3.111 \(\int \frac{\sqrt{2}+x^2}{1+\sqrt{2} x^2+x^4} \, dx\)

Optimal. Leaf size=172 \[ -\frac{1}{4} \sqrt{1-\frac{1}{\sqrt{2}}} \log \left (x^2-\sqrt{2-\sqrt{2}} x+1\right )+\frac{1}{4} \sqrt{1-\frac{1}{\sqrt{2}}} \log \left (x^2+\sqrt{2-\sqrt{2}} x+1\right )-\frac{\tan ^{-1}\left (\frac{\sqrt{2-\sqrt{2}}-2 x}{\sqrt{2+\sqrt{2}}}\right )}{2 \sqrt{2-\sqrt{2}}}+\frac{\tan ^{-1}\left (\frac{2 x+\sqrt{2-\sqrt{2}}}{\sqrt{2+\sqrt{2}}}\right )}{2 \sqrt{2-\sqrt{2}}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.339839, antiderivative size = 172, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ -\frac{1}{4} \sqrt{\frac{1}{2} \left (2-\sqrt{2}\right )} \log \left (x^2-\sqrt{2-\sqrt{2}} x+1\right )+\frac{1}{4} \sqrt{\frac{1}{2} \left (2-\sqrt{2}\right )} \log \left (x^2+\sqrt{2-\sqrt{2}} x+1\right )-\frac{\tan ^{-1}\left (\frac{\sqrt{2-\sqrt{2}}-2 x}{\sqrt{2+\sqrt{2}}}\right )}{2 \sqrt{2-\sqrt{2}}}+\frac{\tan ^{-1}\left (\frac{2 x+\sqrt{2-\sqrt{2}}}{\sqrt{2+\sqrt{2}}}\right )}{2 \sqrt{2-\sqrt{2}}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 28.8526, size = 240, normalized size = 1.4 \[ \frac{\left (- \frac{\sqrt{2}}{2} + \frac{1}{2}\right ) \log{\left (x^{2} - x \sqrt{- \sqrt{2} + 2} + 1 \right )}}{2 \sqrt{- \sqrt{2} + 2}} - \frac{\left (- \frac{\sqrt{2}}{2} + \frac{1}{2}\right ) \log{\left (x^{2} + x \sqrt{- \sqrt{2} + 2} + 1 \right )}}{2 \sqrt{- \sqrt{2} + 2}} + \frac{\left (\frac{\left (- \sqrt{2} + 1\right ) \sqrt{- \sqrt{2} + 2}}{2} + \sqrt{2} \sqrt{- \sqrt{2} + 2}\right ) \operatorname{atan}{\left (\frac{2 x - \sqrt{- \sqrt{2} + 2}}{\sqrt{\sqrt{2} + 2}} \right )}}{\sqrt{- \sqrt{2} + 2} \sqrt{\sqrt{2} + 2}} + \frac{\left (\frac{\left (- \sqrt{2} + 1\right ) \sqrt{- \sqrt{2} + 2}}{2} + \sqrt{2} \sqrt{- \sqrt{2} + 2}\right ) \operatorname{atan}{\left (\frac{2 x + \sqrt{- \sqrt{2} + 2}}{\sqrt{\sqrt{2} + 2}} \right )}}{\sqrt{- \sqrt{2} + 2} \sqrt{\sqrt{2} + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [C]  time = 0.0545709, size = 53, normalized size = 0.31 \[ \frac{\sqrt{1-i} \tan ^{-1}\left (\frac{\sqrt [4]{2} x}{\sqrt{1-i}}\right )+\sqrt{1+i} \tan ^{-1}\left (\frac{\sqrt [4]{2} x}{\sqrt{1+i}}\right )}{2^{3/4}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[1 - I]*ArcTan[(2^(1/4)*x)/Sqrt[1 - I]] + Sqrt[1 + I]*ArcTan[(2^(1/4)*x)/Sq
rt[1 + I]])/2^(3/4)

_______________________________________________________________________________________

Maple [A]  time = 0.083, size = 199, normalized size = 1.2 \[ -{\frac{\sqrt{2}\sqrt{2-\sqrt{2}}\ln \left ( 1+{x}^{2}-x\sqrt{2-\sqrt{2}} \right ) }{8}}+{\frac{\sqrt{2}}{2\,\sqrt{2+\sqrt{2}}}\arctan \left ({\frac{2\,x-\sqrt{2-\sqrt{2}}}{\sqrt{2+\sqrt{2}}}} \right ) }+{\frac{1}{2\,\sqrt{2+\sqrt{2}}}\arctan \left ({\frac{2\,x-\sqrt{2-\sqrt{2}}}{\sqrt{2+\sqrt{2}}}} \right ) }+{\frac{\sqrt{2}\sqrt{2-\sqrt{2}}\ln \left ( 1+{x}^{2}+x\sqrt{2-\sqrt{2}} \right ) }{8}}+{\frac{\sqrt{2}}{2\,\sqrt{2+\sqrt{2}}}\arctan \left ({\frac{2\,x+\sqrt{2-\sqrt{2}}}{\sqrt{2+\sqrt{2}}}} \right ) }+{\frac{1}{2\,\sqrt{2+\sqrt{2}}}\arctan \left ({\frac{2\,x+\sqrt{2-\sqrt{2}}}{\sqrt{2+\sqrt{2}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2} + \sqrt{2}}{x^{4} + \sqrt{2} x^{2} + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((x^2 + sqrt(2))/(x^4 + sqrt(2)*x^2 + 1), x)

_______________________________________________________________________________________

Fricas [A]  time = 0.338771, size = 595, normalized size = 3.46 \[ \frac{\sqrt{2}{\left ({\left (\sqrt{2} - 2\right )} \log \left (-\frac{34 \, x^{2} + \sqrt{2}{\left (41 \, \sqrt{2} x - 58 \, x\right )} \sqrt{\frac{\sqrt{2} - 2}{2 \, \sqrt{2} - 3}} - 24 \, \sqrt{2}{\left (x^{2} + 1\right )} + 34}{2 \,{\left (12 \, \sqrt{2} - 17\right )}}\right ) -{\left (\sqrt{2} - 2\right )} \log \left (-\frac{34 \, x^{2} - \sqrt{2}{\left (41 \, \sqrt{2} x - 58 \, x\right )} \sqrt{\frac{\sqrt{2} - 2}{2 \, \sqrt{2} - 3}} - 24 \, \sqrt{2}{\left (x^{2} + 1\right )} + 34}{2 \,{\left (12 \, \sqrt{2} - 17\right )}}\right ) + 4 \, \sqrt{2} \arctan \left (\frac{\sqrt{2}}{\sqrt{2} \sqrt{\frac{1}{2}}{\left (\sqrt{2} - 2\right )} \sqrt{-\frac{34 \, x^{2} + \sqrt{2}{\left (41 \, \sqrt{2} x - 58 \, x\right )} \sqrt{\frac{\sqrt{2} - 2}{2 \, \sqrt{2} - 3}} - 24 \, \sqrt{2}{\left (x^{2} + 1\right )} + 34}{12 \, \sqrt{2} - 17}} \sqrt{\frac{\sqrt{2} - 2}{2 \, \sqrt{2} - 3}} + \sqrt{2}{\left (\sqrt{2} x - 2 \, x\right )} \sqrt{\frac{\sqrt{2} - 2}{2 \, \sqrt{2} - 3}} - \sqrt{2} + 2}\right ) + 4 \, \sqrt{2} \arctan \left (\frac{\sqrt{2}}{\sqrt{2} \sqrt{\frac{1}{2}}{\left (\sqrt{2} - 2\right )} \sqrt{-\frac{34 \, x^{2} - \sqrt{2}{\left (41 \, \sqrt{2} x - 58 \, x\right )} \sqrt{\frac{\sqrt{2} - 2}{2 \, \sqrt{2} - 3}} - 24 \, \sqrt{2}{\left (x^{2} + 1\right )} + 34}{12 \, \sqrt{2} - 17}} \sqrt{\frac{\sqrt{2} - 2}{2 \, \sqrt{2} - 3}} + \sqrt{2}{\left (\sqrt{2} x - 2 \, x\right )} \sqrt{\frac{\sqrt{2} - 2}{2 \, \sqrt{2} - 3}} + \sqrt{2} - 2}\right )\right )}}{8 \,{\left (\sqrt{2} - 1\right )} \sqrt{\frac{\sqrt{2} - 2}{2 \, \sqrt{2} - 3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*sqrt(2)*((sqrt(2) - 2)*log(-1/2*(34*x^2 + sqrt(2)*(41*sqrt(2)*x - 58*x)*sqrt
((sqrt(2) - 2)/(2*sqrt(2) - 3)) - 24*sqrt(2)*(x^2 + 1) + 34)/(12*sqrt(2) - 17))
- (sqrt(2) - 2)*log(-1/2*(34*x^2 - sqrt(2)*(41*sqrt(2)*x - 58*x)*sqrt((sqrt(2) -
2)/(2*sqrt(2) - 3)) - 24*sqrt(2)*(x^2 + 1) + 34)/(12*sqrt(2) - 17)) + 4*sqrt(2)
*arctan(sqrt(2)/(sqrt(2)*sqrt(1/2)*(sqrt(2) - 2)*sqrt(-(34*x^2 + sqrt(2)*(41*sqr
t(2)*x - 58*x)*sqrt((sqrt(2) - 2)/(2*sqrt(2) - 3)) - 24*sqrt(2)*(x^2 + 1) + 34)/
(12*sqrt(2) - 17))*sqrt((sqrt(2) - 2)/(2*sqrt(2) - 3)) + sqrt(2)*(sqrt(2)*x - 2*
x)*sqrt((sqrt(2) - 2)/(2*sqrt(2) - 3)) - sqrt(2) + 2)) + 4*sqrt(2)*arctan(sqrt(2
)/(sqrt(2)*sqrt(1/2)*(sqrt(2) - 2)*sqrt(-(34*x^2 - sqrt(2)*(41*sqrt(2)*x - 58*x)
*sqrt((sqrt(2) - 2)/(2*sqrt(2) - 3)) - 24*sqrt(2)*(x^2 + 1) + 34)/(12*sqrt(2) -
17))*sqrt((sqrt(2) - 2)/(2*sqrt(2) - 3)) + sqrt(2)*(sqrt(2)*x - 2*x)*sqrt((sqrt(
2) - 2)/(2*sqrt(2) - 3)) + sqrt(2) - 2)))/((sqrt(2) - 1)*sqrt((sqrt(2) - 2)/(2*s
qrt(2) - 3)))

_______________________________________________________________________________________

Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: PolynomialError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: PolynomialError

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2} + \sqrt{2}}{x^{4} + \sqrt{2} x^{2} + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((x^2 + sqrt(2))/(x^4 + sqrt(2)*x^2 + 1), x)