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

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

[Out]

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

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Rubi [A]  time = 0.0595416, 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 = {} \[ \int \frac{1}{x \left (c+a^2 c x^2\right )^2 \sqrt{\tan ^{-1}(a x)}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 1.35283, size = 0, normalized size = 0. \[ \int \frac{1}{x \left (c+a^2 c x^2\right )^2 \sqrt{\tan ^{-1}(a x)}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.688, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x \left ({a}^{2}c{x}^{2}+c \right ) ^{2}}{\frac{1}{\sqrt{\arctan \left ( ax \right ) }}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(1/x/(a^2*c*x^2+c)^2/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(1/x/(a^2*c*x^2+c)^2/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(1/x/(a^2*c*x^2+c)^2/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{1}{a^{4} x^{5} \sqrt{\operatorname{atan}{\left (a x \right )}} + 2 a^{2} x^{3} \sqrt{\operatorname{atan}{\left (a x \right )}} + x \sqrt{\operatorname{atan}{\left (a x \right )}}}\, dx}{c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(a**4*x**5*sqrt(atan(a*x)) + 2*a**2*x**3*sqrt(atan(a*x)) + x*sqrt(atan(a*x))), x)/c**2

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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