\(\int \frac {1}{x^2 (c+a^2 c x^2)^3 \arctan (a x)} \, dx\) [495]

Optimal result

Integrand size = 22, antiderivative size = 22 \[ \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^3 \arctan (a x)} \, dx=\text {Int}\left (\frac {1}{x^2 \left (c+a^2 c x^2\right )^3 \arctan (a x)},x\right ) \]

[Out]

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

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

[In]

Int[1/(x^2*(c + a^2*c*x^2)^3*ArcTan[a*x]),x]

[Out]

Defer[Int][1/(x^2*(c + a^2*c*x^2)^3*ArcTan[a*x]), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^3 \arctan (a x)} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 1.12 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^3 \arctan (a x)} \, dx=\int \frac {1}{x^2 \left (c+a^2 c x^2\right )^3 \arctan (a x)} \, dx \]

[In]

Integrate[1/(x^2*(c + a^2*c*x^2)^3*ArcTan[a*x]),x]

[Out]

Integrate[1/(x^2*(c + a^2*c*x^2)^3*ArcTan[a*x]), x]

Maple [N/A] (verified)

Not integrable

Time = 4.13 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00

\[\int \frac {1}{x^{2} \left (a^{2} c \,x^{2}+c \right )^{3} \arctan \left (a x \right )}d x\]

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.23 (sec) , antiderivative size = 51, normalized size of antiderivative = 2.32 \[ \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^3 \arctan (a x)} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} + c\right )}^{3} x^{2} \arctan \left (a x\right )} \,d x } \]

[In]

integrate(1/x^2/(a^2*c*x^2+c)^3/arctan(a*x),x, algorithm="fricas")

[Out]

integral(1/((a^6*c^3*x^8 + 3*a^4*c^3*x^6 + 3*a^2*c^3*x^4 + c^3*x^2)*arctan(a*x)), x)

Sympy [N/A]

Not integrable

Time = 1.73 (sec) , antiderivative size = 53, normalized size of antiderivative = 2.41 \[ \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^3 \arctan (a x)} \, dx=\frac {\int \frac {1}{a^{6} x^{8} \operatorname {atan}{\left (a x \right )} + 3 a^{4} x^{6} \operatorname {atan}{\left (a x \right )} + 3 a^{2} x^{4} \operatorname {atan}{\left (a x \right )} + x^{2} \operatorname {atan}{\left (a x \right )}}\, dx}{c^{3}} \]

[In]

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

[Out]

Integral(1/(a**6*x**8*atan(a*x) + 3*a**4*x**6*atan(a*x) + 3*a**2*x**4*atan(a*x) + x**2*atan(a*x)), x)/c**3

Maxima [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^3 \arctan (a x)} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} + c\right )}^{3} x^{2} \arctan \left (a x\right )} \,d x } \]

[In]

integrate(1/x^2/(a^2*c*x^2+c)^3/arctan(a*x),x, algorithm="maxima")

[Out]

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

Giac [N/A]

Not integrable

Time = 53.32 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.14 \[ \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^3 \arctan (a x)} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} + c\right )}^{3} x^{2} \arctan \left (a x\right )} \,d x } \]

[In]

integrate(1/x^2/(a^2*c*x^2+c)^3/arctan(a*x),x, algorithm="giac")

[Out]

sage0*x

Mupad [N/A]

Not integrable

Time = 0.42 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^3 \arctan (a x)} \, dx=\int \frac {1}{x^2\,\mathrm {atan}\left (a\,x\right )\,{\left (c\,a^2\,x^2+c\right )}^3} \,d x \]

[In]

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

[Out]

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