3.1.96 \(\int \cot ^{-1}(\frac {1}{x}) \, dx\) [96]

Optimal. Leaf size=17 \[ x \cot ^{-1}\left (\frac {1}{x}\right )-\frac {1}{2} \log \left (1+x^2\right ) \]

[Out]

x*arccot(1/x)-1/2*ln(x^2+1)

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Rubi [A]
time = 0.00, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {4931, 269, 266} \begin {gather*} x \cot ^{-1}\left (\frac {1}{x}\right )-\frac {1}{2} \log \left (x^2+1\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[ArcCot[x^(-1)],x]

[Out]

x*ArcCot[x^(-1)] - Log[1 + x^2]/2

Rule 266

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 269

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(m + n*p)*(b + a/x^n)^p, x] /; FreeQ[{a, b, m
, n}, x] && IntegerQ[p] && NegQ[n]

Rule 4931

Int[((a_.) + ArcCot[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*ArcCot[c*x^n])^p, x] + Dist[b*c
*n*p, Int[x^n*((a + b*ArcCot[c*x^n])^(p - 1)/(1 + c^2*x^(2*n))), x], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[p, 0
] && (EqQ[n, 1] || EqQ[p, 1])

Rubi steps

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

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Mathematica [A]
time = 0.00, size = 17, normalized size = 1.00 \begin {gather*} x \cot ^{-1}\left (\frac {1}{x}\right )-\frac {1}{2} \log \left (1+x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[ArcCot[x^(-1)],x]

[Out]

x*ArcCot[x^(-1)] - Log[1 + x^2]/2

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Maple [A]
time = 0.06, size = 20, normalized size = 1.18

method	result	size
derivativedivides	\(x \,\mathrm {arccot}\left (\frac {1}{x}\right )-\frac {\ln \left (\frac {1}{x^{2}}+1\right )}{2}+\ln \left (\frac {1}{x}\right )\)	\(20\)
default	\(x \,\mathrm {arccot}\left (\frac {1}{x}\right )-\frac {\ln \left (\frac {1}{x^{2}}+1\right )}{2}+\ln \left (\frac {1}{x}\right )\)	\(20\)
risch	\(\frac {i x \ln \left (i+x \right )}{2}-\frac {i \ln \left (x -i\right ) x}{2}-\frac {\pi \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (i \left (x -\RootOf \left (\textit {\_Z}^{2}+1, \mathit {index} =1\right )\right )\right ) \mathrm {csgn}\left (\frac {i \left (x -\RootOf \left (\textit {\_Z}^{2}+1, \mathit {index} =1\right )\right )}{x}\right ) x}{4}+\frac {\pi \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (i \left (x +\RootOf \left (\textit {\_Z}^{2}+1, \mathit {index} =1\right )\right )\right ) \mathrm {csgn}\left (\frac {i \left (x +\RootOf \left (\textit {\_Z}^{2}+1, \mathit {index} =1\right )\right )}{x}\right ) x}{4}+\frac {\pi \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i \left (x -\RootOf \left (\textit {\_Z}^{2}+1, \mathit {index} =1\right )\right )}{x}\right )^{2} x}{4}-\frac {\pi \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i \left (x +\RootOf \left (\textit {\_Z}^{2}+1, \mathit {index} =1\right )\right )}{x}\right )^{2} x}{4}+\frac {\pi \,\mathrm {csgn}\left (i \left (x -\RootOf \left (\textit {\_Z}^{2}+1, \mathit {index} =1\right )\right )\right ) \mathrm {csgn}\left (\frac {i \left (x -\RootOf \left (\textit {\_Z}^{2}+1, \mathit {index} =1\right )\right )}{x}\right )^{2} x}{4}-\frac {\pi \,\mathrm {csgn}\left (i \left (x +\RootOf \left (\textit {\_Z}^{2}+1, \mathit {index} =1\right )\right )\right ) \mathrm {csgn}\left (\frac {i \left (x +\RootOf \left (\textit {\_Z}^{2}+1, \mathit {index} =1\right )\right )}{x}\right )^{2} x}{4}-\frac {\pi \mathrm {csgn}\left (\frac {i \left (x -\RootOf \left (\textit {\_Z}^{2}+1, \mathit {index} =1\right )\right )}{x}\right )^{3} x}{4}+\frac {\pi \mathrm {csgn}\left (\frac {i \left (x +\RootOf \left (\textit {\_Z}^{2}+1, \mathit {index} =1\right )\right )}{x}\right )^{3} x}{4}+\frac {\pi x}{2}-\frac {\ln \left (x^{2}+1\right )}{2}\)	\(320\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccot(1/x),x,method=_RETURNVERBOSE)

[Out]

x*arccot(1/x)-1/2*ln(1/x^2+1)+ln(1/x)

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Maxima [A]
time = 0.25, size = 15, normalized size = 0.88 \begin {gather*} x \operatorname {arccot}\left (\frac {1}{x}\right ) - \frac {1}{2} \, \log \left (x^{2} + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*arccot(1/x) - 1/2*log(x^2 + 1)

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Fricas [A]
time = 3.34, size = 15, normalized size = 0.88 \begin {gather*} x \operatorname {arccot}\left (\frac {1}{x}\right ) - \frac {1}{2} \, \log \left (x^{2} + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*arccot(1/x) - 1/2*log(x^2 + 1)

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Sympy [A]
time = 0.07, size = 14, normalized size = 0.82 \begin {gather*} x \operatorname {acot}{\left (\frac {1}{x} \right )} - \frac {\log {\left (x^{2} + 1 \right )}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acot(1/x),x)

[Out]

x*acot(1/x) - log(x**2 + 1)/2

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Giac [A]
time = 0.41, size = 13, normalized size = 0.76 \begin {gather*} x \arctan \left (x\right ) - \frac {1}{2} \, \log \left (x^{2} + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*arctan(x) - 1/2*log(x^2 + 1)

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Mupad [B]
time = 0.06, size = 15, normalized size = 0.88 \begin {gather*} x\,\mathrm {acot}\left (\frac {1}{x}\right )-\frac {\ln \left (x^2+1\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(acot(1/x),x)

[Out]

x*acot(1/x) - log(x^2 + 1)/2

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