\(\int \frac {(c+a^2 c x^2)^2}{x \arctan (a x)^3} \, dx\) [615]

Optimal result

Integrand size = 22, antiderivative size = 22 \[ \int \frac {\left (c+a^2 c x^2\right )^2}{x \arctan (a x)^3} \, dx=\text {Int}\left (\frac {\left (c+a^2 c x^2\right )^2}{x \arctan (a x)^3},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.04 (sec) , antiderivative size = 22, 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 = {} \[ \int \frac {\left (c+a^2 c x^2\right )^2}{x \arctan (a x)^3} \, dx=\int \frac {\left (c+a^2 c x^2\right )^2}{x \arctan (a x)^3} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 1.04 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {\left (c+a^2 c x^2\right )^2}{x \arctan (a x)^3} \, dx=\int \frac {\left (c+a^2 c x^2\right )^2}{x \arctan (a x)^3} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 182.06 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.68 \[ \int \frac {\left (c+a^2 c x^2\right )^2}{x \arctan (a x)^3} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{2}}{x \arctan \left (a x\right )^{3}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 2.21 (sec) , antiderivative size = 44, normalized size of antiderivative = 2.00 \[ \int \frac {\left (c+a^2 c x^2\right )^2}{x \arctan (a x)^3} \, dx=c^{2} \left (\int \frac {1}{x \operatorname {atan}^{3}{\left (a x \right )}}\, dx + \int \frac {2 a^{2} x}{\operatorname {atan}^{3}{\left (a x \right )}}\, dx + \int \frac {a^{4} x^{3}}{\operatorname {atan}^{3}{\left (a x \right )}}\, dx\right ) \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.42 (sec) , antiderivative size = 168, normalized size of antiderivative = 7.64 \[ \int \frac {\left (c+a^2 c x^2\right )^2}{x \arctan (a x)^3} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{2}}{x \arctan \left (a x\right )^{3}} \,d x } \]

[In]

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

[Out]

-1/2*(a^7*c^2*x^7 + 3*a^5*c^2*x^5 + 3*a^3*c^2*x^3 - 2*x^2*arctan(a*x)^2*integrate((15*a^8*c^2*x^8 + 28*a^6*c^2
*x^6 + 12*a^4*c^2*x^4 + c^2)/(x^3*arctan(a*x)), x) + a*c^2*x + (5*a^8*c^2*x^8 + 14*a^6*c^2*x^6 + 12*a^4*c^2*x^
4 + 2*a^2*c^2*x^2 - c^2)*arctan(a*x))/(a^2*x^2*arctan(a*x)^2)

Giac [N/A]

Not integrable

Time = 88.92 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.14 \[ \int \frac {\left (c+a^2 c x^2\right )^2}{x \arctan (a x)^3} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{2}}{x \arctan \left (a x\right )^{3}} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [N/A]

Not integrable

Time = 0.42 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {\left (c+a^2 c x^2\right )^2}{x \arctan (a x)^3} \, dx=\int \frac {{\left (c\,a^2\,x^2+c\right )}^2}{x\,{\mathrm {atan}\left (a\,x\right )}^3} \,d x \]

[In]

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

[Out]

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