3.79 \(\int \frac{4+x^2+3 x^4+5 x^6}{x \left (2+3 x^2+x^4\right )^2} \, dx\)

Optimal. Leaf size=44 \[ -\frac{9}{2} \log \left (x^2+1\right )+4 \log \left (x^2+2\right )-\frac{12 x^2+11}{2 \left (x^4+3 x^2+2\right )}+\log (x) \]

[Out]

-(11 + 12*x^2)/(2*(2 + 3*x^2 + x^4)) + Log[x] - (9*Log[1 + x^2])/2 + 4*Log[2 + x
^2]

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Rubi [A]  time = 0.140799, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097 \[ -\frac{9}{2} \log \left (x^2+1\right )+4 \log \left (x^2+2\right )-\frac{12 x^2+11}{2 \left (x^4+3 x^2+2\right )}+\log (x) \]

Antiderivative was successfully verified.

[In]  Int[(4 + x^2 + 3*x^4 + 5*x^6)/(x*(2 + 3*x^2 + x^4)^2),x]

[Out]

-(11 + 12*x^2)/(2*(2 + 3*x^2 + x^4)) + Log[x] - (9*Log[1 + x^2])/2 + 4*Log[2 + x
^2]

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Rubi in Sympy [A]  time = 22.1618, size = 56, normalized size = 1.27 \[ - \frac{5 x^{2}}{2 \left (x^{4} + 3 x^{2} + 2\right )} + \frac{\log{\left (x^{2} \right )}}{2} - \frac{9 \log{\left (x^{2} + 1 \right )}}{2} + 4 \log{\left (x^{2} + 2 \right )} - \frac{3}{2 \left (x^{2} + 2\right )} - \frac{2}{x^{2} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((5*x**6+3*x**4+x**2+4)/x/(x**4+3*x**2+2)**2,x)

[Out]

-5*x**2/(2*(x**4 + 3*x**2 + 2)) + log(x**2)/2 - 9*log(x**2 + 1)/2 + 4*log(x**2 +
2) - 3/(2*(x**2 + 2)) - 2/(x**2 + 1)

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Mathematica [A]  time = 0.0351933, size = 44, normalized size = 1. \[ -\frac{9}{2} \log \left (x^2+1\right )+4 \log \left (x^2+2\right )+\frac{-12 x^2-11}{2 \left (x^4+3 x^2+2\right )}+\log (x) \]

Antiderivative was successfully verified.

[In]  Integrate[(4 + x^2 + 3*x^4 + 5*x^6)/(x*(2 + 3*x^2 + x^4)^2),x]

[Out]

(-11 - 12*x^2)/(2*(2 + 3*x^2 + x^4)) + Log[x] - (9*Log[1 + x^2])/2 + 4*Log[2 + x
^2]

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Maple [A]  time = 0.027, size = 38, normalized size = 0.9 \[ \ln \left ( x \right ) +4\,\ln \left ({x}^{2}+2 \right ) -{\frac{13}{2\,{x}^{2}+4}}+{\frac{1}{2\,{x}^{2}+2}}-{\frac{9\,\ln \left ({x}^{2}+1 \right ) }{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((5*x^6+3*x^4+x^2+4)/x/(x^4+3*x^2+2)^2,x)

[Out]

ln(x)+4*ln(x^2+2)-13/2/(x^2+2)+1/2/(x^2+1)-9/2*ln(x^2+1)

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Maxima [A]  time = 0.721556, size = 59, normalized size = 1.34 \[ -\frac{12 \, x^{2} + 11}{2 \,{\left (x^{4} + 3 \, x^{2} + 2\right )}} + 4 \, \log \left (x^{2} + 2\right ) - \frac{9}{2} \, \log \left (x^{2} + 1\right ) + \frac{1}{2} \, \log \left (x^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x^6 + 3*x^4 + x^2 + 4)/((x^4 + 3*x^2 + 2)^2*x),x, algorithm="maxima")

[Out]

-1/2*(12*x^2 + 11)/(x^4 + 3*x^2 + 2) + 4*log(x^2 + 2) - 9/2*log(x^2 + 1) + 1/2*l
og(x^2)

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Fricas [A]  time = 0.260401, size = 96, normalized size = 2.18 \[ -\frac{12 \, x^{2} - 8 \,{\left (x^{4} + 3 \, x^{2} + 2\right )} \log \left (x^{2} + 2\right ) + 9 \,{\left (x^{4} + 3 \, x^{2} + 2\right )} \log \left (x^{2} + 1\right ) - 2 \,{\left (x^{4} + 3 \, x^{2} + 2\right )} \log \left (x\right ) + 11}{2 \,{\left (x^{4} + 3 \, x^{2} + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x^6 + 3*x^4 + x^2 + 4)/((x^4 + 3*x^2 + 2)^2*x),x, algorithm="fricas")

[Out]

-1/2*(12*x^2 - 8*(x^4 + 3*x^2 + 2)*log(x^2 + 2) + 9*(x^4 + 3*x^2 + 2)*log(x^2 +
1) - 2*(x^4 + 3*x^2 + 2)*log(x) + 11)/(x^4 + 3*x^2 + 2)

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Sympy [A]  time = 0.453258, size = 39, normalized size = 0.89 \[ - \frac{12 x^{2} + 11}{2 x^{4} + 6 x^{2} + 4} + \log{\left (x \right )} - \frac{9 \log{\left (x^{2} + 1 \right )}}{2} + 4 \log{\left (x^{2} + 2 \right )} \]

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)

[Out]

-(12*x**2 + 11)/(2*x**4 + 6*x**2 + 4) + log(x) - 9*log(x**2 + 1)/2 + 4*log(x**2
+ 2)

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GIAC/XCAS [A]  time = 0.272368, size = 63, normalized size = 1.43 \[ \frac{x^{4} - 21 \, x^{2} - 20}{4 \,{\left (x^{4} + 3 \, x^{2} + 2\right )}} + 4 \,{\rm ln}\left (x^{2} + 2\right ) - \frac{9}{2} \,{\rm ln}\left (x^{2} + 1\right ) + \frac{1}{2} \,{\rm ln}\left (x^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x^6 + 3*x^4 + x^2 + 4)/((x^4 + 3*x^2 + 2)^2*x),x, algorithm="giac")

[Out]

1/4*(x^4 - 21*x^2 - 20)/(x^4 + 3*x^2 + 2) + 4*ln(x^2 + 2) - 9/2*ln(x^2 + 1) + 1/
2*ln(x^2)