3.8.48 \(\int \frac {\sqrt {\text {ArcTan}(a x)}}{x^2 (c+a^2 c x^2)^{3/2}} \, dx\) [748]

Optimal. Leaf size=29 \[ \text {Int}\left (\frac {\sqrt {\text {ArcTan}(a x)}}{x^2 \left (c+a^2 c x^2\right )^{3/2}},x\right ) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.08, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\sqrt {\text {ArcTan}(a x)}}{x^2 \left (c+a^2 c x^2\right )^{3/2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin {align*} \int \frac {\sqrt {\tan ^{-1}(a x)}}{x^2 \left (c+a^2 c x^2\right )^{3/2}} \, dx &=\int \frac {\sqrt {\tan ^{-1}(a x)}}{x^2 \left (c+a^2 c x^2\right )^{3/2}} \, dx\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 3.63, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {\text {ArcTan}(a x)}}{x^2 \left (c+a^2 c x^2\right )^{3/2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]
time = 0.81, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {\arctan \left (a x \right )}}{x^{2} \left (a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arctan(a*x)^(1/2)/x^2/(a^2*c*x^2+c)^(3/2),x)

[Out]

int(arctan(a*x)^(1/2)/x^2/(a^2*c*x^2+c)^(3/2),x)

________________________________________________________________________________________

Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctan(a*x)^(1/2)/x^2/(a^2*c*x^2+c)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: Error executing code in Maxima: expt: undefined: 0 to a negative e
xponent.

________________________________________________________________________________________

Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctan(a*x)^(1/2)/x^2/(a^2*c*x^2+c)^(3/2),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {\operatorname {atan}{\left (a x \right )}}}{x^{2} \left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(atan(a*x)**(1/2)/x**2/(a**2*c*x**2+c)**(3/2),x)

[Out]

Integral(sqrt(atan(a*x))/(x**2*(c*(a**2*x**2 + 1))**(3/2)), x)

________________________________________________________________________________________

Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctan(a*x)^(1/2)/x^2/(a^2*c*x^2+c)^(3/2),x, algorithm="giac")

[Out]

sage0*x

________________________________________________________________________________________

Mupad [A]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {\sqrt {\mathrm {atan}\left (a\,x\right )}}{x^2\,{\left (c\,a^2\,x^2+c\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(atan(a*x)^(1/2)/(x^2*(c + a^2*c*x^2)^(3/2)),x)

[Out]

int(atan(a*x)^(1/2)/(x^2*(c + a^2*c*x^2)^(3/2)), x)

________________________________________________________________________________________