3.6 \(\int \cot ^{-1}(a x) \, dx\)

Optimal. Leaf size=24 \[ \frac{\log \left (a^2 x^2+1\right )}{2 a}+x \cot ^{-1}(a x) \]

[Out]

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

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Rubi [A]  time = 0.0059009, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 4, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {4847, 260} \[ \frac{\log \left (a^2 x^2+1\right )}{2 a}+x \cot ^{-1}(a x) \]

Antiderivative was successfully verified.

[In]

Int[ArcCot[a*x],x]

[Out]

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

Rule 4847

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

Rule 260

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]

Rubi steps

\begin{align*} \int \cot ^{-1}(a x) \, dx &=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]  time = 0.0027033, size = 24, normalized size = 1. \[ \frac{\log \left (a^2 x^2+1\right )}{2 a}+x \cot ^{-1}(a x) \]

Antiderivative was successfully verified.

[In]

Integrate[ArcCot[a*x],x]

[Out]

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

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Maple [A]  time = 0.036, size = 23, normalized size = 1. \begin{align*} x{\rm arccot} \left (ax\right )+{\frac{\ln \left ({a}^{2}{x}^{2}+1 \right ) }{2\,a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccot(a*x),x)

[Out]

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

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Maxima [A]  time = 0.972293, size = 32, normalized size = 1.33 \begin{align*} \frac{2 \, a x \operatorname{arccot}\left (a x\right ) + \log \left (a^{2} x^{2} + 1\right )}{2 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.02604, size = 62, normalized size = 2.58 \begin{align*} \frac{2 \, a x \operatorname{arccot}\left (a x\right ) + \log \left (a^{2} x^{2} + 1\right )}{2 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.257776, size = 24, normalized size = 1. \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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Giac [A]  time = 1.12187, size = 35, normalized size = 1.46 \begin{align*} x \arctan \left (\frac{1}{a x}\right ) + \frac{\log \left (a^{2} x^{2} + 1\right )}{2 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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