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

Optimal. Leaf size=31 \[ -\frac{\tan ^{-1}(a x)}{2 a^2}+\frac{1}{2} x^2 \cot ^{-1}(a x)+\frac{x}{2 a} \]

[Out]

x/(2*a) + (x^2*ArcCot[a*x])/2 - ArcTan[a*x]/(2*a^2)

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Rubi [A]  time = 0.0115336, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {4853, 321, 203} \[ -\frac{\tan ^{-1}(a x)}{2 a^2}+\frac{1}{2} x^2 \cot ^{-1}(a x)+\frac{x}{2 a} \]

Antiderivative was successfully verified.

[In]

Int[x*ArcCot[a*x],x]

[Out]

x/(2*a) + (x^2*ArcCot[a*x])/2 - ArcTan[a*x]/(2*a^2)

Rule 4853

Int[((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcCo
t[c*x])^p)/(d*(m + 1)), x] + Dist[(b*c*p)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcCot[c*x])^(p - 1))/(1 + c^
2*x^2), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[p, 0] && (EqQ[p, 1] || IntegerQ[m]) && NeQ[m, -1]

Rule 321

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

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rubi steps

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

Mathematica [A]  time = 0.0017257, size = 31, normalized size = 1. \[ -\frac{\tan ^{-1}(a x)}{2 a^2}+\frac{1}{2} x^2 \cot ^{-1}(a x)+\frac{x}{2 a} \]

Antiderivative was successfully verified.

[In]

Integrate[x*ArcCot[a*x],x]

[Out]

x/(2*a) + (x^2*ArcCot[a*x])/2 - ArcTan[a*x]/(2*a^2)

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Maple [A]  time = 0.039, size = 26, normalized size = 0.8 \begin{align*}{\frac{x}{2\,a}}+{\frac{{x}^{2}{\rm arccot} \left (ax\right )}{2}}-{\frac{\arctan \left ( ax \right ) }{2\,{a}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*arccot(a*x),x)

[Out]

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

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Maxima [A]  time = 1.48985, size = 38, normalized size = 1.23 \begin{align*} \frac{1}{2} \, x^{2} \operatorname{arccot}\left (a x\right ) + \frac{1}{2} \, a{\left (\frac{x}{a^{2}} - \frac{\arctan \left (a x\right )}{a^{3}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^2*arccot(a*x) + 1/2*a*(x/a^2 - arctan(a*x)/a^3)

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Fricas [A]  time = 1.90484, size = 58, normalized size = 1.87 \begin{align*} \frac{a x +{\left (a^{2} x^{2} + 1\right )} \operatorname{arccot}\left (a x\right )}{2 \, a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.413212, size = 31, normalized size = 1. \begin{align*} \begin{cases} \frac{x^{2} \operatorname{acot}{\left (a x \right )}}{2} + \frac{x}{2 a} + \frac{\operatorname{acot}{\left (a x \right )}}{2 a^{2}} & \text{for}\: a \neq 0 \\\frac{\pi x^{2}}{4} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*acot(a*x),x)

[Out]

Piecewise((x**2*acot(a*x)/2 + x/(2*a) + acot(a*x)/(2*a**2), Ne(a, 0)), (pi*x**2/4, True))

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Giac [A]  time = 1.11481, size = 43, normalized size = 1.39 \begin{align*} \frac{1}{2} \, x^{2} \arctan \left (\frac{1}{a x}\right ) + \frac{1}{2} \, a{\left (\frac{x}{a^{2}} - \frac{\arctan \left (a x\right )}{a^{3}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^2*arctan(1/(a*x)) + 1/2*a*(x/a^2 - arctan(a*x)/a^3)