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

Optimal. Leaf size=240 \[ \frac{1}{360} a^4 c^3 x^8+\frac{71 a^2 c^3 x^6}{7560}-\frac{107 c^3 x^2}{12600 a^2}-\frac{26 c^3 \log \left (a^2 x^2+1\right )}{1575 a^4}+\frac{1}{10} a^6 c^3 x^{10} \tan ^{-1}(a x)^2-\frac{1}{45} a^5 c^3 x^9 \tan ^{-1}(a x)+\frac{3}{8} a^4 c^3 x^8 \tan ^{-1}(a x)^2-\frac{11}{140} a^3 c^3 x^7 \tan ^{-1}(a x)+\frac{1}{2} a^2 c^3 x^6 \tan ^{-1}(a x)^2+\frac{c^3 x \tan ^{-1}(a x)}{20 a^3}-\frac{c^3 \tan ^{-1}(a x)^2}{40 a^4}-\frac{9}{100} a c^3 x^5 \tan ^{-1}(a x)+\frac{1}{4} c^3 x^4 \tan ^{-1}(a x)^2-\frac{c^3 x^3 \tan ^{-1}(a x)}{60 a}+\frac{53 c^3 x^4}{6300} \]

[Out]

(-107*c^3*x^2)/(12600*a^2) + (53*c^3*x^4)/6300 + (71*a^2*c^3*x^6)/7560 + (a^4*c^3*x^8)/360 + (c^3*x*ArcTan[a*x
])/(20*a^3) - (c^3*x^3*ArcTan[a*x])/(60*a) - (9*a*c^3*x^5*ArcTan[a*x])/100 - (11*a^3*c^3*x^7*ArcTan[a*x])/140
- (a^5*c^3*x^9*ArcTan[a*x])/45 - (c^3*ArcTan[a*x]^2)/(40*a^4) + (c^3*x^4*ArcTan[a*x]^2)/4 + (a^2*c^3*x^6*ArcTa
n[a*x]^2)/2 + (3*a^4*c^3*x^8*ArcTan[a*x]^2)/8 + (a^6*c^3*x^10*ArcTan[a*x]^2)/10 - (26*c^3*Log[1 + a^2*x^2])/(1
575*a^4)

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Rubi [A]  time = 1.22695, antiderivative size = 240, normalized size of antiderivative = 1., number of steps used = 72, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {4948, 4852, 4916, 266, 43, 4846, 260, 4884} \[ \frac{1}{360} a^4 c^3 x^8+\frac{71 a^2 c^3 x^6}{7560}-\frac{107 c^3 x^2}{12600 a^2}-\frac{26 c^3 \log \left (a^2 x^2+1\right )}{1575 a^4}+\frac{1}{10} a^6 c^3 x^{10} \tan ^{-1}(a x)^2-\frac{1}{45} a^5 c^3 x^9 \tan ^{-1}(a x)+\frac{3}{8} a^4 c^3 x^8 \tan ^{-1}(a x)^2-\frac{11}{140} a^3 c^3 x^7 \tan ^{-1}(a x)+\frac{1}{2} a^2 c^3 x^6 \tan ^{-1}(a x)^2+\frac{c^3 x \tan ^{-1}(a x)}{20 a^3}-\frac{c^3 \tan ^{-1}(a x)^2}{40 a^4}-\frac{9}{100} a c^3 x^5 \tan ^{-1}(a x)+\frac{1}{4} c^3 x^4 \tan ^{-1}(a x)^2-\frac{c^3 x^3 \tan ^{-1}(a x)}{60 a}+\frac{53 c^3 x^4}{6300} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-107*c^3*x^2)/(12600*a^2) + (53*c^3*x^4)/6300 + (71*a^2*c^3*x^6)/7560 + (a^4*c^3*x^8)/360 + (c^3*x*ArcTan[a*x
])/(20*a^3) - (c^3*x^3*ArcTan[a*x])/(60*a) - (9*a*c^3*x^5*ArcTan[a*x])/100 - (11*a^3*c^3*x^7*ArcTan[a*x])/140
- (a^5*c^3*x^9*ArcTan[a*x])/45 - (c^3*ArcTan[a*x]^2)/(40*a^4) + (c^3*x^4*ArcTan[a*x]^2)/4 + (a^2*c^3*x^6*ArcTa
n[a*x]^2)/2 + (3*a^4*c^3*x^8*ArcTan[a*x]^2)/8 + (a^6*c^3*x^10*ArcTan[a*x]^2)/10 - (26*c^3*Log[1 + a^2*x^2])/(1
575*a^4)

Rule 4948

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(q_), x_Symbol] :> Int[Ex
pandIntegrand[(f*x)^m*(d + e*x^2)^q*(a + b*ArcTan[c*x])^p, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e,
c^2*d] && IGtQ[p, 0] && IGtQ[q, 1] && (EqQ[p, 1] || IntegerQ[m])

Rule 4852

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcTa
n[c*x])^p)/(d*(m + 1)), x] - Dist[(b*c*p)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcTan[c*x])^(p - 1))/(1 + c^
2*x^2), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[p, 0] && (EqQ[p, 1] || IntegerQ[m]) && NeQ[m, -1]

Rule 4916

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

Rule 266

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

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 4846

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

Rule 260

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

Rule 4884

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

Rubi steps

\begin{align*} \int x^3 \left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^2 \, dx &=\int \left (c^3 x^3 \tan ^{-1}(a x)^2+3 a^2 c^3 x^5 \tan ^{-1}(a x)^2+3 a^4 c^3 x^7 \tan ^{-1}(a x)^2+a^6 c^3 x^9 \tan ^{-1}(a x)^2\right ) \, dx\\ &=c^3 \int x^3 \tan ^{-1}(a x)^2 \, dx+\left (3 a^2 c^3\right ) \int x^5 \tan ^{-1}(a x)^2 \, dx+\left (3 a^4 c^3\right ) \int x^7 \tan ^{-1}(a x)^2 \, dx+\left (a^6 c^3\right ) \int x^9 \tan ^{-1}(a x)^2 \, dx\\ &=\frac{1}{4} c^3 x^4 \tan ^{-1}(a x)^2+\frac{1}{2} a^2 c^3 x^6 \tan ^{-1}(a x)^2+\frac{3}{8} a^4 c^3 x^8 \tan ^{-1}(a x)^2+\frac{1}{10} a^6 c^3 x^{10} \tan ^{-1}(a x)^2-\frac{1}{2} \left (a c^3\right ) \int \frac{x^4 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\left (a^3 c^3\right ) \int \frac{x^6 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac{1}{4} \left (3 a^5 c^3\right ) \int \frac{x^8 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac{1}{5} \left (a^7 c^3\right ) \int \frac{x^{10} \tan ^{-1}(a x)}{1+a^2 x^2} \, dx\\ &=\frac{1}{4} c^3 x^4 \tan ^{-1}(a x)^2+\frac{1}{2} a^2 c^3 x^6 \tan ^{-1}(a x)^2+\frac{3}{8} a^4 c^3 x^8 \tan ^{-1}(a x)^2+\frac{1}{10} a^6 c^3 x^{10} \tan ^{-1}(a x)^2-\frac{c^3 \int x^2 \tan ^{-1}(a x) \, dx}{2 a}+\frac{c^3 \int \frac{x^2 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{2 a}-\left (a c^3\right ) \int x^4 \tan ^{-1}(a x) \, dx+\left (a c^3\right ) \int \frac{x^4 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac{1}{4} \left (3 a^3 c^3\right ) \int x^6 \tan ^{-1}(a x) \, dx+\frac{1}{4} \left (3 a^3 c^3\right ) \int \frac{x^6 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac{1}{5} \left (a^5 c^3\right ) \int x^8 \tan ^{-1}(a x) \, dx+\frac{1}{5} \left (a^5 c^3\right ) \int \frac{x^8 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx\\ &=-\frac{c^3 x^3 \tan ^{-1}(a x)}{6 a}-\frac{1}{5} a c^3 x^5 \tan ^{-1}(a x)-\frac{3}{28} a^3 c^3 x^7 \tan ^{-1}(a x)-\frac{1}{45} a^5 c^3 x^9 \tan ^{-1}(a x)+\frac{1}{4} c^3 x^4 \tan ^{-1}(a x)^2+\frac{1}{2} a^2 c^3 x^6 \tan ^{-1}(a x)^2+\frac{3}{8} a^4 c^3 x^8 \tan ^{-1}(a x)^2+\frac{1}{10} a^6 c^3 x^{10} \tan ^{-1}(a x)^2+\frac{1}{6} c^3 \int \frac{x^3}{1+a^2 x^2} \, dx+\frac{c^3 \int \tan ^{-1}(a x) \, dx}{2 a^3}-\frac{c^3 \int \frac{\tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{2 a^3}+\frac{c^3 \int x^2 \tan ^{-1}(a x) \, dx}{a}-\frac{c^3 \int \frac{x^2 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{a}+\frac{1}{4} \left (3 a c^3\right ) \int x^4 \tan ^{-1}(a x) \, dx-\frac{1}{4} \left (3 a c^3\right ) \int \frac{x^4 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx+\frac{1}{5} \left (a^2 c^3\right ) \int \frac{x^5}{1+a^2 x^2} \, dx+\frac{1}{5} \left (a^3 c^3\right ) \int x^6 \tan ^{-1}(a x) \, dx-\frac{1}{5} \left (a^3 c^3\right ) \int \frac{x^6 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx+\frac{1}{28} \left (3 a^4 c^3\right ) \int \frac{x^7}{1+a^2 x^2} \, dx+\frac{1}{45} \left (a^6 c^3\right ) \int \frac{x^9}{1+a^2 x^2} \, dx\\ &=\frac{c^3 x \tan ^{-1}(a x)}{2 a^3}+\frac{c^3 x^3 \tan ^{-1}(a x)}{6 a}-\frac{1}{20} a c^3 x^5 \tan ^{-1}(a x)-\frac{11}{140} a^3 c^3 x^7 \tan ^{-1}(a x)-\frac{1}{45} a^5 c^3 x^9 \tan ^{-1}(a x)-\frac{c^3 \tan ^{-1}(a x)^2}{4 a^4}+\frac{1}{4} c^3 x^4 \tan ^{-1}(a x)^2+\frac{1}{2} a^2 c^3 x^6 \tan ^{-1}(a x)^2+\frac{3}{8} a^4 c^3 x^8 \tan ^{-1}(a x)^2+\frac{1}{10} a^6 c^3 x^{10} \tan ^{-1}(a x)^2+\frac{1}{12} c^3 \operatorname{Subst}\left (\int \frac{x}{1+a^2 x} \, dx,x,x^2\right )-\frac{1}{3} c^3 \int \frac{x^3}{1+a^2 x^2} \, dx-\frac{c^3 \int \tan ^{-1}(a x) \, dx}{a^3}+\frac{c^3 \int \frac{\tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{a^3}-\frac{c^3 \int \frac{x}{1+a^2 x^2} \, dx}{2 a^2}-\frac{\left (3 c^3\right ) \int x^2 \tan ^{-1}(a x) \, dx}{4 a}+\frac{\left (3 c^3\right ) \int \frac{x^2 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{4 a}-\frac{1}{5} \left (a c^3\right ) \int x^4 \tan ^{-1}(a x) \, dx+\frac{1}{5} \left (a c^3\right ) \int \frac{x^4 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx+\frac{1}{10} \left (a^2 c^3\right ) \operatorname{Subst}\left (\int \frac{x^2}{1+a^2 x} \, dx,x,x^2\right )-\frac{1}{20} \left (3 a^2 c^3\right ) \int \frac{x^5}{1+a^2 x^2} \, dx-\frac{1}{35} \left (a^4 c^3\right ) \int \frac{x^7}{1+a^2 x^2} \, dx+\frac{1}{56} \left (3 a^4 c^3\right ) \operatorname{Subst}\left (\int \frac{x^3}{1+a^2 x} \, dx,x,x^2\right )+\frac{1}{90} \left (a^6 c^3\right ) \operatorname{Subst}\left (\int \frac{x^4}{1+a^2 x} \, dx,x,x^2\right )\\ &=-\frac{c^3 x \tan ^{-1}(a x)}{2 a^3}-\frac{c^3 x^3 \tan ^{-1}(a x)}{12 a}-\frac{9}{100} a c^3 x^5 \tan ^{-1}(a x)-\frac{11}{140} a^3 c^3 x^7 \tan ^{-1}(a x)-\frac{1}{45} a^5 c^3 x^9 \tan ^{-1}(a x)+\frac{c^3 \tan ^{-1}(a x)^2}{4 a^4}+\frac{1}{4} c^3 x^4 \tan ^{-1}(a x)^2+\frac{1}{2} a^2 c^3 x^6 \tan ^{-1}(a x)^2+\frac{3}{8} a^4 c^3 x^8 \tan ^{-1}(a x)^2+\frac{1}{10} a^6 c^3 x^{10} \tan ^{-1}(a x)^2-\frac{c^3 \log \left (1+a^2 x^2\right )}{4 a^4}+\frac{1}{12} c^3 \operatorname{Subst}\left (\int \left (\frac{1}{a^2}-\frac{1}{a^2 \left (1+a^2 x\right )}\right ) \, dx,x,x^2\right )-\frac{1}{6} c^3 \operatorname{Subst}\left (\int \frac{x}{1+a^2 x} \, dx,x,x^2\right )+\frac{1}{4} c^3 \int \frac{x^3}{1+a^2 x^2} \, dx+\frac{\left (3 c^3\right ) \int \tan ^{-1}(a x) \, dx}{4 a^3}-\frac{\left (3 c^3\right ) \int \frac{\tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{4 a^3}+\frac{c^3 \int \frac{x}{1+a^2 x^2} \, dx}{a^2}+\frac{c^3 \int x^2 \tan ^{-1}(a x) \, dx}{5 a}-\frac{c^3 \int \frac{x^2 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{5 a}+\frac{1}{25} \left (a^2 c^3\right ) \int \frac{x^5}{1+a^2 x^2} \, dx-\frac{1}{40} \left (3 a^2 c^3\right ) \operatorname{Subst}\left (\int \frac{x^2}{1+a^2 x} \, dx,x,x^2\right )+\frac{1}{10} \left (a^2 c^3\right ) \operatorname{Subst}\left (\int \left (-\frac{1}{a^4}+\frac{x}{a^2}+\frac{1}{a^4 \left (1+a^2 x\right )}\right ) \, dx,x,x^2\right )-\frac{1}{70} \left (a^4 c^3\right ) \operatorname{Subst}\left (\int \frac{x^3}{1+a^2 x} \, dx,x,x^2\right )+\frac{1}{56} \left (3 a^4 c^3\right ) \operatorname{Subst}\left (\int \left (\frac{1}{a^6}-\frac{x}{a^4}+\frac{x^2}{a^2}-\frac{1}{a^6 \left (1+a^2 x\right )}\right ) \, dx,x,x^2\right )+\frac{1}{90} \left (a^6 c^3\right ) \operatorname{Subst}\left (\int \left (-\frac{1}{a^8}+\frac{x}{a^6}-\frac{x^2}{a^4}+\frac{x^3}{a^2}+\frac{1}{a^8 \left (1+a^2 x\right )}\right ) \, dx,x,x^2\right )\\ &=\frac{13 c^3 x^2}{504 a^2}+\frac{29 c^3 x^4}{1008}+\frac{107 a^2 c^3 x^6}{7560}+\frac{1}{360} a^4 c^3 x^8+\frac{c^3 x \tan ^{-1}(a x)}{4 a^3}-\frac{c^3 x^3 \tan ^{-1}(a x)}{60 a}-\frac{9}{100} a c^3 x^5 \tan ^{-1}(a x)-\frac{11}{140} a^3 c^3 x^7 \tan ^{-1}(a x)-\frac{1}{45} a^5 c^3 x^9 \tan ^{-1}(a x)-\frac{c^3 \tan ^{-1}(a x)^2}{8 a^4}+\frac{1}{4} c^3 x^4 \tan ^{-1}(a x)^2+\frac{1}{2} a^2 c^3 x^6 \tan ^{-1}(a x)^2+\frac{3}{8} a^4 c^3 x^8 \tan ^{-1}(a x)^2+\frac{1}{10} a^6 c^3 x^{10} \tan ^{-1}(a x)^2+\frac{113 c^3 \log \left (1+a^2 x^2\right )}{504 a^4}-\frac{1}{15} c^3 \int \frac{x^3}{1+a^2 x^2} \, dx+\frac{1}{8} c^3 \operatorname{Subst}\left (\int \frac{x}{1+a^2 x} \, dx,x,x^2\right )-\frac{1}{6} c^3 \operatorname{Subst}\left (\int \left (\frac{1}{a^2}-\frac{1}{a^2 \left (1+a^2 x\right )}\right ) \, dx,x,x^2\right )-\frac{c^3 \int \tan ^{-1}(a x) \, dx}{5 a^3}+\frac{c^3 \int \frac{\tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{5 a^3}-\frac{\left (3 c^3\right ) \int \frac{x}{1+a^2 x^2} \, dx}{4 a^2}+\frac{1}{50} \left (a^2 c^3\right ) \operatorname{Subst}\left (\int \frac{x^2}{1+a^2 x} \, dx,x,x^2\right )-\frac{1}{40} \left (3 a^2 c^3\right ) \operatorname{Subst}\left (\int \left (-\frac{1}{a^4}+\frac{x}{a^2}+\frac{1}{a^4 \left (1+a^2 x\right )}\right ) \, dx,x,x^2\right )-\frac{1}{70} \left (a^4 c^3\right ) \operatorname{Subst}\left (\int \left (\frac{1}{a^6}-\frac{x}{a^4}+\frac{x^2}{a^2}-\frac{1}{a^6 \left (1+a^2 x\right )}\right ) \, dx,x,x^2\right )\\ &=-\frac{101 c^3 x^2}{1260 a^2}-\frac{c^3 x^4}{630}+\frac{71 a^2 c^3 x^6}{7560}+\frac{1}{360} a^4 c^3 x^8+\frac{c^3 x \tan ^{-1}(a x)}{20 a^3}-\frac{c^3 x^3 \tan ^{-1}(a x)}{60 a}-\frac{9}{100} a c^3 x^5 \tan ^{-1}(a x)-\frac{11}{140} a^3 c^3 x^7 \tan ^{-1}(a x)-\frac{1}{45} a^5 c^3 x^9 \tan ^{-1}(a x)-\frac{c^3 \tan ^{-1}(a x)^2}{40 a^4}+\frac{1}{4} c^3 x^4 \tan ^{-1}(a x)^2+\frac{1}{2} a^2 c^3 x^6 \tan ^{-1}(a x)^2+\frac{3}{8} a^4 c^3 x^8 \tan ^{-1}(a x)^2+\frac{1}{10} a^6 c^3 x^{10} \tan ^{-1}(a x)^2-\frac{113 c^3 \log \left (1+a^2 x^2\right )}{2520 a^4}-\frac{1}{30} c^3 \operatorname{Subst}\left (\int \frac{x}{1+a^2 x} \, dx,x,x^2\right )+\frac{1}{8} c^3 \operatorname{Subst}\left (\int \left (\frac{1}{a^2}-\frac{1}{a^2 \left (1+a^2 x\right )}\right ) \, dx,x,x^2\right )+\frac{c^3 \int \frac{x}{1+a^2 x^2} \, dx}{5 a^2}+\frac{1}{50} \left (a^2 c^3\right ) \operatorname{Subst}\left (\int \left (-\frac{1}{a^4}+\frac{x}{a^2}+\frac{1}{a^4 \left (1+a^2 x\right )}\right ) \, dx,x,x^2\right )\\ &=\frac{313 c^3 x^2}{12600 a^2}+\frac{53 c^3 x^4}{6300}+\frac{71 a^2 c^3 x^6}{7560}+\frac{1}{360} a^4 c^3 x^8+\frac{c^3 x \tan ^{-1}(a x)}{20 a^3}-\frac{c^3 x^3 \tan ^{-1}(a x)}{60 a}-\frac{9}{100} a c^3 x^5 \tan ^{-1}(a x)-\frac{11}{140} a^3 c^3 x^7 \tan ^{-1}(a x)-\frac{1}{45} a^5 c^3 x^9 \tan ^{-1}(a x)-\frac{c^3 \tan ^{-1}(a x)^2}{40 a^4}+\frac{1}{4} c^3 x^4 \tan ^{-1}(a x)^2+\frac{1}{2} a^2 c^3 x^6 \tan ^{-1}(a x)^2+\frac{3}{8} a^4 c^3 x^8 \tan ^{-1}(a x)^2+\frac{1}{10} a^6 c^3 x^{10} \tan ^{-1}(a x)^2-\frac{157 c^3 \log \left (1+a^2 x^2\right )}{3150 a^4}-\frac{1}{30} c^3 \operatorname{Subst}\left (\int \left (\frac{1}{a^2}-\frac{1}{a^2 \left (1+a^2 x\right )}\right ) \, dx,x,x^2\right )\\ &=-\frac{107 c^3 x^2}{12600 a^2}+\frac{53 c^3 x^4}{6300}+\frac{71 a^2 c^3 x^6}{7560}+\frac{1}{360} a^4 c^3 x^8+\frac{c^3 x \tan ^{-1}(a x)}{20 a^3}-\frac{c^3 x^3 \tan ^{-1}(a x)}{60 a}-\frac{9}{100} a c^3 x^5 \tan ^{-1}(a x)-\frac{11}{140} a^3 c^3 x^7 \tan ^{-1}(a x)-\frac{1}{45} a^5 c^3 x^9 \tan ^{-1}(a x)-\frac{c^3 \tan ^{-1}(a x)^2}{40 a^4}+\frac{1}{4} c^3 x^4 \tan ^{-1}(a x)^2+\frac{1}{2} a^2 c^3 x^6 \tan ^{-1}(a x)^2+\frac{3}{8} a^4 c^3 x^8 \tan ^{-1}(a x)^2+\frac{1}{10} a^6 c^3 x^{10} \tan ^{-1}(a x)^2-\frac{26 c^3 \log \left (1+a^2 x^2\right )}{1575 a^4}\\ \end{align*}

Mathematica [A]  time = 0.0938761, size = 126, normalized size = 0.52 \[ \frac{c^3 \left (105 a^8 x^8+355 a^6 x^6+318 a^4 x^4-321 a^2 x^2-624 \log \left (a^2 x^2+1\right )-6 a x \left (140 a^8 x^8+495 a^6 x^6+567 a^4 x^4+105 a^2 x^2-315\right ) \tan ^{-1}(a x)+945 \left (a^2 x^2+1\right )^4 \left (4 a^2 x^2-1\right ) \tan ^{-1}(a x)^2\right )}{37800 a^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c^3*(-321*a^2*x^2 + 318*a^4*x^4 + 355*a^6*x^6 + 105*a^8*x^8 - 6*a*x*(-315 + 105*a^2*x^2 + 567*a^4*x^4 + 495*a
^6*x^6 + 140*a^8*x^8)*ArcTan[a*x] + 945*(1 + a^2*x^2)^4*(-1 + 4*a^2*x^2)*ArcTan[a*x]^2 - 624*Log[1 + a^2*x^2])
)/(37800*a^4)

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Maple [A]  time = 0.033, size = 211, normalized size = 0.9 \begin{align*} -{\frac{107\,{c}^{3}{x}^{2}}{12600\,{a}^{2}}}+{\frac{53\,{c}^{3}{x}^{4}}{6300}}+{\frac{71\,{a}^{2}{c}^{3}{x}^{6}}{7560}}+{\frac{{a}^{4}{c}^{3}{x}^{8}}{360}}+{\frac{{c}^{3}x\arctan \left ( ax \right ) }{20\,{a}^{3}}}-{\frac{{c}^{3}{x}^{3}\arctan \left ( ax \right ) }{60\,a}}-{\frac{9\,a{c}^{3}{x}^{5}\arctan \left ( ax \right ) }{100}}-{\frac{11\,{a}^{3}{c}^{3}{x}^{7}\arctan \left ( ax \right ) }{140}}-{\frac{{a}^{5}{c}^{3}{x}^{9}\arctan \left ( ax \right ) }{45}}-{\frac{{c}^{3} \left ( \arctan \left ( ax \right ) \right ) ^{2}}{40\,{a}^{4}}}+{\frac{{c}^{3}{x}^{4} \left ( \arctan \left ( ax \right ) \right ) ^{2}}{4}}+{\frac{{a}^{2}{c}^{3}{x}^{6} \left ( \arctan \left ( ax \right ) \right ) ^{2}}{2}}+{\frac{3\,{a}^{4}{c}^{3}{x}^{8} \left ( \arctan \left ( ax \right ) \right ) ^{2}}{8}}+{\frac{{a}^{6}{c}^{3}{x}^{10} \left ( \arctan \left ( ax \right ) \right ) ^{2}}{10}}-{\frac{26\,{c}^{3}\ln \left ({a}^{2}{x}^{2}+1 \right ) }{1575\,{a}^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-107/12600*c^3*x^2/a^2+53/6300*c^3*x^4+71/7560*a^2*c^3*x^6+1/360*a^4*c^3*x^8+1/20*c^3*x*arctan(a*x)/a^3-1/60*c
^3*x^3*arctan(a*x)/a-9/100*a*c^3*x^5*arctan(a*x)-11/140*a^3*c^3*x^7*arctan(a*x)-1/45*a^5*c^3*x^9*arctan(a*x)-1
/40*c^3*arctan(a*x)^2/a^4+1/4*c^3*x^4*arctan(a*x)^2+1/2*a^2*c^3*x^6*arctan(a*x)^2+3/8*a^4*c^3*x^8*arctan(a*x)^
2+1/10*a^6*c^3*x^10*arctan(a*x)^2-26/1575*c^3*ln(a^2*x^2+1)/a^4

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Maxima [A]  time = 1.52462, size = 273, normalized size = 1.14 \begin{align*} -\frac{1}{6300} \, a{\left (\frac{315 \, c^{3} \arctan \left (a x\right )}{a^{5}} + \frac{140 \, a^{8} c^{3} x^{9} + 495 \, a^{6} c^{3} x^{7} + 567 \, a^{4} c^{3} x^{5} + 105 \, a^{2} c^{3} x^{3} - 315 \, c^{3} x}{a^{4}}\right )} \arctan \left (a x\right ) + \frac{1}{40} \,{\left (4 \, a^{6} c^{3} x^{10} + 15 \, a^{4} c^{3} x^{8} + 20 \, a^{2} c^{3} x^{6} + 10 \, c^{3} x^{4}\right )} \arctan \left (a x\right )^{2} + \frac{105 \, a^{8} c^{3} x^{8} + 355 \, a^{6} c^{3} x^{6} + 318 \, a^{4} c^{3} x^{4} - 321 \, a^{2} c^{3} x^{2} + 945 \, c^{3} \arctan \left (a x\right )^{2} - 624 \, c^{3} \log \left (a^{2} x^{2} + 1\right )}{37800 \, a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6300*a*(315*c^3*arctan(a*x)/a^5 + (140*a^8*c^3*x^9 + 495*a^6*c^3*x^7 + 567*a^4*c^3*x^5 + 105*a^2*c^3*x^3 -
315*c^3*x)/a^4)*arctan(a*x) + 1/40*(4*a^6*c^3*x^10 + 15*a^4*c^3*x^8 + 20*a^2*c^3*x^6 + 10*c^3*x^4)*arctan(a*x)
^2 + 1/37800*(105*a^8*c^3*x^8 + 355*a^6*c^3*x^6 + 318*a^4*c^3*x^4 - 321*a^2*c^3*x^2 + 945*c^3*arctan(a*x)^2 -
624*c^3*log(a^2*x^2 + 1))/a^4

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Fricas [A]  time = 2.22803, size = 417, normalized size = 1.74 \begin{align*} \frac{105 \, a^{8} c^{3} x^{8} + 355 \, a^{6} c^{3} x^{6} + 318 \, a^{4} c^{3} x^{4} - 321 \, a^{2} c^{3} x^{2} - 624 \, c^{3} \log \left (a^{2} x^{2} + 1\right ) + 945 \,{\left (4 \, a^{10} c^{3} x^{10} + 15 \, a^{8} c^{3} x^{8} + 20 \, a^{6} c^{3} x^{6} + 10 \, a^{4} c^{3} x^{4} - c^{3}\right )} \arctan \left (a x\right )^{2} - 6 \,{\left (140 \, a^{9} c^{3} x^{9} + 495 \, a^{7} c^{3} x^{7} + 567 \, a^{5} c^{3} x^{5} + 105 \, a^{3} c^{3} x^{3} - 315 \, a c^{3} x\right )} \arctan \left (a x\right )}{37800 \, a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/37800*(105*a^8*c^3*x^8 + 355*a^6*c^3*x^6 + 318*a^4*c^3*x^4 - 321*a^2*c^3*x^2 - 624*c^3*log(a^2*x^2 + 1) + 94
5*(4*a^10*c^3*x^10 + 15*a^8*c^3*x^8 + 20*a^6*c^3*x^6 + 10*a^4*c^3*x^4 - c^3)*arctan(a*x)^2 - 6*(140*a^9*c^3*x^
9 + 495*a^7*c^3*x^7 + 567*a^5*c^3*x^5 + 105*a^3*c^3*x^3 - 315*a*c^3*x)*arctan(a*x))/a^4

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Sympy [A]  time = 7.61995, size = 241, normalized size = 1. \begin{align*} \begin{cases} \frac{a^{6} c^{3} x^{10} \operatorname{atan}^{2}{\left (a x \right )}}{10} - \frac{a^{5} c^{3} x^{9} \operatorname{atan}{\left (a x \right )}}{45} + \frac{3 a^{4} c^{3} x^{8} \operatorname{atan}^{2}{\left (a x \right )}}{8} + \frac{a^{4} c^{3} x^{8}}{360} - \frac{11 a^{3} c^{3} x^{7} \operatorname{atan}{\left (a x \right )}}{140} + \frac{a^{2} c^{3} x^{6} \operatorname{atan}^{2}{\left (a x \right )}}{2} + \frac{71 a^{2} c^{3} x^{6}}{7560} - \frac{9 a c^{3} x^{5} \operatorname{atan}{\left (a x \right )}}{100} + \frac{c^{3} x^{4} \operatorname{atan}^{2}{\left (a x \right )}}{4} + \frac{53 c^{3} x^{4}}{6300} - \frac{c^{3} x^{3} \operatorname{atan}{\left (a x \right )}}{60 a} - \frac{107 c^{3} x^{2}}{12600 a^{2}} + \frac{c^{3} x \operatorname{atan}{\left (a x \right )}}{20 a^{3}} - \frac{26 c^{3} \log{\left (x^{2} + \frac{1}{a^{2}} \right )}}{1575 a^{4}} - \frac{c^{3} \operatorname{atan}^{2}{\left (a x \right )}}{40 a^{4}} & \text{for}\: a \neq 0 \\0 & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(a**2*c*x**2+c)**3*atan(a*x)**2,x)

[Out]

Piecewise((a**6*c**3*x**10*atan(a*x)**2/10 - a**5*c**3*x**9*atan(a*x)/45 + 3*a**4*c**3*x**8*atan(a*x)**2/8 + a
**4*c**3*x**8/360 - 11*a**3*c**3*x**7*atan(a*x)/140 + a**2*c**3*x**6*atan(a*x)**2/2 + 71*a**2*c**3*x**6/7560 -
9*a*c**3*x**5*atan(a*x)/100 + c**3*x**4*atan(a*x)**2/4 + 53*c**3*x**4/6300 - c**3*x**3*atan(a*x)/(60*a) - 107
*c**3*x**2/(12600*a**2) + c**3*x*atan(a*x)/(20*a**3) - 26*c**3*log(x**2 + a**(-2))/(1575*a**4) - c**3*atan(a*x
)**2/(40*a**4), Ne(a, 0)), (0, True))

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Giac [A]  time = 1.17823, size = 267, normalized size = 1.11 \begin{align*} \frac{1}{40} \,{\left (4 \, a^{6} c^{3} x^{10} + 15 \, a^{4} c^{3} x^{8} + 20 \, a^{2} c^{3} x^{6} + 10 \, c^{3} x^{4}\right )} \arctan \left (a x\right )^{2} - \frac{840 \, a^{9} c^{3} x^{9} \arctan \left (a x\right ) - 105 \, a^{8} c^{3} x^{8} + 2970 \, a^{7} c^{3} x^{7} \arctan \left (a x\right ) - 355 \, a^{6} c^{3} x^{6} + 3402 \, a^{5} c^{3} x^{5} \arctan \left (a x\right ) - 318 \, a^{4} c^{3} x^{4} + 630 \, a^{3} c^{3} x^{3} \arctan \left (a x\right ) + 321 \, a^{2} c^{3} x^{2} - 1890 \, a c^{3} x \arctan \left (a x\right ) + 945 \, c^{3} \arctan \left (a x\right )^{2} + 624 \, c^{3} \log \left (a^{2} x^{2} + 1\right )}{37800 \, a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/40*(4*a^6*c^3*x^10 + 15*a^4*c^3*x^8 + 20*a^2*c^3*x^6 + 10*c^3*x^4)*arctan(a*x)^2 - 1/37800*(840*a^9*c^3*x^9*
arctan(a*x) - 105*a^8*c^3*x^8 + 2970*a^7*c^3*x^7*arctan(a*x) - 355*a^6*c^3*x^6 + 3402*a^5*c^3*x^5*arctan(a*x)
- 318*a^4*c^3*x^4 + 630*a^3*c^3*x^3*arctan(a*x) + 321*a^2*c^3*x^2 - 1890*a*c^3*x*arctan(a*x) + 945*c^3*arctan(
a*x)^2 + 624*c^3*log(a^2*x^2 + 1))/a^4