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

Optimal. Leaf size=21 \[ \text{Unintegrable}\left (\left (a^2 c x^2+c\right ) \sqrt{\tan ^{-1}(a x)},x\right ) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0131612, 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 = {} \[ \int \left (c+a^2 c x^2\right ) \sqrt{\tan ^{-1}(a x)} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 3.10737, size = 0, normalized size = 0. \[ \int \left (c+a^2 c x^2\right ) \sqrt{\tan ^{-1}(a x)} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.256, size = 0, normalized size = 0. \begin{align*} \int \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((a^2*c*x^2+c)*arctan(a*x)^(1/2),x)

[Out]

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

________________________________________________________________________________________

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, algorithm="maxima")

[Out]

Exception raised: RuntimeError

________________________________________________________________________________________

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, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\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((a^2*c*x^2+c)*arctan(a*x)^(1/2),x, algorithm="giac")

[Out]

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