\(\int \frac {x}{(c+a^2 c x^2) \arctan (a x)^2} \, dx\) [545]

Optimal result

Integrand size = 20, antiderivative size = 20 \[ \int \frac {x}{\left (c+a^2 c x^2\right ) \arctan (a x)^2} \, dx=-\frac {x}{a c \arctan (a x)}+\frac {\text {Int}\left (\frac {1}{\arctan (a x)},x\right )}{a c} \]

[Out]

-x/a/c/arctan(a*x)+Unintegrable(1/arctan(a*x),x)/a/c

Rubi [N/A]

Not integrable

Time = 0.04 (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 = {} \[ \int \frac {x}{\left (c+a^2 c x^2\right ) \arctan (a x)^2} \, dx=\int \frac {x}{\left (c+a^2 c x^2\right ) \arctan (a x)^2} \, dx \]

[In]

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

[Out]

-(x/(a*c*ArcTan[a*x])) + Defer[Int][ArcTan[a*x]^(-1), x]/(a*c)

Rubi steps \begin{align*} \text {integral}& = -\frac {x}{a c \arctan (a x)}+\frac {\int \frac {1}{\arctan (a x)} \, dx}{a c} \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.38 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {x}{\left (c+a^2 c x^2\right ) \arctan (a x)^2} \, dx=\int \frac {x}{\left (c+a^2 c x^2\right ) \arctan (a x)^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.95 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

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

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 0.62 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.20 \[ \int \frac {x}{\left (c+a^2 c x^2\right ) \arctan (a x)^2} \, dx=\frac {\int \frac {x}{a^{2} x^{2} \operatorname {atan}^{2}{\left (a x \right )} + \operatorname {atan}^{2}{\left (a x \right )}}\, dx}{c} \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.50 \[ \int \frac {x}{\left (c+a^2 c x^2\right ) \arctan (a x)^2} \, dx=\int { \frac {x}{{\left (a^{2} c x^{2} + c\right )} \arctan \left (a x\right )^{2}} \,d x } \]

[In]

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

[Out]

(arctan(a*x)*integrate(1/arctan(a*x), x) - x)/(a*c*arctan(a*x))

Giac [N/A]

Not integrable

Time = 33.98 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.15 \[ \int \frac {x}{\left (c+a^2 c x^2\right ) \arctan (a x)^2} \, dx=\int { \frac {x}{{\left (a^{2} c x^{2} + c\right )} \arctan \left (a x\right )^{2}} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [N/A]

Not integrable

Time = 0.38 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {x}{\left (c+a^2 c x^2\right ) \arctan (a x)^2} \, dx=\int \frac {x}{{\mathrm {atan}\left (a\,x\right )}^2\,\left (c\,a^2\,x^2+c\right )} \,d x \]

[In]

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

[Out]

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