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

Optimal. Leaf size=64 \[ \frac{1}{2} a \text{Unintegrable}\left (\frac{1}{x \sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}},x\right )-\frac{\sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}}{c x} \]

[Out]

-((Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]])/(c*x)) + (a*Unintegrable[1/(x*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]])
, x])/2

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Rubi [A]  time = 0.21421, 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 = {} \[ \int \frac{\sqrt{\tan ^{-1}(a x)}}{x^2 \sqrt{c+a^2 c x^2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

-((Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]])/(c*x)) + (a*Defer[Int][1/(x*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]),
x])/2

Rubi steps

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

Mathematica [A]  time = 1.73093, size = 0, normalized size = 0. \[ \int \frac{\sqrt{\tan ^{-1}(a x)}}{x^2 \sqrt{c+a^2 c x^2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.801, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2}}\sqrt{\arctan \left ( ax \right ) }{\frac{1}{\sqrt{{a}^{2}c{x}^{2}+c}}}}\, dx \end{align*}

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)^(1/2),x)

[Out]

int(arctan(a*x)^(1/2)/x^2/(a^2*c*x^2+c)^(1/2),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^2/(a^2*c*x^2+c)^(1/2),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^2/(a^2*c*x^2+c)^(1/2),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^{2} \sqrt{c \left (a^{2} x^{2} + 1\right )}}\, dx \end{align*}

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)**(1/2),x)

[Out]

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

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

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)^(1/2),x, algorithm="giac")

[Out]

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