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

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

[Out]

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

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Rubi [A]  time = 0.109298, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b+4}+4 x}{\sqrt{4-b}}\right )}{\sqrt{4-b}}-\frac{\tan ^{-1}\left (\frac{\sqrt{b+4}-4 x}{\sqrt{4-b}}\right )}{\sqrt{4-b}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 12.5832, size = 49, normalized size = 0.74 \[ \frac{\operatorname{atan}{\left (\frac{4 x - \sqrt{b + 4}}{\sqrt{- b + 4}} \right )}}{\sqrt{- b + 4}} + \frac{\operatorname{atan}{\left (\frac{4 x + \sqrt{b + 4}}{\sqrt{- b + 4}} \right )}}{\sqrt{- b + 4}} \]

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]

atan((4*x - sqrt(b + 4))/sqrt(-b + 4))/sqrt(-b + 4) + atan((4*x + sqrt(b + 4))/s
qrt(-b + 4))/sqrt(-b + 4)

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Mathematica [B]  time = 0.095841, size = 134, normalized size = 2.03 \[ \frac{\frac{\left (\sqrt{b^2-16}+b+4\right ) \tan ^{-1}\left (\frac{2 \sqrt{2} x}{\sqrt{-\sqrt{b^2-16}-b}}\right )}{\sqrt{-\sqrt{b^2-16}-b}}+\frac{\left (\sqrt{b^2-16}-b-4\right ) \tan ^{-1}\left (\frac{2 \sqrt{2} x}{\sqrt{\sqrt{b^2-16}-b}}\right )}{\sqrt{\sqrt{b^2-16}-b}}}{\sqrt{2} \sqrt{b^2-16}} \]

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]]])/Sq
rt[-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 = 0.034, size = 277, 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 = 0.294226, size = 1, normalized size = 0.02 \[ \left [\frac{\log \left (-\frac{4 \,{\left (b - 4\right )} x^{3} - 2 \,{\left (b - 4\right )} x -{\left (4 \, x^{4} +{\left (b - 8\right )} x^{2} + 1\right )} \sqrt{b - 4}}{4 \, x^{4} - b x^{2} + 1}\right )}{2 \, \sqrt{b - 4}}, \frac{\arctan \left (\frac{4 \, x^{3} -{\left (b - 2\right )} x}{\sqrt{-b + 4}}\right ) - \arctan \left (\frac{2 \, \sqrt{-b + 4} x}{b - 4}\right )}{\sqrt{-b + 4}}\right ] \]

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)) - ar
ctan(2*sqrt(-b + 4)*x/(b - 4)))/sqrt(-b + 4)]

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Sympy [A]  time = 0.833746, size = 83, normalized size = 1.26 \[ \frac{\sqrt{\frac{1}{b - 4}} \log{\left (x^{2} + x \left (- \frac{b \sqrt{\frac{1}{b - 4}}}{2} + 2 \sqrt{\frac{1}{b - 4}}\right ) - \frac{1}{2} \right )}}{2} - \frac{\sqrt{\frac{1}{b - 4}} \log{\left (x^{2} + x \left (\frac{b \sqrt{\frac{1}{b - 4}}}{2} - 2 \sqrt{\frac{1}{b - 4}}\right ) - \frac{1}{2} \right )}}{2} \]

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 = 0.318356, 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