3.297 \(\int \frac{3+x^2+x^3}{\left (2+x^2\right )^2} \, dx\)

Optimal. Leaf size=42 \[ \frac{x+4}{4 \left (x^2+2\right )}+\frac{1}{2} \log \left (x^2+2\right )+\frac{5 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{4 \sqrt{2}} \]

[Out]

(4 + x)/(4*(2 + x^2)) + (5*ArcTan[x/Sqrt[2]])/(4*Sqrt[2]) + Log[2 + x^2]/2

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Rubi [A]  time = 0.0449307, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{x+4}{4 \left (x^2+2\right )}+\frac{1}{2} \log \left (x^2+2\right )+\frac{5 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{4 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]  Int[(3 + x^2 + x^3)/(2 + x^2)^2,x]

[Out]

(4 + x)/(4*(2 + x^2)) + (5*ArcTan[x/Sqrt[2]])/(4*Sqrt[2]) + Log[2 + x^2]/2

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Rubi in Sympy [A]  time = 8.94224, size = 39, normalized size = 0.93 \[ \frac{x \left (- 2 x + 1\right )}{4 \left (x^{2} + 2\right )} + \frac{\log{\left (x^{2} + 2 \right )}}{2} + \frac{5 \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*(-2*x + 1)/(4*(x**2 + 2)) + log(x**2 + 2)/2 + 5*sqrt(2)*atan(sqrt(2)*x/2)/8

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Mathematica [A]  time = 0.0313027, size = 42, normalized size = 1. \[ \frac{x+4}{4 \left (x^2+2\right )}+\frac{1}{2} \log \left (x^2+2\right )+\frac{5 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{4 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(3 + x^2 + x^3)/(2 + x^2)^2,x]

[Out]

(4 + x)/(4*(2 + x^2)) + (5*ArcTan[x/Sqrt[2]])/(4*Sqrt[2]) + Log[2 + x^2]/2

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Maple [A]  time = 0.007, size = 35, normalized size = 0.8 \[{\frac{1}{{x}^{2}+2} \left ({\frac{x}{4}}+1 \right ) }+{\frac{\ln \left ({x}^{2}+2 \right ) }{2}}+{\frac{5\,\sqrt{2}}{8}\arctan \left ({\frac{\sqrt{2}x}{2}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^3+x^2+3)/(x^2+2)^2,x)

[Out]

(1/4*x+1)/(x^2+2)+1/2*ln(x^2+2)+5/8*arctan(1/2*2^(1/2)*x)*2^(1/2)

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Maxima [A]  time = 0.871203, size = 45, normalized size = 1.07 \[ \frac{5}{8} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) + \frac{x + 4}{4 \,{\left (x^{2} + 2\right )}} + \frac{1}{2} \, \log \left (x^{2} + 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5/8*sqrt(2)*arctan(1/2*sqrt(2)*x) + 1/4*(x + 4)/(x^2 + 2) + 1/2*log(x^2 + 2)

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Fricas [A]  time = 0.253075, size = 68, normalized size = 1.62 \[ \frac{\sqrt{2}{\left (2 \, \sqrt{2}{\left (x^{2} + 2\right )} \log \left (x^{2} + 2\right ) + 5 \,{\left (x^{2} + 2\right )} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) + \sqrt{2}{\left (x + 4\right )}\right )}}{8 \,{\left (x^{2} + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*sqrt(2)*(2*sqrt(2)*(x^2 + 2)*log(x^2 + 2) + 5*(x^2 + 2)*arctan(1/2*sqrt(2)*x
) + sqrt(2)*(x + 4))/(x^2 + 2)

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Sympy [A]  time = 0.291692, size = 36, normalized size = 0.86 \[ \frac{x + 4}{4 x^{2} + 8} + \frac{\log{\left (x^{2} + 2 \right )}}{2} + \frac{5 \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(x + 4)/(4*x**2 + 8) + log(x**2 + 2)/2 + 5*sqrt(2)*atan(sqrt(2)*x/2)/8

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GIAC/XCAS [A]  time = 0.262728, size = 45, normalized size = 1.07 \[ \frac{5}{8} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) + \frac{x + 4}{4 \,{\left (x^{2} + 2\right )}} + \frac{1}{2} \,{\rm ln}\left (x^{2} + 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5/8*sqrt(2)*arctan(1/2*sqrt(2)*x) + 1/4*(x + 4)/(x^2 + 2) + 1/2*ln(x^2 + 2)