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

Optimal. Leaf size=97 \[ \frac{x}{4 \left (x^4+1\right )}-\frac{3 \log \left (x^2-\sqrt{2} x+1\right )}{16 \sqrt{2}}+\frac{3 \log \left (x^2+\sqrt{2} x+1\right )}{16 \sqrt{2}}-\frac{3 \tan ^{-1}\left (1-\sqrt{2} x\right )}{8 \sqrt{2}}+\frac{3 \tan ^{-1}\left (\sqrt{2} x+1\right )}{8 \sqrt{2}} \]

[Out]

x/(4*(1 + x^4)) - (3*ArcTan[1 - Sqrt[2]*x])/(8*Sqrt[2]) + (3*ArcTan[1 + Sqrt[2]*
x])/(8*Sqrt[2]) - (3*Log[1 - Sqrt[2]*x + x^2])/(16*Sqrt[2]) + (3*Log[1 + Sqrt[2]
*x + x^2])/(16*Sqrt[2])

_______________________________________________________________________________________

Rubi [A]  time = 0.0930661, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667 \[ \frac{x}{4 \left (x^4+1\right )}-\frac{3 \log \left (x^2-\sqrt{2} x+1\right )}{16 \sqrt{2}}+\frac{3 \log \left (x^2+\sqrt{2} x+1\right )}{16 \sqrt{2}}-\frac{3 \tan ^{-1}\left (1-\sqrt{2} x\right )}{8 \sqrt{2}}+\frac{3 \tan ^{-1}\left (\sqrt{2} x+1\right )}{8 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]  Int[(1 + 2*x^4 + x^8)^(-1),x]

[Out]

x/(4*(1 + x^4)) - (3*ArcTan[1 - Sqrt[2]*x])/(8*Sqrt[2]) + (3*ArcTan[1 + Sqrt[2]*
x])/(8*Sqrt[2]) - (3*Log[1 - Sqrt[2]*x + x^2])/(16*Sqrt[2]) + (3*Log[1 + Sqrt[2]
*x + x^2])/(16*Sqrt[2])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 14.3421, size = 88, normalized size = 0.91 \[ \frac{x}{4 \left (x^{4} + 1\right )} - \frac{3 \sqrt{2} \log{\left (x^{2} - \sqrt{2} x + 1 \right )}}{32} + \frac{3 \sqrt{2} \log{\left (x^{2} + \sqrt{2} x + 1 \right )}}{32} + \frac{3 \sqrt{2} \operatorname{atan}{\left (\sqrt{2} x - 1 \right )}}{16} + \frac{3 \sqrt{2} \operatorname{atan}{\left (\sqrt{2} x + 1 \right )}}{16} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x/(4*(x**4 + 1)) - 3*sqrt(2)*log(x**2 - sqrt(2)*x + 1)/32 + 3*sqrt(2)*log(x**2 +
sqrt(2)*x + 1)/32 + 3*sqrt(2)*atan(sqrt(2)*x - 1)/16 + 3*sqrt(2)*atan(sqrt(2)*x
+ 1)/16

_______________________________________________________________________________________

Mathematica [A]  time = 0.0848057, size = 91, normalized size = 0.94 \[ \frac{1}{32} \left (\frac{8 x}{x^4+1}-3 \sqrt{2} \log \left (x^2-\sqrt{2} x+1\right )+3 \sqrt{2} \log \left (x^2+\sqrt{2} x+1\right )-6 \sqrt{2} \tan ^{-1}\left (1-\sqrt{2} x\right )+6 \sqrt{2} \tan ^{-1}\left (\sqrt{2} x+1\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

((8*x)/(1 + x^4) - 6*Sqrt[2]*ArcTan[1 - Sqrt[2]*x] + 6*Sqrt[2]*ArcTan[1 + Sqrt[2
]*x] - 3*Sqrt[2]*Log[1 - Sqrt[2]*x + x^2] + 3*Sqrt[2]*Log[1 + Sqrt[2]*x + x^2])/
32

_______________________________________________________________________________________

Maple [A]  time = 0.005, size = 68, normalized size = 0.7 \[{\frac{x}{4\,{x}^{4}+4}}+{\frac{3\,\arctan \left ( 1+\sqrt{2}x \right ) \sqrt{2}}{16}}+{\frac{3\,\arctan \left ( \sqrt{2}x-1 \right ) \sqrt{2}}{16}}+{\frac{3\,\sqrt{2}}{32}\ln \left ({\frac{1+{x}^{2}+\sqrt{2}x}{1+{x}^{2}-\sqrt{2}x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(x^8+2*x^4+1),x)

[Out]

1/4*x/(x^4+1)+3/16*arctan(1+2^(1/2)*x)*2^(1/2)+3/16*arctan(2^(1/2)*x-1)*2^(1/2)+
3/32*2^(1/2)*ln((1+x^2+2^(1/2)*x)/(1+x^2-2^(1/2)*x))

_______________________________________________________________________________________

Maxima [A]  time = 0.86165, size = 111, normalized size = 1.14 \[ \frac{3}{16} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x + \sqrt{2}\right )}\right ) + \frac{3}{16} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x - \sqrt{2}\right )}\right ) + \frac{3}{32} \, \sqrt{2} \log \left (x^{2} + \sqrt{2} x + 1\right ) - \frac{3}{32} \, \sqrt{2} \log \left (x^{2} - \sqrt{2} x + 1\right ) + \frac{x}{4 \,{\left (x^{4} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/16*sqrt(2)*arctan(1/2*sqrt(2)*(2*x + sqrt(2))) + 3/16*sqrt(2)*arctan(1/2*sqrt(
2)*(2*x - sqrt(2))) + 3/32*sqrt(2)*log(x^2 + sqrt(2)*x + 1) - 3/32*sqrt(2)*log(x
^2 - sqrt(2)*x + 1) + 1/4*x/(x^4 + 1)

_______________________________________________________________________________________

Fricas [A]  time = 0.281308, size = 174, normalized size = 1.79 \[ -\frac{12 \, \sqrt{2}{\left (x^{4} + 1\right )} \arctan \left (\frac{1}{\sqrt{2} x + \sqrt{2} \sqrt{x^{2} + \sqrt{2} x + 1} + 1}\right ) + 12 \, \sqrt{2}{\left (x^{4} + 1\right )} \arctan \left (\frac{1}{\sqrt{2} x + \sqrt{2} \sqrt{x^{2} - \sqrt{2} x + 1} - 1}\right ) - 3 \, \sqrt{2}{\left (x^{4} + 1\right )} \log \left (x^{2} + \sqrt{2} x + 1\right ) + 3 \, \sqrt{2}{\left (x^{4} + 1\right )} \log \left (x^{2} - \sqrt{2} x + 1\right ) - 8 \, x}{32 \,{\left (x^{4} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/32*(12*sqrt(2)*(x^4 + 1)*arctan(1/(sqrt(2)*x + sqrt(2)*sqrt(x^2 + sqrt(2)*x +
1) + 1)) + 12*sqrt(2)*(x^4 + 1)*arctan(1/(sqrt(2)*x + sqrt(2)*sqrt(x^2 - sqrt(2
)*x + 1) - 1)) - 3*sqrt(2)*(x^4 + 1)*log(x^2 + sqrt(2)*x + 1) + 3*sqrt(2)*(x^4 +
1)*log(x^2 - sqrt(2)*x + 1) - 8*x)/(x^4 + 1)

_______________________________________________________________________________________

Sympy [A]  time = 0.511355, size = 88, normalized size = 0.91 \[ \frac{x}{4 x^{4} + 4} - \frac{3 \sqrt{2} \log{\left (x^{2} - \sqrt{2} x + 1 \right )}}{32} + \frac{3 \sqrt{2} \log{\left (x^{2} + \sqrt{2} x + 1 \right )}}{32} + \frac{3 \sqrt{2} \operatorname{atan}{\left (\sqrt{2} x - 1 \right )}}{16} + \frac{3 \sqrt{2} \operatorname{atan}{\left (\sqrt{2} x + 1 \right )}}{16} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x**8+2*x**4+1),x)

[Out]

x/(4*x**4 + 4) - 3*sqrt(2)*log(x**2 - sqrt(2)*x + 1)/32 + 3*sqrt(2)*log(x**2 + s
qrt(2)*x + 1)/32 + 3*sqrt(2)*atan(sqrt(2)*x - 1)/16 + 3*sqrt(2)*atan(sqrt(2)*x +
1)/16

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.276658, size = 111, normalized size = 1.14 \[ \frac{3}{16} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x + \sqrt{2}\right )}\right ) + \frac{3}{16} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x - \sqrt{2}\right )}\right ) + \frac{3}{32} \, \sqrt{2}{\rm ln}\left (x^{2} + \sqrt{2} x + 1\right ) - \frac{3}{32} \, \sqrt{2}{\rm ln}\left (x^{2} - \sqrt{2} x + 1\right ) + \frac{x}{4 \,{\left (x^{4} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/16*sqrt(2)*arctan(1/2*sqrt(2)*(2*x + sqrt(2))) + 3/16*sqrt(2)*arctan(1/2*sqrt(
2)*(2*x - sqrt(2))) + 3/32*sqrt(2)*ln(x^2 + sqrt(2)*x + 1) - 3/32*sqrt(2)*ln(x^2
- sqrt(2)*x + 1) + 1/4*x/(x^4 + 1)