Optimal. Leaf size=58 \[ -\frac{23 \tan ^{-1}\left (\frac{x^2+1}{\sqrt{2}}\right )}{8 \sqrt{2}}+\frac{25 \left (x^2+1\right )}{8 \left (x^4+2 x^2+3\right )}+\frac{5}{4} \log \left (x^4+2 x^2+3\right ) \]
[Out]
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Rubi [A] time = 0.118904, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207 \[ -\frac{23 \tan ^{-1}\left (\frac{x^2+1}{\sqrt{2}}\right )}{8 \sqrt{2}}+\frac{25 \left (x^2+1\right )}{8 \left (x^4+2 x^2+3\right )}+\frac{5}{4} \log \left (x^4+2 x^2+3\right ) \]
Antiderivative was successfully verified.
[In] Int[(x*(4 + x^2 + 3*x^4 + 5*x^6))/(3 + 2*x^2 + x^4)^2,x]
[Out]
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Rubi in Sympy [A] time = 17.5046, size = 53, normalized size = 0.91 \[ \frac{0.03125 \left (100 x^{2} + 100\right )}{x^{4} + 2 x^{2} + 3} + \frac{5 \log{\left (x^{4} + 2 x^{2} + 3 \right )}}{4} - 1.4375 \sqrt{2} \operatorname{atan}{\left (\sqrt{2} \left (\frac{x^{2}}{2} + \frac{1}{2}\right ) \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x*(5*x**6+3*x**4+x**2+4)/(x**4+2*x**2+3)**2,x)
[Out]
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Mathematica [A] time = 0.0383752, size = 58, normalized size = 1. \[ -\frac{23 \tan ^{-1}\left (\frac{x^2+1}{\sqrt{2}}\right )}{8 \sqrt{2}}+\frac{25 \left (x^2+1\right )}{8 \left (x^4+2 x^2+3\right )}+\frac{5}{4} \log \left (x^4+2 x^2+3\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(x*(4 + x^2 + 3*x^4 + 5*x^6))/(3 + 2*x^2 + x^4)^2,x]
[Out]
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Maple [A] time = 0.012, size = 54, normalized size = 0.9 \[{\frac{1}{2\,{x}^{4}+4\,{x}^{2}+6} \left ({\frac{25\,{x}^{2}}{4}}+{\frac{25}{4}} \right ) }+{\frac{5\,\ln \left ({x}^{4}+2\,{x}^{2}+3 \right ) }{4}}-{\frac{23\,\sqrt{2}}{16}\arctan \left ({\frac{ \left ( 2\,{x}^{2}+2 \right ) \sqrt{2}}{4}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x*(5*x^6+3*x^4+x^2+4)/(x^4+2*x^2+3)^2,x)
[Out]
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Maxima [A] time = 0.783841, size = 66, normalized size = 1.14 \[ -\frac{23}{16} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (x^{2} + 1\right )}\right ) + \frac{25 \,{\left (x^{2} + 1\right )}}{8 \,{\left (x^{4} + 2 \, x^{2} + 3\right )}} + \frac{5}{4} \, \log \left (x^{4} + 2 \, x^{2} + 3\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x^6 + 3*x^4 + x^2 + 4)*x/(x^4 + 2*x^2 + 3)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.274401, size = 104, normalized size = 1.79 \[ \frac{\sqrt{2}{\left (10 \, \sqrt{2}{\left (x^{4} + 2 \, x^{2} + 3\right )} \log \left (x^{4} + 2 \, x^{2} + 3\right ) - 23 \,{\left (x^{4} + 2 \, x^{2} + 3\right )} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (x^{2} + 1\right )}\right ) + 25 \, \sqrt{2}{\left (x^{2} + 1\right )}\right )}}{16 \,{\left (x^{4} + 2 \, x^{2} + 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x^6 + 3*x^4 + x^2 + 4)*x/(x^4 + 2*x^2 + 3)^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.442492, size = 60, normalized size = 1.03 \[ \frac{25 x^{2} + 25}{8 x^{4} + 16 x^{2} + 24} + \frac{5 \log{\left (x^{4} + 2 x^{2} + 3 \right )}}{4} - \frac{23 \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x^{2}}{2} + \frac{\sqrt{2}}{2} \right )}}{16} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*(5*x**6+3*x**4+x**2+4)/(x**4+2*x**2+3)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.272747, size = 66, normalized size = 1.14 \[ -\frac{23}{16} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (x^{2} + 1\right )}\right ) + \frac{25 \,{\left (x^{2} + 1\right )}}{8 \,{\left (x^{4} + 2 \, x^{2} + 3\right )}} + \frac{5}{4} \,{\rm ln}\left (x^{4} + 2 \, x^{2} + 3\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x^6 + 3*x^4 + x^2 + 4)*x/(x^4 + 2*x^2 + 3)^2,x, algorithm="giac")
[Out]