Optimal. Leaf size=42 \[ 4 \log \left (x^2+1\right )-\frac{3}{2} \log \left (x^2+2\right )+\frac{25 x^2+24}{2 \left (x^4+3 x^2+2\right )} \]
[Out]
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Rubi [A] time = 0.084288, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.138 \[ 4 \log \left (x^2+1\right )-\frac{3}{2} \log \left (x^2+2\right )+\frac{25 x^2+24}{2 \left (x^4+3 x^2+2\right )} \]
Antiderivative was successfully verified.
[In] Int[(x*(4 + x^2 + 3*x^4 + 5*x^6))/(2 + 3*x^2 + x^4)^2,x]
[Out]
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Rubi in Sympy [A] time = 17.3863, size = 48, normalized size = 1.14 \[ \frac{0.25 \left (50 x^{2} + 48\right )}{x^{4} + 3 x^{2} + 2} + 2.75 \log{\left (x^{2} + 1 \right )} - 2.75 \log{\left (x^{2} + 2 \right )} + \frac{5 \log{\left (x^{4} + 3 x^{2} + 2 \right )}}{4} \]
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+3*x**2+2)**2,x)
[Out]
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Mathematica [A] time = 0.0312979, size = 42, normalized size = 1. \[ 4 \log \left (x^2+1\right )-\frac{3}{2} \log \left (x^2+2\right )+\frac{25 x^2+24}{2 \left (x^4+3 x^2+2\right )} \]
Antiderivative was successfully verified.
[In] Integrate[(x*(4 + x^2 + 3*x^4 + 5*x^6))/(2 + 3*x^2 + x^4)^2,x]
[Out]
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Maple [A] time = 0.022, size = 36, normalized size = 0.9 \[ -{\frac{3\,\ln \left ({x}^{2}+2 \right ) }{2}}+13\, \left ({x}^{2}+2 \right ) ^{-1}-{\frac{1}{2\,{x}^{2}+2}}+4\,\ln \left ({x}^{2}+1 \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x*(5*x^6+3*x^4+x^2+4)/(x^4+3*x^2+2)^2,x)
[Out]
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Maxima [A] time = 0.725112, size = 51, normalized size = 1.21 \[ \frac{25 \, x^{2} + 24}{2 \,{\left (x^{4} + 3 \, x^{2} + 2\right )}} - \frac{3}{2} \, \log \left (x^{2} + 2\right ) + 4 \, \log \left (x^{2} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x^6 + 3*x^4 + x^2 + 4)*x/(x^4 + 3*x^2 + 2)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.252317, size = 77, normalized size = 1.83 \[ \frac{25 \, x^{2} - 3 \,{\left (x^{4} + 3 \, x^{2} + 2\right )} \log \left (x^{2} + 2\right ) + 8 \,{\left (x^{4} + 3 \, x^{2} + 2\right )} \log \left (x^{2} + 1\right ) + 24}{2 \,{\left (x^{4} + 3 \, x^{2} + 2\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x^6 + 3*x^4 + x^2 + 4)*x/(x^4 + 3*x^2 + 2)^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.394701, size = 36, normalized size = 0.86 \[ \frac{25 x^{2} + 24}{2 x^{4} + 6 x^{2} + 4} + 4 \log{\left (x^{2} + 1 \right )} - \frac{3 \log{\left (x^{2} + 2 \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*(5*x**6+3*x**4+x**2+4)/(x**4+3*x**2+2)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.274234, size = 54, normalized size = 1.29 \[ \frac{25 \, x^{2} + 24}{2 \,{\left (x^{2} + 2\right )}{\left (x^{2} + 1\right )}} - \frac{3}{2} \,{\rm ln}\left (x^{2} + 2\right ) + 4 \,{\rm ln}\left (x^{2} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x^6 + 3*x^4 + x^2 + 4)*x/(x^4 + 3*x^2 + 2)^2,x, algorithm="giac")
[Out]