\(\int x^m \cot ^{-1}(a x)^2 \, dx\) [35]

Optimal result

Integrand size = 10, antiderivative size = 10 \[ \int x^m \cot ^{-1}(a x)^2 \, dx=\text {Int}\left (x^m \cot ^{-1}(a x)^2,x\right ) \]

[Out]

Unintegrable(x^m*arccot(a*x)^2,x)

Rubi [N/A]

Not integrable

Time = 0.01 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int x^m \cot ^{-1}(a x)^2 \, dx=\int x^m \cot ^{-1}(a x)^2 \, dx \]

[In]

Int[x^m*ArcCot[a*x]^2,x]

[Out]

Defer[Int][x^m*ArcCot[a*x]^2, x]

Rubi steps \begin{align*} \text {integral}& = \int x^m \cot ^{-1}(a x)^2 \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.64 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int x^m \cot ^{-1}(a x)^2 \, dx=\int x^m \cot ^{-1}(a x)^2 \, dx \]

[In]

Integrate[x^m*ArcCot[a*x]^2,x]

[Out]

Integrate[x^m*ArcCot[a*x]^2, x]

Maple [N/A] (verified)

Not integrable

Time = 0.53 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00

\[\int x^{m} \operatorname {arccot}\left (a x \right )^{2}d x\]

[In]

int(x^m*arccot(a*x)^2,x)

[Out]

int(x^m*arccot(a*x)^2,x)

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int x^m \cot ^{-1}(a x)^2 \, dx=\int { x^{m} \operatorname {arccot}\left (a x\right )^{2} \,d x } \]

[In]

integrate(x^m*arccot(a*x)^2,x, algorithm="fricas")

[Out]

integral(x^m*arccot(a*x)^2, x)

Sympy [N/A]

Not integrable

Time = 0.98 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int x^m \cot ^{-1}(a x)^2 \, dx=\int x^{m} \operatorname {acot}^{2}{\left (a x \right )}\, dx \]

[In]

integrate(x**m*acot(a*x)**2,x)

[Out]

Integral(x**m*acot(a*x)**2, x)

Maxima [N/A]

Not integrable

Time = 1.57 (sec) , antiderivative size = 183, normalized size of antiderivative = 18.30 \[ \int x^m \cot ^{-1}(a x)^2 \, dx=\int { x^{m} \operatorname {arccot}\left (a x\right )^{2} \,d x } \]

[In]

integrate(x^m*arccot(a*x)^2,x, algorithm="maxima")

[Out]

1/16*(4*x*x^m*arctan2(1, a*x)^2 - x*x^m*log(a^2*x^2 + 1)^2 + 16*(m + 1)*integrate(1/16*(4*a^2*x^2*x^m*log(a^2*
x^2 + 1) + ((a^2*m + a^2)*x^2 + m + 1)*x^m*log(a^2*x^2 + 1)^2 + 4*(3*(a^2*m*arctan2(1, a*x)^2 + a^2*arctan2(1,
a*x)^2)*x^2 + 2*a*x*arctan2(1, a*x) + 3*m*arctan2(1, a*x)^2 + 3*arctan2(1, a*x)^2)*x^m)/((a^2*m + a^2)*x^2 +
m + 1), x))/(m + 1)

Giac [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int x^m \cot ^{-1}(a x)^2 \, dx=\int { x^{m} \operatorname {arccot}\left (a x\right )^{2} \,d x } \]

[In]

integrate(x^m*arccot(a*x)^2,x, algorithm="giac")

[Out]

integrate(x^m*arccot(a*x)^2, x)

Mupad [N/A]

Not integrable

Time = 0.69 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int x^m \cot ^{-1}(a x)^2 \, dx=\int x^m\,{\mathrm {acot}\left (a\,x\right )}^2 \,d x \]

[In]

int(x^m*acot(a*x)^2,x)

[Out]

int(x^m*acot(a*x)^2, x)