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

Optimal. Leaf size=26 \[ \text{Unintegrable}\left (\frac{\sqrt{\tan ^{-1}(a x)}}{x \left (a^2 c x^2+c\right )^2},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.0595246, 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 = {} \[ \int \frac{\sqrt{\tan ^{-1}(a x)}}{x \left (c+a^2 c x^2\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 1.68618, size = 0, normalized size = 0. \[ \int \frac{\sqrt{\tan ^{-1}(a x)}}{x \left (c+a^2 c x^2\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.54, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x \left ({a}^{2}c{x}^{2}+c \right ) ^{2}}\sqrt{\arctan \left ( ax \right ) }}\, 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)^2,x)

[Out]

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

[Out]

Exception raised: UnboundLocalError

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{\sqrt{\operatorname{atan}{\left (a x \right )}}}{a^{4} x^{5} + 2 a^{2} x^{3} + x}\, dx}{c^{2}} \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)**2,x)

[Out]

Integral(sqrt(atan(a*x))/(a**4*x**5 + 2*a**2*x**3 + x), x)/c**2

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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 )}^{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)^2,x, algorithm="giac")

[Out]

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