3.32 \(\int \frac{1+\left (1+\sqrt{3}\right ) x^4}{1-x^4+x^8} \, dx\)

Optimal. Leaf size=164 \[ -\frac{1}{4} \sqrt{2+\sqrt{3}} \log \left (x^2-\sqrt{2-\sqrt{3}} x+1\right )+\frac{1}{4} \sqrt{2+\sqrt{3}} \log \left (x^2+\sqrt{2-\sqrt{3}} x+1\right )-\frac{1}{2} \sqrt{2+\sqrt{3}} \tan ^{-1}\left (\frac{\sqrt{2+\sqrt{3}}-2 x}{\sqrt{2-\sqrt{3}}}\right )+\frac{1}{2} \sqrt{2+\sqrt{3}} \tan ^{-1}\left (\frac{2 x+\sqrt{2+\sqrt{3}}}{\sqrt{2-\sqrt{3}}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.210683, antiderivative size = 164, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{1}{4} \sqrt{2+\sqrt{3}} \log \left (x^2-\sqrt{2-\sqrt{3}} x+1\right )+\frac{1}{4} \sqrt{2+\sqrt{3}} \log \left (x^2+\sqrt{2-\sqrt{3}} x+1\right )-\frac{1}{2} \sqrt{2+\sqrt{3}} \tan ^{-1}\left (\frac{\sqrt{2+\sqrt{3}}-2 x}{\sqrt{2-\sqrt{3}}}\right )+\frac{1}{2} \sqrt{2+\sqrt{3}} \tan ^{-1}\left (\frac{2 x+\sqrt{2+\sqrt{3}}}{\sqrt{2-\sqrt{3}}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 44.5854, size = 165, normalized size = 1.01 \[ - \frac{\log{\left (x^{2} - x \sqrt{- \sqrt{3} + 2} + 1 \right )}}{4 \sqrt{- \sqrt{3} + 2}} + \frac{\log{\left (x^{2} + x \sqrt{- \sqrt{3} + 2} + 1 \right )}}{4 \sqrt{- \sqrt{3} + 2}} + \frac{\operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{2 \sqrt{3} x}{3} - \frac{\sqrt{3 \sqrt{3} + 6}}{3}\right )}{\sqrt{- \sqrt{3} + 2}} \right )}}{2 \sqrt{- \sqrt{3} + 2}} + \frac{\operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{2 \sqrt{3} x}{3} + \frac{\sqrt{3 \sqrt{3} + 6}}{3}\right )}{\sqrt{- \sqrt{3} + 2}} \right )}}{2 \sqrt{- \sqrt{3} + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-log(x**2 - x*sqrt(-sqrt(3) + 2) + 1)/(4*sqrt(-sqrt(3) + 2)) + log(x**2 + x*sqrt
(-sqrt(3) + 2) + 1)/(4*sqrt(-sqrt(3) + 2)) + atan(sqrt(3)*(2*sqrt(3)*x/3 - sqrt(
3*sqrt(3) + 6)/3)/sqrt(-sqrt(3) + 2))/(2*sqrt(-sqrt(3) + 2)) + atan(sqrt(3)*(2*s
qrt(3)*x/3 + sqrt(3*sqrt(3) + 6)/3)/sqrt(-sqrt(3) + 2))/(2*sqrt(-sqrt(3) + 2))

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Mathematica [C]  time = 0.0534861, size = 72, normalized size = 0.44 \[ \frac{1}{4} \text{RootSum}\left [\text{$\#$1}^8-\text{$\#$1}^4+1\&,\frac{\sqrt{3} \text{$\#$1}^4 \log (x-\text{$\#$1})+\text{$\#$1}^4 \log (x-\text{$\#$1})+\log (x-\text{$\#$1})}{2 \text{$\#$1}^7-\text{$\#$1}^3}\&\right ] \]

Antiderivative was successfully verified.

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

[Out]

RootSum[1 - #1^4 + #1^8 & , (Log[x - #1] + Log[x - #1]*#1^4 + Sqrt[3]*Log[x - #1
]*#1^4)/(-#1^3 + 2*#1^7) & ]/4

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Maple [C]  time = 0.054, size = 62, normalized size = 0.4 \[{\frac{1}{8}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{8}-{{\it \_Z}}^{4}+1 \right ) }{\frac{ \left ( 2\,{{\it \_R}}^{4}+2\,\sqrt{3}{{\it \_R}}^{4}+ \left ( 1+\sqrt{3} \right ) \left ( \sqrt{3}-1 \right ) \right ) \ln \left ( x-{\it \_R} \right ) }{2\,{{\it \_R}}^{7}-{{\it \_R}}^{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((x^4*(sqrt(3) + 1) + 1)/(x^8 - x^4 + 1), x)

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: NotImplementedError

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: PolynomialError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: PolynomialError

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GIAC/XCAS [A]  time = 0.297529, size = 166, normalized size = 1.01 \[ \frac{1}{4} \,{\left (\sqrt{6} + \sqrt{2}\right )} \arctan \left (\frac{4 \, x + \sqrt{6} + \sqrt{2}}{\sqrt{6} - \sqrt{2}}\right ) + \frac{1}{4} \,{\left (\sqrt{6} + \sqrt{2}\right )} \arctan \left (\frac{4 \, x - \sqrt{6} - \sqrt{2}}{\sqrt{6} - \sqrt{2}}\right ) + \frac{1}{8} \,{\left (\sqrt{6} + \sqrt{2}\right )}{\rm ln}\left (x^{2} + \frac{1}{2} \, x{\left (\sqrt{6} - \sqrt{2}\right )} + 1\right ) - \frac{1}{8} \,{\left (\sqrt{6} + \sqrt{2}\right )}{\rm ln}\left (x^{2} - \frac{1}{2} \, x{\left (\sqrt{6} - \sqrt{2}\right )} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*(sqrt(6) + sqrt(2))*arctan((4*x + sqrt(6) + sqrt(2))/(sqrt(6) - sqrt(2))) +
1/4*(sqrt(6) + sqrt(2))*arctan((4*x - sqrt(6) - sqrt(2))/(sqrt(6) - sqrt(2))) +
1/8*(sqrt(6) + sqrt(2))*ln(x^2 + 1/2*x*(sqrt(6) - sqrt(2)) + 1) - 1/8*(sqrt(6) +
sqrt(2))*ln(x^2 - 1/2*x*(sqrt(6) - sqrt(2)) + 1)