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

Optimal. Leaf size=160 \[ -\frac{1}{4} \sqrt{1+\frac{1}{\sqrt{2}}} \log \left (x^2-\sqrt{2+\sqrt{2}} x+1\right )+\frac{1}{4} \sqrt{1+\frac{1}{\sqrt{2}}} \log \left (x^2+\sqrt{2+\sqrt{2}} x+1\right )-\frac{\tan ^{-1}\left (\frac{\sqrt{2+\sqrt{2}}-2 x}{\sqrt{2-\sqrt{2}}}\right )}{2 \sqrt{2+\sqrt{2}}}+\frac{\tan ^{-1}\left (\frac{2 x+\sqrt{2+\sqrt{2}}}{\sqrt{2-\sqrt{2}}}\right )}{2 \sqrt{2+\sqrt{2}}} \]

[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 = 0.342478, antiderivative size = 160, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172 \[ -\frac{1}{4} \sqrt{\frac{1}{2} \left (2+\sqrt{2}\right )} \log \left (x^2-\sqrt{2+\sqrt{2}} x+1\right )+\frac{1}{4} \sqrt{\frac{1}{2} \left (2+\sqrt{2}\right )} \log \left (x^2+\sqrt{2+\sqrt{2}} x+1\right )-\frac{\tan ^{-1}\left (\frac{\sqrt{2+\sqrt{2}}-2 x}{\sqrt{2-\sqrt{2}}}\right )}{2 \sqrt{2+\sqrt{2}}}+\frac{\tan ^{-1}\left (\frac{2 x+\sqrt{2+\sqrt{2}}}{\sqrt{2-\sqrt{2}}}\right )}{2 \sqrt{2+\sqrt{2}}} \]

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 = 29.8419, size = 240, normalized size = 1.5 \[ - \frac{\left (\frac{1}{2} + \frac{\sqrt{2}}{2}\right ) \log{\left (x^{2} - x \sqrt{\sqrt{2} + 2} + 1 \right )}}{2 \sqrt{\sqrt{2} + 2}} + \frac{\left (\frac{1}{2} + \frac{\sqrt{2}}{2}\right ) \log{\left (x^{2} + x \sqrt{\sqrt{2} + 2} + 1 \right )}}{2 \sqrt{\sqrt{2} + 2}} + \frac{\left (- \frac{\left (1 + \sqrt{2}\right ) \sqrt{\sqrt{2} + 2}}{2} + \sqrt{2} \sqrt{\sqrt{2} + 2}\right ) \operatorname{atan}{\left (\frac{2 x - \sqrt{\sqrt{2} + 2}}{\sqrt{- \sqrt{2} + 2}} \right )}}{\sqrt{- \sqrt{2} + 2} \sqrt{\sqrt{2} + 2}} + \frac{\left (- \frac{\left (1 + \sqrt{2}\right ) \sqrt{\sqrt{2} + 2}}{2} + \sqrt{2} \sqrt{\sqrt{2} + 2}\right ) \operatorname{atan}{\left (\frac{2 x + \sqrt{\sqrt{2} + 2}}{\sqrt{- \sqrt{2} + 2}} \right )}}{\sqrt{- \sqrt{2} + 2} \sqrt{\sqrt{2} + 2}} \]

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]

-(1/2 + sqrt(2)/2)*log(x**2 - x*sqrt(sqrt(2) + 2) + 1)/(2*sqrt(sqrt(2) + 2)) + (
1/2 + sqrt(2)/2)*log(x**2 + x*sqrt(sqrt(2) + 2) + 1)/(2*sqrt(sqrt(2) + 2)) + (-(
1 + sqrt(2))*sqrt(sqrt(2) + 2)/2 + sqrt(2)*sqrt(sqrt(2) + 2))*atan((2*x - sqrt(s
qrt(2) + 2))/sqrt(-sqrt(2) + 2))/(sqrt(-sqrt(2) + 2)*sqrt(sqrt(2) + 2)) + (-(1 +
sqrt(2))*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 = 0.0694501, size = 53, normalized size = 0.33 \[ \frac{\sqrt{-1-i} \tan ^{-1}\left (\frac{\sqrt [4]{2} x}{\sqrt{-1-i}}\right )+\sqrt{-1+i} \tan ^{-1}\left (\frac{\sqrt [4]{2} x}{\sqrt{-1+i}}\right )}{2^{3/4}} \]

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)
/Sqrt[-1 + I]])/2^(3/4)

_______________________________________________________________________________________

Maple [A]  time = 0.089, size = 199, normalized size = 1.2 \[ -{\frac{\sqrt{2}\sqrt{2+\sqrt{2}}\ln \left ( 1+{x}^{2}-x\sqrt{2+\sqrt{2}} \right ) }{8}}-{\frac{1}{2\,\sqrt{2-\sqrt{2}}}\arctan \left ({\frac{2\,x-\sqrt{2+\sqrt{2}}}{\sqrt{2-\sqrt{2}}}} \right ) }+{\frac{\sqrt{2}}{2\,\sqrt{2-\sqrt{2}}}\arctan \left ({\frac{2\,x-\sqrt{2+\sqrt{2}}}{\sqrt{2-\sqrt{2}}}} \right ) }+{\frac{\sqrt{2}\sqrt{2+\sqrt{2}}\ln \left ( 1+{x}^{2}+x\sqrt{2+\sqrt{2}} \right ) }{8}}-{\frac{1}{2\,\sqrt{2-\sqrt{2}}}\arctan \left ({\frac{2\,x+\sqrt{2+\sqrt{2}}}{\sqrt{2-\sqrt{2}}}} \right ) }+{\frac{\sqrt{2}}{2\,\sqrt{2-\sqrt{2}}}\arctan \left ({\frac{2\,x+\sqrt{2+\sqrt{2}}}{\sqrt{2-\sqrt{2}}}} \right ) } \]

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))+1/2/(2-2^(1/2))^(1/2)*arcta
n((2*x-(2+2^(1/2))^(1/2))/(2-2^(1/2))^(1/2))*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))+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)

_______________________________________________________________________________________

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

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]

Exception raised: NotImplementedError

_______________________________________________________________________________________

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)