3.679 \(\int \frac{x^m}{(c+a^2 c x^2)^3 \tan ^{-1}(a x)^3} \, dx\)

Optimal. Leaf size=24 \[ \text{Unintegrable}\left (\frac{x^m}{\left (a^2 c x^2+c\right )^3 \tan ^{-1}(a x)^3},x\right ) \]

[Out]

Unintegrable[x^m/((c + a^2*c*x^2)^3*ArcTan[a*x]^3), x]

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Rubi [A]  time = 0.0630917, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{x^m}{\left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^3} \, dx \]

Verification is Not applicable to the result.

[In]

Int[x^m/((c + a^2*c*x^2)^3*ArcTan[a*x]^3),x]

[Out]

Defer[Int][x^m/((c + a^2*c*x^2)^3*ArcTan[a*x]^3), x]

Rubi steps

\begin{align*} \int \frac{x^m}{\left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^3} \, dx &=\int \frac{x^m}{\left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^3} \, dx\\ \end{align*}

Mathematica [A]  time = 0.510576, size = 0, normalized size = 0. \[ \int \frac{x^m}{\left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^3} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[x^m/((c + a^2*c*x^2)^3*ArcTan[a*x]^3),x]

[Out]

Integrate[x^m/((c + a^2*c*x^2)^3*ArcTan[a*x]^3), x]

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Maple [A]  time = 1.22, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{m}}{ \left ({a}^{2}c{x}^{2}+c \right ) ^{3} \left ( \arctan \left ( ax \right ) \right ) ^{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m/(a^2*c*x^2+c)^3/arctan(a*x)^3,x)

[Out]

int(x^m/(a^2*c*x^2+c)^3/arctan(a*x)^3,x)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{{\left (a^{6} c^{3} x^{5} + 2 \, a^{4} c^{3} x^{3} + a^{2} c^{3} x\right )} \arctan \left (a x\right )^{2} \int \frac{{\left ({\left (a^{4} m^{2} - 7 \, a^{4} m + 12 \, a^{4}\right )} x^{4} + 2 \,{\left (a^{2} m^{2} - 4 \, a^{2} m - 2 \, a^{2}\right )} x^{2} + m^{2} - m\right )} x^{m}}{{\left (a^{8} c^{3} x^{8} + 3 \, a^{6} c^{3} x^{6} + 3 \, a^{4} c^{3} x^{4} + a^{2} c^{3} x^{2}\right )} \arctan \left (a x\right )}\,{d x} - a x x^{m} -{\left ({\left (a^{2} m - 4 \, a^{2}\right )} x^{2} + m\right )} x^{m} \arctan \left (a x\right )}{2 \,{\left (a^{6} c^{3} x^{5} + 2 \, a^{4} c^{3} x^{3} + a^{2} c^{3} x\right )} \arctan \left (a x\right )^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m/(a^2*c*x^2+c)^3/arctan(a*x)^3,x, algorithm="maxima")

[Out]

1/2*(2*(a^6*c^3*x^5 + 2*a^4*c^3*x^3 + a^2*c^3*x)*arctan(a*x)^2*integrate(1/2*((a^4*m^2 - 7*a^4*m + 12*a^4)*x^4
+ 2*(a^2*m^2 - 4*a^2*m - 2*a^2)*x^2 + m^2 - m)*x^m/((a^8*c^3*x^8 + 3*a^6*c^3*x^6 + 3*a^4*c^3*x^4 + a^2*c^3*x^
2)*arctan(a*x)), x) - a*x*x^m - ((a^2*m - 4*a^2)*x^2 + m)*x^m*arctan(a*x))/((a^6*c^3*x^5 + 2*a^4*c^3*x^3 + a^2
*c^3*x)*arctan(a*x)^2)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x^{m}}{{\left (a^{6} c^{3} x^{6} + 3 \, a^{4} c^{3} x^{4} + 3 \, a^{2} c^{3} x^{2} + c^{3}\right )} \arctan \left (a x\right )^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m/(a^2*c*x^2+c)^3/arctan(a*x)^3,x, algorithm="fricas")

[Out]

integral(x^m/((a^6*c^3*x^6 + 3*a^4*c^3*x^4 + 3*a^2*c^3*x^2 + c^3)*arctan(a*x)^3), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**m/(a**2*c*x**2+c)**3/atan(a*x)**3,x)

[Out]

Timed out

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{m}}{{\left (a^{2} c x^{2} + c\right )}^{3} \arctan \left (a x\right )^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m/(a^2*c*x^2+c)^3/arctan(a*x)^3,x, algorithm="giac")

[Out]

integrate(x^m/((a^2*c*x^2 + c)^3*arctan(a*x)^3), x)