\(\int \frac {1}{x (c+a^2 c x^2)^3 \arctan (a x)} \, dx\) [494]

Optimal result

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

[Out]

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

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 0.75 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {1}{x \left (c+a^2 c x^2\right )^3 \arctan (a x)} \, dx=\int \frac {1}{x \left (c+a^2 c x^2\right )^3 \arctan (a x)} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 4.45 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 49, normalized size of antiderivative = 2.23 \[ \int \frac {1}{x \left (c+a^2 c x^2\right )^3 \arctan (a x)} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} + c\right )}^{3} x \arctan \left (a x\right )} \,d x } \]

[In]

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

[Out]

integral(1/((a^6*c^3*x^7 + 3*a^4*c^3*x^5 + 3*a^2*c^3*x^3 + c^3*x)*arctan(a*x)), x)

Sympy [N/A]

Not integrable

Time = 1.62 (sec) , antiderivative size = 51, normalized size of antiderivative = 2.32 \[ \int \frac {1}{x \left (c+a^2 c x^2\right )^3 \arctan (a x)} \, dx=\frac {\int \frac {1}{a^{6} x^{7} \operatorname {atan}{\left (a x \right )} + 3 a^{4} x^{5} \operatorname {atan}{\left (a x \right )} + 3 a^{2} x^{3} \operatorname {atan}{\left (a x \right )} + x \operatorname {atan}{\left (a x \right )}}\, dx}{c^{3}} \]

[In]

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

[Out]

Integral(1/(a**6*x**7*atan(a*x) + 3*a**4*x**5*atan(a*x) + 3*a**2*x**3*atan(a*x) + x*atan(a*x)), x)/c**3

Maxima [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {1}{x \left (c+a^2 c x^2\right )^3 \arctan (a x)} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} + c\right )}^{3} x \arctan \left (a x\right )} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 49.84 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.14 \[ \int \frac {1}{x \left (c+a^2 c x^2\right )^3 \arctan (a x)} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} + c\right )}^{3} x \arctan \left (a x\right )} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [N/A]

Not integrable

Time = 0.40 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {1}{x \left (c+a^2 c x^2\right )^3 \arctan (a x)} \, dx=\int \frac {1}{x\,\mathrm {atan}\left (a\,x\right )\,{\left (c\,a^2\,x^2+c\right )}^3} \,d x \]

[In]

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

[Out]

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