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

Optimal. Leaf size=62 \[ \frac{\log \left (\sqrt{2-b} x+x^2+1\right )}{2 \sqrt{2-b}}-\frac{\log \left (-\sqrt{2-b} x+x^2+1\right )}{2 \sqrt{2-b}} \]

[Out]

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

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Rubi [A]  time = 0.0556812, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{\log \left (\sqrt{2-b} x+x^2+1\right )}{2 \sqrt{2-b}}-\frac{\log \left (-\sqrt{2-b} x+x^2+1\right )}{2 \sqrt{2-b}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 13.903, size = 46, normalized size = 0.74 \[ - \frac{\log{\left (x^{2} - x \sqrt{- b + 2} + 1 \right )}}{2 \sqrt{- b + 2}} + \frac{\log{\left (x^{2} + x \sqrt{- b + 2} + 1 \right )}}{2 \sqrt{- b + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [B]  time = 0.125112, size = 125, normalized size = 2.02 \[ \frac{\frac{\left (-\sqrt{b^2-4}+b+2\right ) \tan ^{-1}\left (\frac{\sqrt{2} x}{\sqrt{b-\sqrt{b^2-4}}}\right )}{\sqrt{b-\sqrt{b^2-4}}}-\frac{\left (\sqrt{b^2-4}+b+2\right ) \tan ^{-1}\left (\frac{\sqrt{2} x}{\sqrt{\sqrt{b^2-4}+b}}\right )}{\sqrt{\sqrt{b^2-4}+b}}}{\sqrt{2} \sqrt{b^2-4}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.024, size = 279, normalized size = 4.5 \[ 2\,{\frac{1}{\sqrt{ \left ( b-2 \right ) \left ( 2+b \right ) }\sqrt{-2\,\sqrt{ \left ( b-2 \right ) \left ( 2+b \right ) }+2\,b}}\arctan \left ( 2\,{\frac{x}{\sqrt{-2\,\sqrt{ \left ( b-2 \right ) \left ( 2+b \right ) }+2\,b}}} \right ) }-{1\arctan \left ( 2\,{\frac{x}{\sqrt{-2\,\sqrt{ \left ( b-2 \right ) \left ( 2+b \right ) }+2\,b}}} \right ){\frac{1}{\sqrt{-2\,\sqrt{ \left ( b-2 \right ) \left ( 2+b \right ) }+2\,b}}}}+{b\arctan \left ( 2\,{\frac{x}{\sqrt{-2\,\sqrt{ \left ( b-2 \right ) \left ( 2+b \right ) }+2\,b}}} \right ){\frac{1}{\sqrt{ \left ( b-2 \right ) \left ( 2+b \right ) }}}{\frac{1}{\sqrt{-2\,\sqrt{ \left ( b-2 \right ) \left ( 2+b \right ) }+2\,b}}}}-2\,{\frac{1}{\sqrt{ \left ( b-2 \right ) \left ( 2+b \right ) }\sqrt{2\,\sqrt{ \left ( b-2 \right ) \left ( 2+b \right ) }+2\,b}}\arctan \left ( 2\,{\frac{x}{\sqrt{2\,\sqrt{ \left ( b-2 \right ) \left ( 2+b \right ) }+2\,b}}} \right ) }-{1\arctan \left ( 2\,{\frac{x}{\sqrt{2\,\sqrt{ \left ( b-2 \right ) \left ( 2+b \right ) }+2\,b}}} \right ){\frac{1}{\sqrt{2\,\sqrt{ \left ( b-2 \right ) \left ( 2+b \right ) }+2\,b}}}}-{b\arctan \left ( 2\,{\frac{x}{\sqrt{2\,\sqrt{ \left ( b-2 \right ) \left ( 2+b \right ) }+2\,b}}} \right ){\frac{1}{\sqrt{ \left ( b-2 \right ) \left ( 2+b \right ) }}}{\frac{1}{\sqrt{2\,\sqrt{ \left ( b-2 \right ) \left ( 2+b \right ) }+2\,b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-x^2+1)/(x^4+b*x^2+1),x)

[Out]

2/((b-2)*(2+b))^(1/2)/(-2*((b-2)*(2+b))^(1/2)+2*b)^(1/2)*arctan(2*x/(-2*((b-2)*(
2+b))^(1/2)+2*b)^(1/2))-1/(-2*((b-2)*(2+b))^(1/2)+2*b)^(1/2)*arctan(2*x/(-2*((b-
2)*(2+b))^(1/2)+2*b)^(1/2))+1/((b-2)*(2+b))^(1/2)/(-2*((b-2)*(2+b))^(1/2)+2*b)^(
1/2)*arctan(2*x/(-2*((b-2)*(2+b))^(1/2)+2*b)^(1/2))*b-2/((b-2)*(2+b))^(1/2)/(2*(
(b-2)*(2+b))^(1/2)+2*b)^(1/2)*arctan(2*x/(2*((b-2)*(2+b))^(1/2)+2*b)^(1/2))-1/(2
*((b-2)*(2+b))^(1/2)+2*b)^(1/2)*arctan(2*x/(2*((b-2)*(2+b))^(1/2)+2*b)^(1/2))-1/
((b-2)*(2+b))^(1/2)/(2*((b-2)*(2+b))^(1/2)+2*b)^(1/2)*arctan(2*x/(2*((b-2)*(2+b)
)^(1/2)+2*b)^(1/2))*b

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.292533, size = 1, normalized size = 0.02 \[ \left [\frac{\log \left (-\frac{2 \,{\left (b - 2\right )} x^{3} + 2 \,{\left (b - 2\right )} x -{\left (x^{4} -{\left (b - 4\right )} x^{2} + 1\right )} \sqrt{-b + 2}}{x^{4} + b x^{2} + 1}\right )}{2 \, \sqrt{-b + 2}}, \frac{\arctan \left (\frac{x^{3} +{\left (b - 1\right )} x}{\sqrt{b - 2}}\right ) - \arctan \left (\frac{x}{\sqrt{b - 2}}\right )}{\sqrt{b - 2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/2*log(-(2*(b - 2)*x^3 + 2*(b - 2)*x - (x^4 - (b - 4)*x^2 + 1)*sqrt(-b + 2))/(
x^4 + b*x^2 + 1))/sqrt(-b + 2), (arctan((x^3 + (b - 1)*x)/sqrt(b - 2)) - arctan(
x/sqrt(b - 2)))/sqrt(b - 2)]

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Sympy [A]  time = 0.736163, size = 87, normalized size = 1.4 \[ \frac{\sqrt{- \frac{1}{b - 2}} \log{\left (x^{2} + x \left (- b \sqrt{- \frac{1}{b - 2}} + 2 \sqrt{- \frac{1}{b - 2}}\right ) + 1 \right )}}{2} - \frac{\sqrt{- \frac{1}{b - 2}} \log{\left (x^{2} + x \left (b \sqrt{- \frac{1}{b - 2}} - 2 \sqrt{- \frac{1}{b - 2}}\right ) + 1 \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError