\(\int \frac {c+a^2 c x^2}{x \arctan (a x)} \, dx\) [468]

Optimal result

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

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.03 (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 = {} \[ \int \frac {c+a^2 c x^2}{x \arctan (a x)} \, dx=\int \frac {c+a^2 c x^2}{x \arctan (a x)} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 0.57 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {c+a^2 c x^2}{x \arctan (a x)} \, dx=\int \frac {c+a^2 c x^2}{x \arctan (a x)} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 8.87 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

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

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 0.92 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {c+a^2 c x^2}{x \arctan (a x)} \, dx=c \left (\int \frac {1}{x \operatorname {atan}{\left (a x \right )}}\, dx + \int \frac {a^{2} x}{\operatorname {atan}{\left (a x \right )}}\, dx\right ) \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {c+a^2 c x^2}{x \arctan (a x)} \, dx=\int { \frac {a^{2} c x^{2} + c}{x \arctan \left (a x\right )} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 30.29 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.15 \[ \int \frac {c+a^2 c x^2}{x \arctan (a x)} \, dx=\int { \frac {a^{2} c x^{2} + c}{x \arctan \left (a x\right )} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [N/A]

Not integrable

Time = 0.44 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {c+a^2 c x^2}{x \arctan (a x)} \, dx=\int \frac {c\,a^2\,x^2+c}{x\,\mathrm {atan}\left (a\,x\right )} \,d x \]

[In]

int((c + a^2*c*x^2)/(x*atan(a*x)),x)

[Out]

int((c + a^2*c*x^2)/(x*atan(a*x)), x)