\(\int x^2 (c+a^2 c x^2)^3 \arctan (a x)^3 \, dx\) [380]

Optimal result

Integrand size = 22, antiderivative size = 389 \[ \int x^2 \left (c+a^2 c x^2\right )^3 \arctan (a x)^3 \, dx=-\frac {107 c^3 x^2}{7560 a}-\frac {11 a c^3 x^4}{1260}-\frac {1}{504} a^3 c^3 x^6-\frac {47 c^3 x \arctan (a x)}{1260 a^2}+\frac {239 c^3 x^3 \arctan (a x)}{3780}+\frac {59 a^2 c^3 x^5 \arctan (a x)}{1260}+\frac {1}{84} a^4 c^3 x^7 \arctan (a x)+\frac {47 c^3 \arctan (a x)^2}{2520 a^3}-\frac {8 c^3 x^2 \arctan (a x)^2}{105 a}-\frac {89}{420} a c^3 x^4 \arctan (a x)^2-\frac {10}{63} a^3 c^3 x^6 \arctan (a x)^2-\frac {1}{24} a^5 c^3 x^8 \arctan (a x)^2-\frac {16 i c^3 \arctan (a x)^3}{315 a^3}+\frac {1}{3} c^3 x^3 \arctan (a x)^3+\frac {3}{5} a^2 c^3 x^5 \arctan (a x)^3+\frac {3}{7} a^4 c^3 x^7 \arctan (a x)^3+\frac {1}{9} a^6 c^3 x^9 \arctan (a x)^3-\frac {16 c^3 \arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )}{105 a^3}+\frac {31 c^3 \log \left (1+a^2 x^2\right )}{945 a^3}-\frac {16 i c^3 \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{105 a^3}-\frac {8 c^3 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )}{105 a^3} \]

[Out]

-107/7560*c^3*x^2/a-11/1260*a*c^3*x^4-1/504*a^3*c^3*x^6-47/1260*c^3*x*arctan(a*x)/a^2+239/3780*c^3*x^3*arctan(
a*x)+59/1260*a^2*c^3*x^5*arctan(a*x)+1/84*a^4*c^3*x^7*arctan(a*x)+47/2520*c^3*arctan(a*x)^2/a^3-8/105*c^3*x^2*
arctan(a*x)^2/a-89/420*a*c^3*x^4*arctan(a*x)^2-10/63*a^3*c^3*x^6*arctan(a*x)^2-1/24*a^5*c^3*x^8*arctan(a*x)^2-
16/315*I*c^3*arctan(a*x)^3/a^3+1/3*c^3*x^3*arctan(a*x)^3+3/5*a^2*c^3*x^5*arctan(a*x)^3+3/7*a^4*c^3*x^7*arctan(
a*x)^3+1/9*a^6*c^3*x^9*arctan(a*x)^3-16/105*c^3*arctan(a*x)^2*ln(2/(1+I*a*x))/a^3+31/945*c^3*ln(a^2*x^2+1)/a^3
-16/105*I*c^3*arctan(a*x)*polylog(2,1-2/(1+I*a*x))/a^3-8/105*c^3*polylog(3,1-2/(1+I*a*x))/a^3

Rubi [A] (verified)

Time = 2.17 (sec) , antiderivative size = 389, normalized size of antiderivative = 1.00, number of steps used = 132, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.545, Rules used = {5068, 4946, 5036, 4930, 266, 5004, 5040, 4964, 5114, 6745, 272, 45} \[ \int x^2 \left (c+a^2 c x^2\right )^3 \arctan (a x)^3 \, dx=\frac {1}{9} a^6 c^3 x^9 \arctan (a x)^3-\frac {1}{24} a^5 c^3 x^8 \arctan (a x)^2+\frac {3}{7} a^4 c^3 x^7 \arctan (a x)^3+\frac {1}{84} a^4 c^3 x^7 \arctan (a x)-\frac {16 i c^3 \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )}{105 a^3}-\frac {10}{63} a^3 c^3 x^6 \arctan (a x)^2-\frac {16 i c^3 \arctan (a x)^3}{315 a^3}+\frac {47 c^3 \arctan (a x)^2}{2520 a^3}-\frac {16 c^3 \arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )}{105 a^3}-\frac {8 c^3 \operatorname {PolyLog}\left (3,1-\frac {2}{i a x+1}\right )}{105 a^3}-\frac {1}{504} a^3 c^3 x^6+\frac {3}{5} a^2 c^3 x^5 \arctan (a x)^3+\frac {59 a^2 c^3 x^5 \arctan (a x)}{1260}-\frac {47 c^3 x \arctan (a x)}{1260 a^2}+\frac {31 c^3 \log \left (a^2 x^2+1\right )}{945 a^3}-\frac {89}{420} a c^3 x^4 \arctan (a x)^2+\frac {1}{3} c^3 x^3 \arctan (a x)^3+\frac {239 c^3 x^3 \arctan (a x)}{3780}-\frac {8 c^3 x^2 \arctan (a x)^2}{105 a}-\frac {11 a c^3 x^4}{1260}-\frac {107 c^3 x^2}{7560 a} \]

[In]

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

[Out]

(-107*c^3*x^2)/(7560*a) - (11*a*c^3*x^4)/1260 - (a^3*c^3*x^6)/504 - (47*c^3*x*ArcTan[a*x])/(1260*a^2) + (239*c
^3*x^3*ArcTan[a*x])/3780 + (59*a^2*c^3*x^5*ArcTan[a*x])/1260 + (a^4*c^3*x^7*ArcTan[a*x])/84 + (47*c^3*ArcTan[a
*x]^2)/(2520*a^3) - (8*c^3*x^2*ArcTan[a*x]^2)/(105*a) - (89*a*c^3*x^4*ArcTan[a*x]^2)/420 - (10*a^3*c^3*x^6*Arc
Tan[a*x]^2)/63 - (a^5*c^3*x^8*ArcTan[a*x]^2)/24 - (((16*I)/315)*c^3*ArcTan[a*x]^3)/a^3 + (c^3*x^3*ArcTan[a*x]^
3)/3 + (3*a^2*c^3*x^5*ArcTan[a*x]^3)/5 + (3*a^4*c^3*x^7*ArcTan[a*x]^3)/7 + (a^6*c^3*x^9*ArcTan[a*x]^3)/9 - (16
*c^3*ArcTan[a*x]^2*Log[2/(1 + I*a*x)])/(105*a^3) + (31*c^3*Log[1 + a^2*x^2])/(945*a^3) - (((16*I)/105)*c^3*Arc
Tan[a*x]*PolyLog[2, 1 - 2/(1 + I*a*x)])/a^3 - (8*c^3*PolyLog[3, 1 - 2/(1 + I*a*x)])/(105*a^3)

Rule 45

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 266

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 272

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 4930

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

Rule 4946

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

Rule 4964

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

Rule 5004

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]

Rule 5036

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 5040

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

Rule 5068

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 5114

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

Rule 6745

Int[(u_)*PolyLog[n_, v_], x_Symbol] :> With[{w = DerivativeDivides[v, u*v, x]}, Simp[w*PolyLog[n + 1, v], x] /
;  !FalseQ[w]] /; FreeQ[n, x]

Rubi steps \begin{align*} \text {integral}& = \int \left (c^3 x^2 \arctan (a x)^3+3 a^2 c^3 x^4 \arctan (a x)^3+3 a^4 c^3 x^6 \arctan (a x)^3+a^6 c^3 x^8 \arctan (a x)^3\right ) \, dx \\ & = c^3 \int x^2 \arctan (a x)^3 \, dx+\left (3 a^2 c^3\right ) \int x^4 \arctan (a x)^3 \, dx+\left (3 a^4 c^3\right ) \int x^6 \arctan (a x)^3 \, dx+\left (a^6 c^3\right ) \int x^8 \arctan (a x)^3 \, dx \\ & = \frac {1}{3} c^3 x^3 \arctan (a x)^3+\frac {3}{5} a^2 c^3 x^5 \arctan (a x)^3+\frac {3}{7} a^4 c^3 x^7 \arctan (a x)^3+\frac {1}{9} a^6 c^3 x^9 \arctan (a x)^3-\left (a c^3\right ) \int \frac {x^3 \arctan (a x)^2}{1+a^2 x^2} \, dx-\frac {1}{5} \left (9 a^3 c^3\right ) \int \frac {x^5 \arctan (a x)^2}{1+a^2 x^2} \, dx-\frac {1}{7} \left (9 a^5 c^3\right ) \int \frac {x^7 \arctan (a x)^2}{1+a^2 x^2} \, dx-\frac {1}{3} \left (a^7 c^3\right ) \int \frac {x^9 \arctan (a x)^2}{1+a^2 x^2} \, dx \\ & = \frac {1}{3} c^3 x^3 \arctan (a x)^3+\frac {3}{5} a^2 c^3 x^5 \arctan (a x)^3+\frac {3}{7} a^4 c^3 x^7 \arctan (a x)^3+\frac {1}{9} a^6 c^3 x^9 \arctan (a x)^3-\frac {c^3 \int x \arctan (a x)^2 \, dx}{a}+\frac {c^3 \int \frac {x \arctan (a x)^2}{1+a^2 x^2} \, dx}{a}-\frac {1}{5} \left (9 a c^3\right ) \int x^3 \arctan (a x)^2 \, dx+\frac {1}{5} \left (9 a c^3\right ) \int \frac {x^3 \arctan (a x)^2}{1+a^2 x^2} \, dx-\frac {1}{7} \left (9 a^3 c^3\right ) \int x^5 \arctan (a x)^2 \, dx+\frac {1}{7} \left (9 a^3 c^3\right ) \int \frac {x^5 \arctan (a x)^2}{1+a^2 x^2} \, dx-\frac {1}{3} \left (a^5 c^3\right ) \int x^7 \arctan (a x)^2 \, dx+\frac {1}{3} \left (a^5 c^3\right ) \int \frac {x^7 \arctan (a x)^2}{1+a^2 x^2} \, dx \\ & = -\frac {c^3 x^2 \arctan (a x)^2}{2 a}-\frac {9}{20} a c^3 x^4 \arctan (a x)^2-\frac {3}{14} a^3 c^3 x^6 \arctan (a x)^2-\frac {1}{24} a^5 c^3 x^8 \arctan (a x)^2-\frac {i c^3 \arctan (a x)^3}{3 a^3}+\frac {1}{3} c^3 x^3 \arctan (a x)^3+\frac {3}{5} a^2 c^3 x^5 \arctan (a x)^3+\frac {3}{7} a^4 c^3 x^7 \arctan (a x)^3+\frac {1}{9} a^6 c^3 x^9 \arctan (a x)^3+c^3 \int \frac {x^2 \arctan (a x)}{1+a^2 x^2} \, dx-\frac {c^3 \int \frac {\arctan (a x)^2}{i-a x} \, dx}{a^2}+\frac {\left (9 c^3\right ) \int x \arctan (a x)^2 \, dx}{5 a}-\frac {\left (9 c^3\right ) \int \frac {x \arctan (a x)^2}{1+a^2 x^2} \, dx}{5 a}+\frac {1}{7} \left (9 a c^3\right ) \int x^3 \arctan (a x)^2 \, dx-\frac {1}{7} \left (9 a c^3\right ) \int \frac {x^3 \arctan (a x)^2}{1+a^2 x^2} \, dx+\frac {1}{10} \left (9 a^2 c^3\right ) \int \frac {x^4 \arctan (a x)}{1+a^2 x^2} \, dx+\frac {1}{3} \left (a^3 c^3\right ) \int x^5 \arctan (a x)^2 \, dx-\frac {1}{3} \left (a^3 c^3\right ) \int \frac {x^5 \arctan (a x)^2}{1+a^2 x^2} \, dx+\frac {1}{7} \left (3 a^4 c^3\right ) \int \frac {x^6 \arctan (a x)}{1+a^2 x^2} \, dx+\frac {1}{12} \left (a^6 c^3\right ) \int \frac {x^8 \arctan (a x)}{1+a^2 x^2} \, dx \\ & = \frac {2 c^3 x^2 \arctan (a x)^2}{5 a}-\frac {9}{70} a c^3 x^4 \arctan (a x)^2-\frac {10}{63} a^3 c^3 x^6 \arctan (a x)^2-\frac {1}{24} a^5 c^3 x^8 \arctan (a x)^2+\frac {4 i c^3 \arctan (a x)^3}{15 a^3}+\frac {1}{3} c^3 x^3 \arctan (a x)^3+\frac {3}{5} a^2 c^3 x^5 \arctan (a x)^3+\frac {3}{7} a^4 c^3 x^7 \arctan (a x)^3+\frac {1}{9} a^6 c^3 x^9 \arctan (a x)^3-\frac {c^3 \arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )}{a^3}+\frac {1}{10} \left (9 c^3\right ) \int x^2 \arctan (a x) \, dx-\frac {1}{10} \left (9 c^3\right ) \int \frac {x^2 \arctan (a x)}{1+a^2 x^2} \, dx-\frac {1}{5} \left (9 c^3\right ) \int \frac {x^2 \arctan (a x)}{1+a^2 x^2} \, dx+\frac {c^3 \int \arctan (a x) \, dx}{a^2}-\frac {c^3 \int \frac {\arctan (a x)}{1+a^2 x^2} \, dx}{a^2}+\frac {\left (9 c^3\right ) \int \frac {\arctan (a x)^2}{i-a x} \, dx}{5 a^2}+\frac {\left (2 c^3\right ) \int \frac {\arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{a^2}-\frac {\left (9 c^3\right ) \int x \arctan (a x)^2 \, dx}{7 a}+\frac {\left (9 c^3\right ) \int \frac {x \arctan (a x)^2}{1+a^2 x^2} \, dx}{7 a}-\frac {1}{3} \left (a c^3\right ) \int x^3 \arctan (a x)^2 \, dx+\frac {1}{3} \left (a c^3\right ) \int \frac {x^3 \arctan (a x)^2}{1+a^2 x^2} \, dx+\frac {1}{7} \left (3 a^2 c^3\right ) \int x^4 \arctan (a x) \, dx-\frac {1}{7} \left (3 a^2 c^3\right ) \int \frac {x^4 \arctan (a x)}{1+a^2 x^2} \, dx-\frac {1}{14} \left (9 a^2 c^3\right ) \int \frac {x^4 \arctan (a x)}{1+a^2 x^2} \, dx+\frac {1}{12} \left (a^4 c^3\right ) \int x^6 \arctan (a x) \, dx-\frac {1}{12} \left (a^4 c^3\right ) \int \frac {x^6 \arctan (a x)}{1+a^2 x^2} \, dx-\frac {1}{9} \left (a^4 c^3\right ) \int \frac {x^6 \arctan (a x)}{1+a^2 x^2} \, dx \\ & = \frac {c^3 x \arctan (a x)}{a^2}+\frac {3}{10} c^3 x^3 \arctan (a x)+\frac {3}{35} a^2 c^3 x^5 \arctan (a x)+\frac {1}{84} a^4 c^3 x^7 \arctan (a x)-\frac {c^3 \arctan (a x)^2}{2 a^3}-\frac {17 c^3 x^2 \arctan (a x)^2}{70 a}-\frac {89}{420} a c^3 x^4 \arctan (a x)^2-\frac {10}{63} a^3 c^3 x^6 \arctan (a x)^2-\frac {1}{24} a^5 c^3 x^8 \arctan (a x)^2-\frac {17 i c^3 \arctan (a x)^3}{105 a^3}+\frac {1}{3} c^3 x^3 \arctan (a x)^3+\frac {3}{5} a^2 c^3 x^5 \arctan (a x)^3+\frac {3}{7} a^4 c^3 x^7 \arctan (a x)^3+\frac {1}{9} a^6 c^3 x^9 \arctan (a x)^3+\frac {4 c^3 \arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )}{5 a^3}-\frac {i c^3 \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{a^3}-\frac {1}{7} \left (3 c^3\right ) \int x^2 \arctan (a x) \, dx+\frac {1}{7} \left (3 c^3\right ) \int \frac {x^2 \arctan (a x)}{1+a^2 x^2} \, dx-\frac {1}{14} \left (9 c^3\right ) \int x^2 \arctan (a x) \, dx+\frac {1}{14} \left (9 c^3\right ) \int \frac {x^2 \arctan (a x)}{1+a^2 x^2} \, dx+\frac {1}{7} \left (9 c^3\right ) \int \frac {x^2 \arctan (a x)}{1+a^2 x^2} \, dx+\frac {\left (i c^3\right ) \int \frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{a^2}-\frac {\left (9 c^3\right ) \int \arctan (a x) \, dx}{10 a^2}+\frac {\left (9 c^3\right ) \int \frac {\arctan (a x)}{1+a^2 x^2} \, dx}{10 a^2}-\frac {\left (9 c^3\right ) \int \frac {\arctan (a x)^2}{i-a x} \, dx}{7 a^2}-\frac {\left (9 c^3\right ) \int \arctan (a x) \, dx}{5 a^2}+\frac {\left (9 c^3\right ) \int \frac {\arctan (a x)}{1+a^2 x^2} \, dx}{5 a^2}-\frac {\left (18 c^3\right ) \int \frac {\arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{5 a^2}+\frac {c^3 \int x \arctan (a x)^2 \, dx}{3 a}-\frac {c^3 \int \frac {x \arctan (a x)^2}{1+a^2 x^2} \, dx}{3 a}-\frac {c^3 \int \frac {x}{1+a^2 x^2} \, dx}{a}-\frac {1}{10} \left (3 a c^3\right ) \int \frac {x^3}{1+a^2 x^2} \, dx-\frac {1}{12} \left (a^2 c^3\right ) \int x^4 \arctan (a x) \, dx+\frac {1}{12} \left (a^2 c^3\right ) \int \frac {x^4 \arctan (a x)}{1+a^2 x^2} \, dx-\frac {1}{9} \left (a^2 c^3\right ) \int x^4 \arctan (a x) \, dx+\frac {1}{9} \left (a^2 c^3\right ) \int \frac {x^4 \arctan (a x)}{1+a^2 x^2} \, dx+\frac {1}{6} \left (a^2 c^3\right ) \int \frac {x^4 \arctan (a x)}{1+a^2 x^2} \, dx-\frac {1}{35} \left (3 a^3 c^3\right ) \int \frac {x^5}{1+a^2 x^2} \, dx-\frac {1}{84} \left (a^5 c^3\right ) \int \frac {x^7}{1+a^2 x^2} \, dx \\ & = -\frac {17 c^3 x \arctan (a x)}{10 a^2}-\frac {2}{35} c^3 x^3 \arctan (a x)+\frac {59 a^2 c^3 x^5 \arctan (a x)}{1260}+\frac {1}{84} a^4 c^3 x^7 \arctan (a x)+\frac {17 c^3 \arctan (a x)^2}{20 a^3}-\frac {8 c^3 x^2 \arctan (a x)^2}{105 a}-\frac {89}{420} a c^3 x^4 \arctan (a x)^2-\frac {10}{63} a^3 c^3 x^6 \arctan (a x)^2-\frac {1}{24} a^5 c^3 x^8 \arctan (a x)^2-\frac {16 i c^3 \arctan (a x)^3}{315 a^3}+\frac {1}{3} c^3 x^3 \arctan (a x)^3+\frac {3}{5} a^2 c^3 x^5 \arctan (a x)^3+\frac {3}{7} a^4 c^3 x^7 \arctan (a x)^3+\frac {1}{9} a^6 c^3 x^9 \arctan (a x)^3-\frac {17 c^3 \arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )}{35 a^3}-\frac {c^3 \log \left (1+a^2 x^2\right )}{2 a^3}+\frac {4 i c^3 \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{5 a^3}-\frac {c^3 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )}{2 a^3}+\frac {1}{12} c^3 \int x^2 \arctan (a x) \, dx-\frac {1}{12} c^3 \int \frac {x^2 \arctan (a x)}{1+a^2 x^2} \, dx+\frac {1}{9} c^3 \int x^2 \arctan (a x) \, dx-\frac {1}{9} c^3 \int \frac {x^2 \arctan (a x)}{1+a^2 x^2} \, dx+\frac {1}{6} c^3 \int x^2 \arctan (a x) \, dx-\frac {1}{6} c^3 \int \frac {x^2 \arctan (a x)}{1+a^2 x^2} \, dx-\frac {1}{3} c^3 \int \frac {x^2 \arctan (a x)}{1+a^2 x^2} \, dx-\frac {\left (9 i c^3\right ) \int \frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{5 a^2}+\frac {c^3 \int \frac {\arctan (a x)^2}{i-a x} \, dx}{3 a^2}+\frac {\left (3 c^3\right ) \int \arctan (a x) \, dx}{7 a^2}-\frac {\left (3 c^3\right ) \int \frac {\arctan (a x)}{1+a^2 x^2} \, dx}{7 a^2}+\frac {\left (9 c^3\right ) \int \arctan (a x) \, dx}{14 a^2}-\frac {\left (9 c^3\right ) \int \frac {\arctan (a x)}{1+a^2 x^2} \, dx}{14 a^2}+\frac {\left (9 c^3\right ) \int \arctan (a x) \, dx}{7 a^2}-\frac {\left (9 c^3\right ) \int \frac {\arctan (a x)}{1+a^2 x^2} \, dx}{7 a^2}+\frac {\left (18 c^3\right ) \int \frac {\arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{7 a^2}+\frac {\left (9 c^3\right ) \int \frac {x}{1+a^2 x^2} \, dx}{10 a}+\frac {\left (9 c^3\right ) \int \frac {x}{1+a^2 x^2} \, dx}{5 a}+\frac {1}{7} \left (a c^3\right ) \int \frac {x^3}{1+a^2 x^2} \, dx-\frac {1}{20} \left (3 a c^3\right ) \text {Subst}\left (\int \frac {x}{1+a^2 x} \, dx,x,x^2\right )+\frac {1}{14} \left (3 a c^3\right ) \int \frac {x^3}{1+a^2 x^2} \, dx+\frac {1}{60} \left (a^3 c^3\right ) \int \frac {x^5}{1+a^2 x^2} \, dx+\frac {1}{45} \left (a^3 c^3\right ) \int \frac {x^5}{1+a^2 x^2} \, dx-\frac {1}{70} \left (3 a^3 c^3\right ) \text {Subst}\left (\int \frac {x^2}{1+a^2 x} \, dx,x,x^2\right )-\frac {1}{168} \left (a^5 c^3\right ) \text {Subst}\left (\int \frac {x^3}{1+a^2 x} \, dx,x,x^2\right ) \\ & = \frac {23 c^3 x \arctan (a x)}{35 a^2}+\frac {239 c^3 x^3 \arctan (a x)}{3780}+\frac {59 a^2 c^3 x^5 \arctan (a x)}{1260}+\frac {1}{84} a^4 c^3 x^7 \arctan (a x)-\frac {23 c^3 \arctan (a x)^2}{70 a^3}-\frac {8 c^3 x^2 \arctan (a x)^2}{105 a}-\frac {89}{420} a c^3 x^4 \arctan (a x)^2-\frac {10}{63} a^3 c^3 x^6 \arctan (a x)^2-\frac {1}{24} a^5 c^3 x^8 \arctan (a x)^2-\frac {16 i c^3 \arctan (a x)^3}{315 a^3}+\frac {1}{3} c^3 x^3 \arctan (a x)^3+\frac {3}{5} a^2 c^3 x^5 \arctan (a x)^3+\frac {3}{7} a^4 c^3 x^7 \arctan (a x)^3+\frac {1}{9} a^6 c^3 x^9 \arctan (a x)^3-\frac {16 c^3 \arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )}{105 a^3}+\frac {17 c^3 \log \left (1+a^2 x^2\right )}{20 a^3}-\frac {17 i c^3 \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{35 a^3}+\frac {2 c^3 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )}{5 a^3}+\frac {\left (9 i c^3\right ) \int \frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{7 a^2}-\frac {c^3 \int \arctan (a x) \, dx}{12 a^2}+\frac {c^3 \int \frac {\arctan (a x)}{1+a^2 x^2} \, dx}{12 a^2}-\frac {c^3 \int \arctan (a x) \, dx}{9 a^2}+\frac {c^3 \int \frac {\arctan (a x)}{1+a^2 x^2} \, dx}{9 a^2}-\frac {c^3 \int \arctan (a x) \, dx}{6 a^2}+\frac {c^3 \int \frac {\arctan (a x)}{1+a^2 x^2} \, dx}{6 a^2}-\frac {c^3 \int \arctan (a x) \, dx}{3 a^2}+\frac {c^3 \int \frac {\arctan (a x)}{1+a^2 x^2} \, dx}{3 a^2}-\frac {\left (2 c^3\right ) \int \frac {\arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{3 a^2}-\frac {\left (3 c^3\right ) \int \frac {x}{1+a^2 x^2} \, dx}{7 a}-\frac {\left (9 c^3\right ) \int \frac {x}{1+a^2 x^2} \, dx}{14 a}-\frac {\left (9 c^3\right ) \int \frac {x}{1+a^2 x^2} \, dx}{7 a}-\frac {1}{36} \left (a c^3\right ) \int \frac {x^3}{1+a^2 x^2} \, dx-\frac {1}{27} \left (a c^3\right ) \int \frac {x^3}{1+a^2 x^2} \, dx-\frac {1}{18} \left (a c^3\right ) \int \frac {x^3}{1+a^2 x^2} \, dx+\frac {1}{14} \left (a c^3\right ) \text {Subst}\left (\int \frac {x}{1+a^2 x} \, dx,x,x^2\right )+\frac {1}{28} \left (3 a c^3\right ) \text {Subst}\left (\int \frac {x}{1+a^2 x} \, dx,x,x^2\right )-\frac {1}{20} \left (3 a c^3\right ) \text {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}{120} \left (a^3 c^3\right ) \text {Subst}\left (\int \frac {x^2}{1+a^2 x} \, dx,x,x^2\right )+\frac {1}{90} \left (a^3 c^3\right ) \text {Subst}\left (\int \frac {x^2}{1+a^2 x} \, dx,x,x^2\right )-\frac {1}{70} \left (3 a^3 c^3\right ) \text {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}{168} \left (a^5 c^3\right ) \text {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 {19 c^3 x^2}{168 a}-\frac {31 a c^3 x^4}{1680}-\frac {1}{504} a^3 c^3 x^6-\frac {47 c^3 x \arctan (a x)}{1260 a^2}+\frac {239 c^3 x^3 \arctan (a x)}{3780}+\frac {59 a^2 c^3 x^5 \arctan (a x)}{1260}+\frac {1}{84} a^4 c^3 x^7 \arctan (a x)+\frac {47 c^3 \arctan (a x)^2}{2520 a^3}-\frac {8 c^3 x^2 \arctan (a x)^2}{105 a}-\frac {89}{420} a c^3 x^4 \arctan (a x)^2-\frac {10}{63} a^3 c^3 x^6 \arctan (a x)^2-\frac {1}{24} a^5 c^3 x^8 \arctan (a x)^2-\frac {16 i c^3 \arctan (a x)^3}{315 a^3}+\frac {1}{3} c^3 x^3 \arctan (a x)^3+\frac {3}{5} a^2 c^3 x^5 \arctan (a x)^3+\frac {3}{7} a^4 c^3 x^7 \arctan (a x)^3+\frac {1}{9} a^6 c^3 x^9 \arctan (a x)^3-\frac {16 c^3 \arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )}{105 a^3}-\frac {181 c^3 \log \left (1+a^2 x^2\right )}{840 a^3}-\frac {16 i c^3 \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{105 a^3}-\frac {17 c^3 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )}{70 a^3}-\frac {\left (i c^3\right ) \int \frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{3 a^2}+\frac {c^3 \int \frac {x}{1+a^2 x^2} \, dx}{12 a}+\frac {c^3 \int \frac {x}{1+a^2 x^2} \, dx}{9 a}+\frac {c^3 \int \frac {x}{1+a^2 x^2} \, dx}{6 a}+\frac {c^3 \int \frac {x}{1+a^2 x^2} \, dx}{3 a}-\frac {1}{72} \left (a c^3\right ) \text {Subst}\left (\int \frac {x}{1+a^2 x} \, dx,x,x^2\right )-\frac {1}{54} \left (a c^3\right ) \text {Subst}\left (\int \frac {x}{1+a^2 x} \, dx,x,x^2\right )-\frac {1}{36} \left (a c^3\right ) \text {Subst}\left (\int \frac {x}{1+a^2 x} \, dx,x,x^2\right )+\frac {1}{14} \left (a c^3\right ) \text {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}{28} \left (3 a c^3\right ) \text {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}{120} \left (a^3 c^3\right ) \text {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}{90} \left (a^3 c^3\right ) \text {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 {29 c^3 x^2}{630 a}-\frac {11 a c^3 x^4}{1260}-\frac {1}{504} a^3 c^3 x^6-\frac {47 c^3 x \arctan (a x)}{1260 a^2}+\frac {239 c^3 x^3 \arctan (a x)}{3780}+\frac {59 a^2 c^3 x^5 \arctan (a x)}{1260}+\frac {1}{84} a^4 c^3 x^7 \arctan (a x)+\frac {47 c^3 \arctan (a x)^2}{2520 a^3}-\frac {8 c^3 x^2 \arctan (a x)^2}{105 a}-\frac {89}{420} a c^3 x^4 \arctan (a x)^2-\frac {10}{63} a^3 c^3 x^6 \arctan (a x)^2-\frac {1}{24} a^5 c^3 x^8 \arctan (a x)^2-\frac {16 i c^3 \arctan (a x)^3}{315 a^3}+\frac {1}{3} c^3 x^3 \arctan (a x)^3+\frac {3}{5} a^2 c^3 x^5 \arctan (a x)^3+\frac {3}{7} a^4 c^3 x^7 \arctan (a x)^3+\frac {1}{9} a^6 c^3 x^9 \arctan (a x)^3-\frac {16 c^3 \arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )}{105 a^3}-\frac {23 c^3 \log \left (1+a^2 x^2\right )}{840 a^3}-\frac {16 i c^3 \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{105 a^3}-\frac {8 c^3 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )}{105 a^3}-\frac {1}{72} \left (a c^3\right ) \text {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}{54} \left (a c^3\right ) \text {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}{36} \left (a c^3\right ) \text {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}{7560 a}-\frac {11 a c^3 x^4}{1260}-\frac {1}{504} a^3 c^3 x^6-\frac {47 c^3 x \arctan (a x)}{1260 a^2}+\frac {239 c^3 x^3 \arctan (a x)}{3780}+\frac {59 a^2 c^3 x^5 \arctan (a x)}{1260}+\frac {1}{84} a^4 c^3 x^7 \arctan (a x)+\frac {47 c^3 \arctan (a x)^2}{2520 a^3}-\frac {8 c^3 x^2 \arctan (a x)^2}{105 a}-\frac {89}{420} a c^3 x^4 \arctan (a x)^2-\frac {10}{63} a^3 c^3 x^6 \arctan (a x)^2-\frac {1}{24} a^5 c^3 x^8 \arctan (a x)^2-\frac {16 i c^3 \arctan (a x)^3}{315 a^3}+\frac {1}{3} c^3 x^3 \arctan (a x)^3+\frac {3}{5} a^2 c^3 x^5 \arctan (a x)^3+\frac {3}{7} a^4 c^3 x^7 \arctan (a x)^3+\frac {1}{9} a^6 c^3 x^9 \arctan (a x)^3-\frac {16 c^3 \arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )}{105 a^3}+\frac {31 c^3 \log \left (1+a^2 x^2\right )}{945 a^3}-\frac {16 i c^3 \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{105 a^3}-\frac {8 c^3 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )}{105 a^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.60 (sec) , antiderivative size = 281, normalized size of antiderivative = 0.72 \[ \int x^2 \left (c+a^2 c x^2\right )^3 \arctan (a x)^3 \, dx=\frac {c^3 \left (-56-107 a^2 x^2-66 a^4 x^4-15 a^6 x^6-282 a x \arctan (a x)+478 a^3 x^3 \arctan (a x)+354 a^5 x^5 \arctan (a x)+90 a^7 x^7 \arctan (a x)+141 \arctan (a x)^2-576 a^2 x^2 \arctan (a x)^2-1602 a^4 x^4 \arctan (a x)^2-1200 a^6 x^6 \arctan (a x)^2-315 a^8 x^8 \arctan (a x)^2+384 i \arctan (a x)^3+2520 a^3 x^3 \arctan (a x)^3+4536 a^5 x^5 \arctan (a x)^3+3240 a^7 x^7 \arctan (a x)^3+840 a^9 x^9 \arctan (a x)^3-1152 \arctan (a x)^2 \log \left (1+e^{2 i \arctan (a x)}\right )+248 \log \left (1+a^2 x^2\right )+1152 i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{2 i \arctan (a x)}\right )-576 \operatorname {PolyLog}\left (3,-e^{2 i \arctan (a x)}\right )\right )}{7560 a^3} \]

[In]

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

[Out]

(c^3*(-56 - 107*a^2*x^2 - 66*a^4*x^4 - 15*a^6*x^6 - 282*a*x*ArcTan[a*x] + 478*a^3*x^3*ArcTan[a*x] + 354*a^5*x^
5*ArcTan[a*x] + 90*a^7*x^7*ArcTan[a*x] + 141*ArcTan[a*x]^2 - 576*a^2*x^2*ArcTan[a*x]^2 - 1602*a^4*x^4*ArcTan[a
*x]^2 - 1200*a^6*x^6*ArcTan[a*x]^2 - 315*a^8*x^8*ArcTan[a*x]^2 + (384*I)*ArcTan[a*x]^3 + 2520*a^3*x^3*ArcTan[a
*x]^3 + 4536*a^5*x^5*ArcTan[a*x]^3 + 3240*a^7*x^7*ArcTan[a*x]^3 + 840*a^9*x^9*ArcTan[a*x]^3 - 1152*ArcTan[a*x]
^2*Log[1 + E^((2*I)*ArcTan[a*x])] + 248*Log[1 + a^2*x^2] + (1152*I)*ArcTan[a*x]*PolyLog[2, -E^((2*I)*ArcTan[a*
x])] - 576*PolyLog[3, -E^((2*I)*ArcTan[a*x])]))/(7560*a^3)

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 121.68 (sec) , antiderivative size = 1576, normalized size of antiderivative = 4.05

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1576\)
default	\(\text {Expression too large to display}\)	\(1576\)
parts	\(\text {Expression too large to display}\)	\(1576\)

[In]

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

[Out]

1/a^3*(1/9*c^3*arctan(a*x)^3*a^9*x^9+3/7*c^3*arctan(a*x)^3*a^7*x^7+3/5*c^3*arctan(a*x)^3*a^5*x^5+1/3*c^3*arcta
n(a*x)^3*a^3*x^3-1/105*c^3*(-4*I*Pi*csgn(I/((1+I*a*x)^2/(a^2*x^2+1)+1)^2)*csgn(I*(1+I*a*x)^2/(a^2*x^2+1))*csgn
(I*(1+I*a*x)^2/(a^2*x^2+1)/((1+I*a*x)^2/(a^2*x^2+1)+1)^2)*arctan(a*x)^2-47/24*arctan(a*x)^2+35/8*arctan(a*x)^2
*a^8*x^8+8*x^2*arctan(a*x)^2*a^2+175/4*arctan(a*x)*(a*x-I)^4*(I+a*x)^3-35/4*I*arctan(a*x)*(a*x-I)^6-105/4*arct
an(a*x)*(a*x-I)^5*(I+a*x)^2+35/4*arctan(a*x)*(a*x-I)^6*(I+a*x)+115/6*I*arctan(a*x)*(a*x-I)^4+3*I*arctan(a*x)*(
a*x-I)^2-16*I*arctan(a*x)*polylog(2,-(1+I*a*x)^2/(a^2*x^2+1))-175/4*arctan(a*x)*(a*x-I)^3*(I+a*x)^4+16*arctan(
a*x)^2*ln(2)+4*I*Pi*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1)^2)^3*arctan(a*x)^2-230/3*I*arctan(a*x)*(a*x-I)^3*(I+a*x
)+115*I*arctan(a*x)*(a*x-I)^2*(I+a*x)^2+105/2*I*arctan(a*x)*(a*x-I)^5*(I+a*x)-525/4*I*arctan(a*x)*(a*x-I)^4*(I
+a*x)^2-4*I*Pi*csgn(I*(1+I*a*x)^2/(a^2*x^2+1)/((1+I*a*x)^2/(a^2*x^2+1)+1)^2)^3*arctan(a*x)^2-4*I*Pi*csgn(I*(1+
I*a*x)^2/(a^2*x^2+1))^3*arctan(a*x)^2-525/4*I*arctan(a*x)*(a*x-I)^2*(I+a*x)^4-6*I*arctan(a*x)*(a*x-I)*(I+a*x)+
105/2*I*arctan(a*x)*(a*x-I)*(I+a*x)^5-230/3*I*arctan(a*x)*(a*x-I)*(I+a*x)^3+175*I*arctan(a*x)*(a*x-I)^3*(I+a*x
)^3-53/24*(I+a*x)^4+50/3*a^6*x^6*arctan(a*x)^2+5/24*(I+a*x)^6+4*I*Pi*csgn(I/((1+I*a*x)^2/(a^2*x^2+1)+1)^2)*csg
n(I*(1+I*a*x)^2/(a^2*x^2+1)/((1+I*a*x)^2/(a^2*x^2+1)+1)^2)^2*arctan(a*x)^2-4*I*Pi*csgn(I*(1+I*a*x)/(a^2*x^2+1)
^(1/2))^2*csgn(I*(1+I*a*x)^2/(a^2*x^2+1))*arctan(a*x)^2+8*I*Pi*csgn(I*(1+I*a*x)/(a^2*x^2+1)^(1/2))*csgn(I*(1+I
*a*x)^2/(a^2*x^2+1))^2*arctan(a*x)^2+4*I*Pi*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1))^2*csgn(I*((1+I*a*x)^2/(a^2*x^2
+1)+1)^2)*arctan(a*x)^2+4*I*Pi*csgn(I*(1+I*a*x)^2/(a^2*x^2+1))*csgn(I*(1+I*a*x)^2/(a^2*x^2+1)/((1+I*a*x)^2/(a^
2*x^2+1)+1)^2)^2*arctan(a*x)^2-8*I*Pi*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1))*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1)^2
)^2*arctan(a*x)^2+11/3*arctan(a*x)*(a*x-I)^2*(I+a*x)-35/4*arctan(a*x)*(a*x-I)*(I+a*x)^6-320/3*arctan(a*x)*(a*x
-I)^4*(I+a*x)-640/3*arctan(a*x)*(a*x-I)^2*(I+a*x)^3-11/3*arctan(a*x)*(a*x-I)*(I+a*x)^2+640/3*arctan(a*x)*(a*x-
I)^3*(I+a*x)^2+320/3*arctan(a*x)*(a*x-I)*(I+a*x)^4+105/4*arctan(a*x)*(a*x-I)^2*(I+a*x)^5+89/4*a^4*arctan(a*x)^
2*x^4-5/9*I*(I+a*x)-5/4*I*(I+a*x)^5-11/9*arctan(a*x)*(a*x-I)^3+8*arctan(a*x)*(a*x-I)+64/3*arctan(a*x)*(a*x-I)^
5+8*polylog(3,-(1+I*a*x)^2/(a^2*x^2+1))+62/9*ln((1+I*a*x)^2/(a^2*x^2+1)+1)-5/4*arctan(a*x)*(a*x-I)^7+1/2*I*(I+
a*x)^3-16/3*I*arctan(a*x)^3-8*arctan(a*x)^2*ln(a^2*x^2+1)-8/9*(I+a*x)^2+16*arctan(a*x)^2*ln((1+I*a*x)/(a^2*x^2
+1)^(1/2))))

Fricas [F]

\[ \int x^2 \left (c+a^2 c x^2\right )^3 \arctan (a x)^3 \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{3} x^{2} \arctan \left (a x\right )^{3} \,d x } \]

[In]

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

[Out]

integral((a^6*c^3*x^8 + 3*a^4*c^3*x^6 + 3*a^2*c^3*x^4 + c^3*x^2)*arctan(a*x)^3, x)

Sympy [F]

\[ \int x^2 \left (c+a^2 c x^2\right )^3 \arctan (a x)^3 \, dx=c^{3} \left (\int x^{2} \operatorname {atan}^{3}{\left (a x \right )}\, dx + \int 3 a^{2} x^{4} \operatorname {atan}^{3}{\left (a x \right )}\, dx + \int 3 a^{4} x^{6} \operatorname {atan}^{3}{\left (a x \right )}\, dx + \int a^{6} x^{8} \operatorname {atan}^{3}{\left (a x \right )}\, dx\right ) \]

[In]

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

[Out]

c**3*(Integral(x**2*atan(a*x)**3, x) + Integral(3*a**2*x**4*atan(a*x)**3, x) + Integral(3*a**4*x**6*atan(a*x)*
*3, x) + Integral(a**6*x**8*atan(a*x)**3, x))

Maxima [F]

\[ \int x^2 \left (c+a^2 c x^2\right )^3 \arctan (a x)^3 \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{3} x^{2} \arctan \left (a x\right )^{3} \,d x } \]

[In]

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

[Out]

1/2520*(35*a^6*c^3*x^9 + 135*a^4*c^3*x^7 + 189*a^2*c^3*x^5 + 105*c^3*x^3)*arctan(a*x)^3 - 1/3360*(35*a^6*c^3*x
^9 + 135*a^4*c^3*x^7 + 189*a^2*c^3*x^5 + 105*c^3*x^3)*arctan(a*x)*log(a^2*x^2 + 1)^2 + integrate(1/3360*(2940*
(a^8*c^3*x^10 + 4*a^6*c^3*x^8 + 6*a^4*c^3*x^6 + 4*a^2*c^3*x^4 + c^3*x^2)*arctan(a*x)^3 - 4*(35*a^7*c^3*x^9 + 1
35*a^5*c^3*x^7 + 189*a^3*c^3*x^5 + 105*a*c^3*x^3)*arctan(a*x)^2 + 4*(35*a^8*c^3*x^10 + 135*a^6*c^3*x^8 + 189*a
^4*c^3*x^6 + 105*a^2*c^3*x^4)*arctan(a*x)*log(a^2*x^2 + 1) + (35*a^7*c^3*x^9 + 135*a^5*c^3*x^7 + 189*a^3*c^3*x
^5 + 105*a*c^3*x^3 + 315*(a^8*c^3*x^10 + 4*a^6*c^3*x^8 + 6*a^4*c^3*x^6 + 4*a^2*c^3*x^4 + c^3*x^2)*arctan(a*x))
*log(a^2*x^2 + 1)^2)/(a^2*x^2 + 1), x)

Giac [F]

\[ \int x^2 \left (c+a^2 c x^2\right )^3 \arctan (a x)^3 \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{3} x^{2} \arctan \left (a x\right )^{3} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

Timed out. \[ \int x^2 \left (c+a^2 c x^2\right )^3 \arctan (a x)^3 \, dx=\int x^2\,{\mathrm {atan}\left (a\,x\right )}^3\,{\left (c\,a^2\,x^2+c\right )}^3 \,d x \]

[In]

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

[Out]

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