\(\int \frac {x (c+a^2 c x^2)}{\arctan (a x)^3} \, dx\) [610]

Optimal result

Integrand size = 18, antiderivative size = 18 \[ \int \frac {x \left (c+a^2 c x^2\right )}{\arctan (a x)^3} \, dx=\text {Int}\left (\frac {x \left (c+a^2 c x^2\right )}{\arctan (a x)^3},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.02 (sec) , antiderivative size = 18, 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 = {} \[ \int \frac {x \left (c+a^2 c x^2\right )}{\arctan (a x)^3} \, dx=\int \frac {x \left (c+a^2 c x^2\right )}{\arctan (a x)^3} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 1.00 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {x \left (c+a^2 c x^2\right )}{\arctan (a x)^3} \, dx=\int \frac {x \left (c+a^2 c x^2\right )}{\arctan (a x)^3} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 31.55 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.17 \[ \int \frac {x \left (c+a^2 c x^2\right )}{\arctan (a x)^3} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )} x}{\arctan \left (a x\right )^{3}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 0.82 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.44 \[ \int \frac {x \left (c+a^2 c x^2\right )}{\arctan (a x)^3} \, dx=c \left (\int \frac {x}{\operatorname {atan}^{3}{\left (a x \right )}}\, dx + \int \frac {a^{2} x^{3}}{\operatorname {atan}^{3}{\left (a x \right )}}\, dx\right ) \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.35 (sec) , antiderivative size = 110, normalized size of antiderivative = 6.11 \[ \int \frac {x \left (c+a^2 c x^2\right )}{\arctan (a x)^3} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )} x}{\arctan \left (a x\right )^{3}} \,d x } \]

[In]

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

[Out]

-1/2*(a^5*c*x^5 + 2*a^3*c*x^3 - 2*a^2*arctan(a*x)^2*integrate((15*a^4*c*x^5 + 22*a^2*c*x^3 + 7*c*x)/arctan(a*x
), x) + a*c*x + (5*a^6*c*x^6 + 11*a^4*c*x^4 + 7*a^2*c*x^2 + c)*arctan(a*x))/(a^2*arctan(a*x)^2)

Giac [N/A]

Not integrable

Time = 77.72 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.17 \[ \int \frac {x \left (c+a^2 c x^2\right )}{\arctan (a x)^3} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )} x}{\arctan \left (a x\right )^{3}} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [N/A]

Not integrable

Time = 0.51 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {x \left (c+a^2 c x^2\right )}{\arctan (a x)^3} \, dx=\int \frac {x\,\left (c\,a^2\,x^2+c\right )}{{\mathrm {atan}\left (a\,x\right )}^3} \,d x \]

[In]

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

[Out]

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