\(\int \frac {x^2}{(c+a^2 c x^2) \arctan (a x)^3} \, dx\) [620]

Optimal result

Integrand size = 22, antiderivative size = 22 \[ \int \frac {x^2}{\left (c+a^2 c x^2\right ) \arctan (a x)^3} \, dx=-\frac {x^2}{2 a c \arctan (a x)^2}+\frac {\text {Int}\left (\frac {x}{\arctan (a x)^2},x\right )}{a c} \]

[Out]

-1/2*x^2/a/c/arctan(a*x)^2+Unintegrable(x/arctan(a*x)^2,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 = {} \[ \int \frac {x^2}{\left (c+a^2 c x^2\right ) \arctan (a x)^3} \, dx=\int \frac {x^2}{\left (c+a^2 c x^2\right ) \arctan (a x)^3} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 0.52 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {x^2}{\left (c+a^2 c x^2\right ) \arctan (a x)^3} \, dx=\int \frac {x^2}{\left (c+a^2 c x^2\right ) \arctan (a x)^3} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 6.82 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {x^2}{\left (c+a^2 c x^2\right ) \arctan (a x)^3} \, dx=\int { \frac {x^{2}}{{\left (a^{2} c x^{2} + c\right )} \arctan \left (a x\right )^{3}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 0.69 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.18 \[ \int \frac {x^2}{\left (c+a^2 c x^2\right ) \arctan (a x)^3} \, dx=\frac {\int \frac {x^{2}}{a^{2} x^{2} \operatorname {atan}^{3}{\left (a x \right )} + \operatorname {atan}^{3}{\left (a x \right )}}\, dx}{c} \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.82 \[ \int \frac {x^2}{\left (c+a^2 c x^2\right ) \arctan (a x)^3} \, dx=\int { \frac {x^{2}}{{\left (a^{2} c x^{2} + c\right )} \arctan \left (a x\right )^{3}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 61.92 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.14 \[ \int \frac {x^2}{\left (c+a^2 c x^2\right ) \arctan (a x)^3} \, dx=\int { \frac {x^{2}}{{\left (a^{2} c x^{2} + c\right )} \arctan \left (a x\right )^{3}} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [N/A]

Not integrable

Time = 0.44 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {x^2}{\left (c+a^2 c x^2\right ) \arctan (a x)^3} \, dx=\int \frac {x^2}{{\mathrm {atan}\left (a\,x\right )}^3\,\left (c\,a^2\,x^2+c\right )} \,d x \]

[In]

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

[Out]

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