\(\int \frac {x^m (c+a^2 c x^2)^2}{\arctan (a x)^2} \, dx\) [599]

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 = 0.04 (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 = 0.66 (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 = 7.02 (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 = 0.24 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.68 \[ \int \frac {x^m \left (c+a^2 c x^2\right )^2}{\arctan (a x)^2} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{2} x^{m}}{\arctan \left (a x\right )^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 7.01 (sec) , antiderivative size = 53, normalized size of antiderivative = 2.41 \[ \int \frac {x^m \left (c+a^2 c x^2\right )^2}{\arctan (a x)^2} \, dx=c^{2} \left (\int \frac {x^{m}}{\operatorname {atan}^{2}{\left (a x \right )}}\, dx + \int \frac {2 a^{2} x^{2} x^{m}}{\operatorname {atan}^{2}{\left (a x \right )}}\, dx + \int \frac {a^{4} x^{4} x^{m}}{\operatorname {atan}^{2}{\left (a x \right )}}\, dx\right ) \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.60 (sec) , antiderivative size = 144, normalized size of antiderivative = 6.55 \[ \int \frac {x^m \left (c+a^2 c x^2\right )^2}{\arctan (a x)^2} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{2} x^{m}}{\arctan \left (a x\right )^{2}} \,d x } \]

[In]

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

[Out]

-((a^6*c^2*x^6 + 3*a^4*c^2*x^4 + 3*a^2*c^2*x^2 + c^2)*x^m - arctan(a*x)*integrate(((a^6*c^2*m + 6*a^6*c^2)*x^6
+ 3*(a^4*c^2*m + 4*a^4*c^2)*x^4 + c^2*m + 3*(a^2*c^2*m + 2*a^2*c^2)*x^2)*x^m/(x*arctan(a*x)), x))/(a*arctan(a
*x))

Giac [N/A]

Not integrable

Time = 154.01 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.14 \[ \int \frac {x^m \left (c+a^2 c x^2\right )^2}{\arctan (a x)^2} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{2} x^{m}}{\arctan \left (a x\right )^{2}} \,d x } \]

[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 = 0.40 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {x^m \left (c+a^2 c x^2\right )^2}{\arctan (a x)^2} \, dx=\int \frac {x^m\,{\left (c\,a^2\,x^2+c\right )}^2}{{\mathrm {atan}\left (a\,x\right )}^2} \,d x \]

[In]

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

[Out]

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