Optimal. Leaf size=71 \[ -\frac{2}{9 x^2}-\frac{71 \tan ^{-1}\left (\frac{x^2+1}{\sqrt{2}}\right )}{216 \sqrt{2}}-\frac{25 \left (x^2+5\right )}{72 \left (x^4+2 x^2+3\right )}+\frac{13}{108} \log \left (x^4+2 x^2+3\right )-\frac{13 \log (x)}{27} \]
[Out]
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Rubi [A] time = 0.219128, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.226 \[ -\frac{2}{9 x^2}-\frac{71 \tan ^{-1}\left (\frac{x^2+1}{\sqrt{2}}\right )}{216 \sqrt{2}}-\frac{25 \left (x^2+5\right )}{72 \left (x^4+2 x^2+3\right )}+\frac{13}{108} \log \left (x^4+2 x^2+3\right )-\frac{13 \log (x)}{27} \]
Antiderivative was successfully verified.
[In] Int[(4 + x^2 + 3*x^4 + 5*x^6)/(x^3*(3 + 2*x^2 + x^4)^2),x]
[Out]
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Rubi in Sympy [A] time = 26.4935, size = 85, normalized size = 1.2 \[ - \frac{5 \left (10 x^{2} + 14\right )}{144 \left (x^{4} + 2 x^{2} + 3\right )} - \frac{13 \log{\left (x^{2} \right )}}{54} + \frac{13 \log{\left (x^{4} + 2 x^{2} + 3 \right )}}{108} - \frac{71 \sqrt{2} \operatorname{atan}{\left (\sqrt{2} \left (\frac{x^{2}}{2} + \frac{1}{2}\right ) \right )}}{432} - \frac{5}{4 \left (x^{4} + 2 x^{2} + 3\right )} - \frac{2}{9 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((5*x**6+3*x**4+x**2+4)/x**3/(x**4+2*x**2+3)**2,x)
[Out]
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Mathematica [C] time = 0.0856761, size = 97, normalized size = 1.37 \[ \frac{1}{864} \left (-\frac{192}{x^2}+\sqrt{2} \left (52 \sqrt{2}+71 i\right ) \log \left (x^2-i \sqrt{2}+1\right )+\sqrt{2} \left (52 \sqrt{2}-71 i\right ) \log \left (x^2+i \sqrt{2}+1\right )-\frac{300 \left (x^2+5\right )}{x^4+2 x^2+3}-416 \log (x)\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(4 + x^2 + 3*x^4 + 5*x^6)/(x^3*(3 + 2*x^2 + x^4)^2),x]
[Out]
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Maple [A] time = 0.02, size = 63, normalized size = 0.9 \[ -{\frac{2}{9\,{x}^{2}}}-{\frac{13\,\ln \left ( x \right ) }{27}}+{\frac{1}{54\,{x}^{4}+108\,{x}^{2}+162} \left ( -{\frac{75\,{x}^{2}}{4}}-{\frac{375}{4}} \right ) }+{\frac{13\,\ln \left ({x}^{4}+2\,{x}^{2}+3 \right ) }{108}}-{\frac{71\,\sqrt{2}}{432}\arctan \left ({\frac{ \left ( 2\,{x}^{2}+2 \right ) \sqrt{2}}{4}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((5*x^6+3*x^4+x^2+4)/x^3/(x^4+2*x^2+3)^2,x)
[Out]
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Maxima [A] time = 0.798573, size = 89, normalized size = 1.25 \[ -\frac{71}{432} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (x^{2} + 1\right )}\right ) - \frac{41 \, x^{4} + 157 \, x^{2} + 48}{72 \,{\left (x^{6} + 2 \, x^{4} + 3 \, x^{2}\right )}} + \frac{13}{108} \, \log \left (x^{4} + 2 \, x^{2} + 3\right ) - \frac{13}{54} \, \log \left (x^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x^6 + 3*x^4 + x^2 + 4)/((x^4 + 2*x^2 + 3)^2*x^3),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.271167, size = 158, normalized size = 2.23 \[ \frac{\sqrt{2}{\left (26 \, \sqrt{2}{\left (x^{6} + 2 \, x^{4} + 3 \, x^{2}\right )} \log \left (x^{4} + 2 \, x^{2} + 3\right ) - 104 \, \sqrt{2}{\left (x^{6} + 2 \, x^{4} + 3 \, x^{2}\right )} \log \left (x\right ) - 71 \,{\left (x^{6} + 2 \, x^{4} + 3 \, x^{2}\right )} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (x^{2} + 1\right )}\right ) - 3 \, \sqrt{2}{\left (41 \, x^{4} + 157 \, x^{2} + 48\right )}\right )}}{432 \,{\left (x^{6} + 2 \, x^{4} + 3 \, x^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x^6 + 3*x^4 + x^2 + 4)/((x^4 + 2*x^2 + 3)^2*x^3),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.583051, size = 75, normalized size = 1.06 \[ - \frac{41 x^{4} + 157 x^{2} + 48}{72 x^{6} + 144 x^{4} + 216 x^{2}} - \frac{13 \log{\left (x \right )}}{27} + \frac{13 \log{\left (x^{4} + 2 x^{2} + 3 \right )}}{108} - \frac{71 \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x^{2}}{2} + \frac{\sqrt{2}}{2} \right )}}{432} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x**6+3*x**4+x**2+4)/x**3/(x**4+2*x**2+3)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.271676, size = 89, normalized size = 1.25 \[ -\frac{71}{432} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (x^{2} + 1\right )}\right ) - \frac{41 \, x^{4} + 157 \, x^{2} + 48}{72 \,{\left (x^{6} + 2 \, x^{4} + 3 \, x^{2}\right )}} + \frac{13}{108} \,{\rm ln}\left (x^{4} + 2 \, x^{2} + 3\right ) - \frac{13}{54} \,{\rm ln}\left (x^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x^6 + 3*x^4 + x^2 + 4)/((x^4 + 2*x^2 + 3)^2*x^3),x, algorithm="giac")
[Out]