\(\int \frac {1}{x (c+a^2 c x^2) \arctan (a x)^2} \, dx\) [547]

Optimal result

Integrand size = 22, antiderivative size = 22 \[ \int \frac {1}{x \left (c+a^2 c x^2\right ) \arctan (a x)^2} \, dx=-\frac {1}{a c x \arctan (a x)}-\frac {\text {Int}\left (\frac {1}{x^2 \arctan (a x)},x\right )}{a c} \]

[Out]

-1/a/c/x/arctan(a*x)-Unintegrable(1/x^2/arctan(a*x),x)/a/c

Rubi [N/A]

Not integrable

Time = 0.05 (sec) , antiderivative size = 22, 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 = {} \[ \int \frac {1}{x \left (c+a^2 c x^2\right ) \arctan (a x)^2} \, dx=\int \frac {1}{x \left (c+a^2 c x^2\right ) \arctan (a x)^2} \, dx \]

[In]

Int[1/(x*(c + a^2*c*x^2)*ArcTan[a*x]^2),x]

[Out]

-(1/(a*c*x*ArcTan[a*x])) - Defer[Int][1/(x^2*ArcTan[a*x]), x]/(a*c)

Rubi steps \begin{align*} \text {integral}& = -\frac {1}{a c x \arctan (a x)}-\frac {\int \frac {1}{x^2 \arctan (a x)} \, dx}{a c} \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.40 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {1}{x \left (c+a^2 c x^2\right ) \arctan (a x)^2} \, dx=\int \frac {1}{x \left (c+a^2 c x^2\right ) \arctan (a x)^2} \, dx \]

[In]

Integrate[1/(x*(c + a^2*c*x^2)*ArcTan[a*x]^2),x]

[Out]

Integrate[1/(x*(c + a^2*c*x^2)*ArcTan[a*x]^2), x]

Maple [N/A] (verified)

Not integrable

Time = 1.05 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00

\[\int \frac {1}{x \left (a^{2} c \,x^{2}+c \right ) \arctan \left (a x \right )^{2}}d x\]

[In]

int(1/x/(a^2*c*x^2+c)/arctan(a*x)^2,x)

[Out]

int(1/x/(a^2*c*x^2+c)/arctan(a*x)^2,x)

Fricas [N/A]

Not integrable

Time = 0.23 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.05 \[ \int \frac {1}{x \left (c+a^2 c x^2\right ) \arctan (a x)^2} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} + c\right )} x \arctan \left (a x\right )^{2}} \,d x } \]

[In]

integrate(1/x/(a^2*c*x^2+c)/arctan(a*x)^2,x, algorithm="fricas")

[Out]

integral(1/((a^2*c*x^3 + c*x)*arctan(a*x)^2), x)

Sympy [N/A]

Not integrable

Time = 0.83 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.18 \[ \int \frac {1}{x \left (c+a^2 c x^2\right ) \arctan (a x)^2} \, dx=\frac {\int \frac {1}{a^{2} x^{3} \operatorname {atan}^{2}{\left (a x \right )} + x \operatorname {atan}^{2}{\left (a x \right )}}\, dx}{c} \]

[In]

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

[Out]

Integral(1/(a**2*x**3*atan(a*x)**2 + x*atan(a*x)**2), x)/c

Maxima [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.68 \[ \int \frac {1}{x \left (c+a^2 c x^2\right ) \arctan (a x)^2} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} + c\right )} x \arctan \left (a x\right )^{2}} \,d x } \]

[In]

integrate(1/x/(a^2*c*x^2+c)/arctan(a*x)^2,x, algorithm="maxima")

[Out]

-(x*arctan(a*x)*integrate(1/(x^2*arctan(a*x)), x) + 1)/(a*c*x*arctan(a*x))

Giac [N/A]

Not integrable

Time = 33.95 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.14 \[ \int \frac {1}{x \left (c+a^2 c x^2\right ) \arctan (a x)^2} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} + c\right )} x \arctan \left (a x\right )^{2}} \,d x } \]

[In]

integrate(1/x/(a^2*c*x^2+c)/arctan(a*x)^2,x, algorithm="giac")

[Out]

sage0*x

Mupad [N/A]

Not integrable

Time = 0.40 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {1}{x \left (c+a^2 c x^2\right ) \arctan (a x)^2} \, dx=\int \frac {1}{x\,{\mathrm {atan}\left (a\,x\right )}^2\,\left (c\,a^2\,x^2+c\right )} \,d x \]

[In]

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

[Out]

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