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

Optimal. Leaf size=36 \[ \frac{2 x \sqrt{\tan ^{-1}(a x)}}{a c}-\frac{2 \text{Unintegrable}\left (\sqrt{\tan ^{-1}(a x)},x\right )}{a c} \]

[Out]

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

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Rubi [A]  time = 0.0490175, 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 = {} \[ \int \frac{x}{\left (c+a^2 c x^2\right ) \sqrt{\tan ^{-1}(a x)}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.81425, size = 0, normalized size = 0. \[ \int \frac{x}{\left (c+a^2 c x^2\right ) \sqrt{\tan ^{-1}(a x)}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.148, size = 0, normalized size = 0. \begin{align*} \int{\frac{x}{{a}^{2}c{x}^{2}+c}{\frac{1}{\sqrt{\arctan \left ( ax \right ) }}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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