\(\int \frac {\arctan (x)^n}{1+x^2} \, dx\) [68]

Optimal result

Integrand size = 12, antiderivative size = 12 \[ \int \frac {\arctan (x)^n}{1+x^2} \, dx=\frac {\arctan (x)^{1+n}}{1+n} \]

[Out]

arctan(x)^(1+n)/(1+n)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {5004} \[ \int \frac {\arctan (x)^n}{1+x^2} \, dx=\frac {\arctan (x)^{n+1}}{n+1} \]

[In]

Int[ArcTan[x]^n/(1 + x^2),x]

[Out]

ArcTan[x]^(1 + n)/(1 + n)

Rule 5004

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(a + b*ArcTan[c*x])^(p +
1)/(b*c*d*(p + 1)), x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && NeQ[p, -1]

Rubi steps \begin{align*} \text {integral}& = \frac {\arctan (x)^{1+n}}{1+n} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int \frac {\arctan (x)^n}{1+x^2} \, dx=\frac {\arctan (x)^{1+n}}{1+n} \]

[In]

Integrate[ArcTan[x]^n/(1 + x^2),x]

[Out]

ArcTan[x]^(1 + n)/(1 + n)

Maple [A] (verified)

Time = 0.55 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.08

method	result	size
derivativedivides	\(\frac {\arctan \left (x \right )^{1+n}}{1+n}\)	\(13\)
default	\(\frac {\arctan \left (x \right )^{1+n}}{1+n}\)	\(13\)
risch	\(\frac {i \left (\ln \left (-i x +1\right )-\ln \left (i x +1\right )\right ) \left (\frac {i \left (\ln \left (-i x +1\right )-\ln \left (i x +1\right )\right )}{2}\right )^{n}}{2+2 n}\)	\(48\)

[In]

int(arctan(x)^n/(x^2+1),x,method=_RETURNVERBOSE)

[Out]

arctan(x)^(1+n)/(1+n)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int \frac {\arctan (x)^n}{1+x^2} \, dx=\frac {\arctan \left (x\right )^{n} \arctan \left (x\right )}{n + 1} \]

[In]

integrate(arctan(x)^n/(x^2+1),x, algorithm="fricas")

[Out]

arctan(x)^n*arctan(x)/(n + 1)

Sympy [A] (verification not implemented)

Time = 0.83 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.25 \[ \int \frac {\arctan (x)^n}{1+x^2} \, dx=\begin {cases} \frac {\operatorname {atan}^{n + 1}{\left (x \right )}}{n + 1} & \text {for}\: n \neq -1 \\\log {\left (\operatorname {atan}{\left (x \right )} \right )} & \text {otherwise} \end {cases} \]

[In]

integrate(atan(x)**n/(x**2+1),x)

[Out]

Piecewise((atan(x)**(n + 1)/(n + 1), Ne(n, -1)), (log(atan(x)), True))

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int \frac {\arctan (x)^n}{1+x^2} \, dx=\frac {\arctan \left (x\right )^{n + 1}}{n + 1} \]

[In]

integrate(arctan(x)^n/(x^2+1),x, algorithm="maxima")

[Out]

arctan(x)^(n + 1)/(n + 1)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int \frac {\arctan (x)^n}{1+x^2} \, dx=\frac {\arctan \left (x\right )^{n + 1}}{n + 1} \]

[In]

integrate(arctan(x)^n/(x^2+1),x, algorithm="giac")

[Out]

arctan(x)^(n + 1)/(n + 1)

Mupad [B] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int \frac {\arctan (x)^n}{1+x^2} \, dx=\frac {{\mathrm {atan}\left (x\right )}^{n+1}}{n+1} \]

[In]

int(atan(x)^n/(x^2 + 1),x)

[Out]

atan(x)^(n + 1)/(n + 1)