*x + 2*x^2])/(8*2^(1/4)*Sqrt[5])

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Rubi in Sympy [A]  time = 86.547, size = 542, normalized size = 1.21 \[ \frac{2^{\frac{3}{4}} \left (- 2 \sqrt{5} + 6\right ) \left (- \frac{3 \sqrt{5}}{10} + \frac{1}{2}\right ) \log{\left (2 x^{2} - 2 \sqrt [4]{2} x \sqrt [4]{- \sqrt{5} + 3} + \sqrt{- 2 \sqrt{5} + 6} \right )}}{16 \left (- \sqrt{5} + 3\right )^{\frac{5}{4}}} - \frac{2^{\frac{3}{4}} \left (- 2 \sqrt{5} + 6\right ) \left (- \frac{3 \sqrt{5}}{10} + \frac{1}{2}\right ) \log{\left (2 x^{2} + 2 \sqrt [4]{2} x \sqrt [4]{- \sqrt{5} + 3} + \sqrt{- 2 \sqrt{5} + 6} \right )}}{16 \left (- \sqrt{5} + 3\right )^{\frac{5}{4}}} + \frac{2^{\frac{3}{4}} \left (\frac{1}{2} + \frac{3 \sqrt{5}}{10}\right ) \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 )}}{16 \left (\sqrt{5} + 3\right )^{\frac{5}{4}}} - \frac{2^{\frac{3}{4}} \left (\frac{1}{2} + \frac{3 \sqrt{5}}{10}\right ) \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 )}}{16 \left (\sqrt{5} + 3\right )^{\frac{5}{4}}} + \frac{2^{\frac{3}{4}} \left (- \frac{3 \sqrt{5}}{10} + \frac{1}{2}\right ) \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 )}}{4 \sqrt [4]{- \sqrt{5} + 3}} + \frac{2^{\frac{3}{4}} \left (- \frac{3 \sqrt{5}}{10} + \frac{1}{2}\right ) \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 )}}{4 \sqrt [4]{- \sqrt{5} + 3}} + \frac{2^{\frac{3}{4}} \left (\frac{1}{2} + \frac{3 \sqrt{5}}{10}\right ) \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 )}}{4 \sqrt [4]{\sqrt{5} + 3}} + \frac{2^{\frac{3}{4}} \left (\frac{1}{2} + \frac{3 \sqrt{5}}{10}\right ) \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 )}}{4 \sqrt [4]{\sqrt{5} + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**6/(x**8+3*x**4+1),x)

[Out]

2**(3/4)*(-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))/(16*(-sqrt(5) + 3)**(5/4)) - 2**(3/4)*(-
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))/(16*(-sqrt(5) + 3)**(5/4)) + 2**(3/4)*(1/2 + 3*sqrt
(5)/10)*(2*sqrt(5) + 6)*log(2*x**2 - 2*2**(1/4)*x*(sqrt(5) + 3)**(1/4) + sqrt(2*
sqrt(5) + 6))/(16*(sqrt(5) + 3)**(5/4)) - 2**(3/4)*(1/2 + 3*sqrt(5)/10)*(2*sqrt(
5) + 6)*log(2*x**2 + 2*2**(1/4)*x*(sqrt(5) + 3)**(1/4) + sqrt(2*sqrt(5) + 6))/(1
6*(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))/(4*(-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))/(4*(-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))/(4*(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))/(4*(sqrt(5) + 3)**(1/4))

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Mathematica [C]  time = 0.0171233, size = 41, normalized size = 0.09 \[ \frac{1}{4} \text{RootSum}\left [\text{$\#$1}^8+3 \text{$\#$1}^4+1\&,\frac{\text{$\#$1}^3 \log (x-\text{$\#$1})}{2 \text{$\#$1}^4+3}\&\right ] \]

Antiderivative was successfully verified.

[In]  Integrate[x^6/(1 + 3*x^4 + x^8),x]

[Out]

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

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Maple [C]  time = 0.009, size = 40, normalized size = 0.1 \[{\frac{1}{4}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{8}+3\,{{\it \_Z}}^{4}+1 \right ) }{\frac{{{\it \_R}}^{6}\ln \left ( x-{\it \_R} \right ) }{2\,{{\it \_R}}^{7}+3\,{{\it \_R}}^{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^6/(x^8+3*x^4+1),x)

[Out]

1/4*sum(_R^6/(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. \[ \int \frac{x^{6}}{x^{8} + 3 \, x^{4} + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^6/(x^8 + 3*x^4 + 1),x, algorithm="maxima")

[Out]

integrate(x^6/(x^8 + 3*x^4 + 1), x)

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Fricas [A]  time = 0.310487, size = 1864, normalized size = 4.15 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^6/(x^8 + 3*x^4 + 1),x, algorithm="fricas")

[Out]

1/40*sqrt(5)*sqrt(2)*(4*(1/125)^(1/4)*sqrt(sqrt(5)*(9*sqrt(5) + 20))*(sqrt(5)*(9
*sqrt(5) - 20))^(3/4)*arctan(sqrt(5)*(1/125)^(1/4)*(sqrt(5)*(9*sqrt(5) - 20))^(1
/4)*(9*sqrt(5) - 20)*(3*sqrt(5) + 7)/(2*sqrt(5)*sqrt(2)*sqrt(1/5)*sqrt(sqrt(5)*(
9*sqrt(5) - 20))*x + 2*sqrt(5)*sqrt(2)*sqrt(1/2)*sqrt(1/5)*sqrt(sqrt(5)*(9*sqrt(
5) - 20))*sqrt((sqrt(1/5)*(1/125)^(1/4)*(3*sqrt(5)*sqrt(2)*x - 5*sqrt(2)*x)*(sqr
t(5)*(9*sqrt(5) - 20))^(3/4) + 8*sqrt(5)*x^2 - 18*x^2 + sqrt(1/5)*sqrt(sqrt(5)*(
9*sqrt(5) - 20))*(3*sqrt(5) - 7))/(4*sqrt(5) - 9)) + 5*(1/125)^(1/4)*(sqrt(5)*(9
*sqrt(5) - 20))^(1/4)*(sqrt(5) - 3))) + 4*(1/125)^(1/4)*sqrt(sqrt(5)*(9*sqrt(5)
+ 20))*(sqrt(5)*(9*sqrt(5) - 20))^(3/4)*arctan(sqrt(5)*(1/125)^(1/4)*(sqrt(5)*(9
*sqrt(5) - 20))^(1/4)*(9*sqrt(5) - 20)*(3*sqrt(5) + 7)/(2*sqrt(5)*sqrt(2)*sqrt(1
/5)*sqrt(sqrt(5)*(9*sqrt(5) - 20))*x + 2*sqrt(5)*sqrt(2)*sqrt(1/2)*sqrt(1/5)*sqr
t(sqrt(5)*(9*sqrt(5) - 20))*sqrt(-(sqrt(1/5)*(1/125)^(1/4)*(3*sqrt(5)*sqrt(2)*x
- 5*sqrt(2)*x)*(sqrt(5)*(9*sqrt(5) - 20))^(3/4) - 8*sqrt(5)*x^2 + 18*x^2 - sqrt(
1/5)*sqrt(sqrt(5)*(9*sqrt(5) - 20))*(3*sqrt(5) - 7))/(4*sqrt(5) - 9)) - 5*(1/125
)^(1/4)*(sqrt(5)*(9*sqrt(5) - 20))^(1/4)*(sqrt(5) - 3))) + 4*(1/125)^(1/4)*(sqrt
(5)*(9*sqrt(5) + 20))^(3/4)*sqrt(sqrt(5)*(9*sqrt(5) - 20))*arctan(sqrt(5)*(1/125
)^(1/4)*(sqrt(5)*(9*sqrt(5) + 20))^(1/4)*(9*sqrt(5) + 20)*(3*sqrt(5) - 7)/(2*sqr
t(5)*sqrt(2)*sqrt(1/5)*sqrt(sqrt(5)*(9*sqrt(5) + 20))*x + 2*sqrt(5)*sqrt(2)*sqrt
(1/2)*sqrt(1/5)*sqrt(sqrt(5)*(9*sqrt(5) + 20))*sqrt((sqrt(1/5)*(1/125)^(1/4)*(3*
sqrt(5)*sqrt(2)*x + 5*sqrt(2)*x)*(sqrt(5)*(9*sqrt(5) + 20))^(3/4) + 8*sqrt(5)*x^
2 + 18*x^2 + sqrt(1/5)*sqrt(sqrt(5)*(9*sqrt(5) + 20))*(3*sqrt(5) + 7))/(4*sqrt(5
) + 9)) + 5*(1/125)^(1/4)*(sqrt(5)*(9*sqrt(5) + 20))^(1/4)*(sqrt(5) + 3))) + 4*(
1/125)^(1/4)*(sqrt(5)*(9*sqrt(5) + 20))^(3/4)*sqrt(sqrt(5)*(9*sqrt(5) - 20))*arc
tan(sqrt(5)*(1/125)^(1/4)*(sqrt(5)*(9*sqrt(5) + 20))^(1/4)*(9*sqrt(5) + 20)*(3*s
qrt(5) - 7)/(2*sqrt(5)*sqrt(2)*sqrt(1/5)*sqrt(sqrt(5)*(9*sqrt(5) + 20))*x + 2*sq
rt(5)*sqrt(2)*sqrt(1/2)*sqrt(1/5)*sqrt(sqrt(5)*(9*sqrt(5) + 20))*sqrt(-(sqrt(1/5
)*(1/125)^(1/4)*(3*sqrt(5)*sqrt(2)*x + 5*sqrt(2)*x)*(sqrt(5)*(9*sqrt(5) + 20))^(
3/4) - 8*sqrt(5)*x^2 - 18*x^2 - sqrt(1/5)*sqrt(sqrt(5)*(9*sqrt(5) + 20))*(3*sqrt
(5) + 7))/(4*sqrt(5) + 9)) - 5*(1/125)^(1/4)*(sqrt(5)*(9*sqrt(5) + 20))^(1/4)*(s
qrt(5) + 3))) - (1/125)^(1/4)*(sqrt(5)*(9*sqrt(5) + 20))^(3/4)*sqrt(sqrt(5)*(9*s
qrt(5) - 20))*log(1/2*sqrt(1/5)*(1/125)^(1/4)*(3*sqrt(5)*sqrt(2)*x + 5*sqrt(2)*x
)*(sqrt(5)*(9*sqrt(5) + 20))^(3/4) + 4*sqrt(5)*x^2 + 9*x^2 + 1/2*sqrt(1/5)*sqrt(
sqrt(5)*(9*sqrt(5) + 20))*(3*sqrt(5) + 7)) + (1/125)^(1/4)*(sqrt(5)*(9*sqrt(5) +
20))^(3/4)*sqrt(sqrt(5)*(9*sqrt(5) - 20))*log(-1/2*sqrt(1/5)*(1/125)^(1/4)*(3*s
qrt(5)*sqrt(2)*x + 5*sqrt(2)*x)*(sqrt(5)*(9*sqrt(5) + 20))^(3/4) + 4*sqrt(5)*x^2
+ 9*x^2 + 1/2*sqrt(1/5)*sqrt(sqrt(5)*(9*sqrt(5) + 20))*(3*sqrt(5) + 7)) - (1/12
5)^(1/4)*sqrt(sqrt(5)*(9*sqrt(5) + 20))*(sqrt(5)*(9*sqrt(5) - 20))^(3/4)*log(1/2
*sqrt(1/5)*(1/125)^(1/4)*(3*sqrt(5)*sqrt(2)*x - 5*sqrt(2)*x)*(sqrt(5)*(9*sqrt(5)
- 20))^(3/4) + 4*sqrt(5)*x^2 - 9*x^2 + 1/2*sqrt(1/5)*sqrt(sqrt(5)*(9*sqrt(5) -
20))*(3*sqrt(5) - 7)) + (1/125)^(1/4)*sqrt(sqrt(5)*(9*sqrt(5) + 20))*(sqrt(5)*(9
*sqrt(5) - 20))^(3/4)*log(-1/2*sqrt(1/5)*(1/125)^(1/4)*(3*sqrt(5)*sqrt(2)*x - 5*
sqrt(2)*x)*(sqrt(5)*(9*sqrt(5) - 20))^(3/4) + 4*sqrt(5)*x^2 - 9*x^2 + 1/2*sqrt(1
/5)*sqrt(sqrt(5)*(9*sqrt(5) - 20))*(3*sqrt(5) - 7)))/(sqrt(9*sqrt(5) + 20)*sqrt(
9*sqrt(5) - 20))

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Sympy [A]  time = 3.7956, size = 26, normalized size = 0.06 \[ \operatorname{RootSum}{\left (40960000 t^{8} + 115200 t^{4} + 1, \left ( t \mapsto t \log{\left (- 1792000 t^{7} - 4920 t^{3} + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**6/(x**8+3*x**4+1),x)

[Out]

RootSum(40960000*_t**8 + 115200*_t**4 + 1, Lambda(_t, _t*log(-1792000*_t**7 - 49
20*_t**3 + x)))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{6}}{x^{8} + 3 \, x^{4} + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^6/(x^8 + 3*x^4 + 1),x, algorithm="giac")

[Out]

integrate(x^6/(x^8 + 3*x^4 + 1), x)