\(\int \frac {x^m \arctan (a x)}{(c+a^2 c x^2)^2} \, dx\) [252]

Optimal result

Integrand size = 20, antiderivative size = 20 \[ \int \frac {x^m \arctan (a x)}{\left (c+a^2 c x^2\right )^2} \, dx=\text {Int}\left (\frac {x^m \arctan (a x)}{\left (c+a^2 c x^2\right )^2},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.04 (sec) , antiderivative size = 20, 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 = {} \[ \int \frac {x^m \arctan (a x)}{\left (c+a^2 c x^2\right )^2} \, dx=\int \frac {x^m \arctan (a x)}{\left (c+a^2 c x^2\right )^2} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 0.60 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {x^m \arctan (a x)}{\left (c+a^2 c x^2\right )^2} \, dx=\int \frac {x^m \arctan (a x)}{\left (c+a^2 c x^2\right )^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.94 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.85 \[ \int \frac {x^m \arctan (a x)}{\left (c+a^2 c x^2\right )^2} \, dx=\int { \frac {x^{m} \arctan \left (a x\right )}{{\left (a^{2} c x^{2} + c\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 4.44 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.45 \[ \int \frac {x^m \arctan (a x)}{\left (c+a^2 c x^2\right )^2} \, dx=\frac {\int \frac {x^{m} \operatorname {atan}{\left (a x \right )}}{a^{4} x^{4} + 2 a^{2} x^{2} + 1}\, dx}{c^{2}} \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.41 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {x^m \arctan (a x)}{\left (c+a^2 c x^2\right )^2} \, dx=\int { \frac {x^{m} \arctan \left (a x\right )}{{\left (a^{2} c x^{2} + c\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 73.96 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.15 \[ \int \frac {x^m \arctan (a x)}{\left (c+a^2 c x^2\right )^2} \, dx=\int { \frac {x^{m} \arctan \left (a x\right )}{{\left (a^{2} c x^{2} + c\right )}^{2}} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [N/A]

Not integrable

Time = 0.42 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {x^m \arctan (a x)}{\left (c+a^2 c x^2\right )^2} \, dx=\int \frac {x^m\,\mathrm {atan}\left (a\,x\right )}{{\left (c\,a^2\,x^2+c\right )}^2} \,d x \]

[In]

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

[Out]

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