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

Optimal. Leaf size=35 \[ \frac{\text{Unintegrable}\left (\frac{\sqrt{\tan ^{-1}(a x)}}{x^2},x\right )}{c}-\frac{2 a \tan ^{-1}(a x)^{3/2}}{3 c} \]

[Out]

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

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Rubi [A]  time = 0.103873, 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 = {} \[ \int \frac{\sqrt{\tan ^{-1}(a x)}}{x^2 \left (c+a^2 c x^2\right )} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 1.45247, size = 0, normalized size = 0. \[ \int \frac{\sqrt{\tan ^{-1}(a x)}}{x^2 \left (c+a^2 c x^2\right )} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.248, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2} \left ({a}^{2}c{x}^{2}+c \right ) }\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^2/(a^2*c*x^2+c),x)

[Out]

int(arctan(a*x)^(1/2)/x^2/(a^2*c*x^2+c),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),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),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^{2} x^{4} + x^{2}}\, dx}{c} \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),x)

[Out]

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

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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 )} 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),x, algorithm="giac")

[Out]

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