\(\int \frac {\cot ^{-1}(x)^n}{1+x^2} \, dx\) [52]

Optimal result

Integrand size = 12, antiderivative size = 13 \[ \int \frac {\cot ^{-1}(x)^n}{1+x^2} \, dx=-\frac {\cot ^{-1}(x)^{1+n}}{1+n} \]

[Out]

-arccot(x)^(1+n)/(1+n)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 13, 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 = {5005} \[ \int \frac {\cot ^{-1}(x)^n}{1+x^2} \, dx=-\frac {\cot ^{-1}(x)^{n+1}}{n+1} \]

[In]

Int[ArcCot[x]^n/(1 + x^2),x]

[Out]

-(ArcCot[x]^(1 + n)/(1 + n))

Rule 5005

Int[((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[-(a + b*ArcCot[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 {\cot ^{-1}(x)^{1+n}}{1+n} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int \frac {\cot ^{-1}(x)^n}{1+x^2} \, dx=-\frac {\cot ^{-1}(x)^{1+n}}{1+n} \]

[In]

Integrate[ArcCot[x]^n/(1 + x^2),x]

[Out]

-(ArcCot[x]^(1 + n)/(1 + n))

Maple [A] (verified)

Time = 0.64 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.08

method	result	size
derivativedivides	\(-\frac {\operatorname {arccot}\left (x \right )^{n +1}}{n +1}\)	\(14\)
default	\(-\frac {\operatorname {arccot}\left (x \right )^{n +1}}{n +1}\)	\(14\)
risch	\(-\frac {\left (\pi -i \ln \left (-i \left (i+x \right )\right )+i \ln \left (-i \left (i-x \right )\right )\right ) \left (\pi -i \ln \left (-i \left (i+x \right )\right )+i \ln \left (-i \left (i-x \right )\right )\right )^{n} \left (\frac {1}{2}\right )^{n}}{2 \left (n +1\right )}\)	\(65\)

[In]

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

[Out]

-arccot(x)^(n+1)/(n+1)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int \frac {\cot ^{-1}(x)^n}{1+x^2} \, dx=-\frac {\operatorname {arccot}\left (x\right )^{n} \operatorname {arccot}\left (x\right )}{n + 1} \]

[In]

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

[Out]

-arccot(x)^n*arccot(x)/(n + 1)

Sympy [A] (verification not implemented)

Time = 0.69 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.31 \[ \int \frac {\cot ^{-1}(x)^n}{1+x^2} \, dx=- \begin {cases} \frac {\operatorname {acot}^{n + 1}{\left (x \right )}}{n + 1} & \text {for}\: n \neq -1 \\\log {\left (\operatorname {acot}{\left (x \right )} \right )} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int \frac {\cot ^{-1}(x)^n}{1+x^2} \, dx=-\frac {\operatorname {arccot}\left (x\right )^{n + 1}}{n + 1} \]

[In]

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

[Out]

-arccot(x)^(n + 1)/(n + 1)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.15 \[ \int \frac {\cot ^{-1}(x)^n}{1+x^2} \, dx=-\frac {\arctan \left (\frac {1}{x}\right )^{n + 1}}{n + 1} \]

[In]

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

[Out]

-arctan(1/x)^(n + 1)/(n + 1)

Mupad [B] (verification not implemented)

Time = 0.70 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int \frac {\cot ^{-1}(x)^n}{1+x^2} \, dx=-\frac {{\mathrm {acot}\left (x\right )}^{n+1}}{n+1} \]

[In]

int(acot(x)^n/(x^2 + 1),x)

[Out]

-acot(x)^(n + 1)/(n + 1)