\(\int \frac {x^m \arctan (a x)^2}{(c+a^2 c x^2)^{3/2}} \, dx\) [362]

Optimal result

Integrand size = 24, antiderivative size = 24 \[ \int \frac {x^m \arctan (a x)^2}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\text {Int}\left (\frac {x^m \arctan (a x)^2}{\left (c+a^2 c x^2\right )^{3/2}},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.08 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {x^m \arctan (a x)^2}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\int \frac {x^m \arctan (a x)^2}{\left (c+a^2 c x^2\right )^{3/2}} \, dx \]

[In]

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

[Out]

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {x^m \arctan (a x)^2}{\left (c+a^2 c x^2\right )^{3/2}} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.80 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {x^m \arctan (a x)^2}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\int \frac {x^m \arctan (a x)^2}{\left (c+a^2 c x^2\right )^{3/2}} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 1.22 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92

\[\int \frac {x^{m} \arctan \left (a x \right )^{2}}{\left (a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}d x\]

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 51, normalized size of antiderivative = 2.12 \[ \int \frac {x^m \arctan (a x)^2}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\int { \frac {x^{m} \arctan \left (a x\right )^{2}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 20.65 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {x^m \arctan (a x)^2}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\int \frac {x^{m} \operatorname {atan}^{2}{\left (a x \right )}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {3}{2}}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.43 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {x^m \arctan (a x)^2}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\int { \frac {x^{m} \arctan \left (a x\right )^{2}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 3.96 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.12 \[ \int \frac {x^m \arctan (a x)^2}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\int { \frac {x^{m} \arctan \left (a x\right )^{2}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [N/A]

Not integrable

Time = 0.61 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {x^m \arctan (a x)^2}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\int \frac {x^m\,{\mathrm {atan}\left (a\,x\right )}^2}{{\left (c\,a^2\,x^2+c\right )}^{3/2}} \,d x \]

[In]

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

[Out]

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