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

Optimal. Leaf size=8 \[ -\frac{1}{2} \cot ^{-1}(x)^2 \]

[Out]

-ArcCot[x]^2/2

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Rubi [A]  time = 0.0115701, antiderivative size = 8, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {4885} \[ -\frac{1}{2} \cot ^{-1}(x)^2 \]

Antiderivative was successfully verified.

[In]

Int[ArcCot[x]/(1 + x^2),x]

[Out]

-ArcCot[x]^2/2

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

Mathematica [A]  time = 0.0031552, size = 8, normalized size = 1. \[ -\frac{1}{2} \cot ^{-1}(x)^2 \]

Antiderivative was successfully verified.

[In]

Integrate[ArcCot[x]/(1 + x^2),x]

[Out]

-ArcCot[x]^2/2

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Maple [A]  time = 0.02, size = 7, normalized size = 0.9 \begin{align*} -{\frac{ \left ({\rm arccot} \left (x\right ) \right ) ^{2}}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*arccot(x)^2

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Maxima [A]  time = 0.977856, size = 8, normalized size = 1. \begin{align*} -\frac{1}{2} \, \operatorname{arccot}\left (x\right )^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*arccot(x)^2

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Fricas [A]  time = 1.92167, size = 24, normalized size = 3. \begin{align*} -\frac{1}{2} \, \operatorname{arccot}\left (x\right )^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*arccot(x)^2

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Sympy [A]  time = 0.695324, size = 7, normalized size = 0.88 \begin{align*} - \frac{\operatorname{acot}^{2}{\left (x \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-acot(x)**2/2

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Giac [A]  time = 1.10768, size = 11, normalized size = 1.38 \begin{align*} -\frac{1}{2} \, \arctan \left (\frac{1}{x}\right )^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*arctan(1/x)^2