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

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Rubi [A]  time = 0.02, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, 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 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])

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]

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*}

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Mathematica [A]  time = 0.02, 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))

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fricas [F]  time = 0.62, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{m} \operatorname {arccot}\left (a x\right ), x\right ) \]

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)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{m} \operatorname {arccot}\left (a x\right )\,{d x} \]

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)

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maple [F]  time = 1.66, size = 0, normalized size = 0.00 \[ \int x^{m} \mathrm {arccot}\left (a x \right )\, dx \]

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)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {x x^{m} \arctan \left (1, a x\right ) + {\left (a m + a\right )} \int \frac {x x^{m}}{{\left (a^{2} m + a^{2}\right )} x^{2} + m + 1}\,{d x}}{m + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int x^m\,\mathrm {acot}\left (a\,x\right ) \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x^m*acot(a*x), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{m} \operatorname {acot}{\left (a x \right )}\, dx \]

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)

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