\(\int x^2 (c+a^2 c x^2)^2 \arctan (a x)^3 \, dx\) [372]

Optimal result

Integrand size = 22, antiderivative size = 321 \[ \int x^2 \left (c+a^2 c x^2\right )^2 \arctan (a x)^3 \, dx=-\frac {11 c^2 x^2}{420 a}-\frac {1}{140} a c^2 x^4-\frac {c^2 x \arctan (a x)}{70 a^2}+\frac {17}{210} c^2 x^3 \arctan (a x)+\frac {1}{35} a^2 c^2 x^5 \arctan (a x)+\frac {c^2 \arctan (a x)^2}{140 a^3}-\frac {4 c^2 x^2 \arctan (a x)^2}{35 a}-\frac {27}{140} a c^2 x^4 \arctan (a x)^2-\frac {1}{14} a^3 c^2 x^6 \arctan (a x)^2-\frac {8 i c^2 \arctan (a x)^3}{105 a^3}+\frac {1}{3} c^2 x^3 \arctan (a x)^3+\frac {2}{5} a^2 c^2 x^5 \arctan (a x)^3+\frac {1}{7} a^4 c^2 x^7 \arctan (a x)^3-\frac {8 c^2 \arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )}{35 a^3}+\frac {c^2 \log \left (1+a^2 x^2\right )}{30 a^3}-\frac {8 i c^2 \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{35 a^3}-\frac {4 c^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )}{35 a^3} \]

[Out]

-11/420*c^2*x^2/a-1/140*a*c^2*x^4-1/70*c^2*x*arctan(a*x)/a^2+17/210*c^2*x^3*arctan(a*x)+1/35*a^2*c^2*x^5*arcta
n(a*x)+1/140*c^2*arctan(a*x)^2/a^3-4/35*c^2*x^2*arctan(a*x)^2/a-27/140*a*c^2*x^4*arctan(a*x)^2-1/14*a^3*c^2*x^
6*arctan(a*x)^2-8/105*I*c^2*arctan(a*x)^3/a^3+1/3*c^2*x^3*arctan(a*x)^3+2/5*a^2*c^2*x^5*arctan(a*x)^3+1/7*a^4*
c^2*x^7*arctan(a*x)^3-8/35*c^2*arctan(a*x)^2*ln(2/(1+I*a*x))/a^3+1/30*c^2*ln(a^2*x^2+1)/a^3-8/35*I*c^2*arctan(
a*x)*polylog(2,1-2/(1+I*a*x))/a^3-4/35*c^2*polylog(3,1-2/(1+I*a*x))/a^3

Rubi [A] (verified)

Time = 1.33 (sec) , antiderivative size = 321, normalized size of antiderivative = 1.00, number of steps used = 73, 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} \[ \int x^2 \left (c+a^2 c x^2\right )^2 \arctan (a x)^3 \, dx=\frac {1}{7} a^4 c^2 x^7 \arctan (a x)^3-\frac {8 i c^2 \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )}{35 a^3}-\frac {1}{14} a^3 c^2 x^6 \arctan (a x)^2-\frac {8 i c^2 \arctan (a x)^3}{105 a^3}+\frac {c^2 \arctan (a x)^2}{140 a^3}-\frac {8 c^2 \arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )}{35 a^3}-\frac {4 c^2 \operatorname {PolyLog}\left (3,1-\frac {2}{i a x+1}\right )}{35 a^3}+\frac {2}{5} a^2 c^2 x^5 \arctan (a x)^3+\frac {1}{35} a^2 c^2 x^5 \arctan (a x)-\frac {c^2 x \arctan (a x)}{70 a^2}+\frac {c^2 \log \left (a^2 x^2+1\right )}{30 a^3}-\frac {27}{140} a c^2 x^4 \arctan (a x)^2+\frac {1}{3} c^2 x^3 \arctan (a x)^3+\frac {17}{210} c^2 x^3 \arctan (a x)-\frac {4 c^2 x^2 \arctan (a x)^2}{35 a}-\frac {1}{140} a c^2 x^4-\frac {11 c^2 x^2}{420 a} \]

[In]

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

[Out]

(-11*c^2*x^2)/(420*a) - (a*c^2*x^4)/140 - (c^2*x*ArcTan[a*x])/(70*a^2) + (17*c^2*x^3*ArcTan[a*x])/210 + (a^2*c
^2*x^5*ArcTan[a*x])/35 + (c^2*ArcTan[a*x]^2)/(140*a^3) - (4*c^2*x^2*ArcTan[a*x]^2)/(35*a) - (27*a*c^2*x^4*ArcT
an[a*x]^2)/140 - (a^3*c^2*x^6*ArcTan[a*x]^2)/14 - (((8*I)/105)*c^2*ArcTan[a*x]^3)/a^3 + (c^2*x^3*ArcTan[a*x]^3
)/3 + (2*a^2*c^2*x^5*ArcTan[a*x]^3)/5 + (a^4*c^2*x^7*ArcTan[a*x]^3)/7 - (8*c^2*ArcTan[a*x]^2*Log[2/(1 + I*a*x)
])/(35*a^3) + (c^2*Log[1 + a^2*x^2])/(30*a^3) - (((8*I)/35)*c^2*ArcTan[a*x]*PolyLog[2, 1 - 2/(1 + I*a*x)])/a^3
- (4*c^2*PolyLog[3, 1 - 2/(1 + I*a*x)])/(35*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^2 x^2 \arctan (a x)^3+2 a^2 c^2 x^4 \arctan (a x)^3+a^4 c^2 x^6 \arctan (a x)^3\right ) \, dx \\ & = c^2 \int x^2 \arctan (a x)^3 \, dx+\left (2 a^2 c^2\right ) \int x^4 \arctan (a x)^3 \, dx+\left (a^4 c^2\right ) \int x^6 \arctan (a x)^3 \, dx \\ & = \frac {1}{3} c^2 x^3 \arctan (a x)^3+\frac {2}{5} a^2 c^2 x^5 \arctan (a x)^3+\frac {1}{7} a^4 c^2 x^7 \arctan (a x)^3-\left (a c^2\right ) \int \frac {x^3 \arctan (a x)^2}{1+a^2 x^2} \, dx-\frac {1}{5} \left (6 a^3 c^2\right ) \int \frac {x^5 \arctan (a x)^2}{1+a^2 x^2} \, dx-\frac {1}{7} \left (3 a^5 c^2\right ) \int \frac {x^7 \arctan (a x)^2}{1+a^2 x^2} \, dx \\ & = \frac {1}{3} c^2 x^3 \arctan (a x)^3+\frac {2}{5} a^2 c^2 x^5 \arctan (a x)^3+\frac {1}{7} a^4 c^2 x^7 \arctan (a x)^3-\frac {c^2 \int x \arctan (a x)^2 \, dx}{a}+\frac {c^2 \int \frac {x \arctan (a x)^2}{1+a^2 x^2} \, dx}{a}-\frac {1}{5} \left (6 a c^2\right ) \int x^3 \arctan (a x)^2 \, dx+\frac {1}{5} \left (6 a c^2\right ) \int \frac {x^3 \arctan (a x)^2}{1+a^2 x^2} \, dx-\frac {1}{7} \left (3 a^3 c^2\right ) \int x^5 \arctan (a x)^2 \, dx+\frac {1}{7} \left (3 a^3 c^2\right ) \int \frac {x^5 \arctan (a x)^2}{1+a^2 x^2} \, dx \\ & = -\frac {c^2 x^2 \arctan (a x)^2}{2 a}-\frac {3}{10} a c^2 x^4 \arctan (a x)^2-\frac {1}{14} a^3 c^2 x^6 \arctan (a x)^2-\frac {i c^2 \arctan (a x)^3}{3 a^3}+\frac {1}{3} c^2 x^3 \arctan (a x)^3+\frac {2}{5} a^2 c^2 x^5 \arctan (a x)^3+\frac {1}{7} a^4 c^2 x^7 \arctan (a x)^3+c^2 \int \frac {x^2 \arctan (a x)}{1+a^2 x^2} \, dx-\frac {c^2 \int \frac {\arctan (a x)^2}{i-a x} \, dx}{a^2}+\frac {\left (6 c^2\right ) \int x \arctan (a x)^2 \, dx}{5 a}-\frac {\left (6 c^2\right ) \int \frac {x \arctan (a x)^2}{1+a^2 x^2} \, dx}{5 a}+\frac {1}{7} \left (3 a c^2\right ) \int x^3 \arctan (a x)^2 \, dx-\frac {1}{7} \left (3 a c^2\right ) \int \frac {x^3 \arctan (a x)^2}{1+a^2 x^2} \, dx+\frac {1}{5} \left (3 a^2 c^2\right ) \int \frac {x^4 \arctan (a x)}{1+a^2 x^2} \, dx+\frac {1}{7} \left (a^4 c^2\right ) \int \frac {x^6 \arctan (a x)}{1+a^2 x^2} \, dx \\ & = \frac {c^2 x^2 \arctan (a x)^2}{10 a}-\frac {27}{140} a c^2 x^4 \arctan (a x)^2-\frac {1}{14} a^3 c^2 x^6 \arctan (a x)^2+\frac {i c^2 \arctan (a x)^3}{15 a^3}+\frac {1}{3} c^2 x^3 \arctan (a x)^3+\frac {2}{5} a^2 c^2 x^5 \arctan (a x)^3+\frac {1}{7} a^4 c^2 x^7 \arctan (a x)^3-\frac {c^2 \arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )}{a^3}+\frac {1}{5} \left (3 c^2\right ) \int x^2 \arctan (a x) \, dx-\frac {1}{5} \left (3 c^2\right ) \int \frac {x^2 \arctan (a x)}{1+a^2 x^2} \, dx-\frac {1}{5} \left (6 c^2\right ) \int \frac {x^2 \arctan (a x)}{1+a^2 x^2} \, dx+\frac {c^2 \int \arctan (a x) \, dx}{a^2}-\frac {c^2 \int \frac {\arctan (a x)}{1+a^2 x^2} \, dx}{a^2}+\frac {\left (6 c^2\right ) \int \frac {\arctan (a x)^2}{i-a x} \, dx}{5 a^2}+\frac {\left (2 c^2\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 (3 c^2\right ) \int x \arctan (a x)^2 \, dx}{7 a}+\frac {\left (3 c^2\right ) \int \frac {x \arctan (a x)^2}{1+a^2 x^2} \, dx}{7 a}+\frac {1}{7} \left (a^2 c^2\right ) \int x^4 \arctan (a x) \, dx-\frac {1}{7} \left (a^2 c^2\right ) \int \frac {x^4 \arctan (a x)}{1+a^2 x^2} \, dx-\frac {1}{14} \left (3 a^2 c^2\right ) \int \frac {x^4 \arctan (a x)}{1+a^2 x^2} \, dx \\ & = \frac {c^2 x \arctan (a x)}{a^2}+\frac {1}{5} c^2 x^3 \arctan (a x)+\frac {1}{35} a^2 c^2 x^5 \arctan (a x)-\frac {c^2 \arctan (a x)^2}{2 a^3}-\frac {4 c^2 x^2 \arctan (a x)^2}{35 a}-\frac {27}{140} a c^2 x^4 \arctan (a x)^2-\frac {1}{14} a^3 c^2 x^6 \arctan (a x)^2-\frac {8 i c^2 \arctan (a x)^3}{105 a^3}+\frac {1}{3} c^2 x^3 \arctan (a x)^3+\frac {2}{5} a^2 c^2 x^5 \arctan (a x)^3+\frac {1}{7} a^4 c^2 x^7 \arctan (a x)^3+\frac {c^2 \arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )}{5 a^3}-\frac {i c^2 \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{a^3}-\frac {1}{7} c^2 \int x^2 \arctan (a x) \, dx+\frac {1}{7} c^2 \int \frac {x^2 \arctan (a x)}{1+a^2 x^2} \, dx-\frac {1}{14} \left (3 c^2\right ) \int x^2 \arctan (a x) \, dx+\frac {1}{14} \left (3 c^2\right ) \int \frac {x^2 \arctan (a x)}{1+a^2 x^2} \, dx+\frac {1}{7} \left (3 c^2\right ) \int \frac {x^2 \arctan (a x)}{1+a^2 x^2} \, dx+\frac {\left (i c^2\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 (3 c^2\right ) \int \frac {\arctan (a x)^2}{i-a x} \, dx}{7 a^2}-\frac {\left (3 c^2\right ) \int \arctan (a x) \, dx}{5 a^2}+\frac {\left (3 c^2\right ) \int \frac {\arctan (a x)}{1+a^2 x^2} \, dx}{5 a^2}-\frac {\left (6 c^2\right ) \int \arctan (a x) \, dx}{5 a^2}+\frac {\left (6 c^2\right ) \int \frac {\arctan (a x)}{1+a^2 x^2} \, dx}{5 a^2}-\frac {\left (12 c^2\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^2 \int \frac {x}{1+a^2 x^2} \, dx}{a}-\frac {1}{5} \left (a c^2\right ) \int \frac {x^3}{1+a^2 x^2} \, dx-\frac {1}{35} \left (a^3 c^2\right ) \int \frac {x^5}{1+a^2 x^2} \, dx \\ & = -\frac {4 c^2 x \arctan (a x)}{5 a^2}+\frac {17}{210} c^2 x^3 \arctan (a x)+\frac {1}{35} a^2 c^2 x^5 \arctan (a x)+\frac {2 c^2 \arctan (a x)^2}{5 a^3}-\frac {4 c^2 x^2 \arctan (a x)^2}{35 a}-\frac {27}{140} a c^2 x^4 \arctan (a x)^2-\frac {1}{14} a^3 c^2 x^6 \arctan (a x)^2-\frac {8 i c^2 \arctan (a x)^3}{105 a^3}+\frac {1}{3} c^2 x^3 \arctan (a x)^3+\frac {2}{5} a^2 c^2 x^5 \arctan (a x)^3+\frac {1}{7} a^4 c^2 x^7 \arctan (a x)^3-\frac {8 c^2 \arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )}{35 a^3}-\frac {c^2 \log \left (1+a^2 x^2\right )}{2 a^3}+\frac {i c^2 \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{5 a^3}-\frac {c^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )}{2 a^3}-\frac {\left (6 i c^2\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^2 \int \arctan (a x) \, dx}{7 a^2}-\frac {c^2 \int \frac {\arctan (a x)}{1+a^2 x^2} \, dx}{7 a^2}+\frac {\left (3 c^2\right ) \int \arctan (a x) \, dx}{14 a^2}-\frac {\left (3 c^2\right ) \int \frac {\arctan (a x)}{1+a^2 x^2} \, dx}{14 a^2}+\frac {\left (3 c^2\right ) \int \arctan (a x) \, dx}{7 a^2}-\frac {\left (3 c^2\right ) \int \frac {\arctan (a x)}{1+a^2 x^2} \, dx}{7 a^2}+\frac {\left (6 c^2\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 (3 c^2\right ) \int \frac {x}{1+a^2 x^2} \, dx}{5 a}+\frac {\left (6 c^2\right ) \int \frac {x}{1+a^2 x^2} \, dx}{5 a}+\frac {1}{21} \left (a c^2\right ) \int \frac {x^3}{1+a^2 x^2} \, dx+\frac {1}{14} \left (a c^2\right ) \int \frac {x^3}{1+a^2 x^2} \, dx-\frac {1}{10} \left (a c^2\right ) \text {Subst}\left (\int \frac {x}{1+a^2 x} \, dx,x,x^2\right )-\frac {1}{70} \left (a^3 c^2\right ) \text {Subst}\left (\int \frac {x^2}{1+a^2 x} \, dx,x,x^2\right ) \\ & = -\frac {c^2 x \arctan (a x)}{70 a^2}+\frac {17}{210} c^2 x^3 \arctan (a x)+\frac {1}{35} a^2 c^2 x^5 \arctan (a x)+\frac {c^2 \arctan (a x)^2}{140 a^3}-\frac {4 c^2 x^2 \arctan (a x)^2}{35 a}-\frac {27}{140} a c^2 x^4 \arctan (a x)^2-\frac {1}{14} a^3 c^2 x^6 \arctan (a x)^2-\frac {8 i c^2 \arctan (a x)^3}{105 a^3}+\frac {1}{3} c^2 x^3 \arctan (a x)^3+\frac {2}{5} a^2 c^2 x^5 \arctan (a x)^3+\frac {1}{7} a^4 c^2 x^7 \arctan (a x)^3-\frac {8 c^2 \arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )}{35 a^3}+\frac {2 c^2 \log \left (1+a^2 x^2\right )}{5 a^3}-\frac {8 i c^2 \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{35 a^3}+\frac {c^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )}{10 a^3}+\frac {\left (3 i c^2\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^2 \int \frac {x}{1+a^2 x^2} \, dx}{7 a}-\frac {\left (3 c^2\right ) \int \frac {x}{1+a^2 x^2} \, dx}{14 a}-\frac {\left (3 c^2\right ) \int \frac {x}{1+a^2 x^2} \, dx}{7 a}+\frac {1}{42} \left (a c^2\right ) \text {Subst}\left (\int \frac {x}{1+a^2 x} \, dx,x,x^2\right )+\frac {1}{28} \left (a c^2\right ) \text {Subst}\left (\int \frac {x}{1+a^2 x} \, dx,x,x^2\right )-\frac {1}{10} \left (a c^2\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}{70} \left (a^3 c^2\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 {3 c^2 x^2}{35 a}-\frac {1}{140} a c^2 x^4-\frac {c^2 x \arctan (a x)}{70 a^2}+\frac {17}{210} c^2 x^3 \arctan (a x)+\frac {1}{35} a^2 c^2 x^5 \arctan (a x)+\frac {c^2 \arctan (a x)^2}{140 a^3}-\frac {4 c^2 x^2 \arctan (a x)^2}{35 a}-\frac {27}{140} a c^2 x^4 \arctan (a x)^2-\frac {1}{14} a^3 c^2 x^6 \arctan (a x)^2-\frac {8 i c^2 \arctan (a x)^3}{105 a^3}+\frac {1}{3} c^2 x^3 \arctan (a x)^3+\frac {2}{5} a^2 c^2 x^5 \arctan (a x)^3+\frac {1}{7} a^4 c^2 x^7 \arctan (a x)^3-\frac {8 c^2 \arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )}{35 a^3}+\frac {13 c^2 \log \left (1+a^2 x^2\right )}{140 a^3}-\frac {8 i c^2 \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{35 a^3}-\frac {4 c^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )}{35 a^3}+\frac {1}{42} \left (a c^2\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 (a c^2\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 {11 c^2 x^2}{420 a}-\frac {1}{140} a c^2 x^4-\frac {c^2 x \arctan (a x)}{70 a^2}+\frac {17}{210} c^2 x^3 \arctan (a x)+\frac {1}{35} a^2 c^2 x^5 \arctan (a x)+\frac {c^2 \arctan (a x)^2}{140 a^3}-\frac {4 c^2 x^2 \arctan (a x)^2}{35 a}-\frac {27}{140} a c^2 x^4 \arctan (a x)^2-\frac {1}{14} a^3 c^2 x^6 \arctan (a x)^2-\frac {8 i c^2 \arctan (a x)^3}{105 a^3}+\frac {1}{3} c^2 x^3 \arctan (a x)^3+\frac {2}{5} a^2 c^2 x^5 \arctan (a x)^3+\frac {1}{7} a^4 c^2 x^7 \arctan (a x)^3-\frac {8 c^2 \arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )}{35 a^3}+\frac {c^2 \log \left (1+a^2 x^2\right )}{30 a^3}-\frac {8 i c^2 \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{35 a^3}-\frac {4 c^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )}{35 a^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.96 (sec) , antiderivative size = 233, normalized size of antiderivative = 0.73 \[ \int x^2 \left (c+a^2 c x^2\right )^2 \arctan (a x)^3 \, dx=\frac {c^2 \left (-8-11 a^2 x^2-3 a^4 x^4-6 a x \arctan (a x)+34 a^3 x^3 \arctan (a x)+12 a^5 x^5 \arctan (a x)+3 \arctan (a x)^2-48 a^2 x^2 \arctan (a x)^2-81 a^4 x^4 \arctan (a x)^2-30 a^6 x^6 \arctan (a x)^2+32 i \arctan (a x)^3+140 a^3 x^3 \arctan (a x)^3+168 a^5 x^5 \arctan (a x)^3+60 a^7 x^7 \arctan (a x)^3-96 \arctan (a x)^2 \log \left (1+e^{2 i \arctan (a x)}\right )+14 \log \left (1+a^2 x^2\right )+96 i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{2 i \arctan (a x)}\right )-48 \operatorname {PolyLog}\left (3,-e^{2 i \arctan (a x)}\right )\right )}{420 a^3} \]

[In]

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

[Out]

(c^2*(-8 - 11*a^2*x^2 - 3*a^4*x^4 - 6*a*x*ArcTan[a*x] + 34*a^3*x^3*ArcTan[a*x] + 12*a^5*x^5*ArcTan[a*x] + 3*Ar
cTan[a*x]^2 - 48*a^2*x^2*ArcTan[a*x]^2 - 81*a^4*x^4*ArcTan[a*x]^2 - 30*a^6*x^6*ArcTan[a*x]^2 + (32*I)*ArcTan[a
*x]^3 + 140*a^3*x^3*ArcTan[a*x]^3 + 168*a^5*x^5*ArcTan[a*x]^3 + 60*a^7*x^7*ArcTan[a*x]^3 - 96*ArcTan[a*x]^2*Lo
g[1 + E^((2*I)*ArcTan[a*x])] + 14*Log[1 + a^2*x^2] + (96*I)*ArcTan[a*x]*PolyLog[2, -E^((2*I)*ArcTan[a*x])] - 4
8*PolyLog[3, -E^((2*I)*ArcTan[a*x])]))/(420*a^3)

Maple [C] (warning: unable to verify)

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

Time = 72.00 (sec) , antiderivative size = 1256, normalized size of antiderivative = 3.91

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1256\)
default	\(\text {Expression too large to display}\)	\(1256\)
parts	\(\text {Expression too large to display}\)	\(1256\)

[In]

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

[Out]

1/a^3*(1/7*c^2*arctan(a*x)^3*a^7*x^7+2/5*c^2*arctan(a*x)^3*a^5*x^5+1/3*c^2*arctan(a*x)^3*a^3*x^3-1/35*c^2*(-2*
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)^2-1/4*arctan(a*x)^2+4*x^2*arctan(a*x)^2*a^2+8*arctan(a*x)^2*ln(2)+1/4*(
I+a*x)^4+5/2*a^6*x^6*arctan(a*x)^2+27/4*a^4*arctan(a*x)^2*x^4+4*polylog(3,-(1+I*a*x)^2/(a^2*x^2+1))+7/3*ln((1+
I*a*x)^2/(a^2*x^2+1)+1)-4*arctan(a*x)^2*ln(a^2*x^2+1)-5*I*arctan(a*x)*(a*x-I)^4-8*I*arctan(a*x)*polylog(2,-(1+
I*a*x)^2/(a^2*x^2+1))+5*arctan(a*x)*(a*x-I)^4*(I+a*x)-5*arctan(a*x)*(a*x-I)*(I+a*x)^4+10*arctan(a*x)*(a*x-I)^2
*(I+a*x)^3-2*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+2*I*Pi*csgn(I*
((1+I*a*x)^2/(a^2*x^2+1)+1)^2)^3*arctan(a*x)^2-2*I*Pi*csgn(I*(1+I*a*x)^2/(a^2*x^2+1))^3*arctan(a*x)^2-30*I*arc
tan(a*x)*(a*x-I)^2*(I+a*x)^2+20*I*arctan(a*x)*(a*x-I)*(I+a*x)^3-7/12*(I+a*x)^2-8/3*I*arctan(a*x)^3-5/6*I*(I+a*
x)-I*(I+a*x)^3+43/6*arctan(a*x)*(a*x-I)^3-arctan(a*x)*(a*x-I)^5+20*I*arctan(a*x)*(a*x-I)^3*(I+a*x)-3*I*arctan(
a*x)*(a*x-I)*(I+a*x)+3/2*I*arctan(a*x)*(a*x-I)^2-10*arctan(a*x)*(a*x-I)^3*(I+a*x)^2+43/2*arctan(a*x)*(a*x-I)*(
I+a*x)^2-43/2*arctan(a*x)*(a*x-I)^2*(I+a*x)+8*arctan(a*x)^2*ln((1+I*a*x)/(a^2*x^2+1)^(1/2))+4*arctan(a*x)*(a*x
-I)-2*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+2*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+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))*csgn(I*(1+I*a*x)^2/(a^2*x^2+1))^2*arctan(a*x)^2+2*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+2*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)^2-4*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))

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

1/840*(15*a^4*c^2*x^7 + 42*a^2*c^2*x^5 + 35*c^2*x^3)*arctan(a*x)^3 - 1/1120*(15*a^4*c^2*x^7 + 42*a^2*c^2*x^5 +
35*c^2*x^3)*arctan(a*x)*log(a^2*x^2 + 1)^2 + integrate(1/1120*(980*(a^6*c^2*x^8 + 3*a^4*c^2*x^6 + 3*a^2*c^2*x
^4 + c^2*x^2)*arctan(a*x)^3 - 4*(15*a^5*c^2*x^7 + 42*a^3*c^2*x^5 + 35*a*c^2*x^3)*arctan(a*x)^2 + 4*(15*a^6*c^2
*x^8 + 42*a^4*c^2*x^6 + 35*a^2*c^2*x^4)*arctan(a*x)*log(a^2*x^2 + 1) + (15*a^5*c^2*x^7 + 42*a^3*c^2*x^5 + 35*a
*c^2*x^3 + 105*(a^6*c^2*x^8 + 3*a^4*c^2*x^6 + 3*a^2*c^2*x^4 + c^2*x^2)*arctan(a*x))*log(a^2*x^2 + 1)^2)/(a^2*x
^2 + 1), x)

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

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

[Out]

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