\(\int \frac {x^m (c+a^2 c x^2)}{\arctan (a x)^3} \, dx\) [676]

Optimal result

Integrand size = 20, antiderivative size = 20 \[ \int \frac {x^m \left (c+a^2 c x^2\right )}{\arctan (a x)^3} \, dx=\text {Int}\left (\frac {x^m \left (c+a^2 c x^2\right )}{\arctan (a x)^3},x\right ) \]

[Out]

Unintegrable(x^m*(a^2*c*x^2+c)/arctan(a*x)^3,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 = {} \[ \int \frac {x^m \left (c+a^2 c x^2\right )}{\arctan (a x)^3} \, dx=\int \frac {x^m \left (c+a^2 c x^2\right )}{\arctan (a x)^3} \, dx \]

[In]

Int[(x^m*(c + a^2*c*x^2))/ArcTan[a*x]^3,x]

[Out]

Defer[Int][(x^m*(c + a^2*c*x^2))/ArcTan[a*x]^3, x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {x^m \left (c+a^2 c x^2\right )}{\arctan (a x)^3} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.44 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {x^m \left (c+a^2 c x^2\right )}{\arctan (a x)^3} \, dx=\int \frac {x^m \left (c+a^2 c x^2\right )}{\arctan (a x)^3} \, dx \]

[In]

Integrate[(x^m*(c + a^2*c*x^2))/ArcTan[a*x]^3,x]

[Out]

Integrate[(x^m*(c + a^2*c*x^2))/ArcTan[a*x]^3, x]

Maple [N/A] (verified)

Not integrable

Time = 10.63 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00

\[\int \frac {x^{m} \left (a^{2} c \,x^{2}+c \right )}{\arctan \left (a x \right )^{3}}d x\]

[In]

int(x^m*(a^2*c*x^2+c)/arctan(a*x)^3,x)

[Out]

int(x^m*(a^2*c*x^2+c)/arctan(a*x)^3,x)

Fricas [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {x^m \left (c+a^2 c x^2\right )}{\arctan (a x)^3} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )} x^{m}}{\arctan \left (a x\right )^{3}} \,d x } \]

[In]

integrate(x^m*(a^2*c*x^2+c)/arctan(a*x)^3,x, algorithm="fricas")

[Out]

integral((a^2*c*x^2 + c)*x^m/arctan(a*x)^3, x)

Sympy [N/A]

Not integrable

Time = 6.27 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.55 \[ \int \frac {x^m \left (c+a^2 c x^2\right )}{\arctan (a x)^3} \, dx=c \left (\int \frac {x^{m}}{\operatorname {atan}^{3}{\left (a x \right )}}\, dx + \int \frac {a^{2} x^{2} x^{m}}{\operatorname {atan}^{3}{\left (a x \right )}}\, dx\right ) \]

[In]

integrate(x**m*(a**2*c*x**2+c)/atan(a*x)**3,x)

[Out]

c*(Integral(x**m/atan(a*x)**3, x) + Integral(a**2*x**2*x**m/atan(a*x)**3, x))

Maxima [N/A]

Not integrable

Time = 1.00 (sec) , antiderivative size = 221, normalized size of antiderivative = 11.05 \[ \int \frac {x^m \left (c+a^2 c x^2\right )}{\arctan (a x)^3} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )} x^{m}}{\arctan \left (a x\right )^{3}} \,d x } \]

[In]

integrate(x^m*(a^2*c*x^2+c)/arctan(a*x)^3,x, algorithm="maxima")

[Out]

1/2*(x*arctan(a*x)^2*integrate(((a^6*c*m^2 + 9*a^6*c*m + 20*a^6*c)*x^6 + (3*a^4*c*m^2 + 17*a^4*c*m + 24*a^4*c)
*x^4 + c*m^2 + (3*a^2*c*m^2 + 7*a^2*c*m + 4*a^2*c)*x^2 - c*m)*x^m/(x^2*arctan(a*x)), x) - ((a^6*c*m + 4*a^6*c)
*x^6 + (3*a^4*c*m + 8*a^4*c)*x^4 + (3*a^2*c*m + 4*a^2*c)*x^2 + c*m)*x^m*arctan(a*x) - (a^5*c*x^5 + 2*a^3*c*x^3
+ a*c*x)*x^m)/(a^2*x*arctan(a*x)^2)

Giac [N/A]

Not integrable

Time = 167.10 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.15 \[ \int \frac {x^m \left (c+a^2 c x^2\right )}{\arctan (a x)^3} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )} x^{m}}{\arctan \left (a x\right )^{3}} \,d x } \]

[In]

integrate(x^m*(a^2*c*x^2+c)/arctan(a*x)^3,x, algorithm="giac")

[Out]

sage0*x

Mupad [N/A]

Not integrable

Time = 0.42 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {x^m \left (c+a^2 c x^2\right )}{\arctan (a x)^3} \, dx=\int \frac {x^m\,\left (c\,a^2\,x^2+c\right )}{{\mathrm {atan}\left (a\,x\right )}^3} \,d x \]

[In]

int((x^m*(c + a^2*c*x^2))/atan(a*x)^3,x)

[Out]

int((x^m*(c + a^2*c*x^2))/atan(a*x)^3, x)