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

Optimal. Leaf size=66 \[ \frac{89 \tan ^{-1}\left (\frac{x^2+1}{\sqrt{2}}\right )}{72 \sqrt{2}}+\frac{25 \left (1-x^2\right )}{24 \left (x^4+2 x^2+3\right )}-\frac{1}{9} \log \left (x^4+2 x^2+3\right )+\frac{4 \log (x)}{9} \]

[Out]

(25*(1 - x^2))/(24*(3 + 2*x^2 + x^4)) + (89*ArcTan[(1 + x^2)/Sqrt[2]])/(72*Sqrt[
2]) + (4*Log[x])/9 - Log[3 + 2*x^2 + x^4]/9

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Rubi [A]  time = 0.199302, antiderivative size = 66, 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 \[ \frac{89 \tan ^{-1}\left (\frac{x^2+1}{\sqrt{2}}\right )}{72 \sqrt{2}}+\frac{25 \left (1-x^2\right )}{24 \left (x^4+2 x^2+3\right )}-\frac{1}{9} \log \left (x^4+2 x^2+3\right )+\frac{4 \log (x)}{9} \]

Antiderivative was successfully verified.

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

[Out]

(25*(1 - x^2))/(24*(3 + 2*x^2 + x^4)) + (89*ArcTan[(1 + x^2)/Sqrt[2]])/(72*Sqrt[
2]) + (4*Log[x])/9 - Log[3 + 2*x^2 + x^4]/9

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Rubi in Sympy [A]  time = 25.8509, size = 80, normalized size = 1.21 \[ - \frac{5 x^{2}}{2 \left (x^{4} + 2 x^{2} + 3\right )} + \frac{5 \left (14 x^{2} + 10\right )}{48 \left (x^{4} + 2 x^{2} + 3\right )} + \frac{2 \log{\left (x^{2} \right )}}{9} - \frac{\log{\left (x^{4} + 2 x^{2} + 3 \right )}}{9} + \frac{89 \sqrt{2} \operatorname{atan}{\left (\sqrt{2} \left (\frac{x^{2}}{2} + \frac{1}{2}\right ) \right )}}{144} \]

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+2*x**2+3)**2,x)

[Out]

-5*x**2/(2*(x**4 + 2*x**2 + 3)) + 5*(14*x**2 + 10)/(48*(x**4 + 2*x**2 + 3)) + 2*
log(x**2)/9 - log(x**4 + 2*x**2 + 3)/9 + 89*sqrt(2)*atan(sqrt(2)*(x**2/2 + 1/2))
/144

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Mathematica [C]  time = 0.102723, size = 93, normalized size = 1.41 \[ \frac{1}{288} \left (-\sqrt{2} \left (16 \sqrt{2}+89 i\right ) \log \left (x^2-i \sqrt{2}+1\right )+\sqrt{2} \left (-16 \sqrt{2}+89 i\right ) \log \left (x^2+i \sqrt{2}+1\right )-\frac{300 \left (x^2-1\right )}{x^4+2 x^2+3}+128 \log (x)\right ) \]

Antiderivative was successfully verified.

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

[Out]

((-300*(-1 + x^2))/(3 + 2*x^2 + x^4) + 128*Log[x] - Sqrt[2]*(89*I + 16*Sqrt[2])*
Log[1 - I*Sqrt[2] + x^2] + Sqrt[2]*(89*I - 16*Sqrt[2])*Log[1 + I*Sqrt[2] + x^2])
/288

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Maple [A]  time = 0.016, size = 58, normalized size = 0.9 \[{\frac{4\,\ln \left ( x \right ) }{9}}-{\frac{1}{18\,{x}^{4}+36\,{x}^{2}+54} \left ({\frac{75\,{x}^{2}}{4}}-{\frac{75}{4}} \right ) }-{\frac{\ln \left ({x}^{4}+2\,{x}^{2}+3 \right ) }{9}}+{\frac{89\,\sqrt{2}}{144}\arctan \left ({\frac{ \left ( 2\,{x}^{2}+2 \right ) \sqrt{2}}{4}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4/9*ln(x)-1/18*(75/4*x^2-75/4)/(x^4+2*x^2+3)-1/9*ln(x^4+2*x^2+3)+89/144*2^(1/2)*
arctan(1/4*(2*x^2+2)*2^(1/2))

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Maxima [A]  time = 0.784352, size = 74, normalized size = 1.12 \[ \frac{89}{144} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (x^{2} + 1\right )}\right ) - \frac{25 \,{\left (x^{2} - 1\right )}}{24 \,{\left (x^{4} + 2 \, x^{2} + 3\right )}} - \frac{1}{9} \, \log \left (x^{4} + 2 \, x^{2} + 3\right ) + \frac{2}{9} \, \log \left (x^{2}\right ) \]

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),x, algorithm="maxima")

[Out]

89/144*sqrt(2)*arctan(1/2*sqrt(2)*(x^2 + 1)) - 25/24*(x^2 - 1)/(x^4 + 2*x^2 + 3)
- 1/9*log(x^4 + 2*x^2 + 3) + 2/9*log(x^2)

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Fricas [A]  time = 0.267618, size = 127, normalized size = 1.92 \[ -\frac{\sqrt{2}{\left (8 \, \sqrt{2}{\left (x^{4} + 2 \, x^{2} + 3\right )} \log \left (x^{4} + 2 \, x^{2} + 3\right ) - 32 \, \sqrt{2}{\left (x^{4} + 2 \, x^{2} + 3\right )} \log \left (x\right ) - 89 \,{\left (x^{4} + 2 \, x^{2} + 3\right )} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (x^{2} + 1\right )}\right ) + 75 \, \sqrt{2}{\left (x^{2} - 1\right )}\right )}}{144 \,{\left (x^{4} + 2 \, x^{2} + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/144*sqrt(2)*(8*sqrt(2)*(x^4 + 2*x^2 + 3)*log(x^4 + 2*x^2 + 3) - 32*sqrt(2)*(x
^4 + 2*x^2 + 3)*log(x) - 89*(x^4 + 2*x^2 + 3)*arctan(1/2*sqrt(2)*(x^2 + 1)) + 75
*sqrt(2)*(x^2 - 1))/(x^4 + 2*x^2 + 3)

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Sympy [A]  time = 0.486967, size = 65, normalized size = 0.98 \[ - \frac{25 x^{2} - 25}{24 x^{4} + 48 x^{2} + 72} + \frac{4 \log{\left (x \right )}}{9} - \frac{\log{\left (x^{4} + 2 x^{2} + 3 \right )}}{9} + \frac{89 \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x^{2}}{2} + \frac{\sqrt{2}}{2} \right )}}{144} \]

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)

[Out]

-(25*x**2 - 25)/(24*x**4 + 48*x**2 + 72) + 4*log(x)/9 - log(x**4 + 2*x**2 + 3)/9
+ 89*sqrt(2)*atan(sqrt(2)*x**2/2 + sqrt(2)/2)/144

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GIAC/XCAS [A]  time = 0.271836, size = 84, normalized size = 1.27 \[ \frac{89}{144} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (x^{2} + 1\right )}\right ) + \frac{8 \, x^{4} - 59 \, x^{2} + 99}{72 \,{\left (x^{4} + 2 \, x^{2} + 3\right )}} - \frac{1}{9} \,{\rm ln}\left (x^{4} + 2 \, x^{2} + 3\right ) + \frac{2}{9} \,{\rm ln}\left (x^{2}\right ) \]

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),x, algorithm="giac")

[Out]

89/144*sqrt(2)*arctan(1/2*sqrt(2)*(x^2 + 1)) + 1/72*(8*x^4 - 59*x^2 + 99)/(x^4 +
2*x^2 + 3) - 1/9*ln(x^4 + 2*x^2 + 3) + 2/9*ln(x^2)