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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]