\(\int \frac {x}{(c+a^2 c x^2) \arctan (a x)^{3/2}} \, dx\) [985]

Optimal result

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

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.03 (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 = {} \[ \int \frac {x}{\left (c+a^2 c x^2\right ) \arctan (a x)^{3/2}} \, dx=\int \frac {x}{\left (c+a^2 c x^2\right ) \arctan (a x)^{3/2}} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 0.67 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {x}{\left (c+a^2 c x^2\right ) \arctan (a x)^{3/2}} \, dx=\int \frac {x}{\left (c+a^2 c x^2\right ) \arctan (a x)^{3/2}} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.79 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91

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

[In]

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

[Out]

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

Fricas [F(-2)]

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

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 1.45 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.23 \[ \int \frac {x}{\left (c+a^2 c x^2\right ) \arctan (a x)^{3/2}} \, dx=\frac {\int \frac {x}{a^{2} x^{2} \operatorname {atan}^{\frac {3}{2}}{\left (a x \right )} + \operatorname {atan}^{\frac {3}{2}}{\left (a x \right )}}\, dx}{c} \]

[In]

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

[Out]

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

Maxima [F(-2)]

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

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 123.75 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.14 \[ \int \frac {x}{\left (c+a^2 c x^2\right ) \arctan (a x)^{3/2}} \, dx=\int { \frac {x}{{\left (a^{2} c x^{2} + c\right )} \arctan \left (a x\right )^{\frac {3}{2}}} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [N/A]

Not integrable

Time = 0.46 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {x}{\left (c+a^2 c x^2\right ) \arctan (a x)^{3/2}} \, dx=\int \frac {x}{{\mathrm {atan}\left (a\,x\right )}^{3/2}\,\left (c\,a^2\,x^2+c\right )} \,d x \]

[In]

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

[Out]

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