Optimal. Leaf size=12 \[ \frac{\tan ^{-1}(x)^{n+1}}{n+1} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.0414055, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ \frac{\tan ^{-1}(x)^{n+1}}{n+1} \]
Antiderivative was successfully verified.
[In] Int[ArcTan[x]^n/(1 + x^2),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 3.10798, size = 8, normalized size = 0.67 \[ \frac{\operatorname{atan}^{n + 1}{\left (x \right )}}{n + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(atan(x)**n/(x**2+1),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.00460808, size = 12, normalized size = 1. \[ \frac{\tan ^{-1}(x)^{n+1}}{n+1} \]
Antiderivative was successfully verified.
[In] Integrate[ArcTan[x]^n/(1 + x^2),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.001, size = 13, normalized size = 1.1 \[{\frac{ \left ( \arctan \left ( x \right ) \right ) ^{1+n}}{1+n}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(arctan(x)^n/(x^2+1),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(arctan(x)^n/(x^2 + 1),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.235106, size = 16, normalized size = 1.33 \[ \frac{\arctan \left (x\right )^{n} \arctan \left (x\right )}{n + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(arctan(x)^n/(x^2 + 1),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 9.21979, size = 15, normalized size = 1.25 \[ \begin{cases} \frac{\operatorname{atan}^{n + 1}{\left (x \right )}}{n + 1} & \text{for}\: n \neq -1 \\\log{\left (\operatorname{atan}{\left (x \right )} \right )} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(atan(x)**n/(x**2+1),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.202165, size = 16, normalized size = 1.33 \[ \frac{\arctan \left (x\right )^{n + 1}}{n + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(arctan(x)^n/(x^2 + 1),x, algorithm="giac")
[Out]