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

Optimal. Leaf size=355 \[ \frac{\log \left (x^2-\sqrt{2-\sqrt{3}} x+1\right )}{8 \sqrt{3 \left (2-\sqrt{3}\right )}}-\frac{\log \left (x^2+\sqrt{2-\sqrt{3}} x+1\right )}{8 \sqrt{3 \left (2-\sqrt{3}\right )}}-\frac{\log \left (x^2-\sqrt{2+\sqrt{3}} x+1\right )}{8 \sqrt{3 \left (2+\sqrt{3}\right )}}+\frac{\log \left (x^2+\sqrt{2+\sqrt{3}} x+1\right )}{8 \sqrt{3 \left (2+\sqrt{3}\right )}}+\frac{1}{4} \sqrt{\frac{1}{3} \left (2-\sqrt{3}\right )} \tan ^{-1}\left (\frac{\sqrt{2-\sqrt{3}}-2 x}{\sqrt{2+\sqrt{3}}}\right )-\frac{1}{4} \sqrt{\frac{1}{3} \left (2+\sqrt{3}\right )} \tan ^{-1}\left (\frac{\sqrt{2+\sqrt{3}}-2 x}{\sqrt{2-\sqrt{3}}}\right )-\frac{1}{4} \sqrt{\frac{1}{3} \left (2-\sqrt{3}\right )} \tan ^{-1}\left (\frac{2 x+\sqrt{2-\sqrt{3}}}{\sqrt{2+\sqrt{3}}}\right )+\frac{1}{4} \sqrt{\frac{1}{3} \left (2+\sqrt{3}\right )} \tan ^{-1}\left (\frac{2 x+\sqrt{2+\sqrt{3}}}{\sqrt{2-\sqrt{3}}}\right ) \]

[Out]

(Sqrt[(2 - Sqrt[3])/3]*ArcTan[(Sqrt[2 - Sqrt[3]] - 2*x)/Sqrt[2 + Sqrt[3]]])/4 -
(Sqrt[(2 + Sqrt[3])/3]*ArcTan[(Sqrt[2 + Sqrt[3]] - 2*x)/Sqrt[2 - Sqrt[3]]])/4 -
(Sqrt[(2 - Sqrt[3])/3]*ArcTan[(Sqrt[2 - Sqrt[3]] + 2*x)/Sqrt[2 + Sqrt[3]]])/4 +
(Sqrt[(2 + Sqrt[3])/3]*ArcTan[(Sqrt[2 + Sqrt[3]] + 2*x)/Sqrt[2 - Sqrt[3]]])/4 +
Log[1 - Sqrt[2 - Sqrt[3]]*x + x^2]/(8*Sqrt[3*(2 - Sqrt[3])]) - Log[1 + Sqrt[2 -
Sqrt[3]]*x + x^2]/(8*Sqrt[3*(2 - Sqrt[3])]) - Log[1 - Sqrt[2 + Sqrt[3]]*x + x^2]
/(8*Sqrt[3*(2 + Sqrt[3])]) + Log[1 + Sqrt[2 + Sqrt[3]]*x + x^2]/(8*Sqrt[3*(2 + S
qrt[3])])

_______________________________________________________________________________________

Rubi [A]  time = 0.435571, antiderivative size = 355, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 6, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375 \[ \frac{\log \left (x^2-\sqrt{2-\sqrt{3}} x+1\right )}{8 \sqrt{3 \left (2-\sqrt{3}\right )}}-\frac{\log \left (x^2+\sqrt{2-\sqrt{3}} x+1\right )}{8 \sqrt{3 \left (2-\sqrt{3}\right )}}-\frac{\log \left (x^2-\sqrt{2+\sqrt{3}} x+1\right )}{8 \sqrt{3 \left (2+\sqrt{3}\right )}}+\frac{\log \left (x^2+\sqrt{2+\sqrt{3}} x+1\right )}{8 \sqrt{3 \left (2+\sqrt{3}\right )}}+\frac{1}{4} \sqrt{\frac{1}{3} \left (2-\sqrt{3}\right )} \tan ^{-1}\left (\frac{\sqrt{2-\sqrt{3}}-2 x}{\sqrt{2+\sqrt{3}}}\right )-\frac{1}{4} \sqrt{\frac{1}{3} \left (2+\sqrt{3}\right )} \tan ^{-1}\left (\frac{\sqrt{2+\sqrt{3}}-2 x}{\sqrt{2-\sqrt{3}}}\right )-\frac{1}{4} \sqrt{\frac{1}{3} \left (2-\sqrt{3}\right )} \tan ^{-1}\left (\frac{2 x+\sqrt{2-\sqrt{3}}}{\sqrt{2+\sqrt{3}}}\right )+\frac{1}{4} \sqrt{\frac{1}{3} \left (2+\sqrt{3}\right )} \tan ^{-1}\left (\frac{2 x+\sqrt{2+\sqrt{3}}}{\sqrt{2-\sqrt{3}}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[x^2/(1 - x^4 + x^8),x]

[Out]

(Sqrt[(2 - Sqrt[3])/3]*ArcTan[(Sqrt[2 - Sqrt[3]] - 2*x)/Sqrt[2 + Sqrt[3]]])/4 -
(Sqrt[(2 + Sqrt[3])/3]*ArcTan[(Sqrt[2 + Sqrt[3]] - 2*x)/Sqrt[2 - Sqrt[3]]])/4 -
(Sqrt[(2 - Sqrt[3])/3]*ArcTan[(Sqrt[2 - Sqrt[3]] + 2*x)/Sqrt[2 + Sqrt[3]]])/4 +
(Sqrt[(2 + Sqrt[3])/3]*ArcTan[(Sqrt[2 + Sqrt[3]] + 2*x)/Sqrt[2 - Sqrt[3]]])/4 +
Log[1 - Sqrt[2 - Sqrt[3]]*x + x^2]/(8*Sqrt[3*(2 - Sqrt[3])]) - Log[1 + Sqrt[2 -
Sqrt[3]]*x + x^2]/(8*Sqrt[3*(2 - Sqrt[3])]) - Log[1 - Sqrt[2 + Sqrt[3]]*x + x^2]
/(8*Sqrt[3*(2 + Sqrt[3])]) + Log[1 + Sqrt[2 + Sqrt[3]]*x + x^2]/(8*Sqrt[3*(2 + S
qrt[3])])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 43.0205, size = 311, normalized size = 0.88 \[ \frac{\sqrt{3} \log{\left (x^{2} - x \sqrt{- \sqrt{3} + 2} + 1 \right )}}{24 \sqrt{- \sqrt{3} + 2}} - \frac{\sqrt{3} \log{\left (x^{2} + x \sqrt{- \sqrt{3} + 2} + 1 \right )}}{24 \sqrt{- \sqrt{3} + 2}} - \frac{\sqrt{3} \log{\left (x^{2} - x \sqrt{\sqrt{3} + 2} + 1 \right )}}{24 \sqrt{\sqrt{3} + 2}} + \frac{\sqrt{3} \log{\left (x^{2} + x \sqrt{\sqrt{3} + 2} + 1 \right )}}{24 \sqrt{\sqrt{3} + 2}} + \frac{\sqrt{3} \operatorname{atan}{\left (\frac{2 x - \sqrt{\sqrt{3} + 2}}{\sqrt{- \sqrt{3} + 2}} \right )}}{12 \sqrt{- \sqrt{3} + 2}} + \frac{\sqrt{3} \operatorname{atan}{\left (\frac{2 x + \sqrt{\sqrt{3} + 2}}{\sqrt{- \sqrt{3} + 2}} \right )}}{12 \sqrt{- \sqrt{3} + 2}} - \frac{\sqrt{3} \operatorname{atan}{\left (\frac{2 x - \sqrt{- \sqrt{3} + 2}}{\sqrt{\sqrt{3} + 2}} \right )}}{12 \sqrt{\sqrt{3} + 2}} - \frac{\sqrt{3} \operatorname{atan}{\left (\frac{2 x + \sqrt{- \sqrt{3} + 2}}{\sqrt{\sqrt{3} + 2}} \right )}}{12 \sqrt{\sqrt{3} + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(3)*log(x**2 - x*sqrt(-sqrt(3) + 2) + 1)/(24*sqrt(-sqrt(3) + 2)) - sqrt(3)*l
og(x**2 + x*sqrt(-sqrt(3) + 2) + 1)/(24*sqrt(-sqrt(3) + 2)) - sqrt(3)*log(x**2 -
x*sqrt(sqrt(3) + 2) + 1)/(24*sqrt(sqrt(3) + 2)) + sqrt(3)*log(x**2 + x*sqrt(sqr
t(3) + 2) + 1)/(24*sqrt(sqrt(3) + 2)) + sqrt(3)*atan((2*x - sqrt(sqrt(3) + 2))/s
qrt(-sqrt(3) + 2))/(12*sqrt(-sqrt(3) + 2)) + sqrt(3)*atan((2*x + sqrt(sqrt(3) +
2))/sqrt(-sqrt(3) + 2))/(12*sqrt(-sqrt(3) + 2)) - sqrt(3)*atan((2*x - sqrt(-sqrt
(3) + 2))/sqrt(sqrt(3) + 2))/(12*sqrt(sqrt(3) + 2)) - sqrt(3)*atan((2*x + sqrt(-
sqrt(3) + 2))/sqrt(sqrt(3) + 2))/(12*sqrt(sqrt(3) + 2))

_______________________________________________________________________________________

Mathematica [C]  time = 0.016386, size = 40, normalized size = 0.11 \[ \frac{1}{4} \text{RootSum}\left [\text{$\#$1}^8-\text{$\#$1}^4+1\&,\frac{\log (x-\text{$\#$1})}{2 \text{$\#$1}^5-\text{$\#$1}}\&\right ] \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [C]  time = 0.01, size = 40, normalized size = 0.1 \[{\frac{1}{4}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{8}-{{\it \_Z}}^{4}+1 \right ) }{\frac{{{\it \_R}}^{2}\ln \left ( x-{\it \_R} \right ) }{2\,{{\it \_R}}^{7}-{{\it \_R}}^{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(x^8-x^4+1),x)

[Out]

1/4*sum(_R^2/(2*_R^7-_R^3)*ln(x-_R),_R=RootOf(_Z^8-_Z^4+1))

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{x^{8} - x^{4} + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^2/(x^8 - x^4 + 1), x)

_______________________________________________________________________________________

Fricas [A]  time = 0.284676, size = 1107, normalized size = 3.12 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/24*(4*(7*sqrt(3) + 12)*sqrt((sqrt(3) + 2)/(4*sqrt(3) + 7))*arctan((2*sqrt(3)*
sqrt(2) - 3*sqrt(2))/(2*sqrt(2*x^2 + 2*x*sqrt((sqrt(3) - 2)/(4*sqrt(3) - 7)) + 2
)*(2*sqrt(3) - 3)*sqrt((sqrt(3) - 2)/(4*sqrt(3) - 7)) + 2*(2*sqrt(3)*sqrt(2)*x -
3*sqrt(2)*x)*sqrt((sqrt(3) - 2)/(4*sqrt(3) - 7)) + sqrt(3)*sqrt(2))) + 4*(7*sqr
t(3) + 12)*sqrt((sqrt(3) + 2)/(4*sqrt(3) + 7))*arctan((2*sqrt(3)*sqrt(2) - 3*sqr
t(2))/(2*sqrt(2*x^2 - 2*x*sqrt((sqrt(3) - 2)/(4*sqrt(3) - 7)) + 2)*(2*sqrt(3) -
3)*sqrt((sqrt(3) - 2)/(4*sqrt(3) - 7)) + 2*(2*sqrt(3)*sqrt(2)*x - 3*sqrt(2)*x)*s
qrt((sqrt(3) - 2)/(4*sqrt(3) - 7)) - sqrt(3)*sqrt(2))) + (2*sqrt(3) + 3)*sqrt((s
qrt(3) - 2)/(4*sqrt(3) - 7))*log(2*x^2 + 2*x*sqrt((sqrt(3) + 2)/(4*sqrt(3) + 7))
+ 2) - (2*sqrt(3) + 3)*sqrt((sqrt(3) - 2)/(4*sqrt(3) - 7))*log(2*x^2 - 2*x*sqrt
((sqrt(3) + 2)/(4*sqrt(3) + 7)) + 2) - (2*sqrt(3) + 3)*sqrt((sqrt(3) + 2)/(4*sqr
t(3) + 7))*log(2*x^2 + 2*x*sqrt((sqrt(3) - 2)/(4*sqrt(3) - 7)) + 2) + (2*sqrt(3)
+ 3)*sqrt((sqrt(3) + 2)/(4*sqrt(3) + 7))*log(2*x^2 - 2*x*sqrt((sqrt(3) - 2)/(4*
sqrt(3) - 7)) + 2) - 4*sqrt(3)*sqrt((sqrt(3) - 2)/(4*sqrt(3) - 7))*arctan((2*sqr
t(3)*sqrt(2) + 3*sqrt(2))/(2*sqrt(2*x^2 + 2*x*sqrt((sqrt(3) + 2)/(4*sqrt(3) + 7)
) + 2)*(2*sqrt(3) + 3)*sqrt((sqrt(3) + 2)/(4*sqrt(3) + 7)) + 2*(2*sqrt(3)*sqrt(2
)*x + 3*sqrt(2)*x)*sqrt((sqrt(3) + 2)/(4*sqrt(3) + 7)) + sqrt(3)*sqrt(2))) - 4*s
qrt(3)*sqrt((sqrt(3) - 2)/(4*sqrt(3) - 7))*arctan((2*sqrt(3)*sqrt(2) + 3*sqrt(2)
)/(2*sqrt(2*x^2 - 2*x*sqrt((sqrt(3) + 2)/(4*sqrt(3) + 7)) + 2)*(2*sqrt(3) + 3)*s
qrt((sqrt(3) + 2)/(4*sqrt(3) + 7)) + 2*(2*sqrt(3)*sqrt(2)*x + 3*sqrt(2)*x)*sqrt(
(sqrt(3) + 2)/(4*sqrt(3) + 7)) - sqrt(3)*sqrt(2))))/((sqrt(3) + 2)*sqrt((sqrt(3)
+ 2)/(4*sqrt(3) + 7))*sqrt((sqrt(3) - 2)/(4*sqrt(3) - 7)))

_______________________________________________________________________________________

Sympy [A]  time = 4.89047, size = 26, normalized size = 0.07 \[ \operatorname{RootSum}{\left (5308416 t^{8} - 2304 t^{4} + 1, \left ( t \mapsto t \log{\left (- 442368 t^{7} - 192 t^{3} + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2/(x**8-x**4+1),x)

[Out]

RootSum(5308416*_t**8 - 2304*_t**4 + 1, Lambda(_t, _t*log(-442368*_t**7 - 192*_t
**3 + x)))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.2881, size = 342, normalized size = 0.96 \[ \frac{1}{24} \,{\left (\sqrt{6} - 3 \, \sqrt{2}\right )} \arctan \left (\frac{4 \, x + \sqrt{6} - \sqrt{2}}{\sqrt{6} + \sqrt{2}}\right ) + \frac{1}{24} \,{\left (\sqrt{6} - 3 \, \sqrt{2}\right )} \arctan \left (\frac{4 \, x - \sqrt{6} + \sqrt{2}}{\sqrt{6} + \sqrt{2}}\right ) + \frac{1}{24} \,{\left (\sqrt{6} + 3 \, \sqrt{2}\right )} \arctan \left (\frac{4 \, x + \sqrt{6} + \sqrt{2}}{\sqrt{6} - \sqrt{2}}\right ) + \frac{1}{24} \,{\left (\sqrt{6} + 3 \, \sqrt{2}\right )} \arctan \left (\frac{4 \, x - \sqrt{6} - \sqrt{2}}{\sqrt{6} - \sqrt{2}}\right ) - \frac{1}{48} \,{\left (\sqrt{6} - 3 \, \sqrt{2}\right )}{\rm ln}\left (x^{2} + \frac{1}{2} \, x{\left (\sqrt{6} + \sqrt{2}\right )} + 1\right ) + \frac{1}{48} \,{\left (\sqrt{6} - 3 \, \sqrt{2}\right )}{\rm ln}\left (x^{2} - \frac{1}{2} \, x{\left (\sqrt{6} + \sqrt{2}\right )} + 1\right ) - \frac{1}{48} \,{\left (\sqrt{6} + 3 \, \sqrt{2}\right )}{\rm ln}\left (x^{2} + \frac{1}{2} \, x{\left (\sqrt{6} - \sqrt{2}\right )} + 1\right ) + \frac{1}{48} \,{\left (\sqrt{6} + 3 \, \sqrt{2}\right )}{\rm ln}\left (x^{2} - \frac{1}{2} \, x{\left (\sqrt{6} - \sqrt{2}\right )} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/24*(sqrt(6) - 3*sqrt(2))*arctan((4*x + sqrt(6) - sqrt(2))/(sqrt(6) + sqrt(2)))
+ 1/24*(sqrt(6) - 3*sqrt(2))*arctan((4*x - sqrt(6) + sqrt(2))/(sqrt(6) + sqrt(2
))) + 1/24*(sqrt(6) + 3*sqrt(2))*arctan((4*x + sqrt(6) + sqrt(2))/(sqrt(6) - sqr
t(2))) + 1/24*(sqrt(6) + 3*sqrt(2))*arctan((4*x - sqrt(6) - sqrt(2))/(sqrt(6) -
sqrt(2))) - 1/48*(sqrt(6) - 3*sqrt(2))*ln(x^2 + 1/2*x*(sqrt(6) + sqrt(2)) + 1) +
1/48*(sqrt(6) - 3*sqrt(2))*ln(x^2 - 1/2*x*(sqrt(6) + sqrt(2)) + 1) - 1/48*(sqrt
(6) + 3*sqrt(2))*ln(x^2 + 1/2*x*(sqrt(6) - sqrt(2)) + 1) + 1/48*(sqrt(6) + 3*sqr
t(2))*ln(x^2 - 1/2*x*(sqrt(6) - sqrt(2)) + 1)