\(\int \frac {x^m (c+a^2 c x^2)}{\arctan (a x)^2} \, dx\) [600]

Optimal result

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

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.02 (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 = {} \[ \int \frac {x^m \left (c+a^2 c x^2\right )}{\arctan (a x)^2} \, dx=\int \frac {x^m \left (c+a^2 c x^2\right )}{\arctan (a x)^2} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 0.40 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {x^m \left (c+a^2 c x^2\right )}{\arctan (a x)^2} \, dx=\int \frac {x^m \left (c+a^2 c x^2\right )}{\arctan (a x)^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 5.82 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {x^m \left (c+a^2 c x^2\right )}{\arctan (a x)^2} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )} x^{m}}{\arctan \left (a x\right )^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 3.89 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.55 \[ \int \frac {x^m \left (c+a^2 c x^2\right )}{\arctan (a x)^2} \, dx=c \left (\int \frac {x^{m}}{\operatorname {atan}^{2}{\left (a x \right )}}\, dx + \int \frac {a^{2} x^{2} x^{m}}{\operatorname {atan}^{2}{\left (a x \right )}}\, dx\right ) \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.56 (sec) , antiderivative size = 95, normalized size of antiderivative = 4.75 \[ \int \frac {x^m \left (c+a^2 c x^2\right )}{\arctan (a x)^2} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )} x^{m}}{\arctan \left (a x\right )^{2}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 133.17 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.15 \[ \int \frac {x^m \left (c+a^2 c x^2\right )}{\arctan (a x)^2} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )} x^{m}}{\arctan \left (a x\right )^{2}} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [N/A]

Not integrable

Time = 0.44 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {x^m \left (c+a^2 c x^2\right )}{\arctan (a x)^2} \, dx=\int \frac {x^m\,\left (c\,a^2\,x^2+c\right )}{{\mathrm {atan}\left (a\,x\right )}^2} \,d x \]

[In]

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

[Out]

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