\(\int \frac {1}{x^4 (c+a^2 c x^2) \arctan (a x)^2} \, dx\) [550]

Optimal result

Integrand size = 22, antiderivative size = 22 \[ \int \frac {1}{x^4 \left (c+a^2 c x^2\right ) \arctan (a x)^2} \, dx=-\frac {1}{a c x^4 \arctan (a x)}-\frac {4 \text {Int}\left (\frac {1}{x^5 \arctan (a x)},x\right )}{a c} \]

[Out]

-1/a/c/x^4/arctan(a*x)-4*Unintegrable(1/x^5/arctan(a*x),x)/a/c

Rubi [N/A]

Not integrable

Time = 0.06 (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 = {} \[ \int \frac {1}{x^4 \left (c+a^2 c x^2\right ) \arctan (a x)^2} \, dx=\int \frac {1}{x^4 \left (c+a^2 c x^2\right ) \arctan (a x)^2} \, dx \]

[In]

Int[1/(x^4*(c + a^2*c*x^2)*ArcTan[a*x]^2),x]

[Out]

-(1/(a*c*x^4*ArcTan[a*x])) - (4*Defer[Int][1/(x^5*ArcTan[a*x]), x])/(a*c)

Rubi steps \begin{align*} \text {integral}& = -\frac {1}{a c x^4 \arctan (a x)}-\frac {4 \int \frac {1}{x^5 \arctan (a x)} \, dx}{a c} \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 1.13 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {1}{x^4 \left (c+a^2 c x^2\right ) \arctan (a x)^2} \, dx=\int \frac {1}{x^4 \left (c+a^2 c x^2\right ) \arctan (a x)^2} \, dx \]

[In]

Integrate[1/(x^4*(c + a^2*c*x^2)*ArcTan[a*x]^2),x]

[Out]

Integrate[1/(x^4*(c + a^2*c*x^2)*ArcTan[a*x]^2), x]

Maple [N/A] (verified)

Not integrable

Time = 24.22 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00

\[\int \frac {1}{x^{4} \left (a^{2} c \,x^{2}+c \right ) \arctan \left (a x \right )^{2}}d x\]

[In]

int(1/x^4/(a^2*c*x^2+c)/arctan(a*x)^2,x)

[Out]

int(1/x^4/(a^2*c*x^2+c)/arctan(a*x)^2,x)

Fricas [N/A]

Not integrable

Time = 0.23 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.14 \[ \int \frac {1}{x^4 \left (c+a^2 c x^2\right ) \arctan (a x)^2} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} + c\right )} x^{4} \arctan \left (a x\right )^{2}} \,d x } \]

[In]

integrate(1/x^4/(a^2*c*x^2+c)/arctan(a*x)^2,x, algorithm="fricas")

[Out]

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

Sympy [N/A]

Not integrable

Time = 0.93 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.23 \[ \int \frac {1}{x^4 \left (c+a^2 c x^2\right ) \arctan (a x)^2} \, dx=\frac {\int \frac {1}{a^{2} x^{6} \operatorname {atan}^{2}{\left (a x \right )} + x^{4} \operatorname {atan}^{2}{\left (a x \right )}}\, dx}{c} \]

[In]

integrate(1/x**4/(a**2*c*x**2+c)/atan(a*x)**2,x)

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.82 \[ \int \frac {1}{x^4 \left (c+a^2 c x^2\right ) \arctan (a x)^2} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} + c\right )} x^{4} \arctan \left (a x\right )^{2}} \,d x } \]

[In]

integrate(1/x^4/(a^2*c*x^2+c)/arctan(a*x)^2,x, algorithm="maxima")

[Out]

-(4*x^4*arctan(a*x)*integrate(1/(x^5*arctan(a*x)), x) + 1)/(a*c*x^4*arctan(a*x))

Giac [N/A]

Not integrable

Time = 42.95 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.14 \[ \int \frac {1}{x^4 \left (c+a^2 c x^2\right ) \arctan (a x)^2} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} + c\right )} x^{4} \arctan \left (a x\right )^{2}} \,d x } \]

[In]

integrate(1/x^4/(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 \[ \int \frac {1}{x^4 \left (c+a^2 c x^2\right ) \arctan (a x)^2} \, dx=\int \frac {1}{x^4\,{\mathrm {atan}\left (a\,x\right )}^2\,\left (c\,a^2\,x^2+c\right )} \,d x \]

[In]

int(1/(x^4*atan(a*x)^2*(c + a^2*c*x^2)),x)

[Out]

int(1/(x^4*atan(a*x)^2*(c + a^2*c*x^2)), x)