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

Optimal. Leaf size=237 \[ -\frac{3}{128 a^3 c^3 \left (a^2 x^2+1\right )}+\frac{3}{128 a^3 c^3 \left (a^2 x^2+1\right )^2}+\frac{x \tan ^{-1}(a x)^3}{8 a^2 c^3 \left (a^2 x^2+1\right )}-\frac{x \tan ^{-1}(a x)^3}{4 a^2 c^3 \left (a^2 x^2+1\right )^2}+\frac{3 \tan ^{-1}(a x)^2}{16 a^3 c^3 \left (a^2 x^2+1\right )}-\frac{3 \tan ^{-1}(a x)^2}{16 a^3 c^3 \left (a^2 x^2+1\right )^2}-\frac{3 x \tan ^{-1}(a x)}{64 a^2 c^3 \left (a^2 x^2+1\right )}+\frac{3 x \tan ^{-1}(a x)}{32 a^2 c^3 \left (a^2 x^2+1\right )^2}+\frac{\tan ^{-1}(a x)^4}{32 a^3 c^3}-\frac{3 \tan ^{-1}(a x)^2}{128 a^3 c^3} \]

[Out]

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

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Rubi [A]  time = 0.388572, antiderivative size = 237, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {4964, 4892, 4930, 261, 4900, 4896} \[ -\frac{3}{128 a^3 c^3 \left (a^2 x^2+1\right )}+\frac{3}{128 a^3 c^3 \left (a^2 x^2+1\right )^2}+\frac{x \tan ^{-1}(a x)^3}{8 a^2 c^3 \left (a^2 x^2+1\right )}-\frac{x \tan ^{-1}(a x)^3}{4 a^2 c^3 \left (a^2 x^2+1\right )^2}+\frac{3 \tan ^{-1}(a x)^2}{16 a^3 c^3 \left (a^2 x^2+1\right )}-\frac{3 \tan ^{-1}(a x)^2}{16 a^3 c^3 \left (a^2 x^2+1\right )^2}-\frac{3 x \tan ^{-1}(a x)}{64 a^2 c^3 \left (a^2 x^2+1\right )}+\frac{3 x \tan ^{-1}(a x)}{32 a^2 c^3 \left (a^2 x^2+1\right )^2}+\frac{\tan ^{-1}(a x)^4}{32 a^3 c^3}-\frac{3 \tan ^{-1}(a x)^2}{128 a^3 c^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 4964

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

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 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 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{x^2 \tan ^{-1}(a x)^3}{\left (c+a^2 c x^2\right )^3} \, dx &=-\frac{\int \frac{\tan ^{-1}(a x)^3}{\left (c+a^2 c x^2\right )^3} \, dx}{a^2}+\frac{\int \frac{\tan ^{-1}(a x)^3}{\left (c+a^2 c x^2\right )^2} \, dx}{a^2 c}\\ &=-\frac{3 \tan ^{-1}(a x)^2}{16 a^3 c^3 \left (1+a^2 x^2\right )^2}-\frac{x \tan ^{-1}(a x)^3}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}+\frac{x \tan ^{-1}(a x)^3}{2 a^2 c^3 \left (1+a^2 x^2\right )}+\frac{\tan ^{-1}(a x)^4}{8 a^3 c^3}+\frac{3 \int \frac{\tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^3} \, dx}{8 a^2}-\frac{3 \int \frac{\tan ^{-1}(a x)^3}{\left (c+a^2 c x^2\right )^2} \, dx}{4 a^2 c}-\frac{3 \int \frac{x \tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^2} \, dx}{2 a c}\\ &=\frac{3}{128 a^3 c^3 \left (1+a^2 x^2\right )^2}+\frac{3 x \tan ^{-1}(a x)}{32 a^2 c^3 \left (1+a^2 x^2\right )^2}-\frac{3 \tan ^{-1}(a x)^2}{16 a^3 c^3 \left (1+a^2 x^2\right )^2}+\frac{3 \tan ^{-1}(a x)^2}{4 a^3 c^3 \left (1+a^2 x^2\right )}-\frac{x \tan ^{-1}(a x)^3}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}+\frac{x \tan ^{-1}(a x)^3}{8 a^2 c^3 \left (1+a^2 x^2\right )}+\frac{\tan ^{-1}(a x)^4}{32 a^3 c^3}+\frac{9 \int \frac{\tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^2} \, dx}{32 a^2 c}-\frac{3 \int \frac{\tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^2} \, dx}{2 a^2 c}+\frac{9 \int \frac{x \tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^2} \, dx}{8 a c}\\ &=\frac{3}{128 a^3 c^3 \left (1+a^2 x^2\right )^2}+\frac{3 x \tan ^{-1}(a x)}{32 a^2 c^3 \left (1+a^2 x^2\right )^2}-\frac{39 x \tan ^{-1}(a x)}{64 a^2 c^3 \left (1+a^2 x^2\right )}-\frac{39 \tan ^{-1}(a x)^2}{128 a^3 c^3}-\frac{3 \tan ^{-1}(a x)^2}{16 a^3 c^3 \left (1+a^2 x^2\right )^2}+\frac{3 \tan ^{-1}(a x)^2}{16 a^3 c^3 \left (1+a^2 x^2\right )}-\frac{x \tan ^{-1}(a x)^3}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}+\frac{x \tan ^{-1}(a x)^3}{8 a^2 c^3 \left (1+a^2 x^2\right )}+\frac{\tan ^{-1}(a x)^4}{32 a^3 c^3}+\frac{9 \int \frac{\tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^2} \, dx}{8 a^2 c}-\frac{9 \int \frac{x}{\left (c+a^2 c x^2\right )^2} \, dx}{64 a c}+\frac{3 \int \frac{x}{\left (c+a^2 c x^2\right )^2} \, dx}{4 a c}\\ &=\frac{3}{128 a^3 c^3 \left (1+a^2 x^2\right )^2}-\frac{39}{128 a^3 c^3 \left (1+a^2 x^2\right )}+\frac{3 x \tan ^{-1}(a x)}{32 a^2 c^3 \left (1+a^2 x^2\right )^2}-\frac{3 x \tan ^{-1}(a x)}{64 a^2 c^3 \left (1+a^2 x^2\right )}-\frac{3 \tan ^{-1}(a x)^2}{128 a^3 c^3}-\frac{3 \tan ^{-1}(a x)^2}{16 a^3 c^3 \left (1+a^2 x^2\right )^2}+\frac{3 \tan ^{-1}(a x)^2}{16 a^3 c^3 \left (1+a^2 x^2\right )}-\frac{x \tan ^{-1}(a x)^3}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}+\frac{x \tan ^{-1}(a x)^3}{8 a^2 c^3 \left (1+a^2 x^2\right )}+\frac{\tan ^{-1}(a x)^4}{32 a^3 c^3}-\frac{9 \int \frac{x}{\left (c+a^2 c x^2\right )^2} \, dx}{16 a c}\\ &=\frac{3}{128 a^3 c^3 \left (1+a^2 x^2\right )^2}-\frac{3}{128 a^3 c^3 \left (1+a^2 x^2\right )}+\frac{3 x \tan ^{-1}(a x)}{32 a^2 c^3 \left (1+a^2 x^2\right )^2}-\frac{3 x \tan ^{-1}(a x)}{64 a^2 c^3 \left (1+a^2 x^2\right )}-\frac{3 \tan ^{-1}(a x)^2}{128 a^3 c^3}-\frac{3 \tan ^{-1}(a x)^2}{16 a^3 c^3 \left (1+a^2 x^2\right )^2}+\frac{3 \tan ^{-1}(a x)^2}{16 a^3 c^3 \left (1+a^2 x^2\right )}-\frac{x \tan ^{-1}(a x)^3}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}+\frac{x \tan ^{-1}(a x)^3}{8 a^2 c^3 \left (1+a^2 x^2\right )}+\frac{\tan ^{-1}(a x)^4}{32 a^3 c^3}\\ \end{align*}

Mathematica [A]  time = 0.0727406, size = 111, normalized size = 0.47 \[ \frac{-3 a^2 x^2+4 \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)^4+16 a x \left (a^2 x^2-1\right ) \tan ^{-1}(a x)^3-3 \left (a^4 x^4-6 a^2 x^2+1\right ) \tan ^{-1}(a x)^2+\left (6 a x-6 a^3 x^3\right ) \tan ^{-1}(a x)}{128 a^3 c^3 \left (a^2 x^2+1\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*a^2*x^2 + (6*a*x - 6*a^3*x^3)*ArcTan[a*x] - 3*(1 - 6*a^2*x^2 + a^4*x^4)*ArcTan[a*x]^2 + 16*a*x*(-1 + a^2*x
^2)*ArcTan[a*x]^3 + 4*(1 + a^2*x^2)^2*ArcTan[a*x]^4)/(128*a^3*c^3*(1 + a^2*x^2)^2)

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Maple [A]  time = 0.144, size = 216, normalized size = 0.9 \begin{align*}{\frac{ \left ( \arctan \left ( ax \right ) \right ) ^{3}{x}^{3}}{8\,{c}^{3} \left ({a}^{2}{x}^{2}+1 \right ) ^{2}}}-{\frac{x \left ( \arctan \left ( ax \right ) \right ) ^{3}}{8\,{c}^{3}{a}^{2} \left ({a}^{2}{x}^{2}+1 \right ) ^{2}}}+{\frac{ \left ( \arctan \left ( ax \right ) \right ) ^{4}}{32\,{c}^{3}{a}^{3}}}-{\frac{3\, \left ( \arctan \left ( ax \right ) \right ) ^{2}}{16\,{c}^{3}{a}^{3} \left ({a}^{2}{x}^{2}+1 \right ) ^{2}}}+{\frac{3\, \left ( \arctan \left ( ax \right ) \right ) ^{2}}{16\,{c}^{3}{a}^{3} \left ({a}^{2}{x}^{2}+1 \right ) }}-{\frac{3\,\arctan \left ( ax \right ){x}^{3}}{64\,{c}^{3} \left ({a}^{2}{x}^{2}+1 \right ) ^{2}}}+{\frac{3\,x\arctan \left ( ax \right ) }{64\,{c}^{3}{a}^{2} \left ({a}^{2}{x}^{2}+1 \right ) ^{2}}}-{\frac{3\, \left ( \arctan \left ( ax \right ) \right ) ^{2}}{128\,{c}^{3}{a}^{3}}}+{\frac{3}{128\,{c}^{3}{a}^{3} \left ({a}^{2}{x}^{2}+1 \right ) ^{2}}}-{\frac{3}{128\,{c}^{3}{a}^{3} \left ({a}^{2}{x}^{2}+1 \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.866, size = 451, normalized size = 1.9 \begin{align*} \frac{1}{8} \,{\left (\frac{a^{2} x^{3} - x}{a^{6} c^{3} x^{4} + 2 \, a^{4} c^{3} x^{2} + a^{2} c^{3}} + \frac{\arctan \left (a x\right )}{a^{3} c^{3}}\right )} \arctan \left (a x\right )^{3} + \frac{3 \,{\left (a^{2} x^{2} -{\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{2}\right )} a \arctan \left (a x\right )^{2}}{16 \,{\left (a^{8} c^{3} x^{4} + 2 \, a^{6} c^{3} x^{2} + a^{4} c^{3}\right )}} - \frac{1}{128} \,{\left (\frac{{\left (4 \,{\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{4} + 3 \, a^{2} x^{2} - 3 \,{\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{2}\right )} a^{2}}{a^{10} c^{3} x^{4} + 2 \, a^{8} c^{3} x^{2} + a^{6} c^{3}} + \frac{2 \,{\left (3 \, a^{3} x^{3} - 8 \,{\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{3} - 3 \, a x + 3 \,{\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^{9} c^{3} x^{4} + 2 \, a^{7} c^{3} x^{2} + a^{5} c^{3}}\right )} a \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*((a^2*x^3 - x)/(a^6*c^3*x^4 + 2*a^4*c^3*x^2 + a^2*c^3) + arctan(a*x)/(a^3*c^3))*arctan(a*x)^3 + 3/16*(a^2*
x^2 - (a^4*x^4 + 2*a^2*x^2 + 1)*arctan(a*x)^2)*a*arctan(a*x)^2/(a^8*c^3*x^4 + 2*a^6*c^3*x^2 + a^4*c^3) - 1/128
*((4*(a^4*x^4 + 2*a^2*x^2 + 1)*arctan(a*x)^4 + 3*a^2*x^2 - 3*(a^4*x^4 + 2*a^2*x^2 + 1)*arctan(a*x)^2)*a^2/(a^1
0*c^3*x^4 + 2*a^8*c^3*x^2 + a^6*c^3) + 2*(3*a^3*x^3 - 8*(a^4*x^4 + 2*a^2*x^2 + 1)*arctan(a*x)^3 - 3*a*x + 3*(a
^4*x^4 + 2*a^2*x^2 + 1)*arctan(a*x))*a*arctan(a*x)/(a^9*c^3*x^4 + 2*a^7*c^3*x^2 + a^5*c^3))*a

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Fricas [A]  time = 1.96763, size = 289, normalized size = 1.22 \begin{align*} \frac{4 \,{\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{4} - 3 \, a^{2} x^{2} + 16 \,{\left (a^{3} x^{3} - a x\right )} \arctan \left (a x\right )^{3} - 3 \,{\left (a^{4} x^{4} - 6 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{2} - 6 \,{\left (a^{3} x^{3} - a x\right )} \arctan \left (a x\right )}{128 \,{\left (a^{7} c^{3} x^{4} + 2 \, a^{5} c^{3} x^{2} + a^{3} c^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Integral(x**2*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{x^{2} \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(x^2*arctan(a*x)^3/(a^2*c*x^2+c)^3,x, algorithm="giac")

[Out]

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