\(\int \frac {c+a^2 c x^2}{\arctan (a x)^3} \, dx\) [611]

Optimal result

Integrand size = 17, antiderivative size = 17 \[ \int \frac {c+a^2 c x^2}{\arctan (a x)^3} \, dx=\text {Int}\left (\frac {c+a^2 c x^2}{\arctan (a x)^3},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.01 (sec) , antiderivative size = 17, 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 = {} \[ \int \frac {c+a^2 c x^2}{\arctan (a x)^3} \, dx=\int \frac {c+a^2 c x^2}{\arctan (a x)^3} \, dx \]

[In]

Int[(c + a^2*c*x^2)/ArcTan[a*x]^3,x]

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 1.00 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.12 \[ \int \frac {c+a^2 c x^2}{\arctan (a x)^3} \, dx=\int \frac {c+a^2 c x^2}{\arctan (a x)^3} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 49.15 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.12 \[ \int \frac {c+a^2 c x^2}{\arctan (a x)^3} \, dx=\int { \frac {a^{2} c x^{2} + c}{\arctan \left (a x\right )^{3}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 0.68 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.53 \[ \int \frac {c+a^2 c x^2}{\arctan (a x)^3} \, dx=c \left (\int \frac {a^{2} x^{2}}{\operatorname {atan}^{3}{\left (a x \right )}}\, dx + \int \frac {1}{\operatorname {atan}^{3}{\left (a x \right )}}\, dx\right ) \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.36 (sec) , antiderivative size = 97, normalized size of antiderivative = 5.71 \[ \int \frac {c+a^2 c x^2}{\arctan (a x)^3} \, dx=\int { \frac {a^{2} c x^{2} + c}{\arctan \left (a x\right )^{3}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 74.78 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.18 \[ \int \frac {c+a^2 c x^2}{\arctan (a x)^3} \, dx=\int { \frac {a^{2} c x^{2} + c}{\arctan \left (a x\right )^{3}} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [N/A]

Not integrable

Time = 0.49 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.12 \[ \int \frac {c+a^2 c x^2}{\arctan (a x)^3} \, dx=\int \frac {c\,a^2\,x^2+c}{{\mathrm {atan}\left (a\,x\right )}^3} \,d x \]

[In]

int((c + a^2*c*x^2)/atan(a*x)^3,x)

[Out]

int((c + a^2*c*x^2)/atan(a*x)^3, x)