3.271 \(\int \frac{-4+8 x-4 x^2+4 x^3-x^4+x^5}{\left (2+x^2\right )^3} \, dx\)

Optimal. Leaf size=35 \[ -\frac{1}{\left (x^2+2\right )^2}+\frac{1}{2} \log \left (x^2+2\right )-\frac{\tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{\sqrt{2}} \]

[Out]

-(2 + x^2)^(-2) - ArcTan[x/Sqrt[2]]/Sqrt[2] + Log[2 + x^2]/2

_______________________________________________________________________________________

Rubi [A]  time = 0.0566319, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.161 \[ -\frac{1}{\left (x^2+2\right )^2}+\frac{1}{2} \log \left (x^2+2\right )-\frac{\tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

[In]  Int[(-4 + 8*x - 4*x^2 + 4*x^3 - x^4 + x^5)/(2 + x^2)^3,x]

[Out]

-(2 + x^2)^(-2) - ArcTan[x/Sqrt[2]]/Sqrt[2] + Log[2 + x^2]/2

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 23.9828, size = 46, normalized size = 1.31 \[ \frac{x^{2}}{4 \left (x^{2} + 2\right )} + \frac{x^{2}}{2 \left (x^{2} + 2\right )^{2}} + \frac{\log{\left (x^{2} + 2 \right )}}{2} - \frac{\sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((x**5-x**4+4*x**3-4*x**2+8*x-4)/(x**2+2)**3,x)

[Out]

x**2/(4*(x**2 + 2)) + x**2/(2*(x**2 + 2)**2) + log(x**2 + 2)/2 - sqrt(2)*atan(sq
rt(2)*x/2)/2

_______________________________________________________________________________________

Mathematica [A]  time = 0.0259836, size = 35, normalized size = 1. \[ -\frac{1}{\left (x^2+2\right )^2}+\frac{1}{2} \log \left (x^2+2\right )-\frac{\tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(-4 + 8*x - 4*x^2 + 4*x^3 - x^4 + x^5)/(2 + x^2)^3,x]

[Out]

-(2 + x^2)^(-2) - ArcTan[x/Sqrt[2]]/Sqrt[2] + Log[2 + x^2]/2

_______________________________________________________________________________________

Maple [A]  time = 0.008, size = 31, normalized size = 0.9 \[ - \left ({x}^{2}+2 \right ) ^{-2}+{\frac{\ln \left ({x}^{2}+2 \right ) }{2}}-{\frac{\sqrt{2}}{2}\arctan \left ({\frac{\sqrt{2}x}{2}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^5-x^4+4*x^3-4*x^2+8*x-4)/(x^2+2)^3,x)

[Out]

-1/(x^2+2)^2+1/2*ln(x^2+2)-1/2*arctan(1/2*2^(1/2)*x)*2^(1/2)

_______________________________________________________________________________________

Maxima [A]  time = 0.886028, size = 47, normalized size = 1.34 \[ -\frac{1}{2} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) - \frac{1}{x^{4} + 4 \, x^{2} + 4} + \frac{1}{2} \, \log \left (x^{2} + 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^5 - x^4 + 4*x^3 - 4*x^2 + 8*x - 4)/(x^2 + 2)^3,x, algorithm="maxima")

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.249461, size = 84, normalized size = 2.4 \[ \frac{\sqrt{2}{\left (\sqrt{2}{\left (x^{4} + 4 \, x^{2} + 4\right )} \log \left (x^{2} + 2\right ) - 2 \,{\left (x^{4} + 4 \, x^{2} + 4\right )} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) - 2 \, \sqrt{2}\right )}}{4 \,{\left (x^{4} + 4 \, x^{2} + 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^5 - x^4 + 4*x^3 - 4*x^2 + 8*x - 4)/(x^2 + 2)^3,x, algorithm="fricas")

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 0.363944, size = 36, normalized size = 1.03 \[ \frac{\log{\left (x^{2} + 2 \right )}}{2} - \frac{\sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )}}{2} - \frac{1}{x^{4} + 4 x^{2} + 4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x**5-x**4+4*x**3-4*x**2+8*x-4)/(x**2+2)**3,x)

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.262179, size = 41, normalized size = 1.17 \[ -\frac{1}{2} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) - \frac{1}{{\left (x^{2} + 2\right )}^{2}} + \frac{1}{2} \,{\rm ln}\left (x^{2} + 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^5 - x^4 + 4*x^3 - 4*x^2 + 8*x - 4)/(x^2 + 2)^3,x, algorithm="giac")

[Out]

-1/2*sqrt(2)*arctan(1/2*sqrt(2)*x) - 1/(x^2 + 2)^2 + 1/2*ln(x^2 + 2)