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

Optimal. Leaf size=140 \[ \frac{1}{8} \log \left (x^2-x+1\right )-\frac{1}{8} \log \left (x^2+x+1\right )-\frac{1}{8} \sqrt{3} \log \left (x^2-\sqrt{3} x+1\right )+\frac{1}{8} \sqrt{3} \log \left (x^2+\sqrt{3} x+1\right )-\frac{1}{4} \sqrt{3} \tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )+\frac{1}{4} \tan ^{-1}\left (\sqrt{3}-2 x\right )+\frac{1}{4} \sqrt{3} \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )-\frac{1}{4} \tan ^{-1}\left (2 x+\sqrt{3}\right ) \]

[Out]

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

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Rubi [A]  time = 0.204099, antiderivative size = 140, 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 \[ \frac{1}{8} \log \left (x^2-x+1\right )-\frac{1}{8} \log \left (x^2+x+1\right )-\frac{1}{8} \sqrt{3} \log \left (x^2-\sqrt{3} x+1\right )+\frac{1}{8} \sqrt{3} \log \left (x^2+\sqrt{3} x+1\right )-\frac{1}{4} \sqrt{3} \tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )+\frac{1}{4} \tan ^{-1}\left (\sqrt{3}-2 x\right )+\frac{1}{4} \sqrt{3} \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )-\frac{1}{4} \tan ^{-1}\left (2 x+\sqrt{3}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 40.9273, size = 128, normalized size = 0.91 \[ \frac{\log{\left (x^{2} - x + 1 \right )}}{8} - \frac{\log{\left (x^{2} + x + 1 \right )}}{8} - \frac{\sqrt{3} \log{\left (x^{2} - \sqrt{3} x + 1 \right )}}{8} + \frac{\sqrt{3} \log{\left (x^{2} + \sqrt{3} x + 1 \right )}}{8} + \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x}{3} - \frac{1}{3}\right ) \right )}}{4} + \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x}{3} + \frac{1}{3}\right ) \right )}}{4} - \frac{\operatorname{atan}{\left (2 x - \sqrt{3} \right )}}{4} - \frac{\operatorname{atan}{\left (2 x + \sqrt{3} \right )}}{4} \]

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

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Mathematica [C]  time = 0.300273, size = 129, normalized size = 0.92 \[ \frac{1}{8} \left (\log \left (x^2-x+1\right )-\log \left (x^2+x+1\right )-2 \sqrt{-2-2 i \sqrt{3}} \tan ^{-1}\left (\frac{1}{2} \left (1-i \sqrt{3}\right ) x\right )-2 \sqrt{-2+2 i \sqrt{3}} \tan ^{-1}\left (\frac{1}{2} \left (1+i \sqrt{3}\right ) x\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{2 x-1}{\sqrt{3}}\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )\right ) \]

Warning: Unable to verify antiderivative.

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

[Out]

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

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Maple [A]  time = 0.019, size = 109, normalized size = 0.8 \[ -{\frac{\ln \left ({x}^{2}+x+1 \right ) }{8}}+{\frac{\sqrt{3}}{4}\arctan \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{3}}{3}} \right ) }-{\frac{\ln \left ( 1+{x}^{2}-x\sqrt{3} \right ) \sqrt{3}}{8}}-{\frac{\arctan \left ( 2\,x-\sqrt{3} \right ) }{4}}+{\frac{\ln \left ( 1+{x}^{2}+x\sqrt{3} \right ) \sqrt{3}}{8}}-{\frac{\arctan \left ( 2\,x+\sqrt{3} \right ) }{4}}+{\frac{\ln \left ({x}^{2}-x+1 \right ) }{8}}+{\frac{\sqrt{3}}{4}\arctan \left ({\frac{ \left ( 2\,x-1 \right ) \sqrt{3}}{3}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-x^4+1)/(x^8+x^4+1),x)

[Out]

-1/8*ln(x^2+x+1)+1/4*arctan(1/3*(1+2*x)*3^(1/2))*3^(1/2)-1/8*ln(1+x^2-x*3^(1/2))
*3^(1/2)-1/4*arctan(2*x-3^(1/2))+1/8*ln(1+x^2+x*3^(1/2))*3^(1/2)-1/4*arctan(2*x+
3^(1/2))+1/8*ln(x^2-x+1)+1/4*3^(1/2)*arctan(1/3*(2*x-1)*3^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \frac{1}{4} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) + \frac{1}{4} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) - \frac{1}{2} \, \int \frac{2 \, x^{2} - 1}{x^{4} - x^{2} + 1}\,{d x} - \frac{1}{8} \, \log \left (x^{2} + x + 1\right ) + \frac{1}{8} \, \log \left (x^{2} - x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*sqrt(3)*arctan(1/3*sqrt(3)*(2*x + 1)) + 1/4*sqrt(3)*arctan(1/3*sqrt(3)*(2*x
- 1)) - 1/2*integrate((2*x^2 - 1)/(x^4 - x^2 + 1), x) - 1/8*log(x^2 + x + 1) + 1
/8*log(x^2 - x + 1)

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Fricas [A]  time = 0.28571, size = 190, normalized size = 1.36 \[ \frac{1}{4} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) + \frac{1}{4} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) + \frac{1}{8} \, \sqrt{3} \log \left (x^{2} + \sqrt{3} x + 1\right ) - \frac{1}{8} \, \sqrt{3} \log \left (x^{2} - \sqrt{3} x + 1\right ) + \frac{1}{2} \, \arctan \left (\frac{1}{2 \, x + \sqrt{3} + 2 \, \sqrt{x^{2} + \sqrt{3} x + 1}}\right ) + \frac{1}{2} \, \arctan \left (\frac{1}{2 \, x - \sqrt{3} + 2 \, \sqrt{x^{2} - \sqrt{3} x + 1}}\right ) - \frac{1}{8} \, \log \left (x^{2} + x + 1\right ) + \frac{1}{8} \, \log \left (x^{2} - x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*sqrt(3)*arctan(1/3*sqrt(3)*(2*x + 1)) + 1/4*sqrt(3)*arctan(1/3*sqrt(3)*(2*x
- 1)) + 1/8*sqrt(3)*log(x^2 + sqrt(3)*x + 1) - 1/8*sqrt(3)*log(x^2 - sqrt(3)*x +
1) + 1/2*arctan(1/(2*x + sqrt(3) + 2*sqrt(x^2 + sqrt(3)*x + 1))) + 1/2*arctan(1
/(2*x - sqrt(3) + 2*sqrt(x^2 - sqrt(3)*x + 1))) - 1/8*log(x^2 + x + 1) + 1/8*log
(x^2 - x + 1)

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Sympy [A]  time = 2.68794, size = 148, normalized size = 1.06 \[ - \left (- \frac{1}{8} - \frac{\sqrt{3} i}{8}\right ) \log{\left (x + 1024 \left (- \frac{1}{8} - \frac{\sqrt{3} i}{8}\right )^{5} \right )} - \left (- \frac{1}{8} + \frac{\sqrt{3} i}{8}\right ) \log{\left (x + 1024 \left (- \frac{1}{8} + \frac{\sqrt{3} i}{8}\right )^{5} \right )} - \left (\frac{1}{8} - \frac{\sqrt{3} i}{8}\right ) \log{\left (x + 1024 \left (\frac{1}{8} - \frac{\sqrt{3} i}{8}\right )^{5} \right )} - \left (\frac{1}{8} + \frac{\sqrt{3} i}{8}\right ) \log{\left (x + 1024 \left (\frac{1}{8} + \frac{\sqrt{3} i}{8}\right )^{5} \right )} - \operatorname{RootSum}{\left (256 t^{4} - 16 t^{2} + 1, \left ( t \mapsto t \log{\left (1024 t^{5} + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(-1/8 - sqrt(3)*I/8)*log(x + 1024*(-1/8 - sqrt(3)*I/8)**5) - (-1/8 + sqrt(3)*I/
8)*log(x + 1024*(-1/8 + sqrt(3)*I/8)**5) - (1/8 - sqrt(3)*I/8)*log(x + 1024*(1/8
- sqrt(3)*I/8)**5) - (1/8 + sqrt(3)*I/8)*log(x + 1024*(1/8 + sqrt(3)*I/8)**5) -
RootSum(256*_t**4 - 16*_t**2 + 1, Lambda(_t, _t*log(1024*_t**5 + x)))

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GIAC/XCAS [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="giac")

[Out]

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