3.738 \(\int \frac {\sqrt {\tan ^{-1}(a x)}}{x \sqrt {c+a^2 c x^2}} \, dx\)

Optimal. Leaf size=29 \[ \text {Int}\left (\frac {\sqrt {\tan ^{-1}(a x)}}{x \sqrt {a^2 c x^2+c}},x\right ) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.11, 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 = {} \[ \int \frac {\sqrt {\tan ^{-1}(a x)}}{x \sqrt {c+a^2 c x^2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 1.22, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\tan ^{-1}(a x)}}{x \sqrt {c+a^2 c x^2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sage0*x

________________________________________________________________________________________

maple [A]  time = 1.73, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\arctan \left (a x \right )}}{x \sqrt {a^{2} c \,x^{2}+c}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [A]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {\sqrt {\mathrm {atan}\left (a\,x\right )}}{x\,\sqrt {c\,a^2\,x^2+c}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________