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

Optimal. Leaf size=24 \[ \text{Unintegrable}\left (\frac{\left (a^2 c x^2+c\right ) \sqrt{\tan ^{-1}(a x)}}{x},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.039548, 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 = {} \[ \int \frac{\left (c+a^2 c x^2\right ) \sqrt{\tan ^{-1}(a x)}}{x} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 1.61191, size = 0, normalized size = 0. \[ \int \frac{\left (c+a^2 c x^2\right ) \sqrt{\tan ^{-1}(a x)}}{x} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.401, size = 0, normalized size = 0. \begin{align*} \int{\frac{{a}^{2}c{x}^{2}+c}{x}\sqrt{\arctan \left ( ax \right ) }}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: UnboundLocalError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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