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

Optimal. Leaf size=72 \[ -\frac{2 x}{c \sqrt{a^2 c x^2+c}}+\frac{x \tan ^{-1}(a x)^2}{c \sqrt{a^2 c x^2+c}}+\frac{2 \tan ^{-1}(a x)}{a c \sqrt{a^2 c x^2+c}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0455449, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {4898, 191} \[ -\frac{2 x}{c \sqrt{a^2 c x^2+c}}+\frac{x \tan ^{-1}(a x)^2}{c \sqrt{a^2 c x^2+c}}+\frac{2 \tan ^{-1}(a x)}{a c \sqrt{a^2 c x^2+c}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 4898

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_)/((d_) + (e_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[(b*p*(a + b*ArcTan[
c*x])^(p - 1))/(c*d*Sqrt[d + e*x^2]), x] + (-Dist[b^2*p*(p - 1), Int[(a + b*ArcTan[c*x])^(p - 2)/(d + e*x^2)^(
3/2), x], x] + Simp[(x*(a + b*ArcTan[c*x])^p)/(d*Sqrt[d + e*x^2]), x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e,
c^2*d] && GtQ[p, 1]

Rule 191

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^(p + 1))/a, x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rubi steps

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

Mathematica [A]  time = 0.062047, size = 49, normalized size = 0.68 \[ \frac{\sqrt{a^2 c x^2+c} \left (-2 a x+a x \tan ^{-1}(a x)^2+2 \tan ^{-1}(a x)\right )}{c^2 \left (a^3 x^2+a\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [C]  time = 0.223, size = 114, normalized size = 1.6 \begin{align*}{\frac{ \left ( \left ( \arctan \left ( ax \right ) \right ) ^{2}-2+2\,i\arctan \left ( ax \right ) \right ) \left ( ax-i \right ) }{ \left ( 2\,{a}^{2}{x}^{2}+2 \right ){c}^{2}a}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}+{\frac{ \left ( ax+i \right ) \left ( \left ( \arctan \left ( ax \right ) \right ) ^{2}-2-2\,i\arctan \left ( ax \right ) \right ) }{ \left ( 2\,{a}^{2}{x}^{2}+2 \right ){c}^{2}a}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(arctan(a*x)^2-2+2*I*arctan(a*x))*(a*x-I)*(c*(a*x-I)*(a*x+I))^(1/2)/(a^2*x^2+1)/c^2/a+1/2*(c*(a*x-I)*(a*x+
I))^(1/2)*(a*x+I)*(arctan(a*x)^2-2-2*I*arctan(a*x))/(a^2*x^2+1)/c^2/a

________________________________________________________________________________________

Maxima [A]  time = 1.76498, size = 72, normalized size = 1. \begin{align*} \frac{x \arctan \left (a x\right )^{2}}{\sqrt{a^{2} c x^{2} + c} c} - \frac{2 \,{\left (a x - \arctan \left (a x\right )\right )}}{\sqrt{a^{2} x^{2} + 1} a c^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*arctan(a*x)^2/(sqrt(a^2*c*x^2 + c)*c) - 2*(a*x - arctan(a*x))/(sqrt(a^2*x^2 + 1)*a*c^(3/2))

________________________________________________________________________________________

Fricas [A]  time = 2.29909, size = 117, normalized size = 1.62 \begin{align*} \frac{\sqrt{a^{2} c x^{2} + c}{\left (a x \arctan \left (a x\right )^{2} - 2 \, a x + 2 \, \arctan \left (a x\right )\right )}}{a^{3} c^{2} x^{2} + a c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.2316, size = 97, normalized size = 1.35 \begin{align*} -2 \, a{\left (\frac{x}{\sqrt{a^{2} c x^{2} + c} a c} - \frac{\arctan \left (a x\right )}{\sqrt{a^{2} c x^{2} + c} a^{2} c}\right )} + \frac{x \arctan \left (a x\right )^{2}}{\sqrt{a^{2} c x^{2} + c} c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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