3.5.61 \(\int \frac {x^m \text {ArcTan}(a x)^3}{(c+a^2 c x^2)^2} \, dx\) [461]

Optimal. Leaf size=25 \[ \text {Int}\left (\frac {x^m \text {ArcTan}(a x)^3}{\left (c+a^2 c x^2\right )^2},x\right ) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.05, 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 = {} \begin {gather*} \int \frac {x^m \text {ArcTan}(a x)^3}{\left (c+a^2 c x^2\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]
time = 0.52, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^m \text {ArcTan}(a x)^3}{\left (c+a^2 c x^2\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]
time = 1.71, size = 0, normalized size = 0.00 \[\int \frac {x^{m} \arctan \left (a x \right )^{3}}{\left (a^{2} c \,x^{2}+c \right )^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {x^{m} \operatorname {atan}^{3}{\left (a x \right )}}{a^{4} x^{4} + 2 a^{2} x^{2} + 1}\, dx}{c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**m*atan(a*x)**3/(a**4*x**4 + 2*a**2*x**2 + 1), x)/c**2

________________________________________________________________________________________

Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sage0*x

________________________________________________________________________________________

Mupad [A]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {x^m\,{\mathrm {atan}\left (a\,x\right )}^3}{{\left (c\,a^2\,x^2+c\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________