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

Optimal. Leaf size=169 \[ -\frac{\tan ^{-1}\left (\sqrt [4]{\frac{2}{3+\sqrt{5}}} x\right )}{\sqrt [4]{2} \sqrt{5} \left (3+\sqrt{5}\right )^{3/4}}+\frac{\left (3+\sqrt{5}\right )^{3/4} \tan ^{-1}\left (\sqrt [4]{\frac{1}{2} \left (3+\sqrt{5}\right )} x\right )}{2\ 2^{3/4} \sqrt{5}}-\frac{\tanh ^{-1}\left (\sqrt [4]{\frac{2}{3+\sqrt{5}}} x\right )}{\sqrt [4]{2} \sqrt{5} \left (3+\sqrt{5}\right )^{3/4}}+\frac{\left (3+\sqrt{5}\right )^{3/4} \tanh ^{-1}\left (\sqrt [4]{\frac{1}{2} \left (3+\sqrt{5}\right )} x\right )}{2\ 2^{3/4} \sqrt{5}} \]

[Out]

-(ArcTan[(2/(3 + Sqrt[5]))^(1/4)*x]/(2^(1/4)*Sqrt[5]*(3 + Sqrt[5])^(3/4))) + ((3
+ Sqrt[5])^(3/4)*ArcTan[((3 + Sqrt[5])/2)^(1/4)*x])/(2*2^(3/4)*Sqrt[5]) - ArcTa
nh[(2/(3 + Sqrt[5]))^(1/4)*x]/(2^(1/4)*Sqrt[5]*(3 + Sqrt[5])^(3/4)) + ((3 + Sqrt
[5])^(3/4)*ArcTanh[((3 + Sqrt[5])/2)^(1/4)*x])/(2*2^(3/4)*Sqrt[5])

_______________________________________________________________________________________

Rubi [A]  time = 0.137618, antiderivative size = 169, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{\tan ^{-1}\left (\sqrt [4]{\frac{2}{3+\sqrt{5}}} x\right )}{\sqrt [4]{2} \sqrt{5} \left (3+\sqrt{5}\right )^{3/4}}+\frac{\left (3+\sqrt{5}\right )^{3/4} \tan ^{-1}\left (\sqrt [4]{\frac{1}{2} \left (3+\sqrt{5}\right )} x\right )}{2\ 2^{3/4} \sqrt{5}}-\frac{\tanh ^{-1}\left (\sqrt [4]{\frac{2}{3+\sqrt{5}}} x\right )}{\sqrt [4]{2} \sqrt{5} \left (3+\sqrt{5}\right )^{3/4}}+\frac{\left (3+\sqrt{5}\right )^{3/4} \tanh ^{-1}\left (\sqrt [4]{\frac{1}{2} \left (3+\sqrt{5}\right )} x\right )}{2\ 2^{3/4} \sqrt{5}} \]

Antiderivative was successfully verified.

[In]  Int[(1 - 3*x^4 + x^8)^(-1),x]

[Out]

-(ArcTan[(2/(3 + Sqrt[5]))^(1/4)*x]/(2^(1/4)*Sqrt[5]*(3 + Sqrt[5])^(3/4))) + ((3
+ Sqrt[5])^(3/4)*ArcTan[((3 + Sqrt[5])/2)^(1/4)*x])/(2*2^(3/4)*Sqrt[5]) - ArcTa
nh[(2/(3 + Sqrt[5]))^(1/4)*x]/(2^(1/4)*Sqrt[5]*(3 + Sqrt[5])^(3/4)) + ((3 + Sqrt
[5])^(3/4)*ArcTanh[((3 + Sqrt[5])/2)^(1/4)*x])/(2*2^(3/4)*Sqrt[5])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 15.4095, size = 209, normalized size = 1.24 \[ \frac{\sqrt [4]{2} \sqrt{5} \sqrt{- 2 \sqrt{5} + 6} \operatorname{atan}{\left (\frac{\sqrt [4]{2} x}{\sqrt [4]{- \sqrt{5} + 3}} \right )}}{10 \left (- \sqrt{5} + 3\right )^{\frac{5}{4}}} - \frac{\sqrt [4]{2} \sqrt{5} \sqrt{2 \sqrt{5} + 6} \operatorname{atan}{\left (\frac{\sqrt [4]{2} x}{\sqrt [4]{\sqrt{5} + 3}} \right )}}{10 \left (\sqrt{5} + 3\right )^{\frac{5}{4}}} + \frac{\sqrt [4]{2} \sqrt{5} \sqrt{- 2 \sqrt{5} + 6} \operatorname{atanh}{\left (\frac{\sqrt [4]{2} x}{\sqrt [4]{- \sqrt{5} + 3}} \right )}}{10 \left (- \sqrt{5} + 3\right )^{\frac{5}{4}}} - \frac{\sqrt [4]{2} \sqrt{5} \sqrt{2 \sqrt{5} + 6} \operatorname{atanh}{\left (\frac{\sqrt [4]{2} x}{\sqrt [4]{\sqrt{5} + 3}} \right )}}{10 \left (\sqrt{5} + 3\right )^{\frac{5}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2**(1/4)*sqrt(5)*sqrt(-2*sqrt(5) + 6)*atan(2**(1/4)*x/(-sqrt(5) + 3)**(1/4))/(10
*(-sqrt(5) + 3)**(5/4)) - 2**(1/4)*sqrt(5)*sqrt(2*sqrt(5) + 6)*atan(2**(1/4)*x/(
sqrt(5) + 3)**(1/4))/(10*(sqrt(5) + 3)**(5/4)) + 2**(1/4)*sqrt(5)*sqrt(-2*sqrt(5
) + 6)*atanh(2**(1/4)*x/(-sqrt(5) + 3)**(1/4))/(10*(-sqrt(5) + 3)**(5/4)) - 2**(
1/4)*sqrt(5)*sqrt(2*sqrt(5) + 6)*atanh(2**(1/4)*x/(sqrt(5) + 3)**(1/4))/(10*(sqr
t(5) + 3)**(5/4))

_______________________________________________________________________________________

Mathematica [A]  time = 0.27938, size = 160, normalized size = 0.95 \[ \frac{\frac{\left (1+\sqrt{5}\right ) \tan ^{-1}\left (\sqrt{\frac{2}{\sqrt{5}-1}} x\right )}{\sqrt{\sqrt{5}-1}}-\frac{\left (\sqrt{5}-1\right ) \tan ^{-1}\left (\sqrt{\frac{2}{1+\sqrt{5}}} x\right )}{\sqrt{1+\sqrt{5}}}+\frac{\left (1+\sqrt{5}\right ) \tanh ^{-1}\left (\sqrt{\frac{2}{\sqrt{5}-1}} x\right )}{\sqrt{\sqrt{5}-1}}-\frac{\left (\sqrt{5}-1\right ) \tanh ^{-1}\left (\sqrt{\frac{2}{1+\sqrt{5}}} x\right )}{\sqrt{1+\sqrt{5}}}}{2 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

(((1 + Sqrt[5])*ArcTan[Sqrt[2/(-1 + Sqrt[5])]*x])/Sqrt[-1 + Sqrt[5]] - ((-1 + Sq
rt[5])*ArcTan[Sqrt[2/(1 + Sqrt[5])]*x])/Sqrt[1 + Sqrt[5]] + ((1 + Sqrt[5])*ArcTa
nh[Sqrt[2/(-1 + Sqrt[5])]*x])/Sqrt[-1 + Sqrt[5]] - ((-1 + Sqrt[5])*ArcTanh[Sqrt[
2/(1 + Sqrt[5])]*x])/Sqrt[1 + Sqrt[5]])/(2*Sqrt[10])

_______________________________________________________________________________________

Maple [A]  time = 0.035, size = 206, normalized size = 1.2 \[{\frac{\sqrt{5}}{10\,\sqrt{2\,\sqrt{5}+2}}\arctan \left ( 2\,{\frac{x}{\sqrt{2\,\sqrt{5}+2}}} \right ) }-{\frac{1}{2\,\sqrt{2\,\sqrt{5}+2}}\arctan \left ( 2\,{\frac{x}{\sqrt{2\,\sqrt{5}+2}}} \right ) }+{\frac{1}{2\,\sqrt{-2+2\,\sqrt{5}}}{\it Artanh} \left ( 2\,{\frac{x}{\sqrt{-2+2\,\sqrt{5}}}} \right ) }+{\frac{\sqrt{5}}{10\,\sqrt{-2+2\,\sqrt{5}}}{\it Artanh} \left ( 2\,{\frac{x}{\sqrt{-2+2\,\sqrt{5}}}} \right ) }+{\frac{1}{2\,\sqrt{-2+2\,\sqrt{5}}}\arctan \left ( 2\,{\frac{x}{\sqrt{-2+2\,\sqrt{5}}}} \right ) }+{\frac{\sqrt{5}}{10\,\sqrt{-2+2\,\sqrt{5}}}\arctan \left ( 2\,{\frac{x}{\sqrt{-2+2\,\sqrt{5}}}} \right ) }+{\frac{\sqrt{5}}{10\,\sqrt{2\,\sqrt{5}+2}}{\it Artanh} \left ( 2\,{\frac{x}{\sqrt{2\,\sqrt{5}+2}}} \right ) }-{\frac{1}{2\,\sqrt{2\,\sqrt{5}+2}}{\it Artanh} \left ( 2\,{\frac{x}{\sqrt{2\,\sqrt{5}+2}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(x^8-3*x^4+1),x)

[Out]

1/10*5^(1/2)/(2*5^(1/2)+2)^(1/2)*arctan(2*x/(2*5^(1/2)+2)^(1/2))-1/2/(2*5^(1/2)+
2)^(1/2)*arctan(2*x/(2*5^(1/2)+2)^(1/2))+1/2/(-2+2*5^(1/2))^(1/2)*arctanh(2*x/(-
2+2*5^(1/2))^(1/2))+1/10*5^(1/2)/(-2+2*5^(1/2))^(1/2)*arctanh(2*x/(-2+2*5^(1/2))
^(1/2))+1/2/(-2+2*5^(1/2))^(1/2)*arctan(2*x/(-2+2*5^(1/2))^(1/2))+1/10*5^(1/2)/(
-2+2*5^(1/2))^(1/2)*arctan(2*x/(-2+2*5^(1/2))^(1/2))+1/10*5^(1/2)/(2*5^(1/2)+2)^
(1/2)*arctanh(2*x/(2*5^(1/2)+2)^(1/2))-1/2/(2*5^(1/2)+2)^(1/2)*arctanh(2*x/(2*5^
(1/2)+2)^(1/2))

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/(x^8 - 3*x^4 + 1), x)

_______________________________________________________________________________________

Fricas [A]  time = 0.334191, size = 401, normalized size = 2.37 \[ \frac{1}{5} \, \sqrt{-\sqrt{5}{\left (2 \, \sqrt{5} - 5\right )}} \arctan \left (\frac{\sqrt{-\sqrt{5}{\left (2 \, \sqrt{5} - 5\right )}}{\left (3 \, \sqrt{5} + 5\right )}}{10 \,{\left (\sqrt{\frac{1}{10}} \sqrt{\sqrt{5}{\left (\sqrt{5}{\left (2 \, x^{2} + 1\right )} + 5\right )}} + x\right )}}\right ) - \frac{1}{5} \, \sqrt{\sqrt{5}{\left (2 \, \sqrt{5} + 5\right )}} \arctan \left (\frac{\sqrt{\sqrt{5}{\left (2 \, \sqrt{5} + 5\right )}}{\left (3 \, \sqrt{5} - 5\right )}}{10 \,{\left (\sqrt{\frac{1}{10}} \sqrt{\sqrt{5}{\left (\sqrt{5}{\left (2 \, x^{2} - 1\right )} + 5\right )}} + x\right )}}\right ) - \frac{1}{20} \, \sqrt{-\sqrt{5}{\left (2 \, \sqrt{5} - 5\right )}} \log \left (\frac{1}{10} \, \sqrt{-\sqrt{5}{\left (2 \, \sqrt{5} - 5\right )}}{\left (3 \, \sqrt{5} + 5\right )} + x\right ) + \frac{1}{20} \, \sqrt{-\sqrt{5}{\left (2 \, \sqrt{5} - 5\right )}} \log \left (-\frac{1}{10} \, \sqrt{-\sqrt{5}{\left (2 \, \sqrt{5} - 5\right )}}{\left (3 \, \sqrt{5} + 5\right )} + x\right ) + \frac{1}{20} \, \sqrt{\sqrt{5}{\left (2 \, \sqrt{5} + 5\right )}} \log \left (\frac{1}{10} \, \sqrt{\sqrt{5}{\left (2 \, \sqrt{5} + 5\right )}}{\left (3 \, \sqrt{5} - 5\right )} + x\right ) - \frac{1}{20} \, \sqrt{\sqrt{5}{\left (2 \, \sqrt{5} + 5\right )}} \log \left (-\frac{1}{10} \, \sqrt{\sqrt{5}{\left (2 \, \sqrt{5} + 5\right )}}{\left (3 \, \sqrt{5} - 5\right )} + x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5*sqrt(-sqrt(5)*(2*sqrt(5) - 5))*arctan(1/10*sqrt(-sqrt(5)*(2*sqrt(5) - 5))*(3
*sqrt(5) + 5)/(sqrt(1/10)*sqrt(sqrt(5)*(sqrt(5)*(2*x^2 + 1) + 5)) + x)) - 1/5*sq
rt(sqrt(5)*(2*sqrt(5) + 5))*arctan(1/10*sqrt(sqrt(5)*(2*sqrt(5) + 5))*(3*sqrt(5)
- 5)/(sqrt(1/10)*sqrt(sqrt(5)*(sqrt(5)*(2*x^2 - 1) + 5)) + x)) - 1/20*sqrt(-sqr
t(5)*(2*sqrt(5) - 5))*log(1/10*sqrt(-sqrt(5)*(2*sqrt(5) - 5))*(3*sqrt(5) + 5) +
x) + 1/20*sqrt(-sqrt(5)*(2*sqrt(5) - 5))*log(-1/10*sqrt(-sqrt(5)*(2*sqrt(5) - 5)
)*(3*sqrt(5) + 5) + x) + 1/20*sqrt(sqrt(5)*(2*sqrt(5) + 5))*log(1/10*sqrt(sqrt(5
)*(2*sqrt(5) + 5))*(3*sqrt(5) - 5) + x) - 1/20*sqrt(sqrt(5)*(2*sqrt(5) + 5))*log
(-1/10*sqrt(sqrt(5)*(2*sqrt(5) + 5))*(3*sqrt(5) - 5) + x)

_______________________________________________________________________________________

Sympy [A]  time = 3.2198, size = 53, normalized size = 0.31 \[ \operatorname{RootSum}{\left (6400 t^{4} - 320 t^{2} - 1, \left ( t \mapsto t \log{\left (9600 t^{5} - \frac{47 t}{2} + x \right )} \right )\right )} + \operatorname{RootSum}{\left (6400 t^{4} + 320 t^{2} - 1, \left ( t \mapsto t \log{\left (9600 t^{5} - \frac{47 t}{2} + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x**8-3*x**4+1),x)

[Out]

RootSum(6400*_t**4 - 320*_t**2 - 1, Lambda(_t, _t*log(9600*_t**5 - 47*_t/2 + x))
) + RootSum(6400*_t**4 + 320*_t**2 - 1, Lambda(_t, _t*log(9600*_t**5 - 47*_t/2 +
x)))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.337428, size = 198, normalized size = 1.17 \[ -\frac{1}{10} \, \sqrt{5 \, \sqrt{5} - 10} \arctan \left (\frac{x}{\sqrt{\frac{1}{2} \, \sqrt{5} + \frac{1}{2}}}\right ) + \frac{1}{10} \, \sqrt{5 \, \sqrt{5} + 10} \arctan \left (\frac{x}{\sqrt{\frac{1}{2} \, \sqrt{5} - \frac{1}{2}}}\right ) - \frac{1}{20} \, \sqrt{5 \, \sqrt{5} - 10}{\rm ln}\left ({\left | x + \sqrt{\frac{1}{2} \, \sqrt{5} + \frac{1}{2}} \right |}\right ) + \frac{1}{20} \, \sqrt{5 \, \sqrt{5} - 10}{\rm ln}\left ({\left | x - \sqrt{\frac{1}{2} \, \sqrt{5} + \frac{1}{2}} \right |}\right ) + \frac{1}{20} \, \sqrt{5 \, \sqrt{5} + 10}{\rm ln}\left ({\left | x + \sqrt{\frac{1}{2} \, \sqrt{5} - \frac{1}{2}} \right |}\right ) - \frac{1}{20} \, \sqrt{5 \, \sqrt{5} + 10}{\rm ln}\left ({\left | x - \sqrt{\frac{1}{2} \, \sqrt{5} - \frac{1}{2}} \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/10*sqrt(5*sqrt(5) - 10)*arctan(x/sqrt(1/2*sqrt(5) + 1/2)) + 1/10*sqrt(5*sqrt(
5) + 10)*arctan(x/sqrt(1/2*sqrt(5) - 1/2)) - 1/20*sqrt(5*sqrt(5) - 10)*ln(abs(x
+ sqrt(1/2*sqrt(5) + 1/2))) + 1/20*sqrt(5*sqrt(5) - 10)*ln(abs(x - sqrt(1/2*sqrt
(5) + 1/2))) + 1/20*sqrt(5*sqrt(5) + 10)*ln(abs(x + sqrt(1/2*sqrt(5) - 1/2))) -
1/20*sqrt(5*sqrt(5) + 10)*ln(abs(x - sqrt(1/2*sqrt(5) - 1/2)))