\(\int x^m (c+a^2 c x^2) \arctan (a x)^2 \, dx\) [356]

Optimal result

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

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.03 (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 = {} \[ \int x^m \left (c+a^2 c x^2\right ) \arctan (a x)^2 \, dx=\int x^m \left (c+a^2 c x^2\right ) \arctan (a x)^2 \, dx \]

[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 x^m \left (c+a^2 c x^2\right ) \arctan (a x)^2 \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.72 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int x^m \left (c+a^2 c x^2\right ) \arctan (a x)^2 \, dx=\int x^m \left (c+a^2 c x^2\right ) \arctan (a x)^2 \, dx \]

[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 = 0.92 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00

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

[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 = 0.26 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int x^m \left (c+a^2 c x^2\right ) \arctan (a x)^2 \, dx=\int { {\left (a^{2} c x^{2} + c\right )} x^{m} \arctan \left (a x\right )^{2} \,d x } \]

[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 = 6.43 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.55 \[ \int x^m \left (c+a^2 c x^2\right ) \arctan (a x)^2 \, dx=c \left (\int x^{m} \operatorname {atan}^{2}{\left (a x \right )}\, dx + \int a^{2} x^{2} x^{m} \operatorname {atan}^{2}{\left (a x \right )}\, dx\right ) \]

[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 = 4.67 (sec) , antiderivative size = 386, normalized size of antiderivative = 19.30 \[ \int x^m \left (c+a^2 c x^2\right ) \arctan (a x)^2 \, dx=\int { {\left (a^{2} c x^{2} + c\right )} x^{m} \arctan \left (a x\right )^{2} \,d x } \]

[In]

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

[Out]

1/16*(4*((a^2*c*m + a^2*c)*x^3 + (c*m + 3*c)*x)*x^m*arctan(a*x)^2 - ((a^2*c*m + a^2*c)*x^3 + (c*m + 3*c)*x)*x^
m*log(a^2*x^2 + 1)^2 + 16*(m^2 + 4*m + 3)*integrate(1/16*(12*((a^4*c*m^2 + 4*a^4*c*m + 3*a^4*c)*x^4 + c*m^2 +
2*(a^2*c*m^2 + 4*a^2*c*m + 3*a^2*c)*x^2 + 4*c*m + 3*c)*x^m*arctan(a*x)^2 + ((a^4*c*m^2 + 4*a^4*c*m + 3*a^4*c)*
x^4 + c*m^2 + 2*(a^2*c*m^2 + 4*a^2*c*m + 3*a^2*c)*x^2 + 4*c*m + 3*c)*x^m*log(a^2*x^2 + 1)^2 - 8*((a^3*c*m + a^
3*c)*x^3 + (a*c*m + 3*a*c)*x)*x^m*arctan(a*x) + 4*((a^4*c*m + a^4*c)*x^4 + (a^2*c*m + 3*a^2*c)*x^2)*x^m*log(a^
2*x^2 + 1))/((a^2*m^2 + 4*a^2*m + 3*a^2)*x^2 + m^2 + 4*m + 3), x))/(m^2 + 4*m + 3)

Giac [N/A]

Not integrable

Time = 107.23 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.15 \[ \int x^m \left (c+a^2 c x^2\right ) \arctan (a x)^2 \, dx=\int { {\left (a^{2} c x^{2} + c\right )} x^{m} \arctan \left (a x\right )^{2} \,d x } \]

[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 = 0.47 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int x^m \left (c+a^2 c x^2\right ) \arctan (a x)^2 \, dx=\int x^m\,{\mathrm {atan}\left (a\,x\right )}^2\,\left (c\,a^2\,x^2+c\right ) \,d x \]

[In]

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

[Out]

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