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

Optimal. Leaf size=13 \[ -\frac{\cot ^{-1}(x)^{n+1}}{n+1} \]

[Out]

-(ArcCot[x]^(1 + n)/(1 + n))

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Rubi [A]  time = 0.0235998, antiderivative size = 13, 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, Rules used = {4885} \[ -\frac{\cot ^{-1}(x)^{n+1}}{n+1} \]

Antiderivative was successfully verified.

[In]

Int[ArcCot[x]^n/(1 + x^2),x]

[Out]

-(ArcCot[x]^(1 + n)/(1 + n))

Rule 4885

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*} \int \frac{\cot ^{-1}(x)^n}{1+x^2} \, dx &=-\frac{\cot ^{-1}(x)^{1+n}}{1+n}\\ \end{align*}

Mathematica [A]  time = 0.0070566, size = 13, normalized size = 1. \[ -\frac{\cot ^{-1}(x)^{n+1}}{n+1} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcCot[x]^n/(1 + x^2),x]

[Out]

-(ArcCot[x]^(1 + n)/(1 + n))

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Maple [A]  time = 0.038, size = 14, normalized size = 1.1 \begin{align*} -{\frac{ \left ({\rm arccot} \left (x\right ) \right ) ^{1+n}}{1+n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.99926, size = 43, normalized size = 3.31 \begin{align*} -\frac{\operatorname{arccot}\left (x\right )^{n} \operatorname{arccot}\left (x\right )}{n + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-arccot(x)^n*arccot(x)/(n + 1)

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Sympy [A]  time = 4.60103, size = 17, normalized size = 1.31 \begin{align*} - \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.09028, size = 20, normalized size = 1.54 \begin{align*} -\frac{\arctan \left (\frac{1}{x}\right )^{n + 1}}{n + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-arctan(1/x)^(n + 1)/(n + 1)