\(\int \frac {x (c+a^2 c x^2)^3}{\arctan (a x)^2} \, dx\) [540]

Optimal result

Integrand size = 20, antiderivative size = 20 \[ \int \frac {x \left (c+a^2 c x^2\right )^3}{\arctan (a x)^2} \, dx=\text {Int}\left (\frac {x \left (c+a^2 c x^2\right )^3}{\arctan (a x)^2},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.03 (sec) , antiderivative size = 20, 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 = {} \[ \int \frac {x \left (c+a^2 c x^2\right )^3}{\arctan (a x)^2} \, dx=\int \frac {x \left (c+a^2 c x^2\right )^3}{\arctan (a x)^2} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 0.86 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {x \left (c+a^2 c x^2\right )^3}{\arctan (a x)^2} \, dx=\int \frac {x \left (c+a^2 c x^2\right )^3}{\arctan (a x)^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 95.80 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.23 (sec) , antiderivative size = 47, normalized size of antiderivative = 2.35 \[ \int \frac {x \left (c+a^2 c x^2\right )^3}{\arctan (a x)^2} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{3} x}{\arctan \left (a x\right )^{2}} \,d x } \]

[In]

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

[Out]

integral((a^6*c^3*x^7 + 3*a^4*c^3*x^5 + 3*a^2*c^3*x^3 + c^3*x)/arctan(a*x)^2, x)

Sympy [N/A]

Not integrable

Time = 1.50 (sec) , antiderivative size = 61, normalized size of antiderivative = 3.05 \[ \int \frac {x \left (c+a^2 c x^2\right )^3}{\arctan (a x)^2} \, dx=c^{3} \left (\int \frac {x}{\operatorname {atan}^{2}{\left (a x \right )}}\, dx + \int \frac {3 a^{2} x^{3}}{\operatorname {atan}^{2}{\left (a x \right )}}\, dx + \int \frac {3 a^{4} x^{5}}{\operatorname {atan}^{2}{\left (a x \right )}}\, dx + \int \frac {a^{6} x^{7}}{\operatorname {atan}^{2}{\left (a x \right )}}\, dx\right ) \]

[In]

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

[Out]

c**3*(Integral(x/atan(a*x)**2, x) + Integral(3*a**2*x**3/atan(a*x)**2, x) + Integral(3*a**4*x**5/atan(a*x)**2,
x) + Integral(a**6*x**7/atan(a*x)**2, x))

Maxima [N/A]

Not integrable

Time = 0.36 (sec) , antiderivative size = 123, normalized size of antiderivative = 6.15 \[ \int \frac {x \left (c+a^2 c x^2\right )^3}{\arctan (a x)^2} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{3} x}{\arctan \left (a x\right )^{2}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 70.29 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.15 \[ \int \frac {x \left (c+a^2 c x^2\right )^3}{\arctan (a x)^2} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{3} x}{\arctan \left (a x\right )^{2}} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [N/A]

Not integrable

Time = 0.46 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {x \left (c+a^2 c x^2\right )^3}{\arctan (a x)^2} \, dx=\int \frac {x\,{\left (c\,a^2\,x^2+c\right )}^3}{{\mathrm {atan}\left (a\,x\right )}^2} \,d x \]

[In]

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

[Out]

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