\(\int \frac {(c+a^2 c x^2)^3}{x \arctan (a x)^2} \, dx\) [542]

Optimal result

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

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.03 (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 = {} \[ \int \frac {\left (c+a^2 c x^2\right )^3}{x \arctan (a x)^2} \, dx=\int \frac {\left (c+a^2 c x^2\right )^3}{x \arctan (a x)^2} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 1.11 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {\left (c+a^2 c x^2\right )^3}{x \arctan (a x)^2} \, dx=\int \frac {\left (c+a^2 c x^2\right )^3}{x \arctan (a x)^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 135.07 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 48, normalized size of antiderivative = 2.18 \[ \int \frac {\left (c+a^2 c x^2\right )^3}{x \arctan (a x)^2} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{3}}{x \arctan \left (a x\right )^{2}} \,d x } \]

[In]

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

Sympy [N/A]

Not integrable

Time = 2.67 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.77 \[ \int \frac {\left (c+a^2 c x^2\right )^3}{x \arctan (a x)^2} \, dx=c^{3} \left (\int \frac {1}{x \operatorname {atan}^{2}{\left (a x \right )}}\, dx + \int \frac {3 a^{2} x}{\operatorname {atan}^{2}{\left (a x \right )}}\, dx + \int \frac {3 a^{4} x^{3}}{\operatorname {atan}^{2}{\left (a x \right )}}\, dx + \int \frac {a^{6} x^{5}}{\operatorname {atan}^{2}{\left (a x \right )}}\, dx\right ) \]

[In]

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

[Out]

c**3*(Integral(1/(x*atan(a*x)**2), x) + Integral(3*a**2*x/atan(a*x)**2, x) + Integral(3*a**4*x**3/atan(a*x)**2
, x) + Integral(a**6*x**5/atan(a*x)**2, x))

Maxima [N/A]

Not integrable

Time = 0.38 (sec) , antiderivative size = 130, normalized size of antiderivative = 5.91 \[ \int \frac {\left (c+a^2 c x^2\right )^3}{x \arctan (a x)^2} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{3}}{x \arctan \left (a x\right )^{2}} \,d x } \]

[In]

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

[Out]

-(a^8*c^3*x^8 + 4*a^6*c^3*x^6 + 6*a^4*c^3*x^4 + 4*a^2*c^3*x^2 + c^3 - x*arctan(a*x)*integrate((7*a^8*c^3*x^8 +
20*a^6*c^3*x^6 + 18*a^4*c^3*x^4 + 4*a^2*c^3*x^2 - c^3)/(x^2*arctan(a*x)), x))/(a*x*arctan(a*x))

Giac [N/A]

Not integrable

Time = 70.33 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.14 \[ \int \frac {\left (c+a^2 c x^2\right )^3}{x \arctan (a x)^2} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{3}}{x \arctan \left (a x\right )^{2}} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [N/A]

Not integrable

Time = 0.45 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {\left (c+a^2 c x^2\right )^3}{x \arctan (a x)^2} \, dx=\int \frac {{\left (c\,a^2\,x^2+c\right )}^3}{x\,{\mathrm {atan}\left (a\,x\right )}^2} \,d x \]

[In]

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

[Out]

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