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

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

[Out]

-1/2*arccot(x)^2

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Rubi [A]
time = 0.01, antiderivative size = 8, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {5005} \begin {gather*} -\frac {1}{2} \cot ^{-1}(x)^2 \end {gather*}

Antiderivative was successfully verified.

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

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Mathematica [A]
time = 0.00, size = 8, normalized size = 1.00 \begin {gather*} -\frac {1}{2} \cot ^{-1}(x)^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[ArcCot[x]/(1 + x^2),x]

[Out]

-1/2*ArcCot[x]^2

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Maple [A]
time = 0.11, size = 7, normalized size = 0.88

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*arccot(x)^2

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Maxima [A]
time = 0.26, size = 6, normalized size = 0.75 \begin {gather*} -\frac {1}{2} \, \operatorname {arccot}\left (x\right )^{2} \end {gather*}

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 = 1.86, size = 6, normalized size = 0.75 \begin {gather*} -\frac {1}{2} \, \operatorname {arccot}\left (x\right )^{2} \end {gather*}

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 = 0.53, size = 7, normalized size = 0.88 \begin {gather*} - \frac {\operatorname {acot}^{2}{\left (x \right )}}{2} \end {gather*}

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 = 0.42, size = 8, normalized size = 1.00 \begin {gather*} -\frac {1}{2} \, \arctan \left (\frac {1}{x}\right )^{2} \end {gather*}

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

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Mupad [B]
time = 0.62, size = 6, normalized size = 0.75 \begin {gather*} -\frac {{\mathrm {acot}\left (x\right )}^2}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(acot(x)/(x^2 + 1),x)

[Out]

-acot(x)^2/2

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