Integrand size = 22, antiderivative size = 130 \[ \int \frac {x^2 \arctan (a x)^3}{c+a^2 c x^2} \, dx=\frac {i \arctan (a x)^3}{a^3 c}+\frac {x \arctan (a x)^3}{a^2 c}-\frac {\arctan (a x)^4}{4 a^3 c}+\frac {3 \arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )}{a^3 c}+\frac {3 i \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{a^3 c}+\frac {3 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )}{2 a^3 c} \]
I*arctan(a*x)^3/a^3/c+x*arctan(a*x)^3/a^2/c-1/4*arctan(a*x)^4/a^3/c+3*arct an(a*x)^2*ln(2/(1+I*a*x))/a^3/c+3*I*arctan(a*x)*polylog(2,1-2/(1+I*a*x))/a ^3/c+3/2*polylog(3,1-2/(1+I*a*x))/a^3/c
Time = 0.21 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.72 \[ \int \frac {x^2 \arctan (a x)^3}{c+a^2 c x^2} \, dx=\frac {-\frac {1}{4} \arctan (a x)^2 \left ((4 i-4 a x) \arctan (a x)+\arctan (a x)^2-12 \log \left (1+e^{2 i \arctan (a x)}\right )\right )-3 i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{2 i \arctan (a x)}\right )+\frac {3}{2} \operatorname {PolyLog}\left (3,-e^{2 i \arctan (a x)}\right )}{a^3 c} \]
(-1/4*(ArcTan[a*x]^2*((4*I - 4*a*x)*ArcTan[a*x] + ArcTan[a*x]^2 - 12*Log[1 + E^((2*I)*ArcTan[a*x])])) - (3*I)*ArcTan[a*x]*PolyLog[2, -E^((2*I)*ArcTa n[a*x])] + (3*PolyLog[3, -E^((2*I)*ArcTan[a*x])])/2)/(a^3*c)
Time = 0.83 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.05, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {5451, 27, 5345, 5419, 5455, 5379, 5529, 7164}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {x^2 \arctan (a x)^3}{a^2 c x^2+c} \, dx\) |
| \(\Big \downarrow \) 5451 |
| \(\displaystyle \frac {\int \arctan (a x)^3dx}{a^2 c}-\frac {\int \frac {\arctan (a x)^3}{c \left (a^2 x^2+1\right )}dx}{a^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \arctan (a x)^3dx}{a^2 c}-\frac {\int \frac {\arctan (a x)^3}{a^2 x^2+1}dx}{a^2 c}\) |
| \(\Big \downarrow \) 5345 |
| \(\displaystyle \frac {x \arctan (a x)^3-3 a \int \frac {x \arctan (a x)^2}{a^2 x^2+1}dx}{a^2 c}-\frac {\int \frac {\arctan (a x)^3}{a^2 x^2+1}dx}{a^2 c}\) |
| \(\Big \downarrow \) 5419 |
| \(\displaystyle \frac {x \arctan (a x)^3-3 a \int \frac {x \arctan (a x)^2}{a^2 x^2+1}dx}{a^2 c}-\frac {\arctan (a x)^4}{4 a^3 c}\) |
| \(\Big \downarrow \) 5455 |
| \(\displaystyle -\frac {\arctan (a x)^4}{4 a^3 c}+\frac {x \arctan (a x)^3-3 a \left (-\frac {\int \frac {\arctan (a x)^2}{i-a x}dx}{a}-\frac {i \arctan (a x)^3}{3 a^2}\right )}{a^2 c}\) |
| \(\Big \downarrow \) 5379 |
| \(\displaystyle -\frac {\arctan (a x)^4}{4 a^3 c}+\frac {x \arctan (a x)^3-3 a \left (-\frac {\frac {\arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )}{a}-2 \int \frac {\arctan (a x) \log \left (\frac {2}{i a x+1}\right )}{a^2 x^2+1}dx}{a}-\frac {i \arctan (a x)^3}{3 a^2}\right )}{a^2 c}\) |
| \(\Big \downarrow \) 5529 |
| \(\displaystyle -\frac {\arctan (a x)^4}{4 a^3 c}+\frac {x \arctan (a x)^3-3 a \left (-\frac {\frac {\arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )}{a}-2 \left (\frac {1}{2} i \int \frac {\operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )}{a^2 x^2+1}dx-\frac {i \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )}{2 a}\right )}{a}-\frac {i \arctan (a x)^3}{3 a^2}\right )}{a^2 c}\) |
| \(\Big \downarrow \) 7164 |
| \(\displaystyle -\frac {\arctan (a x)^4}{4 a^3 c}+\frac {x \arctan (a x)^3-3 a \left (-\frac {i \arctan (a x)^3}{3 a^2}-\frac {\frac {\arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )}{a}-2 \left (-\frac {i \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )}{2 a}-\frac {\operatorname {PolyLog}\left (3,1-\frac {2}{i a x+1}\right )}{4 a}\right )}{a}\right )}{a^2 c}\) |
-1/4*ArcTan[a*x]^4/(a^3*c) + (x*ArcTan[a*x]^3 - 3*a*(((-1/3*I)*ArcTan[a*x] ^3)/a^2 - ((ArcTan[a*x]^2*Log[2/(1 + I*a*x)])/a - 2*(((-1/2*I)*ArcTan[a*x] *PolyLog[2, 1 - 2/(1 + I*a*x)])/a - PolyLog[3, 1 - 2/(1 + I*a*x)]/(4*a)))/ a))/(a^2*c)
3.4.89.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + ArcTan[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*ArcTan[c*x^n])^p, x] - Simp[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])
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] + Simp[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 ]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2), x_Symbo l] :> 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]
Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + (e _.)*(x_)^2), x_Symbol] :> Simp[f^2/e Int[(f*x)^(m - 2)*(a + b*ArcTan[c*x] )^p, x], x] - Simp[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]
Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(-I)*((a + b*ArcTan[c*x])^(p + 1)/(b*e*(p + 1))), x] - Si mp[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]
Int[(Log[u_]*((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2 ), x_Symbol] :> Simp[(-I)*(a + b*ArcTan[c*x])^p*(PolyLog[2, 1 - u]/(2*c*d)) , x] + Simp[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]
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]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 14.14 (sec) , antiderivative size = 785, normalized size of antiderivative = 6.04
| method | result | size |
| derivativedivides | \(\frac {\frac {\arctan \left (a x \right )^{3} a x}{c}-\frac {\arctan \left (a x \right )^{4}}{c}-\frac {3 \left (\frac {\arctan \left (a x \right )^{2} \ln \left (a^{2} x^{2}+1\right )}{2}-\arctan \left (a x \right )^{2} \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+\frac {i \arctan \left (a x \right )^{3}}{3}-\frac {\left (-i \pi \,\operatorname {csgn}\left (\frac {i}{\left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}}\right ) \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right ) \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{\left (a^{2} x^{2}+1\right ) \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}}\right )+i \pi \,\operatorname {csgn}\left (\frac {i}{\left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}}\right ) \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{\left (a^{2} x^{2}+1\right ) \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}}\right )^{2}+i \pi {\operatorname {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )\right )}^{2} \operatorname {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}\right )-2 i \pi \,\operatorname {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )\right ) {\operatorname {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}\right )}^{2}+i \pi {\operatorname {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}\right )}^{3}-i \pi {\operatorname {csgn}\left (\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )}^{2} \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )+2 i \pi \,\operatorname {csgn}\left (\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right ) \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )^{2}-i \pi \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )^{3}+i \pi \,\operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right ) \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{\left (a^{2} x^{2}+1\right ) \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}}\right )^{2}-i \pi \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{\left (a^{2} x^{2}+1\right ) \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}}\right )^{3}+4 \ln \left (2\right )\right ) \arctan \left (a x \right )^{2}}{4}+i \arctan \left (a x \right ) \operatorname {polylog}\left (2, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )-\frac {\operatorname {polylog}\left (3, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{2}-\frac {\arctan \left (a x \right )^{4}}{4}\right )}{c}}{a^{3}}\) | \(785\) |
| default | \(\frac {\frac {\arctan \left (a x \right )^{3} a x}{c}-\frac {\arctan \left (a x \right )^{4}}{c}-\frac {3 \left (\frac {\arctan \left (a x \right )^{2} \ln \left (a^{2} x^{2}+1\right )}{2}-\arctan \left (a x \right )^{2} \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+\frac {i \arctan \left (a x \right )^{3}}{3}-\frac {\left (-i \pi \,\operatorname {csgn}\left (\frac {i}{\left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}}\right ) \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right ) \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{\left (a^{2} x^{2}+1\right ) \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}}\right )+i \pi \,\operatorname {csgn}\left (\frac {i}{\left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}}\right ) \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{\left (a^{2} x^{2}+1\right ) \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}}\right )^{2}+i \pi {\operatorname {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )\right )}^{2} \operatorname {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}\right )-2 i \pi \,\operatorname {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )\right ) {\operatorname {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}\right )}^{2}+i \pi {\operatorname {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}\right )}^{3}-i \pi {\operatorname {csgn}\left (\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )}^{2} \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )+2 i \pi \,\operatorname {csgn}\left (\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right ) \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )^{2}-i \pi \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )^{3}+i \pi \,\operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right ) \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{\left (a^{2} x^{2}+1\right ) \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}}\right )^{2}-i \pi \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{\left (a^{2} x^{2}+1\right ) \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}}\right )^{3}+4 \ln \left (2\right )\right ) \arctan \left (a x \right )^{2}}{4}+i \arctan \left (a x \right ) \operatorname {polylog}\left (2, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )-\frac {\operatorname {polylog}\left (3, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{2}-\frac {\arctan \left (a x \right )^{4}}{4}\right )}{c}}{a^{3}}\) | \(785\) |
| parts | \(\frac {x \arctan \left (a x \right )^{3}}{a^{2} c}-\frac {\arctan \left (a x \right )^{4}}{a^{3} c}-\frac {3 \left (\frac {\frac {\arctan \left (a x \right )^{2} \ln \left (a^{2} x^{2}+1\right )}{2}-\arctan \left (a x \right )^{2} \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+\frac {i \arctan \left (a x \right )^{3}}{3}-\frac {\left (-i \pi \,\operatorname {csgn}\left (\frac {i}{\left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}}\right ) \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right ) \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{\left (a^{2} x^{2}+1\right ) \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}}\right )+i \pi \,\operatorname {csgn}\left (\frac {i}{\left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}}\right ) \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{\left (a^{2} x^{2}+1\right ) \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}}\right )^{2}+i \pi {\operatorname {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )\right )}^{2} \operatorname {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}\right )-2 i \pi \,\operatorname {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )\right ) {\operatorname {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}\right )}^{2}+i \pi {\operatorname {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}\right )}^{3}-i \pi {\operatorname {csgn}\left (\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )}^{2} \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )+2 i \pi \,\operatorname {csgn}\left (\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right ) \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )^{2}-i \pi \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )^{3}+i \pi \,\operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right ) \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{\left (a^{2} x^{2}+1\right ) \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}}\right )^{2}-i \pi \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{\left (a^{2} x^{2}+1\right ) \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}}\right )^{3}+4 \ln \left (2\right )\right ) \arctan \left (a x \right )^{2}}{4}+i \arctan \left (a x \right ) \operatorname {polylog}\left (2, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )-\frac {\operatorname {polylog}\left (3, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{2}}{a^{3}}-\frac {\arctan \left (a x \right )^{4}}{4 a^{3}}\right )}{c}\) | \(794\) |
1/a^3*(1/c*arctan(a*x)^3*a*x-1/c*arctan(a*x)^4-3/c*(1/2*arctan(a*x)^2*ln(a ^2*x^2+1)-arctan(a*x)^2*ln((1+I*a*x)/(a^2*x^2+1)^(1/2))+1/3*I*arctan(a*x)^ 3-1/4*(-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)+I*P i*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+I*Pi*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1))^2*c sgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1)^2)-2*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+I*Pi*csgn(I*((1+I*a*x)^2/(a^ 2*x^2+1)+1)^2)^3-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))+2*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-I*Pi*csgn(I*(1+I*a*x)^2/(a^2*x^2+1))^3+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-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+4*ln(2))*arctan(a*x)^2+I*arctan(a*x)*polylog(2,-(1+I*a*x)^2/(a^2*x ^2+1))-1/2*polylog(3,-(1+I*a*x)^2/(a^2*x^2+1))-1/4*arctan(a*x)^4))
\[ \int \frac {x^2 \arctan (a x)^3}{c+a^2 c x^2} \, dx=\int { \frac {x^{2} \arctan \left (a x\right )^{3}}{a^{2} c x^{2} + c} \,d x } \]
\[ \int \frac {x^2 \arctan (a x)^3}{c+a^2 c x^2} \, dx=\frac {\int \frac {x^{2} \operatorname {atan}^{3}{\left (a x \right )}}{a^{2} x^{2} + 1}\, dx}{c} \]
\[ \int \frac {x^2 \arctan (a x)^3}{c+a^2 c x^2} \, dx=\int { \frac {x^{2} \arctan \left (a x\right )^{3}}{a^{2} c x^{2} + c} \,d x } \]
1/1024*(16*(7168*a^2*integrate(1/128*x^2*arctan(a*x)^3/(a^4*c*x^2 + a^2*c) , x) + 768*a^2*integrate(1/128*x^2*arctan(a*x)*log(a^2*x^2 + 1)^2/(a^4*c*x ^2 + a^2*c), x) + 3072*a^2*integrate(1/128*x^2*arctan(a*x)*log(a^2*x^2 + 1 )/(a^4*c*x^2 + a^2*c), x) - 768*a*integrate(1/128*x*arctan(a*x)^2*log(a^2* x^2 + 1)/(a^4*c*x^2 + a^2*c), x) - 192*a*integrate(1/128*x*log(a^2*x^2 + 1 )^3/(a^4*c*x^2 + a^2*c), x) - 3072*a*integrate(1/128*x*arctan(a*x)^2/(a^4* c*x^2 + a^2*c), x) + 768*a*integrate(1/128*x*log(a^2*x^2 + 1)^2/(a^4*c*x^2 + a^2*c), x) + 3*arctan(a*x)^4/(a^3*c) + 384*integrate(1/128*arctan(a*x)* log(a^2*x^2 + 1)^2/(a^4*c*x^2 + a^2*c), x))*a^3*c + 128*a*x*arctan(a*x)^3 - 80*arctan(a*x)^4 + 3*log(a^2*x^2 + 1)^4 - 24*(4*a*x*arctan(a*x) - arctan (a*x)^2)*log(a^2*x^2 + 1)^2)/(a^3*c)
\[ \int \frac {x^2 \arctan (a x)^3}{c+a^2 c x^2} \, dx=\int { \frac {x^{2} \arctan \left (a x\right )^{3}}{a^{2} c x^{2} + c} \,d x } \]
Timed out. \[ \int \frac {x^2 \arctan (a x)^3}{c+a^2 c x^2} \, dx=\int \frac {x^2\,{\mathrm {atan}\left (a\,x\right )}^3}{c\,a^2\,x^2+c} \,d x \]