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

Optimal. Leaf size=331 \[ -\frac{\log \left (x^2-\sqrt{2-\sqrt{3}} x+1\right )}{8 \sqrt{2-\sqrt{3}}}+\frac{\log \left (x^2+\sqrt{2-\sqrt{3}} x+1\right )}{8 \sqrt{2-\sqrt{3}}}-\frac{\log \left (x^2-\sqrt{2+\sqrt{3}} x+1\right )}{8 \sqrt{2+\sqrt{3}}}+\frac{\log \left (x^2+\sqrt{2+\sqrt{3}} x+1\right )}{8 \sqrt{2+\sqrt{3}}}-\frac{1}{4} \sqrt{2-\sqrt{3}} \tan ^{-1}\left (\frac{\sqrt{2-\sqrt{3}}-2 x}{\sqrt{2+\sqrt{3}}}\right )-\frac{1}{4} \sqrt{2+\sqrt{3}} \tan ^{-1}\left (\frac{\sqrt{2+\sqrt{3}}-2 x}{\sqrt{2-\sqrt{3}}}\right )+\frac{1}{4} \sqrt{2-\sqrt{3}} \tan ^{-1}\left (\frac{2 x+\sqrt{2-\sqrt{3}}}{\sqrt{2+\sqrt{3}}}\right )+\frac{1}{4} \sqrt{2+\sqrt{3}} \tan ^{-1}\left (\frac{2 x+\sqrt{2+\sqrt{3}}}{\sqrt{2-\sqrt{3}}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.479201, antiderivative size = 331, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 6, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{\log \left (x^2-\sqrt{2-\sqrt{3}} x+1\right )}{8 \sqrt{2-\sqrt{3}}}+\frac{\log \left (x^2+\sqrt{2-\sqrt{3}} x+1\right )}{8 \sqrt{2-\sqrt{3}}}-\frac{\log \left (x^2-\sqrt{2+\sqrt{3}} x+1\right )}{8 \sqrt{2+\sqrt{3}}}+\frac{\log \left (x^2+\sqrt{2+\sqrt{3}} x+1\right )}{8 \sqrt{2+\sqrt{3}}}-\frac{1}{4} \sqrt{2-\sqrt{3}} \tan ^{-1}\left (\frac{\sqrt{2-\sqrt{3}}-2 x}{\sqrt{2+\sqrt{3}}}\right )-\frac{1}{4} \sqrt{2+\sqrt{3}} \tan ^{-1}\left (\frac{\sqrt{2+\sqrt{3}}-2 x}{\sqrt{2-\sqrt{3}}}\right )+\frac{1}{4} \sqrt{2-\sqrt{3}} \tan ^{-1}\left (\frac{2 x+\sqrt{2-\sqrt{3}}}{\sqrt{2+\sqrt{3}}}\right )+\frac{1}{4} \sqrt{2+\sqrt{3}} \tan ^{-1}\left (\frac{2 x+\sqrt{2+\sqrt{3}}}{\sqrt{2-\sqrt{3}}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 43.0162, size = 270, normalized size = 0.82 \[ - \frac{\log{\left (x^{2} - x \sqrt{- \sqrt{3} + 2} + 1 \right )}}{8 \sqrt{- \sqrt{3} + 2}} + \frac{\log{\left (x^{2} + x \sqrt{- \sqrt{3} + 2} + 1 \right )}}{8 \sqrt{- \sqrt{3} + 2}} - \frac{\log{\left (x^{2} - x \sqrt{\sqrt{3} + 2} + 1 \right )}}{8 \sqrt{\sqrt{3} + 2}} + \frac{\log{\left (x^{2} + x \sqrt{\sqrt{3} + 2} + 1 \right )}}{8 \sqrt{\sqrt{3} + 2}} + \frac{\operatorname{atan}{\left (\frac{2 x - \sqrt{\sqrt{3} + 2}}{\sqrt{- \sqrt{3} + 2}} \right )}}{4 \sqrt{- \sqrt{3} + 2}} + \frac{\operatorname{atan}{\left (\frac{2 x + \sqrt{\sqrt{3} + 2}}{\sqrt{- \sqrt{3} + 2}} \right )}}{4 \sqrt{- \sqrt{3} + 2}} + \frac{\operatorname{atan}{\left (\frac{2 x - \sqrt{- \sqrt{3} + 2}}{\sqrt{\sqrt{3} + 2}} \right )}}{4 \sqrt{\sqrt{3} + 2}} + \frac{\operatorname{atan}{\left (\frac{2 x + \sqrt{- \sqrt{3} + 2}}{\sqrt{\sqrt{3} + 2}} \right )}}{4 \sqrt{\sqrt{3} + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-log(x**2 - x*sqrt(-sqrt(3) + 2) + 1)/(8*sqrt(-sqrt(3) + 2)) + log(x**2 + x*sqrt
(-sqrt(3) + 2) + 1)/(8*sqrt(-sqrt(3) + 2)) - log(x**2 - x*sqrt(sqrt(3) + 2) + 1)
/(8*sqrt(sqrt(3) + 2)) + log(x**2 + x*sqrt(sqrt(3) + 2) + 1)/(8*sqrt(sqrt(3) + 2
)) + atan((2*x - sqrt(sqrt(3) + 2))/sqrt(-sqrt(3) + 2))/(4*sqrt(-sqrt(3) + 2)) +
atan((2*x + sqrt(sqrt(3) + 2))/sqrt(-sqrt(3) + 2))/(4*sqrt(-sqrt(3) + 2)) + ata
n((2*x - sqrt(-sqrt(3) + 2))/sqrt(sqrt(3) + 2))/(4*sqrt(sqrt(3) + 2)) + atan((2*
x + sqrt(-sqrt(3) + 2))/sqrt(sqrt(3) + 2))/(4*sqrt(sqrt(3) + 2))

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Mathematica [C]  time = 0.0235172, size = 55, normalized size = 0.17 \[ \frac{1}{4} \text{RootSum}\left [\text{$\#$1}^8-\text{$\#$1}^4+1\&,\frac{\text{$\#$1}^4 \log (x-\text{$\#$1})+\log (x-\text{$\#$1})}{2 \text{$\#$1}^7-\text{$\#$1}^3}\&\right ] \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [C]  time = 0.011, size = 42, normalized size = 0.1 \[{\frac{1}{4}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{8}-{{\it \_Z}}^{4}+1 \right ) }{\frac{ \left ({{\it \_R}}^{4}+1 \right ) \ln \left ( x-{\it \_R} \right ) }{2\,{{\it \_R}}^{7}-{{\it \_R}}^{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^4+1)/(x^8-x^4+1),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.293455, size = 1048, normalized size = 3.17 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/8*(4*(4*sqrt(3) + 7)*sqrt((sqrt(3) + 2)/(4*sqrt(3) + 7))*arctan((sqrt(3)*sqrt
(2) - 2*sqrt(2))/(2*sqrt(2*x^2 + 2*x*sqrt((sqrt(3) - 2)/(4*sqrt(3) - 7)) + 2)*(s
qrt(3) - 2)*sqrt((sqrt(3) - 2)/(4*sqrt(3) - 7)) + 2*(sqrt(3)*sqrt(2)*x - 2*sqrt(
2)*x)*sqrt((sqrt(3) - 2)/(4*sqrt(3) - 7)) - sqrt(2))) + 4*(4*sqrt(3) + 7)*sqrt((
sqrt(3) + 2)/(4*sqrt(3) + 7))*arctan((sqrt(3)*sqrt(2) - 2*sqrt(2))/(2*sqrt(2*x^2
- 2*x*sqrt((sqrt(3) - 2)/(4*sqrt(3) - 7)) + 2)*(sqrt(3) - 2)*sqrt((sqrt(3) - 2)
/(4*sqrt(3) - 7)) + 2*(sqrt(3)*sqrt(2)*x - 2*sqrt(2)*x)*sqrt((sqrt(3) - 2)/(4*sq
rt(3) - 7)) + sqrt(2))) - (sqrt(3) + 2)*sqrt((sqrt(3) - 2)/(4*sqrt(3) - 7))*log(
2*x^2 + 2*x*sqrt((sqrt(3) + 2)/(4*sqrt(3) + 7)) + 2) + (sqrt(3) + 2)*sqrt((sqrt(
3) - 2)/(4*sqrt(3) - 7))*log(2*x^2 - 2*x*sqrt((sqrt(3) + 2)/(4*sqrt(3) + 7)) + 2
) - (sqrt(3) + 2)*sqrt((sqrt(3) + 2)/(4*sqrt(3) + 7))*log(2*x^2 + 2*x*sqrt((sqrt
(3) - 2)/(4*sqrt(3) - 7)) + 2) + (sqrt(3) + 2)*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((sqrt(3) -
2)/(4*sqrt(3) - 7))*arctan((sqrt(3)*sqrt(2) + 2*sqrt(2))/(2*sqrt(2*x^2 + 2*x*sqr
t((sqrt(3) + 2)/(4*sqrt(3) + 7)) + 2)*(sqrt(3) + 2)*sqrt((sqrt(3) + 2)/(4*sqrt(3
) + 7)) + 2*(sqrt(3)*sqrt(2)*x + 2*sqrt(2)*x)*sqrt((sqrt(3) + 2)/(4*sqrt(3) + 7)
) + sqrt(2))) + 4*sqrt((sqrt(3) - 2)/(4*sqrt(3) - 7))*arctan((sqrt(3)*sqrt(2) +
2*sqrt(2))/(2*sqrt(2*x^2 - 2*x*sqrt((sqrt(3) + 2)/(4*sqrt(3) + 7)) + 2)*(sqrt(3)
+ 2)*sqrt((sqrt(3) + 2)/(4*sqrt(3) + 7)) + 2*(sqrt(3)*sqrt(2)*x + 2*sqrt(2)*x)*
sqrt((sqrt(3) + 2)/(4*sqrt(3) + 7)) - sqrt(2))))/((sqrt(3) + 2)*sqrt((sqrt(3) +
2)/(4*sqrt(3) + 7))*sqrt((sqrt(3) - 2)/(4*sqrt(3) - 7)))

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Sympy [A]  time = 4.85068, size = 20, normalized size = 0.06 \[ \operatorname{RootSum}{\left (65536 t^{8} - 256 t^{4} + 1, \left ( t \mapsto t \log{\left (1024 t^{5} + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

RootSum(65536*_t**8 - 256*_t**4 + 1, Lambda(_t, _t*log(1024*_t**5 + x)))

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GIAC/XCAS [A]  time = 0.283116, size = 331, normalized size = 1. \[ \frac{1}{8} \,{\left (\sqrt{6} - \sqrt{2}\right )} \arctan \left (\frac{4 \, x + \sqrt{6} - \sqrt{2}}{\sqrt{6} + \sqrt{2}}\right ) + \frac{1}{8} \,{\left (\sqrt{6} - \sqrt{2}\right )} \arctan \left (\frac{4 \, x - \sqrt{6} + \sqrt{2}}{\sqrt{6} + \sqrt{2}}\right ) + \frac{1}{8} \,{\left (\sqrt{6} + \sqrt{2}\right )} \arctan \left (\frac{4 \, x + \sqrt{6} + \sqrt{2}}{\sqrt{6} - \sqrt{2}}\right ) + \frac{1}{8} \,{\left (\sqrt{6} + \sqrt{2}\right )} \arctan \left (\frac{4 \, x - \sqrt{6} - \sqrt{2}}{\sqrt{6} - \sqrt{2}}\right ) + \frac{1}{16} \,{\left (\sqrt{6} - \sqrt{2}\right )}{\rm ln}\left (x^{2} + \frac{1}{2} \, x{\left (\sqrt{6} + \sqrt{2}\right )} + 1\right ) - \frac{1}{16} \,{\left (\sqrt{6} - \sqrt{2}\right )}{\rm ln}\left (x^{2} - \frac{1}{2} \, x{\left (\sqrt{6} + \sqrt{2}\right )} + 1\right ) + \frac{1}{16} \,{\left (\sqrt{6} + \sqrt{2}\right )}{\rm ln}\left (x^{2} + \frac{1}{2} \, x{\left (\sqrt{6} - \sqrt{2}\right )} + 1\right ) - \frac{1}{16} \,{\left (\sqrt{6} + \sqrt{2}\right )}{\rm ln}\left (x^{2} - \frac{1}{2} \, x{\left (\sqrt{6} - \sqrt{2}\right )} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*(sqrt(6) - sqrt(2))*arctan((4*x + sqrt(6) - sqrt(2))/(sqrt(6) + sqrt(2))) +
1/8*(sqrt(6) - sqrt(2))*arctan((4*x - sqrt(6) + sqrt(2))/(sqrt(6) + sqrt(2))) +
1/8*(sqrt(6) + sqrt(2))*arctan((4*x + sqrt(6) + sqrt(2))/(sqrt(6) - sqrt(2))) +
1/8*(sqrt(6) + sqrt(2))*arctan((4*x - sqrt(6) - sqrt(2))/(sqrt(6) - sqrt(2))) +
1/16*(sqrt(6) - sqrt(2))*ln(x^2 + 1/2*x*(sqrt(6) + sqrt(2)) + 1) - 1/16*(sqrt(6)
- sqrt(2))*ln(x^2 - 1/2*x*(sqrt(6) + sqrt(2)) + 1) + 1/16*(sqrt(6) + sqrt(2))*l
n(x^2 + 1/2*x*(sqrt(6) - sqrt(2)) + 1) - 1/16*(sqrt(6) + sqrt(2))*ln(x^2 - 1/2*x
*(sqrt(6) - sqrt(2)) + 1)