3.485 \(\int \frac{x}{(1+x)^2 \left (1+x^2\right )} \, dx\)

Optimal. Leaf size=16 \[ \frac{1}{2 (x+1)}+\frac{1}{2} \tan ^{-1}(x) \]

[Out]

1/(2*(1 + x)) + ArcTan[x]/2

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Rubi [A]  time = 0.0374706, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{1}{2 (x+1)}+\frac{1}{2} \tan ^{-1}(x) \]

Antiderivative was successfully verified.

[In]  Int[x/((1 + x)^2*(1 + x^2)),x]

[Out]

1/(2*(1 + x)) + ArcTan[x]/2

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Rubi in Sympy [A]  time = 4.70513, size = 10, normalized size = 0.62 \[ \frac{\operatorname{atan}{\left (x \right )}}{2} + \frac{1}{2 \left (x + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x/(1+x)**2/(x**2+1),x)

[Out]

atan(x)/2 + 1/(2*(x + 1))

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Mathematica [A]  time = 0.0105242, size = 12, normalized size = 0.75 \[ \frac{1}{2} \left (\frac{1}{x+1}+\tan ^{-1}(x)\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x/((1 + x)^2*(1 + x^2)),x]

[Out]

((1 + x)^(-1) + ArcTan[x])/2

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Maple [A]  time = 0.006, size = 13, normalized size = 0.8 \[{\frac{1}{2+2\,x}}+{\frac{\arctan \left ( x \right ) }{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(1+x)^2/(x^2+1),x)

[Out]

1/2/(1+x)+1/2*arctan(x)

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Maxima [A]  time = 0.876112, size = 16, normalized size = 1. \[ \frac{1}{2 \,{\left (x + 1\right )}} + \frac{1}{2} \, \arctan \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/((x^2 + 1)*(x + 1)^2),x, algorithm="maxima")

[Out]

1/2/(x + 1) + 1/2*arctan(x)

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Fricas [A]  time = 0.248934, size = 20, normalized size = 1.25 \[ \frac{{\left (x + 1\right )} \arctan \left (x\right ) + 1}{2 \,{\left (x + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/((x^2 + 1)*(x + 1)^2),x, algorithm="fricas")

[Out]

1/2*((x + 1)*arctan(x) + 1)/(x + 1)

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Sympy [A]  time = 0.242806, size = 10, normalized size = 0.62 \[ \frac{\operatorname{atan}{\left (x \right )}}{2} + \frac{1}{2 x + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(1+x)**2/(x**2+1),x)

[Out]

atan(x)/2 + 1/(2*x + 2)

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GIAC/XCAS [A]  time = 0.264845, size = 43, normalized size = 2.69 \[ -\frac{1}{8} \, \pi - \frac{1}{2} \, \pi \left \lfloor -\frac{\pi - 4 \, \arctan \left (x\right )}{4 \, \pi } + \frac{1}{2} \right \rfloor + \frac{1}{2 \,{\left (x + 1\right )}} + \frac{1}{2} \, \arctan \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/((x^2 + 1)*(x + 1)^2),x, algorithm="giac")

[Out]

-1/8*pi - 1/2*pi*floor(-1/4*(pi - 4*arctan(x))/pi + 1/2) + 1/2/(x + 1) + 1/2*arc
tan(x)