\(\int x^m \cot ^{-1}(a x) \, dx\) [36]

Optimal result

Integrand size = 8, antiderivative size = 57 \[ \int x^m \cot ^{-1}(a x) \, dx=\frac {x^{1+m} \cot ^{-1}(a x)}{1+m}+\frac {a x^{2+m} \operatorname {Hypergeometric2F1}\left (1,\frac {2+m}{2},\frac {4+m}{2},-a^2 x^2\right )}{2+3 m+m^2} \]

[Out]

x^(1+m)*arccot(a*x)/(1+m)+a*x^(2+m)*hypergeom([1, 1+1/2*m],[2+1/2*m],-a^2*x^2)/(m^2+3*m+2)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4947, 371} \[ \int x^m \cot ^{-1}(a x) \, dx=\frac {a x^{m+2} \operatorname {Hypergeometric2F1}\left (1,\frac {m+2}{2},\frac {m+4}{2},-a^2 x^2\right )}{m^2+3 m+2}+\frac {x^{m+1} \cot ^{-1}(a x)}{m+1} \]

[In]

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

[Out]

(x^(1 + m)*ArcCot[a*x])/(1 + m) + (a*x^(2 + m)*Hypergeometric2F1[1, (2 + m)/2, (4 + m)/2, -(a^2*x^2)])/(2 + 3*
m + m^2)

Rule 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

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 {x^{1+m} \cot ^{-1}(a x)}{1+m}+\frac {a \int \frac {x^{1+m}}{1+a^2 x^2} \, dx}{1+m} \\ & = \frac {x^{1+m} \cot ^{-1}(a x)}{1+m}+\frac {a x^{2+m} \operatorname {Hypergeometric2F1}\left (1,\frac {2+m}{2},\frac {4+m}{2},-a^2 x^2\right )}{2+3 m+m^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.91 \[ \int x^m \cot ^{-1}(a x) \, dx=\frac {x^{1+m} \left ((2+m) \cot ^{-1}(a x)+a x \operatorname {Hypergeometric2F1}\left (1,1+\frac {m}{2},2+\frac {m}{2},-a^2 x^2\right )\right )}{(1+m) (2+m)} \]

[In]

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

[Out]

(x^(1 + m)*((2 + m)*ArcCot[a*x] + a*x*Hypergeometric2F1[1, 1 + m/2, 2 + m/2, -(a^2*x^2)]))/((1 + m)*(2 + m))

Maple [F]

\[\int x^{m} \operatorname {arccot}\left (a x \right )d x\]

[In]

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

[Out]

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

Fricas [F]

\[ \int x^m \cot ^{-1}(a x) \, dx=\int { x^{m} \operatorname {arccot}\left (a x\right ) \,d x } \]

[In]

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

[Out]

integral(x^m*arccot(a*x), x)

Sympy [F]

\[ \int x^m \cot ^{-1}(a x) \, dx=\int x^{m} \operatorname {acot}{\left (a x \right )}\, dx \]

[In]

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

[Out]

Integral(x**m*acot(a*x), x)

Maxima [F]

\[ \int x^m \cot ^{-1}(a x) \, dx=\int { x^{m} \operatorname {arccot}\left (a x\right ) \,d x } \]

[In]

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

[Out]

(x*x^m*arctan2(1, a*x) + (a*m + a)*integrate(x*x^m/((a^2*m + a^2)*x^2 + m + 1), x))/(m + 1)

Giac [F]

\[ \int x^m \cot ^{-1}(a x) \, dx=\int { x^{m} \operatorname {arccot}\left (a x\right ) \,d x } \]

[In]

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

[Out]

integrate(x^m*arccot(a*x), x)

Mupad [F(-1)]

Timed out. \[ \int x^m \cot ^{-1}(a x) \, dx=\int x^m\,\mathrm {acot}\left (a\,x\right ) \,d x \]

[In]

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

[Out]

int(x^m*acot(a*x), x)