\(\int \sqrt {\arctan (a x)} \, dx\) [42]

Optimal result

Integrand size = 8, antiderivative size = 8 \[ \int \sqrt {\arctan (a x)} \, dx=\text {Int}\left (\sqrt {\arctan (a x)},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.00 (sec) , antiderivative size = 8, 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 = {} \[ \int \sqrt {\arctan (a x)} \, dx=\int \sqrt {\arctan (a x)} \, dx \]

[In]

Int[Sqrt[ArcTan[a*x]],x]

[Out]

Defer[Int][Sqrt[ArcTan[a*x]], x]

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

Mathematica [N/A]

Not integrable

Time = 1.81 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.25 \[ \int \sqrt {\arctan (a x)} \, dx=\int \sqrt {\arctan (a x)} \, dx \]

[In]

Integrate[Sqrt[ArcTan[a*x]],x]

[Out]

Integrate[Sqrt[ArcTan[a*x]], x]

Maple [N/A] (verified)

Not integrable

Time = 1.64 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.75

\[\int \sqrt {\arctan \left (a x \right )}d x\]

[In]

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

[Out]

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

Fricas [F(-2)]

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

[In]

integrate(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 = 0.32 (sec) , antiderivative size = 8, normalized size of antiderivative = 1.00 \[ \int \sqrt {\arctan (a x)} \, dx=\int \sqrt {\operatorname {atan}{\left (a x \right )}}\, dx \]

[In]

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

[Out]

Integral(sqrt(atan(a*x)), x)

Maxima [F(-2)]

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

[In]

integrate(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 = 53.95 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.38 \[ \int \sqrt {\arctan (a x)} \, dx=\int { \sqrt {\arctan \left (a x\right )} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [N/A]

Not integrable

Time = 0.33 (sec) , antiderivative size = 8, normalized size of antiderivative = 1.00 \[ \int \sqrt {\arctan (a x)} \, dx=\int \sqrt {\mathrm {atan}\left (a\,x\right )} \,d x \]

[In]

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

[Out]

int(atan(a*x)^(1/2), x)