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

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

[Out]

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

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Rubi [A]  time = 0.104, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{2-b}+2 x}{\sqrt{b+2}}\right )}{\sqrt{b+2}}-\frac{\tan ^{-1}\left (\frac{\sqrt{2-b}-2 x}{\sqrt{b+2}}\right )}{\sqrt{b+2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 10.9647, size = 49, normalized size = 0.79 \[ \frac{\operatorname{atan}{\left (\frac{2 x - \sqrt{- b + 2}}{\sqrt{b + 2}} \right )}}{\sqrt{b + 2}} + \frac{\operatorname{atan}{\left (\frac{2 x + \sqrt{- b + 2}}{\sqrt{b + 2}} \right )}}{\sqrt{b + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

atan((2*x - sqrt(-b + 2))/sqrt(b + 2))/sqrt(b + 2) + atan((2*x + sqrt(-b + 2))/s
qrt(b + 2))/sqrt(b + 2)

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Mathematica [A]  time = 0.0935912, size = 124, normalized size = 2. \[ \frac{\frac{\left (\sqrt{b^2-4}-b+2\right ) \tan ^{-1}\left (\frac{\sqrt{2} x}{\sqrt{b-\sqrt{b^2-4}}}\right )}{\sqrt{b-\sqrt{b^2-4}}}+\frac{\left (\sqrt{b^2-4}+b-2\right ) \tan ^{-1}\left (\frac{\sqrt{2} x}{\sqrt{\sqrt{b^2-4}+b}}\right )}{\sqrt{\sqrt{b^2-4}+b}}}{\sqrt{2} \sqrt{b^2-4}} \]

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 = 0.049, size = 277, 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 = 0.289962, 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 = 0.734525, size = 88, normalized size = 1.42 \[ - \frac{\sqrt{- \frac{1}{b + 2}} \log{\left (x^{2} + x \left (- b \sqrt{- \frac{1}{b + 2}} - 2 \sqrt{- \frac{1}{b + 2}}\right ) - 1 \right )}}{2} + \frac{\sqrt{- \frac{1}{b + 2}} \log{\left (x^{2} + x \left (b \sqrt{- \frac{1}{b + 2}} + 2 \sqrt{- \frac{1}{b + 2}}\right ) - 1 \right )}}{2} \]

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