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

Optimal. Leaf size=85 \[ -\frac{\log \left (x^2-\sqrt{2} x+1\right )}{4 \sqrt{2}}+\frac{\log \left (x^2+\sqrt{2} x+1\right )}{4 \sqrt{2}}-\frac{\tan ^{-1}\left (1-\sqrt{2} x\right )}{2 \sqrt{2}}+\frac{\tan ^{-1}\left (\sqrt{2} x+1\right )}{2 \sqrt{2}} \]

[Out]

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

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Rubi [A]  time = 0.0869912, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.389 \[ -\frac{\log \left (x^2-\sqrt{2} x+1\right )}{4 \sqrt{2}}+\frac{\log \left (x^2+\sqrt{2} x+1\right )}{4 \sqrt{2}}-\frac{\tan ^{-1}\left (1-\sqrt{2} x\right )}{2 \sqrt{2}}+\frac{\tan ^{-1}\left (\sqrt{2} x+1\right )}{2 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]  Int[(1 + x^4)/(1 + 2*x^4 + x^8),x]

[Out]

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

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Rubi in Sympy [A]  time = 15.5855, size = 73, normalized size = 0.86 \[ - \frac{\sqrt{2} \log{\left (x^{2} - \sqrt{2} x + 1 \right )}}{8} + \frac{\sqrt{2} \log{\left (x^{2} + \sqrt{2} x + 1 \right )}}{8} + \frac{\sqrt{2} \operatorname{atan}{\left (\sqrt{2} x - 1 \right )}}{4} + \frac{\sqrt{2} \operatorname{atan}{\left (\sqrt{2} x + 1 \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0311356, size = 64, normalized size = 0.75 \[ \frac{-\log \left (x^2-\sqrt{2} x+1\right )+\log \left (x^2+\sqrt{2} x+1\right )-2 \tan ^{-1}\left (1-\sqrt{2} x\right )+2 \tan ^{-1}\left (\sqrt{2} x+1\right )}{4 \sqrt{2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.004, size = 58, normalized size = 0.7 \[{\frac{\arctan \left ( \sqrt{2}x-1 \right ) \sqrt{2}}{4}}+{\frac{\sqrt{2}}{8}\ln \left ({\frac{1+{x}^{2}+\sqrt{2}x}{1+{x}^{2}-\sqrt{2}x}} \right ) }+{\frac{\arctan \left ( 1+\sqrt{2}x \right ) \sqrt{2}}{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^4+1)/(x^8+2*x^4+1),x)

[Out]

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

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Maxima [A]  time = 0.836387, size = 97, normalized size = 1.14 \[ \frac{1}{4} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x + \sqrt{2}\right )}\right ) + \frac{1}{4} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x - \sqrt{2}\right )}\right ) + \frac{1}{8} \, \sqrt{2} \log \left (x^{2} + \sqrt{2} x + 1\right ) - \frac{1}{8} \, \sqrt{2} \log \left (x^{2} - \sqrt{2} x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.288385, size = 131, normalized size = 1.54 \[ -\frac{1}{2} \, \sqrt{2} \arctan \left (\frac{1}{\sqrt{2} x + \sqrt{2} \sqrt{x^{2} + \sqrt{2} x + 1} + 1}\right ) - \frac{1}{2} \, \sqrt{2} \arctan \left (\frac{1}{\sqrt{2} x + \sqrt{2} \sqrt{x^{2} - \sqrt{2} x + 1} - 1}\right ) + \frac{1}{8} \, \sqrt{2} \log \left (x^{2} + \sqrt{2} x + 1\right ) - \frac{1}{8} \, \sqrt{2} \log \left (x^{2} - \sqrt{2} x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*sqrt(2)*arctan(1/(sqrt(2)*x + sqrt(2)*sqrt(x^2 + sqrt(2)*x + 1) + 1)) - 1/2
*sqrt(2)*arctan(1/(sqrt(2)*x + sqrt(2)*sqrt(x^2 - sqrt(2)*x + 1) - 1)) + 1/8*sqr
t(2)*log(x^2 + sqrt(2)*x + 1) - 1/8*sqrt(2)*log(x^2 - sqrt(2)*x + 1)

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Sympy [A]  time = 0.416257, size = 73, normalized size = 0.86 \[ - \frac{\sqrt{2} \log{\left (x^{2} - \sqrt{2} x + 1 \right )}}{8} + \frac{\sqrt{2} \log{\left (x^{2} + \sqrt{2} x + 1 \right )}}{8} + \frac{\sqrt{2} \operatorname{atan}{\left (\sqrt{2} x - 1 \right )}}{4} + \frac{\sqrt{2} \operatorname{atan}{\left (\sqrt{2} x + 1 \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.271182, size = 97, normalized size = 1.14 \[ \frac{1}{4} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x + \sqrt{2}\right )}\right ) + \frac{1}{4} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x - \sqrt{2}\right )}\right ) + \frac{1}{8} \, \sqrt{2}{\rm ln}\left (x^{2} + \sqrt{2} x + 1\right ) - \frac{1}{8} \, \sqrt{2}{\rm ln}\left (x^{2} - \sqrt{2} x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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