\(\int \frac {(c+a^2 c x^2)^3}{\arctan (a x)^3} \, dx\) [617]

Optimal result

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

[Out]

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

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 1.01 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.11 \[ \int \frac {\left (c+a^2 c x^2\right )^3}{\arctan (a x)^3} \, dx=\int \frac {\left (c+a^2 c x^2\right )^3}{\arctan (a x)^3} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 154.42 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 45, normalized size of antiderivative = 2.37 \[ \int \frac {\left (c+a^2 c x^2\right )^3}{\arctan (a x)^3} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{3}}{\arctan \left (a x\right )^{3}} \,d x } \]

[In]

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

Sympy [N/A]

Not integrable

Time = 1.47 (sec) , antiderivative size = 61, normalized size of antiderivative = 3.21 \[ \int \frac {\left (c+a^2 c x^2\right )^3}{\arctan (a x)^3} \, dx=c^{3} \left (\int \frac {3 a^{2} x^{2}}{\operatorname {atan}^{3}{\left (a x \right )}}\, dx + \int \frac {3 a^{4} x^{4}}{\operatorname {atan}^{3}{\left (a x \right )}}\, dx + \int \frac {a^{6} x^{6}}{\operatorname {atan}^{3}{\left (a x \right )}}\, dx + \int \frac {1}{\operatorname {atan}^{3}{\left (a x \right )}}\, dx\right ) \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.41 (sec) , antiderivative size = 181, normalized size of antiderivative = 9.53 \[ \int \frac {\left (c+a^2 c x^2\right )^3}{\arctan (a x)^3} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{3}}{\arctan \left (a x\right )^{3}} \,d x } \]

[In]

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

[Out]

-1/2*(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 - 2*a*arctan(a*x)^2*integrate(4*(9*a^8*c^3*x
^8 + 28*a^6*c^3*x^6 + 30*a^4*c^3*x^4 + 12*a^2*c^3*x^2 + c^3)/arctan(a*x), x) + c^3 + 8*(a^9*c^3*x^9 + 4*a^7*c^
3*x^7 + 6*a^5*c^3*x^5 + 4*a^3*c^3*x^3 + a*c^3*x)*arctan(a*x))/(a*arctan(a*x)^2)

Giac [N/A]

Not integrable

Time = 94.74 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.16 \[ \int \frac {\left (c+a^2 c x^2\right )^3}{\arctan (a x)^3} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{3}}{\arctan \left (a x\right )^{3}} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [N/A]

Not integrable

Time = 0.44 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.11 \[ \int \frac {\left (c+a^2 c x^2\right )^3}{\arctan (a x)^3} \, dx=\int \frac {{\left (c\,a^2\,x^2+c\right )}^3}{{\mathrm {atan}\left (a\,x\right )}^3} \,d x \]

[In]

int((c + a^2*c*x^2)^3/atan(a*x)^3,x)

[Out]

int((c + a^2*c*x^2)^3/atan(a*x)^3, x)