\(\int \frac {c+a^2 c x^2}{x \arctan (a x)^2} \, dx\) [536]

Optimal result

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

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.02 (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 = {} \[ \int \frac {c+a^2 c x^2}{x \arctan (a x)^2} \, dx=\int \frac {c+a^2 c x^2}{x \arctan (a x)^2} \, dx \]

[In]

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

[Out]

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

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

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 29.71 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.23 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {c+a^2 c x^2}{x \arctan (a x)^2} \, dx=\int { \frac {a^{2} c x^{2} + c}{x \arctan \left (a x\right )^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 1.01 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.30 \[ \int \frac {c+a^2 c x^2}{x \arctan (a x)^2} \, dx=c \left (\int \frac {1}{x \operatorname {atan}^{2}{\left (a x \right )}}\, dx + \int \frac {a^{2} x}{\operatorname {atan}^{2}{\left (a x \right )}}\, dx\right ) \]

[In]

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

[Out]

c*(Integral(1/(x*atan(a*x)**2), x) + Integral(a**2*x/atan(a*x)**2, x))

Maxima [N/A]

Not integrable

Time = 0.33 (sec) , antiderivative size = 74, normalized size of antiderivative = 3.70 \[ \int \frac {c+a^2 c x^2}{x \arctan (a x)^2} \, dx=\int { \frac {a^{2} c x^{2} + c}{x \arctan \left (a x\right )^{2}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 57.92 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.15 \[ \int \frac {c+a^2 c x^2}{x \arctan (a x)^2} \, dx=\int { \frac {a^{2} c x^{2} + c}{x \arctan \left (a x\right )^{2}} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [N/A]

Not integrable

Time = 0.45 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {c+a^2 c x^2}{x \arctan (a x)^2} \, dx=\int \frac {c\,a^2\,x^2+c}{x\,{\mathrm {atan}\left (a\,x\right )}^2} \,d x \]

[In]

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

[Out]

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