\(\int x \cot ^{-1}(a x) \, dx\) [5]

Optimal result

Integrand size = 6, antiderivative size = 31 \[ \int x \cot ^{-1}(a x) \, dx=\frac {x}{2 a}+\frac {1}{2} x^2 \cot ^{-1}(a x)-\frac {\arctan (a x)}{2 a^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4947, 327, 209} \[ \int x \cot ^{-1}(a x) \, dx=-\frac {\arctan (a x)}{2 a^2}+\frac {1}{2} x^2 \cot ^{-1}(a x)+\frac {x}{2 a} \]

[In]

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

[Out]

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

Rule 209

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

Rule 327

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 4947

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

Rubi steps \begin{align*} \text {integral}& = \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 {\arctan (a x)}{2 a^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00 \[ \int x \cot ^{-1}(a x) \, dx=\frac {x}{2 a}+\frac {1}{2} x^2 \cot ^{-1}(a x)-\frac {\arctan (a x)}{2 a^2} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.15 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.81

method	result	size
parallelrisch	\(\frac {\operatorname {arccot}\left (a x \right ) a^{2} x^{2}+a x +\operatorname {arccot}\left (a x \right )}{2 a^{2}}\)	\(25\)
derivativedivides	\(\frac {\frac {\operatorname {arccot}\left (a x \right ) a^{2} x^{2}}{2}+\frac {a x}{2}-\frac {\arctan \left (a x \right )}{2}}{a^{2}}\)	\(28\)
default	\(\frac {\frac {\operatorname {arccot}\left (a x \right ) a^{2} x^{2}}{2}+\frac {a x}{2}-\frac {\arctan \left (a x \right )}{2}}{a^{2}}\)	\(28\)
parts	\(\frac {x^{2} \operatorname {arccot}\left (a x \right )}{2}+\frac {a \left (\frac {x}{a^{2}}-\frac {\arctan \left (a x \right )}{a^{3}}\right )}{2}\)	\(29\)
risch	\(\frac {i x^{2} \ln \left (i a x +1\right )}{4}-\frac {i x^{2} \ln \left (-i a x +1\right )}{4}+\frac {\pi \,x^{2}}{4}+\frac {x}{2 a}-\frac {\arctan \left (a x \right )}{2 a^{2}}\)	\(51\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.74 \[ \int x \cot ^{-1}(a x) \, dx=\frac {a x + {\left (a^{2} x^{2} + 1\right )} \operatorname {arccot}\left (a x\right )}{2 \, a^{2}} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00 \[ \int x \cot ^{-1}(a x) \, dx=\begin {cases} \frac {x^{2} \operatorname {acot}{\left (a x \right )}}{2} + \frac {x}{2 a} + \frac {\operatorname {acot}{\left (a x \right )}}{2 a^{2}} & \text {for}\: a \neq 0 \\\frac {\pi x^{2}}{4} & \text {otherwise} \end {cases} \]

[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))

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.90 \[ \int x \cot ^{-1}(a x) \, dx=\frac {1}{2} \, x^{2} \operatorname {arccot}\left (a x\right ) + \frac {1}{2} \, a {\left (\frac {x}{a^{2}} - \frac {\arctan \left (a x\right )}{a^{3}}\right )} \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.16 \[ \int x \cot ^{-1}(a x) \, dx=\frac {1}{2} \, {\left (\frac {x^{2} \arctan \left (\frac {1}{a x}\right )}{a} + \frac {x}{a^{2}} + \frac {\arctan \left (\frac {1}{a x}\right )}{a^{3}}\right )} a \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.80 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.26 \[ \int x \cot ^{-1}(a x) \, dx=\left \{\begin {array}{cl} \frac {\pi \,x^2}{4} & \text {\ if\ \ }a=0\\ \frac {x-\frac {\mathrm {atan}\left (a\,x\right )}{a}}{2\,a}+\frac {x^2\,\mathrm {acot}\left (a\,x\right )}{2} & \text {\ if\ \ }a\neq 0 \end {array}\right . \]

[In]

int(x*acot(a*x),x)

[Out]

piecewise(a == 0, (x^2*pi)/4, a ~= 0, (x - atan(a*x)/a)/(2*a) + (x^2*acot(a*x))/2)