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

Optimal. Leaf size=100 \[ -\frac{6}{a c \sqrt{a^2 c x^2+c}}+\frac{x \tan ^{-1}(a x)^3}{c \sqrt{a^2 c x^2+c}}+\frac{3 \tan ^{-1}(a x)^2}{a c \sqrt{a^2 c x^2+c}}-\frac{6 x \tan ^{-1}(a x)}{c \sqrt{a^2 c x^2+c}} \]

[Out]

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

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Rubi [A]  time = 0.0690659, antiderivative size = 100, 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, 4894} \[ -\frac{6}{a c \sqrt{a^2 c x^2+c}}+\frac{x \tan ^{-1}(a x)^3}{c \sqrt{a^2 c x^2+c}}+\frac{3 \tan ^{-1}(a x)^2}{a c \sqrt{a^2 c x^2+c}}-\frac{6 x \tan ^{-1}(a x)}{c \sqrt{a^2 c x^2+c}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-6/(a*c*Sqrt[c + a^2*c*x^2]) - (6*x*ArcTan[a*x])/(c*Sqrt[c + a^2*c*x^2]) + (3*ArcTan[a*x]^2)/(a*c*Sqrt[c + a^2
*c*x^2]) + (x*ArcTan[a*x]^3)/(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 4894

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

Rubi steps

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

Mathematica [A]  time = 0.0674057, size = 56, normalized size = 0.56 \[ \frac{\sqrt{a^2 c x^2+c} \left (a x \tan ^{-1}(a x)^3+3 \tan ^{-1}(a x)^2-6 a x \tan ^{-1}(a x)-6\right )}{c^2 \left (a^3 x^2+a\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C]  time = 0.236, size = 132, 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 ( 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 ) ^{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}\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)^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)*(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)^3-6*arctan(a*x)-3*I*arctan(a*x)^2+6*I)/(a^2*x^2+1)/c^2/a

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Maxima [A]  time = 2.55137, size = 134, normalized size = 1.34 \begin{align*} \frac{x \arctan \left (a x\right )^{3}}{\sqrt{a^{2} c x^{2} + c} c} - \frac{3 \, a{\left (\frac{2 \, x \arctan \left (a x\right )}{\sqrt{a^{2} x^{2} + 1} a c} - \frac{\arctan \left (a x\right )^{2}}{\sqrt{a^{2} x^{2} + 1} a^{2} c} + \frac{2}{\sqrt{a^{2} x^{2} + 1} a^{2} c}\right )}}{\sqrt{c}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.78762, size = 142, normalized size = 1.42 \begin{align*} \frac{\sqrt{a^{2} c x^{2} + c}{\left (a x \arctan \left (a x\right )^{3} - 6 \, a x \arctan \left (a x\right ) + 3 \, \arctan \left (a x\right )^{2} - 6\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)^3/(a^2*c*x^2+c)^(3/2),x, algorithm="fricas")

[Out]

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

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

[Out]

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

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Giac [A]  time = 1.27381, size = 134, normalized size = 1.34 \begin{align*} \frac{x \arctan \left (a x\right )^{3}}{\sqrt{a^{2} c x^{2} + c} c} - 3 \, a{\left (\frac{2 \, x \arctan \left (a x\right )}{\sqrt{a^{2} c x^{2} + c} a c} - \frac{\arctan \left (a x\right )^{2}}{\sqrt{a^{2} c x^{2} + c} a^{2} c} + \frac{2}{\sqrt{a^{2} c x^{2} + c} a^{2} c}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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