\(\int \frac {x (c+a^2 c x^2)}{\arctan (a x)} \, dx\) [466]

Optimal result

Integrand size = 18, antiderivative size = 18 \[ \int \frac {x \left (c+a^2 c x^2\right )}{\arctan (a x)} \, dx=\text {Int}\left (\frac {x \left (c+a^2 c x^2\right )}{\arctan (a x)},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.02 (sec) , antiderivative size = 18, 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 = {} \[ \int \frac {x \left (c+a^2 c x^2\right )}{\arctan (a x)} \, dx=\int \frac {x \left (c+a^2 c x^2\right )}{\arctan (a x)} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 0.39 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {x \left (c+a^2 c x^2\right )}{\arctan (a x)} \, dx=\int \frac {x \left (c+a^2 c x^2\right )}{\arctan (a x)} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 13.42 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.17 \[ \int \frac {x \left (c+a^2 c x^2\right )}{\arctan (a x)} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )} x}{\arctan \left (a x\right )} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 0.69 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.22 \[ \int \frac {x \left (c+a^2 c x^2\right )}{\arctan (a x)} \, dx=c \left (\int \frac {x}{\operatorname {atan}{\left (a x \right )}}\, dx + \int \frac {a^{2} x^{3}}{\operatorname {atan}{\left (a x \right )}}\, dx\right ) \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.29 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {x \left (c+a^2 c x^2\right )}{\arctan (a x)} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )} x}{\arctan \left (a x\right )} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 30.61 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.17 \[ \int \frac {x \left (c+a^2 c x^2\right )}{\arctan (a x)} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )} x}{\arctan \left (a x\right )} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [N/A]

Not integrable

Time = 0.46 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {x \left (c+a^2 c x^2\right )}{\arctan (a x)} \, dx=\int \frac {x\,\left (c\,a^2\,x^2+c\right )}{\mathrm {atan}\left (a\,x\right )} \,d x \]

[In]

int((x*(c + a^2*c*x^2))/atan(a*x),x)

[Out]

int((x*(c + a^2*c*x^2))/atan(a*x), x)