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

Optimal. Leaf size=97 \[ -\frac{x}{4 \left (x^4+1\right )}-\frac{\log \left (x^2-\sqrt{2} x+1\right )}{16 \sqrt{2}}+\frac{\log \left (x^2+\sqrt{2} x+1\right )}{16 \sqrt{2}}-\frac{\tan ^{-1}\left (1-\sqrt{2} x\right )}{8 \sqrt{2}}+\frac{\tan ^{-1}\left (\sqrt{2} x+1\right )}{8 \sqrt{2}} \]

[Out]

-x/(4*(1 + x^4)) - ArcTan[1 - Sqrt[2]*x]/(8*Sqrt[2]) + ArcTan[1 + Sqrt[2]*x]/(8*
Sqrt[2]) - Log[1 - Sqrt[2]*x + x^2]/(16*Sqrt[2]) + Log[1 + Sqrt[2]*x + x^2]/(16*
Sqrt[2])

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Rubi [A]  time = 0.0992792, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5 \[ -\frac{x}{4 \left (x^4+1\right )}-\frac{\log \left (x^2-\sqrt{2} x+1\right )}{16 \sqrt{2}}+\frac{\log \left (x^2+\sqrt{2} x+1\right )}{16 \sqrt{2}}-\frac{\tan ^{-1}\left (1-\sqrt{2} x\right )}{8 \sqrt{2}}+\frac{\tan ^{-1}\left (\sqrt{2} x+1\right )}{8 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]  Int[x^4/(1 + 2*x^4 + x^8),x]

[Out]

-x/(4*(1 + x^4)) - ArcTan[1 - Sqrt[2]*x]/(8*Sqrt[2]) + ArcTan[1 + Sqrt[2]*x]/(8*
Sqrt[2]) - Log[1 - Sqrt[2]*x + x^2]/(16*Sqrt[2]) + Log[1 + Sqrt[2]*x + x^2]/(16*
Sqrt[2])

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Rubi in Sympy [A]  time = 16.5481, size = 82, normalized size = 0.85 \[ - \frac{x}{4 \left (x^{4} + 1\right )} - \frac{\sqrt{2} \log{\left (x^{2} - \sqrt{2} x + 1 \right )}}{32} + \frac{\sqrt{2} \log{\left (x^{2} + \sqrt{2} x + 1 \right )}}{32} + \frac{\sqrt{2} \operatorname{atan}{\left (\sqrt{2} x - 1 \right )}}{16} + \frac{\sqrt{2} \operatorname{atan}{\left (\sqrt{2} x + 1 \right )}}{16} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x/(4*(x**4 + 1)) - sqrt(2)*log(x**2 - sqrt(2)*x + 1)/32 + sqrt(2)*log(x**2 + sq
rt(2)*x + 1)/32 + sqrt(2)*atan(sqrt(2)*x - 1)/16 + sqrt(2)*atan(sqrt(2)*x + 1)/1
6

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Mathematica [A]  time = 0.15934, size = 90, normalized size = 0.93 \[ \frac{1}{32} \left (-\frac{8 x}{x^4+1}-\sqrt{2} \log \left (x^2-\sqrt{2} x+1\right )+\sqrt{2} \log \left (x^2+\sqrt{2} x+1\right )-2 \sqrt{2} \tan ^{-1}\left (1-\sqrt{2} x\right )+2 \sqrt{2} \tan ^{-1}\left (\sqrt{2} x+1\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

((-8*x)/(1 + x^4) - 2*Sqrt[2]*ArcTan[1 - Sqrt[2]*x] + 2*Sqrt[2]*ArcTan[1 + Sqrt[
2]*x] - Sqrt[2]*Log[1 - Sqrt[2]*x + x^2] + Sqrt[2]*Log[1 + Sqrt[2]*x + x^2])/32

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Maple [A]  time = 0.009, size = 68, normalized size = 0.7 \[ -{\frac{x}{4\,{x}^{4}+4}}+{\frac{\arctan \left ( 1+\sqrt{2}x \right ) \sqrt{2}}{16}}+{\frac{\arctan \left ( \sqrt{2}x-1 \right ) \sqrt{2}}{16}}+{\frac{\sqrt{2}}{32}\ln \left ({\frac{1+{x}^{2}+\sqrt{2}x}{1+{x}^{2}-\sqrt{2}x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(x^8+2*x^4+1),x)

[Out]

-1/4*x/(x^4+1)+1/16*arctan(1+2^(1/2)*x)*2^(1/2)+1/16*arctan(2^(1/2)*x-1)*2^(1/2)
+1/32*2^(1/2)*ln((1+x^2+2^(1/2)*x)/(1+x^2-2^(1/2)*x))

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Maxima [A]  time = 0.853659, size = 111, normalized size = 1.14 \[ \frac{1}{16} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x + \sqrt{2}\right )}\right ) + \frac{1}{16} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x - \sqrt{2}\right )}\right ) + \frac{1}{32} \, \sqrt{2} \log \left (x^{2} + \sqrt{2} x + 1\right ) - \frac{1}{32} \, \sqrt{2} \log \left (x^{2} - \sqrt{2} x + 1\right ) - \frac{x}{4 \,{\left (x^{4} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/16*sqrt(2)*arctan(1/2*sqrt(2)*(2*x + sqrt(2))) + 1/16*sqrt(2)*arctan(1/2*sqrt(
2)*(2*x - sqrt(2))) + 1/32*sqrt(2)*log(x^2 + sqrt(2)*x + 1) - 1/32*sqrt(2)*log(x
^2 - sqrt(2)*x + 1) - 1/4*x/(x^4 + 1)

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Fricas [A]  time = 0.271101, size = 173, normalized size = 1.78 \[ -\frac{4 \, \sqrt{2}{\left (x^{4} + 1\right )} \arctan \left (\frac{1}{\sqrt{2} x + \sqrt{2} \sqrt{x^{2} + \sqrt{2} x + 1} + 1}\right ) + 4 \, \sqrt{2}{\left (x^{4} + 1\right )} \arctan \left (\frac{1}{\sqrt{2} x + \sqrt{2} \sqrt{x^{2} - \sqrt{2} x + 1} - 1}\right ) - \sqrt{2}{\left (x^{4} + 1\right )} \log \left (x^{2} + \sqrt{2} x + 1\right ) + \sqrt{2}{\left (x^{4} + 1\right )} \log \left (x^{2} - \sqrt{2} x + 1\right ) + 8 \, x}{32 \,{\left (x^{4} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/32*(4*sqrt(2)*(x^4 + 1)*arctan(1/(sqrt(2)*x + sqrt(2)*sqrt(x^2 + sqrt(2)*x +
1) + 1)) + 4*sqrt(2)*(x^4 + 1)*arctan(1/(sqrt(2)*x + sqrt(2)*sqrt(x^2 - sqrt(2)*
x + 1) - 1)) - sqrt(2)*(x^4 + 1)*log(x^2 + sqrt(2)*x + 1) + sqrt(2)*(x^4 + 1)*lo
g(x^2 - sqrt(2)*x + 1) + 8*x)/(x^4 + 1)

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Sympy [A]  time = 0.509865, size = 82, normalized size = 0.85 \[ - \frac{x}{4 x^{4} + 4} - \frac{\sqrt{2} \log{\left (x^{2} - \sqrt{2} x + 1 \right )}}{32} + \frac{\sqrt{2} \log{\left (x^{2} + \sqrt{2} x + 1 \right )}}{32} + \frac{\sqrt{2} \operatorname{atan}{\left (\sqrt{2} x - 1 \right )}}{16} + \frac{\sqrt{2} \operatorname{atan}{\left (\sqrt{2} x + 1 \right )}}{16} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x/(4*x**4 + 4) - sqrt(2)*log(x**2 - sqrt(2)*x + 1)/32 + sqrt(2)*log(x**2 + sqrt
(2)*x + 1)/32 + sqrt(2)*atan(sqrt(2)*x - 1)/16 + sqrt(2)*atan(sqrt(2)*x + 1)/16

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GIAC/XCAS [A]  time = 0.296154, size = 111, normalized size = 1.14 \[ \frac{1}{16} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x + \sqrt{2}\right )}\right ) + \frac{1}{16} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x - \sqrt{2}\right )}\right ) + \frac{1}{32} \, \sqrt{2}{\rm ln}\left (x^{2} + \sqrt{2} x + 1\right ) - \frac{1}{32} \, \sqrt{2}{\rm ln}\left (x^{2} - \sqrt{2} x + 1\right ) - \frac{x}{4 \,{\left (x^{4} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/16*sqrt(2)*arctan(1/2*sqrt(2)*(2*x + sqrt(2))) + 1/16*sqrt(2)*arctan(1/2*sqrt(
2)*(2*x - sqrt(2))) + 1/32*sqrt(2)*ln(x^2 + sqrt(2)*x + 1) - 1/32*sqrt(2)*ln(x^2
- sqrt(2)*x + 1) - 1/4*x/(x^4 + 1)