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

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0645185, 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 = {} \[ \int \frac{x^m \tan ^{-1}(a x)^3}{c+a^2 c x^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.864707, size = 0, normalized size = 0. \[ \int \frac{x^m \tan ^{-1}(a x)^3}{c+a^2 c x^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.443, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{m} \left ( \arctan \left ( ax \right ) \right ) ^{3}}{{a}^{2}c{x}^{2}+c}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{x^{m} \operatorname{atan}^{3}{\left (a x \right )}}{a^{2} x^{2} + 1}\, dx}{c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**m*atan(a*x)**3/(a**2*x**2 + 1), x)/c

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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