3.646 \(\int \frac{\sqrt{c+a^2 c x^2}}{x \tan ^{-1}(a x)^3} \, dx\)

Optimal. Leaf size=26 \[ \text{Unintegrable}\left (\frac{\sqrt{a^2 c x^2+c}}{x \tan ^{-1}(a x)^3},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.115324, 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 = {} \[ \int \frac{\sqrt{c+a^2 c x^2}}{x \tan ^{-1}(a x)^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 3.37709, size = 0, normalized size = 0. \[ \int \frac{\sqrt{c+a^2 c x^2}}{x \tan ^{-1}(a x)^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.877, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x \left ( \arctan \left ( ax \right ) \right ) ^{3}}\sqrt{{a}^{2}c{x}^{2}+c}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a^2*c*x^2+c)^(1/2)/x/arctan(a*x)^3,x)

[Out]

int((a^2*c*x^2+c)^(1/2)/x/arctan(a*x)^3,x)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{a^{2} c x^{2} + c}}{x \arctan \left (a x\right )^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a^2*c*x^2+c)^(1/2)/x/arctan(a*x)^3,x, algorithm="maxima")

[Out]

integrate(sqrt(a^2*c*x^2 + c)/(x*arctan(a*x)^3), x)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{a^{2} c x^{2} + c}}{x \arctan \left (a x\right )^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a^2*c*x^2+c)^(1/2)/x/arctan(a*x)^3,x, algorithm="fricas")

[Out]

integral(sqrt(a^2*c*x^2 + c)/(x*arctan(a*x)^3), x)

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{c \left (a^{2} x^{2} + 1\right )}}{x \operatorname{atan}^{3}{\left (a x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a**2*c*x**2+c)**(1/2)/x/atan(a*x)**3,x)

[Out]

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

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{a^{2} c x^{2} + c}}{x \arctan \left (a x\right )^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a^2*c*x^2+c)^(1/2)/x/arctan(a*x)^3,x, algorithm="giac")

[Out]

integrate(sqrt(a^2*c*x^2 + c)/(x*arctan(a*x)^3), x)