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

Optimal. Leaf size=13 \[ \text {Int}\left (x^m \cot ^{-1}(a x)^3,x\right ) \]

[Out]

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

Verification is Not applicable to the result.

[In]

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

[Out]

Defer[Int][x^m*ArcCot[a*x]^3, x]

Rubi steps

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

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Mathematica [A]  time = 0.94, size = 0, normalized size = 0.00 \[ \int x^m \cot ^{-1}(a x)^3 \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

Integrate[x^m*ArcCot[a*x]^3, x]

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fricas [A]  time = 0.55, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{m} \operatorname {arccot}\left (a x\right )^{3}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(x^m*arccot(a*x)^3, x)

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{m} \operatorname {arccot}\left (a x\right )^{3}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^m*arccot(a*x)^3, x)

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maple [A]  time = 1.84, size = 0, normalized size = 0.00 \[ \int x^{m} \mathrm {arccot}\left (a x \right )^{3}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\frac {15}{2} \, x x^{m} \arctan \left (1, a x\right )^{3} - \frac {21}{8} \, x x^{m} \arctan \left (1, a x\right ) \log \left (a^{2} x^{2} + 1\right )^{2} + {\left (m + 1\right )} \int \frac {84 \, a^{2} x^{2} x^{m} \arctan \left (1, a x\right ) \log \left (a^{2} x^{2} + 1\right ) + 21 \, {\left ({\left (a^{2} m \arctan \left (1, a x\right ) + a^{2} \arctan \left (1, a x\right )\right )} x^{2} - a x + m \arctan \left (1, a x\right ) + \arctan \left (1, a x\right )\right )} x^{m} \log \left (a^{2} x^{2} + 1\right )^{2} + 4 \, {\left (45 \, a x \arctan \left (1, a x\right )^{2} + 49 \, m \arctan \left (1, a x\right )^{3} + 49 \, {\left (a^{2} m \arctan \left (1, a x\right )^{3} + a^{2} \arctan \left (1, a x\right )^{3}\right )} x^{2} + 49 \, \arctan \left (1, a x\right )^{3}\right )} x^{m}}{8 \, {\left ({\left (a^{2} m + a^{2}\right )} x^{2} + m + 1\right )}}\,{d x}}{32 \, {\left (m + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [A]  time = 0.00, size = -1, normalized size = -0.08 \[ \int x^m\,{\mathrm {acot}\left (a\,x\right )}^3 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x^m*acot(a*x)^3, x)

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{m} \operatorname {acot}^{3}{\left (a x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**m*acot(a*x)**3, x)

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