\(\int \cot ^{-1}(a x) \, dx\) [6]

Optimal result

Integrand size = 4, antiderivative size = 24 \[ \int \cot ^{-1}(a x) \, dx=x \cot ^{-1}(a x)+\frac {\log \left (1+a^2 x^2\right )}{2 a} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4931, 266} \[ \int \cot ^{-1}(a x) \, dx=\frac {\log \left (a^2 x^2+1\right )}{2 a}+x \cot ^{-1}(a x) \]

[In]

Int[ArcCot[a*x],x]

[Out]

x*ArcCot[a*x] + Log[1 + a^2*x^2]/(2*a)

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

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \cot ^{-1}(a x) \, dx=x \cot ^{-1}(a x)+\frac {\log \left (1+a^2 x^2\right )}{2 a} \]

[In]

Integrate[ArcCot[a*x],x]

[Out]

x*ArcCot[a*x] + Log[1 + a^2*x^2]/(2*a)

Maple [A] (verified)

Time = 0.10 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.96

method	result	size
parts	\(x \,\operatorname {arccot}\left (a x \right )+\frac {\ln \left (a^{2} x^{2}+1\right )}{2 a}\)	\(23\)
derivativedivides	\(\frac {\operatorname {arccot}\left (a x \right ) a x +\frac {\ln \left (a^{2} x^{2}+1\right )}{2}}{a}\)	\(25\)
default	\(\frac {\operatorname {arccot}\left (a x \right ) a x +\frac {\ln \left (a^{2} x^{2}+1\right )}{2}}{a}\)	\(25\)
parallelrisch	\(\frac {2 \,\operatorname {arccot}\left (a x \right ) a x +\ln \left (a^{2} x^{2}+1\right )}{2 a}\)	\(25\)
risch	\(\frac {i x \ln \left (i a x +1\right )}{2}-\frac {i x \ln \left (-i a x +1\right )}{2}+\frac {\pi x}{2}+\frac {\ln \left (-a^{2} x^{2}-1\right )}{2 a}\)	\(46\)

[In]

int(arccot(a*x),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \cot ^{-1}(a x) \, dx=\frac {2 \, a x \operatorname {arccot}\left (a x\right ) + \log \left (a^{2} x^{2} + 1\right )}{2 \, a} \]

[In]

integrate(arccot(a*x),x, algorithm="fricas")

[Out]

1/2*(2*a*x*arccot(a*x) + log(a^2*x^2 + 1))/a

Sympy [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \cot ^{-1}(a x) \, dx=\begin {cases} x \operatorname {acot}{\left (a x \right )} + \frac {\log {\left (a^{2} x^{2} + 1 \right )}}{2 a} & \text {for}\: a \neq 0 \\\frac {\pi x}{2} & \text {otherwise} \end {cases} \]

[In]

integrate(acot(a*x),x)

[Out]

Piecewise((x*acot(a*x) + log(a**2*x**2 + 1)/(2*a), Ne(a, 0)), (pi*x/2, True))

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \cot ^{-1}(a x) \, dx=\frac {2 \, a x \operatorname {arccot}\left (a x\right ) + \log \left (a^{2} x^{2} + 1\right )}{2 \, a} \]

[In]

integrate(arccot(a*x),x, algorithm="maxima")

[Out]

1/2*(2*a*x*arccot(a*x) + log(a^2*x^2 + 1))/a

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 45 vs. \(2 (22) = 44\).

Time = 0.27 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.88 \[ \int \cot ^{-1}(a x) \, dx=\frac {1}{2} \, a {\left (\frac {2 \, x \arctan \left (\frac {1}{a x}\right )}{a} + \frac {\log \left (\frac {1}{a^{2} x^{2}} + 1\right )}{a^{2}} - \frac {\log \left (\frac {1}{a^{2} x^{2}}\right )}{a^{2}}\right )} \]

[In]

integrate(arccot(a*x),x, algorithm="giac")

[Out]

1/2*a*(2*x*arctan(1/(a*x))/a + log(1/(a^2*x^2) + 1)/a^2 - log(1/(a^2*x^2))/a^2)

Mupad [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \cot ^{-1}(a x) \, dx=x\,\mathrm {acot}\left (a\,x\right )+\frac {\ln \left (a^2\,x^2+1\right )}{2\,a} \]

[In]

int(acot(a*x),x)

[Out]

x*acot(a*x) + log(a^2*x^2 + 1)/(2*a)