\(\int \frac {\sqrt {c+a^2 c x^2}}{x \sqrt {\arctan (a x)}} \, dx\) [947]

Optimal result

Integrand size = 26, antiderivative size = 26 \[ \int \frac {\sqrt {c+a^2 c x^2}}{x \sqrt {\arctan (a x)}} \, dx=\text {Int}\left (\frac {\sqrt {c+a^2 c x^2}}{x \sqrt {\arctan (a x)}},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.07 (sec) , antiderivative size = 26, 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 = {} \[ \int \frac {\sqrt {c+a^2 c x^2}}{x \sqrt {\arctan (a x)}} \, dx=\int \frac {\sqrt {c+a^2 c x^2}}{x \sqrt {\arctan (a x)}} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 4.42 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \[ \int \frac {\sqrt {c+a^2 c x^2}}{x \sqrt {\arctan (a x)}} \, dx=\int \frac {\sqrt {c+a^2 c x^2}}{x \sqrt {\arctan (a x)}} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 2.65 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85

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

[In]

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

[Out]

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

Fricas [F(-2)]

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

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 2.09 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \frac {\sqrt {c+a^2 c x^2}}{x \sqrt {\arctan (a x)}} \, dx=\int \frac {\sqrt {c \left (a^{2} x^{2} + 1\right )}}{x \sqrt {\operatorname {atan}{\left (a x \right )}}}\, dx \]

[In]

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

[Out]

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

Maxima [F(-2)]

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

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 129.14 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.12 \[ \int \frac {\sqrt {c+a^2 c x^2}}{x \sqrt {\arctan (a x)}} \, dx=\int { \frac {\sqrt {a^{2} c x^{2} + c}}{x \sqrt {\arctan \left (a x\right )}} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [N/A]

Not integrable

Time = 0.35 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \frac {\sqrt {c+a^2 c x^2}}{x \sqrt {\arctan (a x)}} \, dx=\int \frac {\sqrt {c\,a^2\,x^2+c}}{x\,\sqrt {\mathrm {atan}\left (a\,x\right )}} \,d x \]

[In]

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

[Out]

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