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

Optimal. Leaf size=62 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{4-b}+4 x}{\sqrt{b+4}}\right )}{\sqrt{b+4}}-\frac{\tan ^{-1}\left (\frac{\sqrt{4-b}-4 x}{\sqrt{b+4}}\right )}{\sqrt{b+4}} \]

[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 = 0.107997, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{4-b}+4 x}{\sqrt{b+4}}\right )}{\sqrt{b+4}}-\frac{\tan ^{-1}\left (\frac{\sqrt{4-b}-4 x}{\sqrt{b+4}}\right )}{\sqrt{b+4}} \]

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 = 12.0003, size = 49, normalized size = 0.79 \[ \frac{\operatorname{atan}{\left (\frac{4 x - \sqrt{- b + 4}}{\sqrt{b + 4}} \right )}}{\sqrt{b + 4}} + \frac{\operatorname{atan}{\left (\frac{4 x + \sqrt{- b + 4}}{\sqrt{b + 4}} \right )}}{\sqrt{b + 4}} \]

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 = 0.0978946, size = 126, normalized size = 2.03 \[ \frac{\frac{\left (\sqrt{b^2-16}-b+4\right ) \tan ^{-1}\left (\frac{2 \sqrt{2} x}{\sqrt{b-\sqrt{b^2-16}}}\right )}{\sqrt{b-\sqrt{b^2-16}}}+\frac{\left (\sqrt{b^2-16}+b-4\right ) \tan ^{-1}\left (\frac{2 \sqrt{2} x}{\sqrt{\sqrt{b^2-16}+b}}\right )}{\sqrt{\sqrt{b^2-16}+b}}}{\sqrt{2} \sqrt{b^2-16}} \]

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 = 0.051, size = 277, normalized size = 4.5 \[ -4\,{\frac{1}{\sqrt{ \left ( b-4 \right ) \left ( 4+b \right ) }\sqrt{2\,\sqrt{ \left ( b-4 \right ) \left ( 4+b \right ) }+2\,b}}\arctan \left ( 4\,{\frac{x}{\sqrt{2\,\sqrt{ \left ( b-4 \right ) \left ( 4+b \right ) }+2\,b}}} \right ) }+{1\arctan \left ( 4\,{\frac{x}{\sqrt{2\,\sqrt{ \left ( b-4 \right ) \left ( 4+b \right ) }+2\,b}}} \right ){\frac{1}{\sqrt{2\,\sqrt{ \left ( b-4 \right ) \left ( 4+b \right ) }+2\,b}}}}+{b\arctan \left ( 4\,{\frac{x}{\sqrt{2\,\sqrt{ \left ( b-4 \right ) \left ( 4+b \right ) }+2\,b}}} \right ){\frac{1}{\sqrt{ \left ( b-4 \right ) \left ( 4+b \right ) }}}{\frac{1}{\sqrt{2\,\sqrt{ \left ( b-4 \right ) \left ( 4+b \right ) }+2\,b}}}}+4\,{\frac{1}{\sqrt{ \left ( b-4 \right ) \left ( 4+b \right ) }\sqrt{-2\,\sqrt{ \left ( b-4 \right ) \left ( 4+b \right ) }+2\,b}}\arctan \left ( 4\,{\frac{x}{\sqrt{-2\,\sqrt{ \left ( b-4 \right ) \left ( 4+b \right ) }+2\,b}}} \right ) }+{1\arctan \left ( 4\,{\frac{x}{\sqrt{-2\,\sqrt{ \left ( b-4 \right ) \left ( 4+b \right ) }+2\,b}}} \right ){\frac{1}{\sqrt{-2\,\sqrt{ \left ( b-4 \right ) \left ( 4+b \right ) }+2\,b}}}}-{b\arctan \left ( 4\,{\frac{x}{\sqrt{-2\,\sqrt{ \left ( b-4 \right ) \left ( 4+b \right ) }+2\,b}}} \right ){\frac{1}{\sqrt{ \left ( b-4 \right ) \left ( 4+b \right ) }}}{\frac{1}{\sqrt{-2\,\sqrt{ \left ( b-4 \right ) \left ( 4+b \right ) }+2\,b}}}} \]

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 = 0.278878, 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)) + ar
ctan(2*x/sqrt(b + 4)))/sqrt(b + 4)]

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Sympy [A]  time = 0.792382, size = 95, normalized size = 1.53 \[ - \frac{\sqrt{- \frac{1}{b + 4}} \log{\left (x^{2} + x \left (- \frac{b \sqrt{- \frac{1}{b + 4}}}{2} - 2 \sqrt{- \frac{1}{b + 4}}\right ) - \frac{1}{2} \right )}}{2} + \frac{\sqrt{- \frac{1}{b + 4}} \log{\left (x^{2} + x \left (\frac{b \sqrt{- \frac{1}{b + 4}}}{2} + 2 \sqrt{- \frac{1}{b + 4}}\right ) - \frac{1}{2} \right )}}{2} \]

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 = 0.325165, 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