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

Optimal. Leaf size=145 \[ \frac{1}{20} \sqrt{10 \sqrt{5}-10} \tan ^{-1}\left (\frac{1}{2} \sqrt{2 \sqrt{5}-2} x\right )-\frac{1}{20} \sqrt{10+10 \sqrt{5}} \tan ^{-1}\left (\frac{1}{2} \sqrt{2+2 \sqrt{5}} x\right )-\frac{1}{20} \sqrt{10 \sqrt{5}-10} \tanh ^{-1}\left (\frac{1}{2} \sqrt{2 \sqrt{5}-2} x\right )+\frac{1}{20} \sqrt{10+10 \sqrt{5}} \tanh ^{-1}\left (\frac{1}{2} \sqrt{2+2 \sqrt{5}} x\right ) \]

[Out]

(Sqrt[-10 + 10*Sqrt[5]]*ArcTan[(Sqrt[-2 + 2*Sqrt[5]]*x)/2])/20 - (Sqrt[10 + 10*S
qrt[5]]*ArcTan[(Sqrt[2 + 2*Sqrt[5]]*x)/2])/20 - (Sqrt[-10 + 10*Sqrt[5]]*ArcTanh[
(Sqrt[-2 + 2*Sqrt[5]]*x)/2])/20 + (Sqrt[10 + 10*Sqrt[5]]*ArcTanh[(Sqrt[2 + 2*Sqr
t[5]]*x)/2])/20

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Rubi [A]  time = 0.141482, antiderivative size = 166, normalized size of antiderivative = 1.14, number of steps used = 7, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{\tan ^{-1}\left (\sqrt [4]{\frac{2}{3+\sqrt{5}}} x\right )}{2^{3/4} \sqrt{5} \sqrt [4]{3+\sqrt{5}}}-\frac{\sqrt [4]{\frac{1}{2} \left (3+\sqrt{5}\right )} \tan ^{-1}\left (\sqrt [4]{\frac{1}{2} \left (3+\sqrt{5}\right )} x\right )}{2 \sqrt{5}}-\frac{\tanh ^{-1}\left (\sqrt [4]{\frac{2}{3+\sqrt{5}}} x\right )}{2^{3/4} \sqrt{5} \sqrt [4]{3+\sqrt{5}}}+\frac{\sqrt [4]{\frac{1}{2} \left (3+\sqrt{5}\right )} \tanh ^{-1}\left (\sqrt [4]{\frac{1}{2} \left (3+\sqrt{5}\right )} x\right )}{2 \sqrt{5}} \]

Antiderivative was successfully verified.

[In]  Int[x^2/(1 - 3*x^4 + x^8),x]

[Out]

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

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Rubi in Sympy [A]  time = 23.1541, size = 162, normalized size = 1.12 \[ - \frac{\sqrt [4]{2} \sqrt{5} \operatorname{atan}{\left (\frac{\sqrt [4]{2} x}{\sqrt [4]{- \sqrt{5} + 3}} \right )}}{10 \sqrt [4]{- \sqrt{5} + 3}} + \frac{\sqrt [4]{2} \sqrt{5} \operatorname{atan}{\left (\frac{\sqrt [4]{2} x}{\sqrt [4]{\sqrt{5} + 3}} \right )}}{10 \sqrt [4]{\sqrt{5} + 3}} + \frac{\sqrt [4]{2} \sqrt{5} \operatorname{atanh}{\left (\frac{\sqrt [4]{2} x}{\sqrt [4]{- \sqrt{5} + 3}} \right )}}{10 \sqrt [4]{- \sqrt{5} + 3}} - \frac{\sqrt [4]{2} \sqrt{5} \operatorname{atanh}{\left (\frac{\sqrt [4]{2} x}{\sqrt [4]{\sqrt{5} + 3}} \right )}}{10 \sqrt [4]{\sqrt{5} + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0693973, size = 131, normalized size = 0.9 \[ -\frac{\tan ^{-1}\left (\sqrt{\frac{2}{\sqrt{5}-1}} x\right )}{\sqrt{10 \left (\sqrt{5}-1\right )}}+\frac{\tan ^{-1}\left (\sqrt{\frac{2}{1+\sqrt{5}}} x\right )}{\sqrt{10 \left (1+\sqrt{5}\right )}}+\frac{\tanh ^{-1}\left (\sqrt{\frac{2}{\sqrt{5}-1}} x\right )}{\sqrt{10 \left (\sqrt{5}-1\right )}}-\frac{\tanh ^{-1}\left (\sqrt{\frac{2}{1+\sqrt{5}}} x\right )}{\sqrt{10 \left (1+\sqrt{5}\right )}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.037, size = 110, normalized size = 0.8 \[{\frac{\sqrt{5}}{5\,\sqrt{2\,\sqrt{5}+2}}\arctan \left ( 2\,{\frac{x}{\sqrt{2\,\sqrt{5}+2}}} \right ) }+{\frac{\sqrt{5}}{5\,\sqrt{-2+2\,\sqrt{5}}}{\it Artanh} \left ( 2\,{\frac{x}{\sqrt{-2+2\,\sqrt{5}}}} \right ) }-{\frac{\sqrt{5}}{5\,\sqrt{-2+2\,\sqrt{5}}}\arctan \left ( 2\,{\frac{x}{\sqrt{-2+2\,\sqrt{5}}}} \right ) }-{\frac{\sqrt{5}}{5\,\sqrt{2\,\sqrt{5}+2}}{\it Artanh} \left ( 2\,{\frac{x}{\sqrt{2\,\sqrt{5}+2}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(x^8-3*x^4+1),x)

[Out]

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

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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 = 0.297645, size = 441, normalized size = 3.04 \[ -\frac{1}{5} \, \sqrt{\frac{1}{2}} \sqrt{-\sqrt{5}{\left (\sqrt{5} - 5\right )}} \arctan \left (\frac{\sqrt{\frac{1}{2}} \sqrt{-\sqrt{5}{\left (\sqrt{5} - 5\right )}}{\left (\sqrt{5} + 1\right )}}{2 \,{\left (\sqrt{5} \sqrt{\frac{1}{10}} \sqrt{\sqrt{5}{\left (\sqrt{5}{\left (2 \, x^{2} + 1\right )} + 5\right )}} + \sqrt{5} x\right )}}\right ) + \frac{1}{5} \, \sqrt{\frac{1}{2}} \sqrt{\sqrt{5}{\left (\sqrt{5} + 5\right )}} \arctan \left (\frac{\sqrt{\frac{1}{2}} \sqrt{\sqrt{5}{\left (\sqrt{5} + 5\right )}}{\left (\sqrt{5} - 1\right )}}{2 \,{\left (\sqrt{5} \sqrt{\frac{1}{10}} \sqrt{\sqrt{5}{\left (\sqrt{5}{\left (2 \, x^{2} - 1\right )} + 5\right )}} + \sqrt{5} x\right )}}\right ) - \frac{1}{20} \, \sqrt{\frac{1}{2}} \sqrt{-\sqrt{5}{\left (\sqrt{5} - 5\right )}} \log \left (\frac{1}{2} \, \sqrt{\frac{1}{2}} \sqrt{-\sqrt{5}{\left (\sqrt{5} - 5\right )}}{\left (\sqrt{5} + 1\right )} + \sqrt{5} x\right ) + \frac{1}{20} \, \sqrt{\frac{1}{2}} \sqrt{-\sqrt{5}{\left (\sqrt{5} - 5\right )}} \log \left (-\frac{1}{2} \, \sqrt{\frac{1}{2}} \sqrt{-\sqrt{5}{\left (\sqrt{5} - 5\right )}}{\left (\sqrt{5} + 1\right )} + \sqrt{5} x\right ) + \frac{1}{20} \, \sqrt{\frac{1}{2}} \sqrt{\sqrt{5}{\left (\sqrt{5} + 5\right )}} \log \left (\frac{1}{2} \, \sqrt{\frac{1}{2}} \sqrt{\sqrt{5}{\left (\sqrt{5} + 5\right )}}{\left (\sqrt{5} - 1\right )} + \sqrt{5} x\right ) - \frac{1}{20} \, \sqrt{\frac{1}{2}} \sqrt{\sqrt{5}{\left (\sqrt{5} + 5\right )}} \log \left (-\frac{1}{2} \, \sqrt{\frac{1}{2}} \sqrt{\sqrt{5}{\left (\sqrt{5} + 5\right )}}{\left (\sqrt{5} - 1\right )} + \sqrt{5} x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 3.14754, size = 53, normalized size = 0.37 \[ \operatorname{RootSum}{\left (6400 t^{4} - 80 t^{2} - 1, \left ( t \mapsto t \log{\left (6144000 t^{7} - 2240 t^{3} + x \right )} \right )\right )} + \operatorname{RootSum}{\left (6400 t^{4} + 80 t^{2} - 1, \left ( t \mapsto t \log{\left (6144000 t^{7} - 2240 t^{3} + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

RootSum(6400*_t**4 - 80*_t**2 - 1, Lambda(_t, _t*log(6144000*_t**7 - 2240*_t**3
+ x))) + RootSum(6400*_t**4 + 80*_t**2 - 1, Lambda(_t, _t*log(6144000*_t**7 - 22
40*_t**3 + x)))

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GIAC/XCAS [A]  time = 0.348456, size = 198, normalized size = 1.37 \[ \frac{1}{20} \, \sqrt{10 \, \sqrt{5} - 10} \arctan \left (\frac{x}{\sqrt{\frac{1}{2} \, \sqrt{5} + \frac{1}{2}}}\right ) - \frac{1}{20} \, \sqrt{10 \, \sqrt{5} + 10} \arctan \left (\frac{x}{\sqrt{\frac{1}{2} \, \sqrt{5} - \frac{1}{2}}}\right ) - \frac{1}{40} \, \sqrt{10 \, \sqrt{5} - 10}{\rm ln}\left ({\left | x + \sqrt{\frac{1}{2} \, \sqrt{5} + \frac{1}{2}} \right |}\right ) + \frac{1}{40} \, \sqrt{10 \, \sqrt{5} - 10}{\rm ln}\left ({\left | x - \sqrt{\frac{1}{2} \, \sqrt{5} + \frac{1}{2}} \right |}\right ) + \frac{1}{40} \, \sqrt{10 \, \sqrt{5} + 10}{\rm ln}\left ({\left | x + \sqrt{\frac{1}{2} \, \sqrt{5} - \frac{1}{2}} \right |}\right ) - \frac{1}{40} \, \sqrt{10 \, \sqrt{5} + 10}{\rm ln}\left ({\left | x - \sqrt{\frac{1}{2} \, \sqrt{5} - \frac{1}{2}} \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/20*sqrt(10*sqrt(5) - 10)*arctan(x/sqrt(1/2*sqrt(5) + 1/2)) - 1/20*sqrt(10*sqrt
(5) + 10)*arctan(x/sqrt(1/2*sqrt(5) - 1/2)) - 1/40*sqrt(10*sqrt(5) - 10)*ln(abs(
x + sqrt(1/2*sqrt(5) + 1/2))) + 1/40*sqrt(10*sqrt(5) - 10)*ln(abs(x - sqrt(1/2*s
qrt(5) + 1/2))) + 1/40*sqrt(10*sqrt(5) + 10)*ln(abs(x + sqrt(1/2*sqrt(5) - 1/2))
) - 1/40*sqrt(10*sqrt(5) + 10)*ln(abs(x - sqrt(1/2*sqrt(5) - 1/2)))