\(\int \frac {x^m}{(c+a^2 c x^2)^2 \arctan (a x)^2} \, dx\) [602]

Optimal result

Integrand size = 22, antiderivative size = 22 \[ \int \frac {x^m}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^2} \, dx=\text {Int}\left (\frac {x^m}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^2},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.05 (sec) , antiderivative size = 22, 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 = {} \[ \int \frac {x^m}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^2} \, dx=\int \frac {x^m}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^2} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 0.47 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {x^m}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^2} \, dx=\int \frac {x^m}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 7.84 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.77 \[ \int \frac {x^m}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^2} \, dx=\int { \frac {x^{m}}{{\left (a^{2} c x^{2} + c\right )}^{2} \arctan \left (a x\right )^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 10.11 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.91 \[ \int \frac {x^m}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^2} \, dx=\frac {\int \frac {x^{m}}{a^{4} x^{4} \operatorname {atan}^{2}{\left (a x \right )} + 2 a^{2} x^{2} \operatorname {atan}^{2}{\left (a x \right )} + \operatorname {atan}^{2}{\left (a x \right )}}\, dx}{c^{2}} \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.40 (sec) , antiderivative size = 111, normalized size of antiderivative = 5.05 \[ \int \frac {x^m}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^2} \, dx=\int { \frac {x^{m}}{{\left (a^{2} c x^{2} + c\right )}^{2} \arctan \left (a x\right )^{2}} \,d x } \]

[In]

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

[Out]

((a^3*c^2*x^2 + a*c^2)*arctan(a*x)*integrate(((a^2*m - 2*a^2)*x^2 + m)*x^m/((a^5*c^2*x^5 + 2*a^3*c^2*x^3 + a*c
^2*x)*arctan(a*x)), x) - x^m)/((a^3*c^2*x^2 + a*c^2)*arctan(a*x))

Giac [N/A]

Not integrable

Time = 230.71 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.14 \[ \int \frac {x^m}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^2} \, dx=\int { \frac {x^{m}}{{\left (a^{2} c x^{2} + c\right )}^{2} \arctan \left (a x\right )^{2}} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [N/A]

Not integrable

Time = 0.45 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {x^m}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^2} \, dx=\int \frac {x^m}{{\mathrm {atan}\left (a\,x\right )}^2\,{\left (c\,a^2\,x^2+c\right )}^2} \,d x \]

[In]

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

[Out]

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