\(\int \frac {\arctan (a x)^{3/2}}{x (c+a^2 c x^2)^2} \, dx\) [788]

Optimal result

Integrand size = 24, antiderivative size = 24 \[ \int \frac {\arctan (a x)^{3/2}}{x \left (c+a^2 c x^2\right )^2} \, dx=\text {Int}\left (\frac {\arctan (a x)^{3/2}}{x \left (c+a^2 c x^2\right )^2},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.05 (sec) , antiderivative size = 24, 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 = {} \[ \int \frac {\arctan (a x)^{3/2}}{x \left (c+a^2 c x^2\right )^2} \, dx=\int \frac {\arctan (a x)^{3/2}}{x \left (c+a^2 c x^2\right )^2} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 0.89 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {\arctan (a x)^{3/2}}{x \left (c+a^2 c x^2\right )^2} \, dx=\int \frac {\arctan (a x)^{3/2}}{x \left (c+a^2 c x^2\right )^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 4.59 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92

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

[In]

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

[Out]

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

Fricas [F(-2)]

Exception generated. \[ \int \frac {\arctan (a x)^{3/2}}{x \left (c+a^2 c x^2\right )^2} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

Sympy [N/A]

Not integrable

Time = 2.41 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.21 \[ \int \frac {\arctan (a x)^{3/2}}{x \left (c+a^2 c x^2\right )^2} \, dx=\frac {\int \frac {\operatorname {atan}^{\frac {3}{2}}{\left (a x \right )}}{a^{4} x^{5} + 2 a^{2} x^{3} + x}\, dx}{c^{2}} \]

[In]

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

[Out]

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {\arctan (a x)^{3/2}}{x \left (c+a^2 c x^2\right )^2} \, dx=\text {Exception raised: RuntimeError} \]

[In]

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

[Out]

Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negative exponent.

Giac [N/A]

Not integrable

Time = 82.79 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.12 \[ \int \frac {\arctan (a x)^{3/2}}{x \left (c+a^2 c x^2\right )^2} \, dx=\int { \frac {\arctan \left (a x\right )^{\frac {3}{2}}}{{\left (a^{2} c x^{2} + c\right )}^{2} x} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [N/A]

Not integrable

Time = 0.53 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {\arctan (a x)^{3/2}}{x \left (c+a^2 c x^2\right )^2} \, dx=\int \frac {{\mathrm {atan}\left (a\,x\right )}^{3/2}}{x\,{\left (c\,a^2\,x^2+c\right )}^2} \,d x \]

[In]

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

[Out]

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