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

Optimal. Leaf size=107 \[ -\frac{6 x}{a c \sqrt{a^2 c x^2+c}}-\frac{\tan ^{-1}(a x)^3}{a^2 c \sqrt{a^2 c x^2+c}}+\frac{3 x \tan ^{-1}(a x)^2}{a c \sqrt{a^2 c x^2+c}}+\frac{6 \tan ^{-1}(a x)}{a^2 c \sqrt{a^2 c x^2+c}} \]

[Out]

(-6*x)/(a*c*Sqrt[c + a^2*c*x^2]) + (6*ArcTan[a*x])/(a^2*c*Sqrt[c + a^2*c*x^2]) + (3*x*ArcTan[a*x]^2)/(a*c*Sqrt
[c + a^2*c*x^2]) - ArcTan[a*x]^3/(a^2*c*Sqrt[c + a^2*c*x^2])

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Rubi [A]  time = 0.13112, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {4930, 4898, 191} \[ -\frac{6 x}{a c \sqrt{a^2 c x^2+c}}-\frac{\tan ^{-1}(a x)^3}{a^2 c \sqrt{a^2 c x^2+c}}+\frac{3 x \tan ^{-1}(a x)^2}{a c \sqrt{a^2 c x^2+c}}+\frac{6 \tan ^{-1}(a x)}{a^2 c \sqrt{a^2 c x^2+c}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-6*x)/(a*c*Sqrt[c + a^2*c*x^2]) + (6*ArcTan[a*x])/(a^2*c*Sqrt[c + a^2*c*x^2]) + (3*x*ArcTan[a*x]^2)/(a*c*Sqrt
[c + a^2*c*x^2]) - ArcTan[a*x]^3/(a^2*c*Sqrt[c + a^2*c*x^2])

Rule 4930

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_)*((d_) + (e_.)*(x_)^2)^(q_.), x_Symbol] :> Simp[((d + e*x^2)^
(q + 1)*(a + b*ArcTan[c*x])^p)/(2*e*(q + 1)), x] - Dist[(b*p)/(2*c*(q + 1)), Int[(d + e*x^2)^q*(a + b*ArcTan[c
*x])^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, q}, x] && EqQ[e, c^2*d] && GtQ[p, 0] && NeQ[q, -1]

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{x \tan ^{-1}(a x)^3}{\left (c+a^2 c x^2\right )^{3/2}} \, dx &=-\frac{\tan ^{-1}(a x)^3}{a^2 c \sqrt{c+a^2 c x^2}}+\frac{3 \int \frac{\tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{a}\\ &=\frac{6 \tan ^{-1}(a x)}{a^2 c \sqrt{c+a^2 c x^2}}+\frac{3 x \tan ^{-1}(a x)^2}{a c \sqrt{c+a^2 c x^2}}-\frac{\tan ^{-1}(a x)^3}{a^2 c \sqrt{c+a^2 c x^2}}-\frac{6 \int \frac{1}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{a}\\ &=-\frac{6 x}{a c \sqrt{c+a^2 c x^2}}+\frac{6 \tan ^{-1}(a x)}{a^2 c \sqrt{c+a^2 c x^2}}+\frac{3 x \tan ^{-1}(a x)^2}{a c \sqrt{c+a^2 c x^2}}-\frac{\tan ^{-1}(a x)^3}{a^2 c \sqrt{c+a^2 c x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0934493, size = 61, normalized size = 0.57 \[ \frac{\sqrt{a^2 c x^2+c} \left (-6 a x-\tan ^{-1}(a x)^3+3 a x \tan ^{-1}(a x)^2+6 \tan ^{-1}(a x)\right )}{a^2 c^2 \left (a^2 x^2+1\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[c + a^2*c*x^2]*(-6*a*x + 6*ArcTan[a*x] + 3*a*x*ArcTan[a*x]^2 - ArcTan[a*x]^3))/(a^2*c^2*(1 + a^2*x^2))

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Maple [C]  time = 0.275, size = 134, normalized size = 1.3 \begin{align*} -{\frac{ \left ( \left ( \arctan \left ( ax \right ) \right ) ^{3}-6\,\arctan \left ( ax \right ) +3\,i \left ( \arctan \left ( ax \right ) \right ) ^{2}-6\,i \right ) \left ( 1+iax \right ) }{ \left ( 2\,{a}^{2}{x}^{2}+2 \right ){c}^{2}{a}^{2}}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}+{\frac{ \left ( -1+iax \right ) \left ( \left ( \arctan \left ( ax \right ) \right ) ^{3}-6\,\arctan \left ( ax \right ) -3\,i \left ( \arctan \left ( ax \right ) \right ) ^{2}+6\,i \right ) }{ \left ( 2\,{a}^{2}{x}^{2}+2 \right ){c}^{2}{a}^{2}}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(arctan(a*x)^3-6*arctan(a*x)+3*I*arctan(a*x)^2-6*I)*(1+I*a*x)*(c*(a*x-I)*(a*x+I))^(1/2)/(a^2*x^2+1)/c^2/a
^2+1/2*(c*(a*x-I)*(a*x+I))^(1/2)*(-1+I*a*x)*(arctan(a*x)^3-6*arctan(a*x)-3*I*arctan(a*x)^2+6*I)/(a^2*x^2+1)/c^
2/a^2

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Maxima [A]  time = 3.28509, size = 132, normalized size = 1.23 \begin{align*} \sqrt{c}{\left (\frac{3 \, x \arctan \left (a x\right )^{2}}{\sqrt{a^{2} x^{2} + 1} a c^{2}} - \frac{\arctan \left (a x\right )^{3}}{\sqrt{a^{2} x^{2} + 1} a^{2} c^{2}} - \frac{6 \,{\left (\frac{x}{\sqrt{a^{2} x^{2} + 1}} - \frac{\arctan \left (a x\right )}{\sqrt{a^{2} x^{2} + 1} a}\right )}}{a c^{2}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(c)*(3*x*arctan(a*x)^2/(sqrt(a^2*x^2 + 1)*a*c^2) - arctan(a*x)^3/(sqrt(a^2*x^2 + 1)*a^2*c^2) - 6*(x/sqrt(a
^2*x^2 + 1) - arctan(a*x)/(sqrt(a^2*x^2 + 1)*a))/(a*c^2))

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Fricas [A]  time = 1.75106, size = 144, normalized size = 1.35 \begin{align*} \frac{\sqrt{a^{2} c x^{2} + c}{\left (3 \, a x \arctan \left (a x\right )^{2} - \arctan \left (a x\right )^{3} - 6 \, a x + 6 \, \arctan \left (a x\right )\right )}}{a^{4} c^{2} x^{2} + a^{2} c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(a^2*c*x^2 + c)*(3*a*x*arctan(a*x)^2 - arctan(a*x)^3 - 6*a*x + 6*arctan(a*x))/(a^4*c^2*x^2 + a^2*c^2)

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

[Out]

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

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Giac [A]  time = 1.29754, size = 134, normalized size = 1.25 \begin{align*} \frac{3 \, x \arctan \left (a x\right )^{2}}{\sqrt{a^{2} c x^{2} + c} a c} - \frac{\arctan \left (a x\right )^{3}}{\sqrt{a^{2} c x^{2} + c} a^{2} c} - \frac{6 \, x}{\sqrt{a^{2} c x^{2} + c} a c} + \frac{6 \, \arctan \left (a x\right )}{\sqrt{a^{2} c x^{2} + c} a^{2} c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*x*arctan(a*x)^2/(sqrt(a^2*c*x^2 + c)*a*c) - arctan(a*x)^3/(sqrt(a^2*c*x^2 + c)*a^2*c) - 6*x/(sqrt(a^2*c*x^2
+ c)*a*c) + 6*arctan(a*x)/(sqrt(a^2*c*x^2 + c)*a^2*c)