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

Optimal. Leaf size=199 \[ -\frac{40 x}{27 a c^2 \sqrt{a^2 c x^2+c}}+\frac{2 x \tan ^{-1}(a x)^2}{3 a c^2 \sqrt{a^2 c x^2+c}}+\frac{4 \tan ^{-1}(a x)}{3 a^2 c^2 \sqrt{a^2 c x^2+c}}-\frac{2 x}{27 a c \left (a^2 c x^2+c\right )^{3/2}}-\frac{\tan ^{-1}(a x)^3}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}+\frac{x \tan ^{-1}(a x)^2}{3 a c \left (a^2 c x^2+c\right )^{3/2}}+\frac{2 \tan ^{-1}(a x)}{9 a^2 c \left (a^2 c x^2+c\right )^{3/2}} \]

[Out]

(-2*x)/(27*a*c*(c + a^2*c*x^2)^(3/2)) - (40*x)/(27*a*c^2*Sqrt[c + a^2*c*x^2]) + (2*ArcTan[a*x])/(9*a^2*c*(c +
a^2*c*x^2)^(3/2)) + (4*ArcTan[a*x])/(3*a^2*c^2*Sqrt[c + a^2*c*x^2]) + (x*ArcTan[a*x]^2)/(3*a*c*(c + a^2*c*x^2)
^(3/2)) + (2*x*ArcTan[a*x]^2)/(3*a*c^2*Sqrt[c + a^2*c*x^2]) - ArcTan[a*x]^3/(3*a^2*c*(c + a^2*c*x^2)^(3/2))

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Rubi [A]  time = 0.193512, antiderivative size = 199, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {4930, 4900, 4898, 191, 192} \[ -\frac{40 x}{27 a c^2 \sqrt{a^2 c x^2+c}}+\frac{2 x \tan ^{-1}(a x)^2}{3 a c^2 \sqrt{a^2 c x^2+c}}+\frac{4 \tan ^{-1}(a x)}{3 a^2 c^2 \sqrt{a^2 c x^2+c}}-\frac{2 x}{27 a c \left (a^2 c x^2+c\right )^{3/2}}-\frac{\tan ^{-1}(a x)^3}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}+\frac{x \tan ^{-1}(a x)^2}{3 a c \left (a^2 c x^2+c\right )^{3/2}}+\frac{2 \tan ^{-1}(a x)}{9 a^2 c \left (a^2 c x^2+c\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*x)/(27*a*c*(c + a^2*c*x^2)^(3/2)) - (40*x)/(27*a*c^2*Sqrt[c + a^2*c*x^2]) + (2*ArcTan[a*x])/(9*a^2*c*(c +
a^2*c*x^2)^(3/2)) + (4*ArcTan[a*x])/(3*a^2*c^2*Sqrt[c + a^2*c*x^2]) + (x*ArcTan[a*x]^2)/(3*a*c*(c + a^2*c*x^2)
^(3/2)) + (2*x*ArcTan[a*x]^2)/(3*a*c^2*Sqrt[c + a^2*c*x^2]) - ArcTan[a*x]^3/(3*a^2*c*(c + a^2*c*x^2)^(3/2))

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 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 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]

Rule 192

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

Rubi steps

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

Mathematica [A]  time = 0.0929382, size = 91, normalized size = 0.46 \[ \frac{\sqrt{a^2 c x^2+c} \left (-2 a x \left (20 a^2 x^2+21\right )+9 a x \left (2 a^2 x^2+3\right ) \tan ^{-1}(a x)^2+6 \left (6 a^2 x^2+7\right ) \tan ^{-1}(a x)-9 \tan ^{-1}(a x)^3\right )}{27 c^3 \left (a^3 x^2+a\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C]  time = 0.311, size = 312, normalized size = 1.6 \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 ( i{x}^{3}{a}^{3}+3\,{a}^{2}{x}^{2}-3\,iax-1 \right ) }{216\, \left ({a}^{2}{x}^{2}+1 \right ) ^{2}{c}^{3}{a}^{2}}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}-{\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 ) }{8\,{c}^{3}{a}^{2} \left ({a}^{2}{x}^{2}+1 \right ) }\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 ) }{8\,{c}^{3}{a}^{2} \left ({a}^{2}{x}^{2}+1 \right ) }\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}-{\frac{ \left ( i{x}^{3}{a}^{3}-3\,{a}^{2}{x}^{2}-3\,iax+1 \right ) \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 ) }{216\,{c}^{3}{a}^{2} \left ({a}^{4}{x}^{4}+2\,{a}^{2}{x}^{2}+1 \right ) }\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)^(5/2),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x \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(x*arctan(a*x)^3/(a^2*c*x^2+c)^(5/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.95946, size = 239, normalized size = 1.2 \begin{align*} -\frac{{\left (40 \, a^{3} x^{3} - 9 \,{\left (2 \, a^{3} x^{3} + 3 \, a x\right )} \arctan \left (a x\right )^{2} + 9 \, \arctan \left (a x\right )^{3} + 42 \, a x - 6 \,{\left (6 \, a^{2} x^{2} + 7\right )} \arctan \left (a x\right )\right )} \sqrt{a^{2} c x^{2} + c}}{27 \,{\left (a^{6} c^{3} x^{4} + 2 \, a^{4} c^{3} x^{2} + a^{2} c^{3}\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)^(5/2),x, algorithm="fricas")

[Out]

-1/27*(40*a^3*x^3 - 9*(2*a^3*x^3 + 3*a*x)*arctan(a*x)^2 + 9*arctan(a*x)^3 + 42*a*x - 6*(6*a^2*x^2 + 7)*arctan(
a*x))*sqrt(a^2*c*x^2 + c)/(a^6*c^3*x^4 + 2*a^4*c^3*x^2 + a^2*c^3)

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

[Out]

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

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Giac [A]  time = 1.30652, size = 184, normalized size = 0.92 \begin{align*} \frac{{\left (\frac{2 \, a x^{2}}{c} + \frac{3}{a c}\right )} x \arctan \left (a x\right )^{2}}{3 \,{\left (a^{2} c x^{2} + c\right )}^{\frac{3}{2}}} - \frac{2 \,{\left (\frac{20 \, a x^{2}}{c} + \frac{21}{a c}\right )} x}{27 \,{\left (a^{2} c x^{2} + c\right )}^{\frac{3}{2}}} - \frac{\arctan \left (a x\right )^{3}}{3 \,{\left (a^{2} c x^{2} + c\right )}^{\frac{3}{2}} a^{2} c} + \frac{2 \,{\left (6 \, a^{2} c x^{2} + 7 \, c\right )} \arctan \left (a x\right )}{9 \,{\left (a^{2} c x^{2} + c\right )}^{\frac{3}{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)^(5/2),x, algorithm="giac")

[Out]

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