\(\int \frac {\sqrt {\arctan (a x)}}{x (c+a^2 c x^2)} \, dx\) [703]

Optimal result

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

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.08 (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 = {} \[ \int \frac {\sqrt {\arctan (a x)}}{x \left (c+a^2 c x^2\right )} \, dx=\int \frac {\sqrt {\arctan (a x)}}{x \left (c+a^2 c x^2\right )} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 0.33 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {\sqrt {\arctan (a x)}}{x \left (c+a^2 c x^2\right )} \, dx=\int \frac {\sqrt {\arctan (a x)}}{x \left (c+a^2 c x^2\right )} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 3.86 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92

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

[In]

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

[Out]

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

Fricas [F(-2)]

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

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 0.96 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.79 \[ \int \frac {\sqrt {\arctan (a x)}}{x \left (c+a^2 c x^2\right )} \, dx=\frac {\int \frac {\sqrt {\operatorname {atan}{\left (a x \right )}}}{a^{2} x^{3} + x}\, dx}{c} \]

[In]

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

[Out]

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

Maxima [F(-2)]

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

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 24.11 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.12 \[ \int \frac {\sqrt {\arctan (a x)}}{x \left (c+a^2 c x^2\right )} \, dx=\int { \frac {\sqrt {\arctan \left (a x\right )}}{{\left (a^{2} c x^{2} + c\right )} x} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [N/A]

Not integrable

Time = 0.47 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {\arctan (a x)}}{x \left (c+a^2 c x^2\right )} \, dx=\int \frac {\sqrt {\mathrm {atan}\left (a\,x\right )}}{x\,\left (c\,a^2\,x^2+c\right )} \,d x \]

[In]

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

[Out]

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