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

Optimal. Leaf size=45 \[ \frac{9 \tan ^{-1}\left (\frac{x^2+1}{\sqrt{2}}\right )}{8 \sqrt{2}}+\frac{5-7 x^2}{8 \left (x^4+2 x^2+3\right )} \]

[Out]

(5 - 7*x^2)/(8*(3 + 2*x^2 + x^4)) + (9*ArcTan[(1 + x^2)/Sqrt[2]])/(8*Sqrt[2])

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Rubi [A]  time = 0.111402, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ \frac{9 \tan ^{-1}\left (\frac{x^2+1}{\sqrt{2}}\right )}{8 \sqrt{2}}+\frac{5-7 x^2}{8 \left (x^4+2 x^2+3\right )} \]

Antiderivative was successfully verified.

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

[Out]

(5 - 7*x^2)/(8*(3 + 2*x^2 + x^4)) + (9*ArcTan[(1 + x^2)/Sqrt[2]])/(8*Sqrt[2])

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Rubi in Sympy [A]  time = 23.4117, size = 41, normalized size = 0.91 \[ \frac{- 14 x^{2} + 10}{16 \left (x^{4} + 2 x^{2} + 3\right )} + \frac{9 \sqrt{2} \operatorname{atan}{\left (\sqrt{2} \left (\frac{x^{2}}{2} + \frac{1}{2}\right ) \right )}}{16} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-14*x**2 + 10)/(16*(x**4 + 2*x**2 + 3)) + 9*sqrt(2)*atan(sqrt(2)*(x**2/2 + 1/2)
)/16

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Mathematica [A]  time = 0.0421578, size = 45, normalized size = 1. \[ \frac{9 \tan ^{-1}\left (\frac{x^2+1}{\sqrt{2}}\right )}{8 \sqrt{2}}+\frac{5-7 x^2}{8 \left (x^4+2 x^2+3\right )} \]

Antiderivative was successfully verified.

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

[Out]

(5 - 7*x^2)/(8*(3 + 2*x^2 + x^4)) + (9*ArcTan[(1 + x^2)/Sqrt[2]])/(8*Sqrt[2])

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Maple [A]  time = 0.013, size = 41, normalized size = 0.9 \[{\frac{1}{2\,{x}^{4}+4\,{x}^{2}+6} \left ( -{\frac{7\,{x}^{2}}{4}}+{\frac{5}{4}} \right ) }+{\frac{9\,\sqrt{2}}{16}\arctan \left ({\frac{ \left ( 2\,{x}^{2}+2 \right ) \sqrt{2}}{4}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(-7/4*x^2+5/4)/(x^4+2*x^2+3)+9/16*2^(1/2)*arctan(1/4*(2*x^2+2)*2^(1/2))

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Maxima [A]  time = 0.92412, size = 51, normalized size = 1.13 \[ \frac{9}{16} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (x^{2} + 1\right )}\right ) - \frac{7 \, x^{2} - 5}{8 \,{\left (x^{4} + 2 \, x^{2} + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

9/16*sqrt(2)*arctan(1/2*sqrt(2)*(x^2 + 1)) - 1/8*(7*x^2 - 5)/(x^4 + 2*x^2 + 3)

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Fricas [A]  time = 0.267367, size = 72, normalized size = 1.6 \[ \frac{\sqrt{2}{\left (9 \,{\left (x^{4} + 2 \, x^{2} + 3\right )} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (x^{2} + 1\right )}\right ) - \sqrt{2}{\left (7 \, x^{2} - 5\right )}\right )}}{16 \,{\left (x^{4} + 2 \, x^{2} + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/16*sqrt(2)*(9*(x^4 + 2*x^2 + 3)*arctan(1/2*sqrt(2)*(x^2 + 1)) - sqrt(2)*(7*x^2
- 5))/(x^4 + 2*x^2 + 3)

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Sympy [A]  time = 0.397817, size = 44, normalized size = 0.98 \[ - \frac{7 x^{2} - 5}{8 x^{4} + 16 x^{2} + 24} + \frac{9 \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x^{2}}{2} + \frac{\sqrt{2}}{2} \right )}}{16} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((4*x**5+2*x**3-x)/(x**4+2*x**2+3)**2,x)

[Out]

-(7*x**2 - 5)/(8*x**4 + 16*x**2 + 24) + 9*sqrt(2)*atan(sqrt(2)*x**2/2 + sqrt(2)/
2)/16

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GIAC/XCAS [A]  time = 0.262093, size = 51, normalized size = 1.13 \[ \frac{9}{16} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (x^{2} + 1\right )}\right ) - \frac{7 \, x^{2} - 5}{8 \,{\left (x^{4} + 2 \, x^{2} + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

9/16*sqrt(2)*arctan(1/2*sqrt(2)*(x^2 + 1)) - 1/8*(7*x^2 - 5)/(x^4 + 2*x^2 + 3)