\(\int \frac {x (c+a^2 c x^2)^2}{\arctan (a x)^3} \, dx\) [613]

Optimal result

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

[Out]

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

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 0.84 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {x \left (c+a^2 c x^2\right )^2}{\arctan (a x)^3} \, dx=\int \frac {x \left (c+a^2 c x^2\right )^2}{\arctan (a x)^3} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 145.09 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.80 \[ \int \frac {x \left (c+a^2 c x^2\right )^2}{\arctan (a x)^3} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{2} x}{\arctan \left (a x\right )^{3}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 1.29 (sec) , antiderivative size = 44, normalized size of antiderivative = 2.20 \[ \int \frac {x \left (c+a^2 c x^2\right )^2}{\arctan (a x)^3} \, dx=c^{2} \left (\int \frac {x}{\operatorname {atan}^{3}{\left (a x \right )}}\, dx + \int \frac {2 a^{2} x^{3}}{\operatorname {atan}^{3}{\left (a x \right )}}\, dx + \int \frac {a^{4} x^{5}}{\operatorname {atan}^{3}{\left (a x \right )}}\, dx\right ) \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.39 (sec) , antiderivative size = 164, normalized size of antiderivative = 8.20 \[ \int \frac {x \left (c+a^2 c x^2\right )^2}{\arctan (a x)^3} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{2} x}{\arctan \left (a x\right )^{3}} \,d x } \]

[In]

integrate(x*(a^2*c*x^2+c)^2/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*a^2*arctan(a*x)^2*integrate(2*(14*a^6*c^2*x^7 + 33*a^4*c
^2*x^5 + 24*a^2*c^2*x^3 + 5*c^2*x)/arctan(a*x), x) + a*c^2*x + (7*a^8*c^2*x^8 + 22*a^6*c^2*x^6 + 24*a^4*c^2*x^
4 + 10*a^2*c^2*x^2 + c^2)*arctan(a*x))/(a^2*arctan(a*x)^2)

Giac [N/A]

Not integrable

Time = 87.31 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.15 \[ \int \frac {x \left (c+a^2 c x^2\right )^2}{\arctan (a x)^3} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{2} x}{\arctan \left (a x\right )^{3}} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [N/A]

Not integrable

Time = 0.44 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {x \left (c+a^2 c x^2\right )^2}{\arctan (a x)^3} \, dx=\int \frac {x\,{\left (c\,a^2\,x^2+c\right )}^2}{{\mathrm {atan}\left (a\,x\right )}^3} \,d x \]

[In]

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

[Out]

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