\(\int x^m \cot ^{-1}(a x)^3 \, dx\) [34]

Optimal result

Integrand size = 10, antiderivative size = 10 \[ \int x^m \cot ^{-1}(a x)^3 \, dx=\text {Int}\left (x^m \cot ^{-1}(a x)^3,x\right ) \]

[Out]

Unintegrable(x^m*arccot(a*x)^3,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 = {} \[ \int x^m \cot ^{-1}(a x)^3 \, dx=\int x^m \cot ^{-1}(a x)^3 \, dx \]

[In]

Int[x^m*ArcCot[a*x]^3,x]

[Out]

Defer[Int][x^m*ArcCot[a*x]^3, x]

Rubi steps \begin{align*} \text {integral}& = \int x^m \cot ^{-1}(a x)^3 \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.64 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int x^m \cot ^{-1}(a x)^3 \, dx=\int x^m \cot ^{-1}(a x)^3 \, dx \]

[In]

Integrate[x^m*ArcCot[a*x]^3,x]

[Out]

Integrate[x^m*ArcCot[a*x]^3, x]

Maple [N/A] (verified)

Not integrable

Time = 0.77 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00

\[\int x^{m} \operatorname {arccot}\left (a x \right )^{3}d x\]

[In]

int(x^m*arccot(a*x)^3,x)

[Out]

int(x^m*arccot(a*x)^3,x)

Fricas [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int x^m \cot ^{-1}(a x)^3 \, dx=\int { x^{m} \operatorname {arccot}\left (a x\right )^{3} \,d x } \]

[In]

integrate(x^m*arccot(a*x)^3,x, algorithm="fricas")

[Out]

integral(x^m*arccot(a*x)^3, x)

Sympy [N/A]

Not integrable

Time = 1.95 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int x^m \cot ^{-1}(a x)^3 \, dx=\int x^{m} \operatorname {acot}^{3}{\left (a x \right )}\, dx \]

[In]

integrate(x**m*acot(a*x)**3,x)

[Out]

Integral(x**m*acot(a*x)**3, x)

Maxima [N/A]

Not integrable

Time = 2.29 (sec) , antiderivative size = 221, normalized size of antiderivative = 22.10 \[ \int x^m \cot ^{-1}(a x)^3 \, dx=\int { x^{m} \operatorname {arccot}\left (a x\right )^{3} \,d x } \]

[In]

integrate(x^m*arccot(a*x)^3,x, algorithm="maxima")

[Out]

1/32*(4*x*x^m*arctan2(1, a*x)^3 - 3*x*x^m*arctan2(1, a*x)*log(a^2*x^2 + 1)^2 + 32*(m + 1)*integrate(1/32*(12*a
^2*x^2*x^m*arctan2(1, a*x)*log(a^2*x^2 + 1) + 3*((a^2*m*arctan2(1, a*x) + a^2*arctan2(1, a*x))*x^2 - a*x + m*a
rctan2(1, a*x) + arctan2(1, a*x))*x^m*log(a^2*x^2 + 1)^2 + 4*(3*a*x*arctan2(1, a*x)^2 + 7*m*arctan2(1, a*x)^3
+ 7*(a^2*m*arctan2(1, a*x)^3 + a^2*arctan2(1, a*x)^3)*x^2 + 7*arctan2(1, a*x)^3)*x^m)/((a^2*m + a^2)*x^2 + m +
1), x))/(m + 1)

Giac [N/A]

Not integrable

Time = 0.29 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int x^m \cot ^{-1}(a x)^3 \, dx=\int { x^{m} \operatorname {arccot}\left (a x\right )^{3} \,d x } \]

[In]

integrate(x^m*arccot(a*x)^3,x, algorithm="giac")

[Out]

integrate(x^m*arccot(a*x)^3, x)

Mupad [N/A]

Not integrable

Time = 0.69 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int x^m \cot ^{-1}(a x)^3 \, dx=\int x^m\,{\mathrm {acot}\left (a\,x\right )}^3 \,d x \]

[In]

int(x^m*acot(a*x)^3,x)

[Out]

int(x^m*acot(a*x)^3, x)