Optimal. Leaf size=119 \[ -\frac{1}{8} (2 a-b) \log \left (x^2-x+1\right )+\frac{1}{8} (2 a-b) \log \left (x^2+x+1\right )+\frac{x \left (x^2 (-(a-2 b))+a+b\right )}{6 \left (x^4+x^2+1\right )}-\frac{(4 a+b) \tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{12 \sqrt{3}}+\frac{(4 a+b) \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{12 \sqrt{3}} \]
[Out]
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Rubi [A] time = 0.188073, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{1}{8} (2 a-b) \log \left (x^2-x+1\right )+\frac{1}{8} (2 a-b) \log \left (x^2+x+1\right )+\frac{x \left (x^2 (-(a-2 b))+a+b\right )}{6 \left (x^4+x^2+1\right )}-\frac{(4 a+b) \tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{12 \sqrt{3}}+\frac{(4 a+b) \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{12 \sqrt{3}} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x^2)/(1 + x^2 + x^4)^2,x]
[Out]
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Rubi in Sympy [A] time = 30.9515, size = 109, normalized size = 0.92 \[ \frac{x \left (a + b - x^{2} \left (a - 2 b\right )\right )}{6 \left (x^{4} + x^{2} + 1\right )} - \left (\frac{a}{4} - \frac{b}{8}\right ) \log{\left (x^{2} - x + 1 \right )} + \left (\frac{a}{4} - \frac{b}{8}\right ) \log{\left (x^{2} + x + 1 \right )} + \frac{\sqrt{3} \left (4 a + b\right ) \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x}{3} - \frac{1}{3}\right ) \right )}}{36} + \frac{\sqrt{3} \left (4 a + b\right ) \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x}{3} + \frac{1}{3}\right ) \right )}}{36} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**2+a)/(x**4+x**2+1)**2,x)
[Out]
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Mathematica [C] time = 0.412382, size = 147, normalized size = 1.24 \[ \frac{x \left (-a x^2+a+2 b x^2+b\right )}{6 \left (x^4+x^2+1\right )}-\frac{\left (\left (\sqrt{3}-11 i\right ) a-2 \left (\sqrt{3}-2 i\right ) b\right ) \tan ^{-1}\left (\frac{1}{2} \left (\sqrt{3}-i\right ) x\right )}{6 \sqrt{6+6 i \sqrt{3}}}-\frac{\left (\left (\sqrt{3}+11 i\right ) a-2 \left (\sqrt{3}+2 i\right ) b\right ) \tan ^{-1}\left (\frac{1}{2} \left (\sqrt{3}+i\right ) x\right )}{6 \sqrt{6-6 i \sqrt{3}}} \]
Warning: Unable to verify antiderivative.
[In] Integrate[(a + b*x^2)/(1 + x^2 + x^4)^2,x]
[Out]
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Maple [A] time = 0.016, size = 168, normalized size = 1.4 \[{\frac{1}{4\,{x}^{2}+4\,x+4} \left ( \left ( -{\frac{a}{3}}+{\frac{2\,b}{3}} \right ) x-{\frac{2\,a}{3}}+{\frac{b}{3}} \right ) }+{\frac{\ln \left ({x}^{2}+x+1 \right ) a}{4}}-{\frac{\ln \left ({x}^{2}+x+1 \right ) b}{8}}+{\frac{\sqrt{3}a}{9}\arctan \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{3}}{3}} \right ) }+{\frac{\sqrt{3}b}{36}\arctan \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{3}}{3}} \right ) }-{\frac{1}{4\,{x}^{2}-4\,x+4} \left ( \left ({\frac{a}{3}}-{\frac{2\,b}{3}} \right ) x-{\frac{2\,a}{3}}+{\frac{b}{3}} \right ) }-{\frac{\ln \left ({x}^{2}-x+1 \right ) a}{4}}+{\frac{\ln \left ({x}^{2}-x+1 \right ) b}{8}}+{\frac{\sqrt{3}a}{9}\arctan \left ({\frac{ \left ( 2\,x-1 \right ) \sqrt{3}}{3}} \right ) }+{\frac{\sqrt{3}b}{36}\arctan \left ({\frac{ \left ( 2\,x-1 \right ) \sqrt{3}}{3}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^2+a)/(x^4+x^2+1)^2,x)
[Out]
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Maxima [A] time = 0.842297, size = 142, normalized size = 1.19 \[ \frac{1}{36} \, \sqrt{3}{\left (4 \, a + b\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) + \frac{1}{36} \, \sqrt{3}{\left (4 \, a + b\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) + \frac{1}{8} \,{\left (2 \, a - b\right )} \log \left (x^{2} + x + 1\right ) - \frac{1}{8} \,{\left (2 \, a - b\right )} \log \left (x^{2} - x + 1\right ) - \frac{{\left (a - 2 \, b\right )} x^{3} -{\left (a + b\right )} x}{6 \,{\left (x^{4} + x^{2} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)/(x^4 + x^2 + 1)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.298028, size = 261, normalized size = 2.19 \[ \frac{\sqrt{3}{\left (3 \, \sqrt{3}{\left ({\left (2 \, a - b\right )} x^{4} +{\left (2 \, a - b\right )} x^{2} + 2 \, a - b\right )} \log \left (x^{2} + x + 1\right ) - 3 \, \sqrt{3}{\left ({\left (2 \, a - b\right )} x^{4} +{\left (2 \, a - b\right )} x^{2} + 2 \, a - b\right )} \log \left (x^{2} - x + 1\right ) + 2 \,{\left ({\left (4 \, a + b\right )} x^{4} +{\left (4 \, a + b\right )} x^{2} + 4 \, a + b\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) + 2 \,{\left ({\left (4 \, a + b\right )} x^{4} +{\left (4 \, a + b\right )} x^{2} + 4 \, a + b\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) - 4 \, \sqrt{3}{\left ({\left (a - 2 \, b\right )} x^{3} -{\left (a + b\right )} x\right )}\right )}}{72 \,{\left (x^{4} + x^{2} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)/(x^4 + x^2 + 1)^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 4.76565, size = 876, normalized size = 7.36 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x**2+a)/(x**4+x**2+1)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.271991, size = 147, normalized size = 1.24 \[ \frac{1}{36} \, \sqrt{3}{\left (4 \, a + b\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) + \frac{1}{36} \, \sqrt{3}{\left (4 \, a + b\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) + \frac{1}{8} \,{\left (2 \, a - b\right )}{\rm ln}\left (x^{2} + x + 1\right ) - \frac{1}{8} \,{\left (2 \, a - b\right )}{\rm ln}\left (x^{2} - x + 1\right ) - \frac{a x^{3} - 2 \, b x^{3} - a x - b x}{6 \,{\left (x^{4} + x^{2} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)/(x^4 + x^2 + 1)^2,x, algorithm="giac")
[Out]