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

Optimal. Leaf size=34 \[ -\frac{1}{4 \left (x^2+1\right )}-\frac{x \tan ^{-1}(x)}{2 \left (x^2+1\right )}+\frac{1}{4} \tan ^{-1}(x)^2 \]

[Out]

-1/(4*(1 + x^2)) - (x*ArcTan[x])/(2*(1 + x^2)) + ArcTan[x]^2/4

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Rubi [A]  time = 0.0753278, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ -\frac{1}{4 \left (x^2+1\right )}-\frac{x \tan ^{-1}(x)}{2 \left (x^2+1\right )}+\frac{1}{4} \tan ^{-1}(x)^2 \]

Antiderivative was successfully verified.

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

[Out]

-1/(4*(1 + x^2)) - (x*ArcTan[x])/(2*(1 + x^2)) + ArcTan[x]^2/4

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Rubi in Sympy [A]  time = 5.12067, size = 26, normalized size = 0.76 \[ - \frac{x \operatorname{atan}{\left (x \right )}}{2 \left (x^{2} + 1\right )} + \frac{\operatorname{atan}^{2}{\left (x \right )}}{4} - \frac{1}{4 \left (x^{2} + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0116563, size = 28, normalized size = 0.82 \[ \frac{\left (x^2+1\right ) \tan ^{-1}(x)^2-2 x \tan ^{-1}(x)-1}{4 \left (x^2+1\right )} \]

Antiderivative was successfully verified.

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

[Out]

(-1 - 2*x*ArcTan[x] + (1 + x^2)*ArcTan[x]^2)/(4*(1 + x^2))

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Maple [A]  time = 0.01, size = 29, normalized size = 0.9 \[ -{\frac{1}{4\,{x}^{2}+4}}-{\frac{x\arctan \left ( x \right ) }{2\,{x}^{2}+2}}+{\frac{ \left ( \arctan \left ( x \right ) \right ) ^{2}}{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*arctan(x)/(x^2+1)^2,x)

[Out]

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

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Maxima [A]  time = 1.51529, size = 54, normalized size = 1.59 \[ -\frac{1}{2} \,{\left (\frac{x}{x^{2} + 1} - \arctan \left (x\right )\right )} \arctan \left (x\right ) - \frac{{\left (x^{2} + 1\right )} \arctan \left (x\right )^{2} + 1}{4 \,{\left (x^{2} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.225648, size = 35, normalized size = 1.03 \[ \frac{{\left (x^{2} + 1\right )} \arctan \left (x\right )^{2} - 2 \, x \arctan \left (x\right ) - 1}{4 \,{\left (x^{2} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RecursionError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: RecursionError

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2} \arctan \left (x\right )}{{\left (x^{2} + 1\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^2*arctan(x)/(x^2 + 1)^2, x)