3.455 \(\int \frac{\tan ^{-1}(a x)^3}{(c+a^2 c x^2)^{5/2}} \, dx\)

Optimal. Leaf size=215 \[ -\frac{40}{9 a c^2 \sqrt{a^2 c x^2+c}}+\frac{2 x \tan ^{-1}(a x)^3}{3 c^2 \sqrt{a^2 c x^2+c}}+\frac{2 \tan ^{-1}(a x)^2}{a c^2 \sqrt{a^2 c x^2+c}}-\frac{40 x \tan ^{-1}(a x)}{9 c^2 \sqrt{a^2 c x^2+c}}-\frac{2}{27 a c \left (a^2 c x^2+c\right )^{3/2}}+\frac{x \tan ^{-1}(a x)^3}{3 c \left (a^2 c x^2+c\right )^{3/2}}+\frac{\tan ^{-1}(a x)^2}{3 a c \left (a^2 c x^2+c\right )^{3/2}}-\frac{2 x \tan ^{-1}(a x)}{9 c \left (a^2 c x^2+c\right )^{3/2}} \]

[Out]

-2/(27*a*c*(c + a^2*c*x^2)^(3/2)) - 40/(9*a*c^2*Sqrt[c + a^2*c*x^2]) - (2*x*ArcTan[a*x])/(9*c*(c + a^2*c*x^2)^
(3/2)) - (40*x*ArcTan[a*x])/(9*c^2*Sqrt[c + a^2*c*x^2]) + ArcTan[a*x]^2/(3*a*c*(c + a^2*c*x^2)^(3/2)) + (2*Arc
Tan[a*x]^2)/(a*c^2*Sqrt[c + a^2*c*x^2]) + (x*ArcTan[a*x]^3)/(3*c*(c + a^2*c*x^2)^(3/2)) + (2*x*ArcTan[a*x]^3)/
(3*c^2*Sqrt[c + a^2*c*x^2])

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Rubi [A]  time = 0.178554, antiderivative size = 215, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {4900, 4898, 4894, 4896} \[ -\frac{40}{9 a c^2 \sqrt{a^2 c x^2+c}}+\frac{2 x \tan ^{-1}(a x)^3}{3 c^2 \sqrt{a^2 c x^2+c}}+\frac{2 \tan ^{-1}(a x)^2}{a c^2 \sqrt{a^2 c x^2+c}}-\frac{40 x \tan ^{-1}(a x)}{9 c^2 \sqrt{a^2 c x^2+c}}-\frac{2}{27 a c \left (a^2 c x^2+c\right )^{3/2}}+\frac{x \tan ^{-1}(a x)^3}{3 c \left (a^2 c x^2+c\right )^{3/2}}+\frac{\tan ^{-1}(a x)^2}{3 a c \left (a^2 c x^2+c\right )^{3/2}}-\frac{2 x \tan ^{-1}(a x)}{9 c \left (a^2 c x^2+c\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[ArcTan[a*x]^3/(c + a^2*c*x^2)^(5/2),x]

[Out]

-2/(27*a*c*(c + a^2*c*x^2)^(3/2)) - 40/(9*a*c^2*Sqrt[c + a^2*c*x^2]) - (2*x*ArcTan[a*x])/(9*c*(c + a^2*c*x^2)^
(3/2)) - (40*x*ArcTan[a*x])/(9*c^2*Sqrt[c + a^2*c*x^2]) + ArcTan[a*x]^2/(3*a*c*(c + a^2*c*x^2)^(3/2)) + (2*Arc
Tan[a*x]^2)/(a*c^2*Sqrt[c + a^2*c*x^2]) + (x*ArcTan[a*x]^3)/(3*c*(c + a^2*c*x^2)^(3/2)) + (2*x*ArcTan[a*x]^3)/
(3*c^2*Sqrt[c + a^2*c*x^2])

Rule 4900

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_)*((d_) + (e_.)*(x_)^2)^(q_), x_Symbol] :> Simp[(b*p*(d + e*x^2)^(q
+ 1)*(a + b*ArcTan[c*x])^(p - 1))/(4*c*d*(q + 1)^2), x] + (Dist[(2*q + 3)/(2*d*(q + 1)), Int[(d + e*x^2)^(q +
1)*(a + b*ArcTan[c*x])^p, x], x] - Dist[(b^2*p*(p - 1))/(4*(q + 1)^2), Int[(d + e*x^2)^q*(a + b*ArcTan[c*x])^(
p - 2), x], x] - Simp[(x*(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x])^p)/(2*d*(q + 1)), x]) /; FreeQ[{a, b, c, d, e
}, x] && EqQ[e, c^2*d] && LtQ[q, -1] && GtQ[p, 1] && NeQ[q, -3/2]

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]

Rule 4896

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(q_), x_Symbol] :> Simp[(b*(d + e*x^2)^(q + 1))/(
4*c*d*(q + 1)^2), x] + (Dist[(2*q + 3)/(2*d*(q + 1)), Int[(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x]), x], x] - Si
mp[(x*(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x]))/(2*d*(q + 1)), x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d
] && LtQ[q, -1] && NeQ[q, -3/2]

Rubi steps

\begin{align*} \int \frac{\tan ^{-1}(a x)^3}{\left (c+a^2 c x^2\right )^{5/2}} \, dx &=\frac{\tan ^{-1}(a x)^2}{3 a c \left (c+a^2 c x^2\right )^{3/2}}+\frac{x \tan ^{-1}(a x)^3}{3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac{2}{3} \int \frac{\tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^{5/2}} \, dx+\frac{2 \int \frac{\tan ^{-1}(a x)^3}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{3 c}\\ &=-\frac{2}{27 a c \left (c+a^2 c x^2\right )^{3/2}}-\frac{2 x \tan ^{-1}(a x)}{9 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{\tan ^{-1}(a x)^2}{3 a c \left (c+a^2 c x^2\right )^{3/2}}+\frac{2 \tan ^{-1}(a x)^2}{a c^2 \sqrt{c+a^2 c x^2}}+\frac{x \tan ^{-1}(a x)^3}{3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{2 x \tan ^{-1}(a x)^3}{3 c^2 \sqrt{c+a^2 c x^2}}-\frac{4 \int \frac{\tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{9 c}-\frac{4 \int \frac{\tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{c}\\ &=-\frac{2}{27 a c \left (c+a^2 c x^2\right )^{3/2}}-\frac{40}{9 a c^2 \sqrt{c+a^2 c x^2}}-\frac{2 x \tan ^{-1}(a x)}{9 c \left (c+a^2 c x^2\right )^{3/2}}-\frac{40 x \tan ^{-1}(a x)}{9 c^2 \sqrt{c+a^2 c x^2}}+\frac{\tan ^{-1}(a x)^2}{3 a c \left (c+a^2 c x^2\right )^{3/2}}+\frac{2 \tan ^{-1}(a x)^2}{a c^2 \sqrt{c+a^2 c x^2}}+\frac{x \tan ^{-1}(a x)^3}{3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{2 x \tan ^{-1}(a x)^3}{3 c^2 \sqrt{c+a^2 c x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0847638, size = 104, normalized size = 0.48 \[ \frac{\sqrt{a^2 c x^2+c} \left (-2 \left (60 a^2 x^2+61\right )+9 a x \left (2 a^2 x^2+3\right ) \tan ^{-1}(a x)^3+9 \left (6 a^2 x^2+7\right ) \tan ^{-1}(a x)^2-6 a x \left (20 a^2 x^2+21\right ) \tan ^{-1}(a x)\right )}{27 a c^3 \left (a^2 x^2+1\right )^2} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcTan[a*x]^3/(c + a^2*c*x^2)^(5/2),x]

[Out]

(Sqrt[c + a^2*c*x^2]*(-2*(61 + 60*a^2*x^2) - 6*a*x*(21 + 20*a^2*x^2)*ArcTan[a*x] + 9*(7 + 6*a^2*x^2)*ArcTan[a*
x]^2 + 9*a*x*(3 + 2*a^2*x^2)*ArcTan[a*x]^3))/(27*a*c^3*(1 + a^2*x^2)^2)

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Maple [C]  time = 0.27, size = 308, normalized size = 1.4 \begin{align*} -{\frac{ \left ( 9\,i \left ( \arctan \left ( ax \right ) \right ) ^{2}+9\, \left ( \arctan \left ( ax \right ) \right ) ^{3}-2\,i-6\,\arctan \left ( ax \right ) \right ) \left ({a}^{3}{x}^{3}-3\,i{a}^{2}{x}^{2}-3\,ax+i \right ) }{216\, \left ({a}^{2}{x}^{2}+1 \right ) ^{2}a{c}^{3}}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}+{\frac{ \left ( 3\, \left ( \arctan \left ( ax \right ) \right ) ^{3}-18\,\arctan \left ( ax \right ) +9\,i \left ( \arctan \left ( ax \right ) \right ) ^{2}-18\,i \right ) \left ( ax-i \right ) }{8\,a{c}^{3} \left ({a}^{2}{x}^{2}+1 \right ) }\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}+{\frac{ \left ( 3\,ax+3\,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 ) }{8\,a{c}^{3} \left ({a}^{2}{x}^{2}+1 \right ) }\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}-{\frac{ \left ( -9\,i \left ( \arctan \left ( ax \right ) \right ) ^{2}+9\, \left ( \arctan \left ( ax \right ) \right ) ^{3}+2\,i-6\,\arctan \left ( ax \right ) \right ) \left ({a}^{3}{x}^{3}+3\,i{a}^{2}{x}^{2}-3\,ax-i \right ) }{ \left ( 216\,{a}^{4}{x}^{4}+432\,{a}^{2}{x}^{2}+216 \right ) a{c}^{3}}\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)^(5/2),x)

[Out]

-1/216*(9*I*arctan(a*x)^2+9*arctan(a*x)^3-2*I-6*arctan(a*x))*(a^3*x^3-3*I*a^2*x^2-3*a*x+I)*(c*(a*x-I)*(a*x+I))
^(1/2)/(a^2*x^2+1)^2/a/c^3+3/8*(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)/c^3/a/(a^2*x^2+1)+3/8*(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)/c^3/a/(a^2*x^2+1)-1/216*(-9*I*arctan(a*x)^2+9*arctan(a*x)^3+2*I-6*arctan(a*x))*(c*(a*x-I)*(a*x+I))^(1/2)*
(a^3*x^3+3*I*a^2*x^2-3*a*x-I)/(a^4*x^4+2*a^2*x^2+1)/a/c^3

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\arctan \left (a x\right )^{3}}{{\left (a^{2} c x^{2} + c\right )}^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctan(a*x)^3/(a^2*c*x^2+c)^(5/2),x, algorithm="maxima")

[Out]

integrate(arctan(a*x)^3/(a^2*c*x^2 + c)^(5/2), x)

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Fricas [A]  time = 1.72267, size = 263, normalized size = 1.22 \begin{align*} -\frac{\sqrt{a^{2} c x^{2} + c}{\left (120 \, a^{2} x^{2} - 9 \,{\left (2 \, a^{3} x^{3} + 3 \, a x\right )} \arctan \left (a x\right )^{3} - 9 \,{\left (6 \, a^{2} x^{2} + 7\right )} \arctan \left (a x\right )^{2} + 6 \,{\left (20 \, a^{3} x^{3} + 21 \, a x\right )} \arctan \left (a x\right ) + 122\right )}}{27 \,{\left (a^{5} c^{3} x^{4} + 2 \, a^{3} c^{3} x^{2} + a c^{3}\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)^(5/2),x, algorithm="fricas")

[Out]

-1/27*sqrt(a^2*c*x^2 + c)*(120*a^2*x^2 - 9*(2*a^3*x^3 + 3*a*x)*arctan(a*x)^3 - 9*(6*a^2*x^2 + 7)*arctan(a*x)^2
+ 6*(20*a^3*x^3 + 21*a*x)*arctan(a*x) + 122)/(a^5*c^3*x^4 + 2*a^3*c^3*x^2 + a*c^3)

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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{5}{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)**(5/2),x)

[Out]

Integral(atan(a*x)**3/(c*(a**2*x**2 + 1))**(5/2), x)

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Giac [A]  time = 1.43972, size = 198, normalized size = 0.92 \begin{align*} \frac{{\left (\frac{2 \, a^{2} x^{2}}{c} + \frac{3}{c}\right )} x \arctan \left (a x\right )^{3}}{3 \,{\left (a^{2} c x^{2} + c\right )}^{\frac{3}{2}}} - \frac{2 \,{\left (\frac{20 \, a^{2} x^{2}}{c} + \frac{21}{c}\right )} x \arctan \left (a x\right )}{9 \,{\left (a^{2} c x^{2} + c\right )}^{\frac{3}{2}}} + \frac{{\left (6 \, a^{2} c x^{2} + 7 \, c\right )} \arctan \left (a x\right )^{2}}{3 \,{\left (a^{2} c x^{2} + c\right )}^{\frac{3}{2}} a c^{2}} - \frac{2 \,{\left (60 \, a^{2} c x^{2} + 61 \, c\right )}}{27 \,{\left (a^{2} c x^{2} + c\right )}^{\frac{3}{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)^(5/2),x, algorithm="giac")

[Out]

1/3*(2*a^2*x^2/c + 3/c)*x*arctan(a*x)^3/(a^2*c*x^2 + c)^(3/2) - 2/9*(20*a^2*x^2/c + 21/c)*x*arctan(a*x)/(a^2*c
*x^2 + c)^(3/2) + 1/3*(6*a^2*c*x^2 + 7*c)*arctan(a*x)^2/((a^2*c*x^2 + c)^(3/2)*a*c^2) - 2/27*(60*a^2*c*x^2 + 6
1*c)/((a^2*c*x^2 + c)^(3/2)*a*c^2)