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

Optimal. Leaf size=13 \[ \text {Int}\left (x^m \cot ^{-1}(a x)^2,x\right ) \]

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int x^m \cot ^{-1}(a x)^2 \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

Defer[Int][x^m*ArcCot[a*x]^2, x]

Rubi steps

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

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Mathematica [A]  time = 0.93, size = 0, normalized size = 0.00 \[ \int x^m \cot ^{-1}(a x)^2 \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

Integrate[x^m*ArcCot[a*x]^2, x]

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fricas [A]  time = 0.58, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{m} \operatorname {arccot}\left (a x\right )^{2}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(x^m*arccot(a*x)^2, x)

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{m} \operatorname {arccot}\left (a x\right )^{2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^m*arccot(a*x)^2, x)

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maple [A]  time = 1.35, size = 0, normalized size = 0.00 \[ \int x^{m} \mathrm {arccot}\left (a x \right )^{2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {7 \, x x^{m} \arctan \left (1, a x\right )^{2} - \frac {3}{4} \, x x^{m} \log \left (a^{2} x^{2} + 1\right )^{2} + {\left (m + 1\right )} \int \frac {12 \, a^{2} x^{2} x^{m} \log \left (a^{2} x^{2} + 1\right ) + 3 \, {\left ({\left (a^{2} m + a^{2}\right )} x^{2} + m + 1\right )} x^{m} \log \left (a^{2} x^{2} + 1\right )^{2} + 4 \, {\left (9 \, {\left (a^{2} m \arctan \left (1, a x\right )^{2} + a^{2} \arctan \left (1, a x\right )^{2}\right )} x^{2} + 14 \, a x \arctan \left (1, a x\right ) + 9 \, m \arctan \left (1, a x\right )^{2} + 9 \, \arctan \left (1, a x\right )^{2}\right )} x^{m}}{4 \, {\left ({\left (a^{2} m + a^{2}\right )} x^{2} + m + 1\right )}}\,{d x}}{16 \, {\left (m + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [A]  time = 0.00, size = -1, normalized size = -0.08 \[ \int x^m\,{\mathrm {acot}\left (a\,x\right )}^2 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x^m*acot(a*x)^2, x)

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{m} \operatorname {acot}^{2}{\left (a x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**m*acot(a*x)**2, x)

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