\(\int \frac {(c+a^2 c x^2)^3}{\arctan (a x)} \, dx\) [473]

Optimal result

Integrand size = 19, antiderivative size = 19 \[ \int \frac {\left (c+a^2 c x^2\right )^3}{\arctan (a x)} \, dx=\text {Int}\left (\frac {\left (c+a^2 c x^2\right )^3}{\arctan (a x)},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.02 (sec) , antiderivative size = 19, 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 = {} \[ \int \frac {\left (c+a^2 c x^2\right )^3}{\arctan (a x)} \, dx=\int \frac {\left (c+a^2 c x^2\right )^3}{\arctan (a x)} \, dx \]

[In]

Int[(c + a^2*c*x^2)^3/ArcTan[a*x],x]

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 0.42 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.11 \[ \int \frac {\left (c+a^2 c x^2\right )^3}{\arctan (a x)} \, dx=\int \frac {\left (c+a^2 c x^2\right )^3}{\arctan (a x)} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 59.87 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 45, normalized size of antiderivative = 2.37 \[ \int \frac {\left (c+a^2 c x^2\right )^3}{\arctan (a x)} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{3}}{\arctan \left (a x\right )} \,d x } \]

[In]

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

[Out]

integral((a^6*c^3*x^6 + 3*a^4*c^3*x^4 + 3*a^2*c^3*x^2 + c^3)/arctan(a*x), x)

Sympy [N/A]

Not integrable

Time = 1.35 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.84 \[ \int \frac {\left (c+a^2 c x^2\right )^3}{\arctan (a x)} \, dx=c^{3} \left (\int \frac {3 a^{2} x^{2}}{\operatorname {atan}{\left (a x \right )}}\, dx + \int \frac {3 a^{4} x^{4}}{\operatorname {atan}{\left (a x \right )}}\, dx + \int \frac {a^{6} x^{6}}{\operatorname {atan}{\left (a x \right )}}\, dx + \int \frac {1}{\operatorname {atan}{\left (a x \right )}}\, dx\right ) \]

[In]

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

[Out]

c**3*(Integral(3*a**2*x**2/atan(a*x), x) + Integral(3*a**4*x**4/atan(a*x), x) + Integral(a**6*x**6/atan(a*x),
x) + Integral(1/atan(a*x), x))

Maxima [N/A]

Not integrable

Time = 0.29 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.11 \[ \int \frac {\left (c+a^2 c x^2\right )^3}{\arctan (a x)} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{3}}{\arctan \left (a x\right )} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 39.04 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.16 \[ \int \frac {\left (c+a^2 c x^2\right )^3}{\arctan (a x)} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{3}}{\arctan \left (a x\right )} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [N/A]

Not integrable

Time = 0.45 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.11 \[ \int \frac {\left (c+a^2 c x^2\right )^3}{\arctan (a x)} \, dx=\int \frac {{\left (c\,a^2\,x^2+c\right )}^3}{\mathrm {atan}\left (a\,x\right )} \,d x \]

[In]

int((c + a^2*c*x^2)^3/atan(a*x),x)

[Out]

int((c + a^2*c*x^2)^3/atan(a*x), x)