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

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]

(25*(1 + x^2))/(8*(3 + 2*x^2 + x^4)) - (23*ArcTan[(1 + x^2)/Sqrt[2]])/(8*Sqrt[2]
) + (5*Log[3 + 2*x^2 + x^4])/4

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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]

(25*(1 + x^2))/(8*(3 + 2*x^2 + x^4)) - (23*ArcTan[(1 + x^2)/Sqrt[2]])/(8*Sqrt[2]
) + (5*Log[3 + 2*x^2 + x^4])/4

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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]

0.03125*(100*x**2 + 100)/(x**4 + 2*x**2 + 3) + 5*log(x**4 + 2*x**2 + 3)/4 - 1.43
75*sqrt(2)*atan(sqrt(2)*(x**2/2 + 1/2))

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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]

(25*(1 + x^2))/(8*(3 + 2*x^2 + x^4)) - (23*ArcTan[(1 + x^2)/Sqrt[2]])/(8*Sqrt[2]
) + (5*Log[3 + 2*x^2 + x^4])/4

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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]

1/2*(25/4*x^2+25/4)/(x^4+2*x^2+3)+5/4*ln(x^4+2*x^2+3)-23/16*2^(1/2)*arctan(1/4*(
2*x^2+2)*2^(1/2))

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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]

-23/16*sqrt(2)*arctan(1/2*sqrt(2)*(x^2 + 1)) + 25/8*(x^2 + 1)/(x^4 + 2*x^2 + 3)
+ 5/4*log(x^4 + 2*x^2 + 3)

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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]

1/16*sqrt(2)*(10*sqrt(2)*(x^4 + 2*x^2 + 3)*log(x^4 + 2*x^2 + 3) - 23*(x^4 + 2*x^
2 + 3)*arctan(1/2*sqrt(2)*(x^2 + 1)) + 25*sqrt(2)*(x^2 + 1))/(x^4 + 2*x^2 + 3)

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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]

(25*x**2 + 25)/(8*x**4 + 16*x**2 + 24) + 5*log(x**4 + 2*x**2 + 3)/4 - 23*sqrt(2)
*atan(sqrt(2)*x**2/2 + sqrt(2)/2)/16

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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]

-23/16*sqrt(2)*arctan(1/2*sqrt(2)*(x^2 + 1)) + 25/8*(x^2 + 1)/(x^4 + 2*x^2 + 3)
+ 5/4*ln(x^4 + 2*x^2 + 3)