\(\int \frac {\cot ^{-1}(x)}{1+x^2} \, dx\) [41]

Optimal result

Integrand size = 10, antiderivative size = 8 \[ \int \frac {\cot ^{-1}(x)}{1+x^2} \, dx=-\frac {1}{2} \cot ^{-1}(x)^2 \]

[Out]

-1/2*arccot(x)^2

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 8, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {5005} \[ \int \frac {\cot ^{-1}(x)}{1+x^2} \, dx=-\frac {1}{2} \cot ^{-1}(x)^2 \]

[In]

Int[ArcCot[x]/(1 + x^2),x]

[Out]

-1/2*ArcCot[x]^2

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 {1}{2} \cot ^{-1}(x)^2 \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 8, normalized size of antiderivative = 1.00 \[ \int \frac {\cot ^{-1}(x)}{1+x^2} \, dx=-\frac {1}{2} \cot ^{-1}(x)^2 \]

[In]

Integrate[ArcCot[x]/(1 + x^2),x]

[Out]

-1/2*ArcCot[x]^2

Maple [A] (verified)

Time = 0.52 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.88

method	result	size
derivativedivides	\(-\frac {\operatorname {arccot}\left (x \right )^{2}}{2}\)	\(7\)
default	\(-\frac {\operatorname {arccot}\left (x \right )^{2}}{2}\)	\(7\)
parts	\(\operatorname {arccot}\left (x \right ) \arctan \left (x \right )+\frac {\arctan \left (x \right )^{2}}{2}\)	\(13\)
risch	\(\frac {\ln \left (i x +1\right )^{2}}{8}-\frac {\ln \left (-i x +1\right ) \ln \left (i x +1\right )}{4}+\frac {\ln \left (-i x +1\right )^{2}}{8}+\frac {\pi \arctan \left (x \right )}{2}\)	\(45\)

[In]

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

[Out]

-1/2*arccot(x)^2

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.75 \[ \int \frac {\cot ^{-1}(x)}{1+x^2} \, dx=-\frac {1}{2} \, \operatorname {arccot}\left (x\right )^{2} \]

[In]

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

[Out]

-1/2*arccot(x)^2

Sympy [A] (verification not implemented)

Time = 0.32 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.88 \[ \int \frac {\cot ^{-1}(x)}{1+x^2} \, dx=- \frac {\operatorname {acot}^{2}{\left (x \right )}}{2} \]

[In]

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

[Out]

-acot(x)**2/2

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.75 \[ \int \frac {\cot ^{-1}(x)}{1+x^2} \, dx=-\frac {1}{2} \, \operatorname {arccot}\left (x\right )^{2} \]

[In]

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

[Out]

-1/2*arccot(x)^2

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 8, normalized size of antiderivative = 1.00 \[ \int \frac {\cot ^{-1}(x)}{1+x^2} \, dx=-\frac {1}{2} \, \arctan \left (\frac {1}{x}\right )^{2} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.68 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.75 \[ \int \frac {\cot ^{-1}(x)}{1+x^2} \, dx=-\frac {{\mathrm {acot}\left (x\right )}^2}{2} \]

[In]

int(acot(x)/(x^2 + 1),x)

[Out]

-acot(x)^2/2