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

Optimal. Leaf size=225 \[ -\frac{45}{128 a c^3 \left (a^2 x^2+1\right )}-\frac{3}{128 a c^3 \left (a^2 x^2+1\right )^2}+\frac{3 x \tan ^{-1}(a x)^3}{8 c^3 \left (a^2 x^2+1\right )}+\frac{x \tan ^{-1}(a x)^3}{4 c^3 \left (a^2 x^2+1\right )^2}+\frac{9 \tan ^{-1}(a x)^2}{16 a c^3 \left (a^2 x^2+1\right )}+\frac{3 \tan ^{-1}(a x)^2}{16 a c^3 \left (a^2 x^2+1\right )^2}-\frac{45 x \tan ^{-1}(a x)}{64 c^3 \left (a^2 x^2+1\right )}-\frac{3 x \tan ^{-1}(a x)}{32 c^3 \left (a^2 x^2+1\right )^2}+\frac{3 \tan ^{-1}(a x)^4}{32 a c^3}-\frac{45 \tan ^{-1}(a x)^2}{128 a c^3} \]

[Out]

-3/(128*a*c^3*(1 + a^2*x^2)^2) - 45/(128*a*c^3*(1 + a^2*x^2)) - (3*x*ArcTan[a*x])/(32*c^3*(1 + a^2*x^2)^2) - (
45*x*ArcTan[a*x])/(64*c^3*(1 + a^2*x^2)) - (45*ArcTan[a*x]^2)/(128*a*c^3) + (3*ArcTan[a*x]^2)/(16*a*c^3*(1 + a
^2*x^2)^2) + (9*ArcTan[a*x]^2)/(16*a*c^3*(1 + a^2*x^2)) + (x*ArcTan[a*x]^3)/(4*c^3*(1 + a^2*x^2)^2) + (3*x*Arc
Tan[a*x]^3)/(8*c^3*(1 + a^2*x^2)) + (3*ArcTan[a*x]^4)/(32*a*c^3)

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Rubi [A]  time = 0.196444, antiderivative size = 225, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {4900, 4892, 4930, 261, 4896} \[ -\frac{45}{128 a c^3 \left (a^2 x^2+1\right )}-\frac{3}{128 a c^3 \left (a^2 x^2+1\right )^2}+\frac{3 x \tan ^{-1}(a x)^3}{8 c^3 \left (a^2 x^2+1\right )}+\frac{x \tan ^{-1}(a x)^3}{4 c^3 \left (a^2 x^2+1\right )^2}+\frac{9 \tan ^{-1}(a x)^2}{16 a c^3 \left (a^2 x^2+1\right )}+\frac{3 \tan ^{-1}(a x)^2}{16 a c^3 \left (a^2 x^2+1\right )^2}-\frac{45 x \tan ^{-1}(a x)}{64 c^3 \left (a^2 x^2+1\right )}-\frac{3 x \tan ^{-1}(a x)}{32 c^3 \left (a^2 x^2+1\right )^2}+\frac{3 \tan ^{-1}(a x)^4}{32 a c^3}-\frac{45 \tan ^{-1}(a x)^2}{128 a c^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-3/(128*a*c^3*(1 + a^2*x^2)^2) - 45/(128*a*c^3*(1 + a^2*x^2)) - (3*x*ArcTan[a*x])/(32*c^3*(1 + a^2*x^2)^2) - (
45*x*ArcTan[a*x])/(64*c^3*(1 + a^2*x^2)) - (45*ArcTan[a*x]^2)/(128*a*c^3) + (3*ArcTan[a*x]^2)/(16*a*c^3*(1 + a
^2*x^2)^2) + (9*ArcTan[a*x]^2)/(16*a*c^3*(1 + a^2*x^2)) + (x*ArcTan[a*x]^3)/(4*c^3*(1 + a^2*x^2)^2) + (3*x*Arc
Tan[a*x]^3)/(8*c^3*(1 + a^2*x^2)) + (3*ArcTan[a*x]^4)/(32*a*c^3)

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 4892

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

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 261

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

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 )^3} \, dx &=\frac{3 \tan ^{-1}(a x)^2}{16 a c^3 \left (1+a^2 x^2\right )^2}+\frac{x \tan ^{-1}(a x)^3}{4 c^3 \left (1+a^2 x^2\right )^2}-\frac{3}{8} \int \frac{\tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^3} \, dx+\frac{3 \int \frac{\tan ^{-1}(a x)^3}{\left (c+a^2 c x^2\right )^2} \, dx}{4 c}\\ &=-\frac{3}{128 a c^3 \left (1+a^2 x^2\right )^2}-\frac{3 x \tan ^{-1}(a x)}{32 c^3 \left (1+a^2 x^2\right )^2}+\frac{3 \tan ^{-1}(a x)^2}{16 a c^3 \left (1+a^2 x^2\right )^2}+\frac{x \tan ^{-1}(a x)^3}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac{3 x \tan ^{-1}(a x)^3}{8 c^3 \left (1+a^2 x^2\right )}+\frac{3 \tan ^{-1}(a x)^4}{32 a c^3}-\frac{9 \int \frac{\tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^2} \, dx}{32 c}-\frac{(9 a) \int \frac{x \tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^2} \, dx}{8 c}\\ &=-\frac{3}{128 a c^3 \left (1+a^2 x^2\right )^2}-\frac{3 x \tan ^{-1}(a x)}{32 c^3 \left (1+a^2 x^2\right )^2}-\frac{9 x \tan ^{-1}(a x)}{64 c^3 \left (1+a^2 x^2\right )}-\frac{9 \tan ^{-1}(a x)^2}{128 a c^3}+\frac{3 \tan ^{-1}(a x)^2}{16 a c^3 \left (1+a^2 x^2\right )^2}+\frac{9 \tan ^{-1}(a x)^2}{16 a c^3 \left (1+a^2 x^2\right )}+\frac{x \tan ^{-1}(a x)^3}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac{3 x \tan ^{-1}(a x)^3}{8 c^3 \left (1+a^2 x^2\right )}+\frac{3 \tan ^{-1}(a x)^4}{32 a c^3}-\frac{9 \int \frac{\tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^2} \, dx}{8 c}+\frac{(9 a) \int \frac{x}{\left (c+a^2 c x^2\right )^2} \, dx}{64 c}\\ &=-\frac{3}{128 a c^3 \left (1+a^2 x^2\right )^2}-\frac{9}{128 a c^3 \left (1+a^2 x^2\right )}-\frac{3 x \tan ^{-1}(a x)}{32 c^3 \left (1+a^2 x^2\right )^2}-\frac{45 x \tan ^{-1}(a x)}{64 c^3 \left (1+a^2 x^2\right )}-\frac{45 \tan ^{-1}(a x)^2}{128 a c^3}+\frac{3 \tan ^{-1}(a x)^2}{16 a c^3 \left (1+a^2 x^2\right )^2}+\frac{9 \tan ^{-1}(a x)^2}{16 a c^3 \left (1+a^2 x^2\right )}+\frac{x \tan ^{-1}(a x)^3}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac{3 x \tan ^{-1}(a x)^3}{8 c^3 \left (1+a^2 x^2\right )}+\frac{3 \tan ^{-1}(a x)^4}{32 a c^3}+\frac{(9 a) \int \frac{x}{\left (c+a^2 c x^2\right )^2} \, dx}{16 c}\\ &=-\frac{3}{128 a c^3 \left (1+a^2 x^2\right )^2}-\frac{45}{128 a c^3 \left (1+a^2 x^2\right )}-\frac{3 x \tan ^{-1}(a x)}{32 c^3 \left (1+a^2 x^2\right )^2}-\frac{45 x \tan ^{-1}(a x)}{64 c^3 \left (1+a^2 x^2\right )}-\frac{45 \tan ^{-1}(a x)^2}{128 a c^3}+\frac{3 \tan ^{-1}(a x)^2}{16 a c^3 \left (1+a^2 x^2\right )^2}+\frac{9 \tan ^{-1}(a x)^2}{16 a c^3 \left (1+a^2 x^2\right )}+\frac{x \tan ^{-1}(a x)^3}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac{3 x \tan ^{-1}(a x)^3}{8 c^3 \left (1+a^2 x^2\right )}+\frac{3 \tan ^{-1}(a x)^4}{32 a c^3}\\ \end{align*}

Mathematica [A]  time = 0.0564496, size = 114, normalized size = 0.51 \[ -\frac{45 a^2 x^2-12 \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)^4-16 a x \left (3 a^2 x^2+5\right ) \tan ^{-1}(a x)^3+3 \left (15 a^4 x^4+6 a^2 x^2-17\right ) \tan ^{-1}(a x)^2+6 a x \left (15 a^2 x^2+17\right ) \tan ^{-1}(a x)+48}{128 a c^3 \left (a^2 x^2+1\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(48 + 45*a^2*x^2 + 6*a*x*(17 + 15*a^2*x^2)*ArcTan[a*x] + 3*(-17 + 6*a^2*x^2 + 15*a^4*x^4)*ArcTan[a*x]^2 - 16*
a*x*(5 + 3*a^2*x^2)*ArcTan[a*x]^3 - 12*(1 + a^2*x^2)^2*ArcTan[a*x]^4)/(128*a*c^3*(1 + a^2*x^2)^2)

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Maple [A]  time = 0.302, size = 211, normalized size = 0.9 \begin{align*}{\frac{x \left ( \arctan \left ( ax \right ) \right ) ^{3}}{4\,{c}^{3} \left ({a}^{2}{x}^{2}+1 \right ) ^{2}}}+{\frac{3\,x \left ( \arctan \left ( ax \right ) \right ) ^{3}}{8\,{c}^{3} \left ({a}^{2}{x}^{2}+1 \right ) }}+{\frac{3\, \left ( \arctan \left ( ax \right ) \right ) ^{4}}{32\,a{c}^{3}}}+{\frac{3\, \left ( \arctan \left ( ax \right ) \right ) ^{2}}{16\,a{c}^{3} \left ({a}^{2}{x}^{2}+1 \right ) ^{2}}}+{\frac{9\, \left ( \arctan \left ( ax \right ) \right ) ^{2}}{16\,a{c}^{3} \left ({a}^{2}{x}^{2}+1 \right ) }}-{\frac{45\,{a}^{2}\arctan \left ( ax \right ){x}^{3}}{64\,{c}^{3} \left ({a}^{2}{x}^{2}+1 \right ) ^{2}}}-{\frac{51\,x\arctan \left ( ax \right ) }{64\,{c}^{3} \left ({a}^{2}{x}^{2}+1 \right ) ^{2}}}-{\frac{45\, \left ( \arctan \left ( ax \right ) \right ) ^{2}}{128\,a{c}^{3}}}-{\frac{3}{128\,a{c}^{3} \left ({a}^{2}{x}^{2}+1 \right ) ^{2}}}-{\frac{45}{128\,a{c}^{3} \left ({a}^{2}{x}^{2}+1 \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,x)

[Out]

1/4*x*arctan(a*x)^3/c^3/(a^2*x^2+1)^2+3/8*x*arctan(a*x)^3/c^3/(a^2*x^2+1)+3/32*arctan(a*x)^4/a/c^3+3/16*arctan
(a*x)^2/a/c^3/(a^2*x^2+1)^2+9/16*arctan(a*x)^2/a/c^3/(a^2*x^2+1)-45/64*a^2/c^3*arctan(a*x)*x^3/(a^2*x^2+1)^2-5
1/64*x*arctan(a*x)/c^3/(a^2*x^2+1)^2-45/128*arctan(a*x)^2/a/c^3-3/128/a/c^3/(a^2*x^2+1)^2-45/128/a/c^3/(a^2*x^
2+1)

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Maxima [A]  time = 1.88486, size = 452, normalized size = 2.01 \begin{align*} \frac{1}{8} \,{\left (\frac{3 \, a^{2} x^{3} + 5 \, x}{a^{4} c^{3} x^{4} + 2 \, a^{2} c^{3} x^{2} + c^{3}} + \frac{3 \, \arctan \left (a x\right )}{a c^{3}}\right )} \arctan \left (a x\right )^{3} + \frac{3 \,{\left (3 \, a^{2} x^{2} - 3 \,{\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{2} + 4\right )} a \arctan \left (a x\right )^{2}}{16 \,{\left (a^{6} c^{3} x^{4} + 2 \, a^{4} c^{3} x^{2} + a^{2} c^{3}\right )}} - \frac{3}{128} \,{\left (\frac{{\left (4 \,{\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{4} + 15 \, a^{2} x^{2} - 15 \,{\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{2} + 16\right )} a^{2}}{a^{8} c^{3} x^{4} + 2 \, a^{6} c^{3} x^{2} + a^{4} c^{3}} + \frac{2 \,{\left (15 \, a^{3} x^{3} - 8 \,{\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{3} + 17 \, a x + 15 \,{\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )\right )} a \arctan \left (a x\right )}{a^{7} c^{3} x^{4} + 2 \, a^{5} c^{3} x^{2} + a^{3} c^{3}}\right )} a \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,x, algorithm="maxima")

[Out]

1/8*((3*a^2*x^3 + 5*x)/(a^4*c^3*x^4 + 2*a^2*c^3*x^2 + c^3) + 3*arctan(a*x)/(a*c^3))*arctan(a*x)^3 + 3/16*(3*a^
2*x^2 - 3*(a^4*x^4 + 2*a^2*x^2 + 1)*arctan(a*x)^2 + 4)*a*arctan(a*x)^2/(a^6*c^3*x^4 + 2*a^4*c^3*x^2 + a^2*c^3)
- 3/128*((4*(a^4*x^4 + 2*a^2*x^2 + 1)*arctan(a*x)^4 + 15*a^2*x^2 - 15*(a^4*x^4 + 2*a^2*x^2 + 1)*arctan(a*x)^2
+ 16)*a^2/(a^8*c^3*x^4 + 2*a^6*c^3*x^2 + a^4*c^3) + 2*(15*a^3*x^3 - 8*(a^4*x^4 + 2*a^2*x^2 + 1)*arctan(a*x)^3
+ 17*a*x + 15*(a^4*x^4 + 2*a^2*x^2 + 1)*arctan(a*x))*a*arctan(a*x)/(a^7*c^3*x^4 + 2*a^5*c^3*x^2 + a^3*c^3))*a

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Fricas [A]  time = 2.0007, size = 315, normalized size = 1.4 \begin{align*} \frac{12 \,{\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{4} - 45 \, a^{2} x^{2} + 16 \,{\left (3 \, a^{3} x^{3} + 5 \, a x\right )} \arctan \left (a x\right )^{3} - 3 \,{\left (15 \, a^{4} x^{4} + 6 \, a^{2} x^{2} - 17\right )} \arctan \left (a x\right )^{2} - 6 \,{\left (15 \, a^{3} x^{3} + 17 \, a x\right )} \arctan \left (a x\right ) - 48}{128 \,{\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)^3,x, algorithm="fricas")

[Out]

1/128*(12*(a^4*x^4 + 2*a^2*x^2 + 1)*arctan(a*x)^4 - 45*a^2*x^2 + 16*(3*a^3*x^3 + 5*a*x)*arctan(a*x)^3 - 3*(15*
a^4*x^4 + 6*a^2*x^2 - 17)*arctan(a*x)^2 - 6*(15*a^3*x^3 + 17*a*x)*arctan(a*x) - 48)/(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*} \frac{\int \frac{\operatorname{atan}^{3}{\left (a x \right )}}{a^{6} x^{6} + 3 a^{4} x^{4} + 3 a^{2} x^{2} + 1}\, dx}{c^{3}} \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,x)

[Out]

Integral(atan(a*x)**3/(a**6*x**6 + 3*a**4*x**4 + 3*a**2*x**2 + 1), x)/c**3

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Giac [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 )}^{3}}\,{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)^3,x, algorithm="giac")

[Out]

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