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

Optimal. Leaf size=66 \[ \frac{\log \left (\sqrt{4-b} x+2 x^2+1\right )}{2 \sqrt{4-b}}-\frac{\log \left (-\sqrt{4-b} x+2 x^2+1\right )}{2 \sqrt{4-b}} \]

[Out]

-Log[1 - Sqrt[4 - b]*x + 2*x^2]/(2*Sqrt[4 - b]) + Log[1 + Sqrt[4 - b]*x + 2*x^2]
/(2*Sqrt[4 - b])

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Rubi [A]  time = 0.0629816, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{\log \left (\sqrt{4-b} x+2 x^2+1\right )}{2 \sqrt{4-b}}-\frac{\log \left (-\sqrt{4-b} x+2 x^2+1\right )}{2 \sqrt{4-b}} \]

Antiderivative was successfully verified.

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

[Out]

-Log[1 - Sqrt[4 - b]*x + 2*x^2]/(2*Sqrt[4 - b]) + Log[1 + Sqrt[4 - b]*x + 2*x^2]
/(2*Sqrt[4 - b])

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Rubi in Sympy [A]  time = 16.1353, size = 53, normalized size = 0.8 \[ - \frac{\log{\left (x^{2} - \frac{x \sqrt{- b + 4}}{2} + \frac{1}{2} \right )}}{2 \sqrt{- b + 4}} + \frac{\log{\left (x^{2} + \frac{x \sqrt{- b + 4}}{2} + \frac{1}{2} \right )}}{2 \sqrt{- b + 4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-log(x**2 - x*sqrt(-b + 4)/2 + 1/2)/(2*sqrt(-b + 4)) + log(x**2 + x*sqrt(-b + 4)
/2 + 1/2)/(2*sqrt(-b + 4))

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Mathematica [A]  time = 0.122538, size = 127, normalized size = 1.92 \[ \frac{\frac{\left (-\sqrt{b^2-16}+b+4\right ) \tan ^{-1}\left (\frac{2 \sqrt{2} x}{\sqrt{b-\sqrt{b^2-16}}}\right )}{\sqrt{b-\sqrt{b^2-16}}}-\frac{\left (\sqrt{b^2-16}+b+4\right ) \tan ^{-1}\left (\frac{2 \sqrt{2} x}{\sqrt{\sqrt{b^2-16}+b}}\right )}{\sqrt{\sqrt{b^2-16}+b}}}{\sqrt{2} \sqrt{b^2-16}} \]

Antiderivative was successfully verified.

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

[Out]

(((4 + b - Sqrt[-16 + b^2])*ArcTan[(2*Sqrt[2]*x)/Sqrt[b - Sqrt[-16 + b^2]]])/Sqr
t[b - Sqrt[-16 + b^2]] - ((4 + b + Sqrt[-16 + b^2])*ArcTan[(2*Sqrt[2]*x)/Sqrt[b
+ Sqrt[-16 + b^2]]])/Sqrt[b + Sqrt[-16 + b^2]])/(Sqrt[2]*Sqrt[-16 + b^2])

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Maple [B]  time = 0.022, size = 279, normalized size = 4.2 \[ -4\,{\frac{1}{\sqrt{ \left ( b-4 \right ) \left ( 4+b \right ) }\sqrt{2\,\sqrt{ \left ( b-4 \right ) \left ( 4+b \right ) }+2\,b}}\arctan \left ( 4\,{\frac{x}{\sqrt{2\,\sqrt{ \left ( b-4 \right ) \left ( 4+b \right ) }+2\,b}}} \right ) }-{1\arctan \left ( 4\,{\frac{x}{\sqrt{2\,\sqrt{ \left ( b-4 \right ) \left ( 4+b \right ) }+2\,b}}} \right ){\frac{1}{\sqrt{2\,\sqrt{ \left ( b-4 \right ) \left ( 4+b \right ) }+2\,b}}}}-{b\arctan \left ( 4\,{\frac{x}{\sqrt{2\,\sqrt{ \left ( b-4 \right ) \left ( 4+b \right ) }+2\,b}}} \right ){\frac{1}{\sqrt{ \left ( b-4 \right ) \left ( 4+b \right ) }}}{\frac{1}{\sqrt{2\,\sqrt{ \left ( b-4 \right ) \left ( 4+b \right ) }+2\,b}}}}+4\,{\frac{1}{\sqrt{ \left ( b-4 \right ) \left ( 4+b \right ) }\sqrt{-2\,\sqrt{ \left ( b-4 \right ) \left ( 4+b \right ) }+2\,b}}\arctan \left ( 4\,{\frac{x}{\sqrt{-2\,\sqrt{ \left ( b-4 \right ) \left ( 4+b \right ) }+2\,b}}} \right ) }-{1\arctan \left ( 4\,{\frac{x}{\sqrt{-2\,\sqrt{ \left ( b-4 \right ) \left ( 4+b \right ) }+2\,b}}} \right ){\frac{1}{\sqrt{-2\,\sqrt{ \left ( b-4 \right ) \left ( 4+b \right ) }+2\,b}}}}+{b\arctan \left ( 4\,{\frac{x}{\sqrt{-2\,\sqrt{ \left ( b-4 \right ) \left ( 4+b \right ) }+2\,b}}} \right ){\frac{1}{\sqrt{ \left ( b-4 \right ) \left ( 4+b \right ) }}}{\frac{1}{\sqrt{-2\,\sqrt{ \left ( b-4 \right ) \left ( 4+b \right ) }+2\,b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.290522, size = 1, normalized size = 0.02 \[ \left [\frac{\log \left (-\frac{4 \,{\left (b - 4\right )} x^{3} + 2 \,{\left (b - 4\right )} x -{\left (4 \, x^{4} -{\left (b - 8\right )} x^{2} + 1\right )} \sqrt{-b + 4}}{4 \, x^{4} + b x^{2} + 1}\right )}{2 \, \sqrt{-b + 4}}, \frac{\arctan \left (\frac{4 \, x^{3} +{\left (b - 2\right )} x}{\sqrt{b - 4}}\right ) - \arctan \left (\frac{2 \, x}{\sqrt{b - 4}}\right )}{\sqrt{b - 4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/2*log(-(4*(b - 4)*x^3 + 2*(b - 4)*x - (4*x^4 - (b - 8)*x^2 + 1)*sqrt(-b + 4))
/(4*x^4 + b*x^2 + 1))/sqrt(-b + 4), (arctan((4*x^3 + (b - 2)*x)/sqrt(b - 4)) - a
rctan(2*x/sqrt(b - 4)))/sqrt(b - 4)]

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Sympy [A]  time = 0.783422, size = 94, normalized size = 1.42 \[ \frac{\sqrt{- \frac{1}{b - 4}} \log{\left (x^{2} + x \left (- \frac{b \sqrt{- \frac{1}{b - 4}}}{2} + 2 \sqrt{- \frac{1}{b - 4}}\right ) + \frac{1}{2} \right )}}{2} - \frac{\sqrt{- \frac{1}{b - 4}} \log{\left (x^{2} + x \left (\frac{b \sqrt{- \frac{1}{b - 4}}}{2} - 2 \sqrt{- \frac{1}{b - 4}}\right ) + \frac{1}{2} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(-1/(b - 4))*log(x**2 + x*(-b*sqrt(-1/(b - 4))/2 + 2*sqrt(-1/(b - 4))) + 1/2
)/2 - sqrt(-1/(b - 4))*log(x**2 + x*(b*sqrt(-1/(b - 4))/2 - 2*sqrt(-1/(b - 4)))
+ 1/2)/2

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GIAC/XCAS [A]  time = 0.326921, size = 1, normalized size = 0.02 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done