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

Optimal. Leaf size=28 \[ -\frac{1}{2} \log \left (x^2+1\right )+\log (x)-\frac{1}{2} \tan ^{-1}(x)^2-\frac{\tan ^{-1}(x)}{x} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.110146, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.538 \[ -\frac{1}{2} \log \left (x^2+1\right )+\log (x)-\frac{1}{2} \tan ^{-1}(x)^2-\frac{\tan ^{-1}(x)}{x} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 6.72582, size = 26, normalized size = 0.93 \[ \frac{\log{\left (x^{2} \right )}}{2} - \frac{\log{\left (x^{2} + 1 \right )}}{2} - \frac{\operatorname{atan}^{2}{\left (x \right )}}{2} - \frac{\operatorname{atan}{\left (x \right )}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.00826388, size = 28, normalized size = 1. \[ -\frac{1}{2} \log \left (x^2+1\right )+\log (x)-\frac{1}{2} \tan ^{-1}(x)^2-\frac{\tan ^{-1}(x)}{x} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.012, size = 25, normalized size = 0.9 \[ -{\frac{\arctan \left ( x \right ) }{x}}-{\frac{ \left ( \arctan \left ( x \right ) \right ) ^{2}}{2}}+\ln \left ( x \right ) -{\frac{\ln \left ({x}^{2}+1 \right ) }{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(arctan(x)/x^2/(x^2+1),x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 1.50925, size = 36, normalized size = 1.29 \[ -{\left (\frac{1}{x} + \arctan \left (x\right )\right )} \arctan \left (x\right ) + \frac{1}{2} \, \arctan \left (x\right )^{2} - \frac{1}{2} \, \log \left (x^{2} + 1\right ) + \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.223709, size = 39, normalized size = 1.39 \[ -\frac{x \arctan \left (x\right )^{2} + x \log \left (x^{2} + 1\right ) - 2 \, x \log \left (x\right ) + 2 \, \arctan \left (x\right )}{2 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 0.993274, size = 22, normalized size = 0.79 \[ \log{\left (x \right )} - \frac{\log{\left (x^{2} + 1 \right )}}{2} - \frac{\operatorname{atan}^{2}{\left (x \right )}}{2} - \frac{\operatorname{atan}{\left (x \right )}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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