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

Optimal. Leaf size=57 \[ \frac{a x^{m+2} \, _2F_1\left (1,\frac{m+2}{2};\frac{m+4}{2};-a^2 x^2\right )}{m^2+3 m+2}+\frac{x^{m+1} \cot ^{-1}(a x)}{m+1} \]

[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)

________________________________________________________________________________________

Rubi [A]  time = 0.020079, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {4853, 364} \[ \frac{a x^{m+2} \, _2F_1\left (1,\frac{m+2}{2};\frac{m+4}{2};-a^2 x^2\right )}{m^2+3 m+2}+\frac{x^{m+1} \cot ^{-1}(a x)}{m+1} \]

Antiderivative was successfully verified.

[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 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 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rubi steps

\begin{align*} \int x^m \cot ^{-1}(a x) \, dx &=\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} \, _2F_1\left (1,\frac{2+m}{2};\frac{4+m}{2};-a^2 x^2\right )}{2+3 m+m^2}\\ \end{align*}

Mathematica [A]  time = 0.0198512, size = 52, normalized size = 0.91 \[ \frac{x^{m+1} \left (a x \, _2F_1\left (1,\frac{m}{2}+1;\frac{m}{2}+2;-a^2 x^2\right )+(m+2) \cot ^{-1}(a x)\right )}{(m+1) (m+2)} \]

Antiderivative was successfully verified.

[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]  time = 1.026, size = 0, normalized size = 0. \begin{align*} \int{x}^{m}{\rm arccot} \left (ax\right )\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (x^{m} \operatorname{arccot}\left (a x\right ), x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(x^m*arccot(a*x), x)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{m} \operatorname{acot}{\left (a x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**m*acot(a*x), x)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{m} \operatorname{arccot}\left (a x\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^m*arccot(a*x), x)