3.747 \(\int \frac{\sqrt{\tan ^{-1}(a x)}}{x (c+a^2 c x^2)^{3/2}} \, dx\)

Optimal. Leaf size=28 \[ \text{Unintegrable}\left (\frac{\sqrt{\tan ^{-1}(a x)}}{x \left (a^2 c x^2+c\right )^{3/2}},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.116241, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{\sqrt{\tan ^{-1}(a x)}}{x \left (c+a^2 c x^2\right )^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 1.94337, size = 0, normalized size = 0. \[ \int \frac{\sqrt{\tan ^{-1}(a x)}}{x \left (c+a^2 c x^2\right )^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.733, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x}\sqrt{\arctan \left ( ax \right ) } \left ({a}^{2}c{x}^{2}+c \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: UnboundLocalError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{\arctan \left (a x\right )}}{{\left (a^{2} c x^{2} + c\right )}^{\frac{3}{2}} x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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