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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]