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

Optimal. Leaf size=62 \[ \frac{\sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}}{a^2 c}-\frac{\text{Unintegrable}\left (\frac{1}{\sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}},x\right )}{2 a} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.108974, 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 = {} \[ \int \frac{x \sqrt{\tan ^{-1}(a x)}}{\sqrt{c+a^2 c x^2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.79745, size = 0, normalized size = 0. \[ \int \frac{x \sqrt{\tan ^{-1}(a x)}}{\sqrt{c+a^2 c x^2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 1.201, size = 0, normalized size = 0. \begin{align*} \int{x\sqrt{\arctan \left ( ax \right ) }{\frac{1}{\sqrt{{a}^{2}c{x}^{2}+c}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: UnboundLocalError

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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