Integrand size = 22, antiderivative size = 447 \[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^3}{x} \, dx=-\frac {13}{30} a c^3 x-\frac {1}{60} a^3 c^3 x^3+\frac {13}{30} c^3 \arctan (a x)+\frac {29}{60} a^2 c^3 x^2 \arctan (a x)+\frac {1}{20} a^4 c^3 x^4 \arctan (a x)-\frac {34}{15} i c^3 \arctan (a x)^2-\frac {11}{4} a c^3 x \arctan (a x)^2-\frac {7}{12} a^3 c^3 x^3 \arctan (a x)^2-\frac {1}{10} a^5 c^3 x^5 \arctan (a x)^2+\frac {11}{12} c^3 \arctan (a x)^3+\frac {3}{2} a^2 c^3 x^2 \arctan (a x)^3+\frac {3}{4} a^4 c^3 x^4 \arctan (a x)^3+\frac {1}{6} a^6 c^3 x^6 \arctan (a x)^3+2 c^3 \arctan (a x)^3 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )-\frac {68}{15} c^3 \arctan (a x) \log \left (\frac {2}{1+i a x}\right )-\frac {34}{15} i c^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )-\frac {3}{2} i c^3 \arctan (a x)^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )+\frac {3}{2} i c^3 \arctan (a x)^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i a x}\right )-\frac {3}{2} c^3 \arctan (a x) \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )+\frac {3}{2} c^3 \arctan (a x) \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i a x}\right )+\frac {3}{4} i c^3 \operatorname {PolyLog}\left (4,1-\frac {2}{1+i a x}\right )-\frac {3}{4} i c^3 \operatorname {PolyLog}\left (4,-1+\frac {2}{1+i a x}\right ) \]
-13/30*a*c^3*x-1/60*a^3*c^3*x^3+13/30*c^3*arctan(a*x)+29/60*a^2*c^3*x^2*ar ctan(a*x)+1/20*a^4*c^3*x^4*arctan(a*x)-34/15*I*c^3*arctan(a*x)^2-11/4*a*c^ 3*x*arctan(a*x)^2-7/12*a^3*c^3*x^3*arctan(a*x)^2-1/10*a^5*c^3*x^5*arctan(a *x)^2+11/12*c^3*arctan(a*x)^3+3/2*a^2*c^3*x^2*arctan(a*x)^3+3/4*a^4*c^3*x^ 4*arctan(a*x)^3+1/6*a^6*c^3*x^6*arctan(a*x)^3-2*c^3*arctan(a*x)^3*arctanh( -1+2/(1+I*a*x))-68/15*c^3*arctan(a*x)*ln(2/(1+I*a*x))+3/2*I*c^3*arctan(a*x )^2*polylog(2,-1+2/(1+I*a*x))-3/2*I*c^3*arctan(a*x)^2*polylog(2,1-2/(1+I*a *x))-3/4*I*c^3*polylog(4,-1+2/(1+I*a*x))-3/2*c^3*arctan(a*x)*polylog(3,1-2 /(1+I*a*x))+3/2*c^3*arctan(a*x)*polylog(3,-1+2/(1+I*a*x))+3/4*I*c^3*polylo g(4,1-2/(1+I*a*x))-34/15*I*c^3*polylog(2,1-2/(1+I*a*x))
Time = 0.82 (sec) , antiderivative size = 350, normalized size of antiderivative = 0.78 \[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^3}{x} \, dx=\frac {1}{960} c^3 \left (-15 i \pi ^4-416 a x-16 a^3 x^3+416 \arctan (a x)+464 a^2 x^2 \arctan (a x)+48 a^4 x^4 \arctan (a x)+2176 i \arctan (a x)^2-2640 a x \arctan (a x)^2-560 a^3 x^3 \arctan (a x)^2-96 a^5 x^5 \arctan (a x)^2+880 \arctan (a x)^3+1440 a^2 x^2 \arctan (a x)^3+720 a^4 x^4 \arctan (a x)^3+160 a^6 x^6 \arctan (a x)^3+480 i \arctan (a x)^4+960 \arctan (a x)^3 \log \left (1-e^{-2 i \arctan (a x)}\right )-4352 \arctan (a x) \log \left (1+e^{2 i \arctan (a x)}\right )-960 \arctan (a x)^3 \log \left (1+e^{2 i \arctan (a x)}\right )+1440 i \arctan (a x)^2 \operatorname {PolyLog}\left (2,e^{-2 i \arctan (a x)}\right )+32 i \left (68+45 \arctan (a x)^2\right ) \operatorname {PolyLog}\left (2,-e^{2 i \arctan (a x)}\right )+1440 \arctan (a x) \operatorname {PolyLog}\left (3,e^{-2 i \arctan (a x)}\right )-1440 \arctan (a x) \operatorname {PolyLog}\left (3,-e^{2 i \arctan (a x)}\right )-720 i \operatorname {PolyLog}\left (4,e^{-2 i \arctan (a x)}\right )-720 i \operatorname {PolyLog}\left (4,-e^{2 i \arctan (a x)}\right )\right ) \]
(c^3*((-15*I)*Pi^4 - 416*a*x - 16*a^3*x^3 + 416*ArcTan[a*x] + 464*a^2*x^2* ArcTan[a*x] + 48*a^4*x^4*ArcTan[a*x] + (2176*I)*ArcTan[a*x]^2 - 2640*a*x*A rcTan[a*x]^2 - 560*a^3*x^3*ArcTan[a*x]^2 - 96*a^5*x^5*ArcTan[a*x]^2 + 880* ArcTan[a*x]^3 + 1440*a^2*x^2*ArcTan[a*x]^3 + 720*a^4*x^4*ArcTan[a*x]^3 + 1 60*a^6*x^6*ArcTan[a*x]^3 + (480*I)*ArcTan[a*x]^4 + 960*ArcTan[a*x]^3*Log[1 - E^((-2*I)*ArcTan[a*x])] - 4352*ArcTan[a*x]*Log[1 + E^((2*I)*ArcTan[a*x] )] - 960*ArcTan[a*x]^3*Log[1 + E^((2*I)*ArcTan[a*x])] + (1440*I)*ArcTan[a* x]^2*PolyLog[2, E^((-2*I)*ArcTan[a*x])] + (32*I)*(68 + 45*ArcTan[a*x]^2)*P olyLog[2, -E^((2*I)*ArcTan[a*x])] + 1440*ArcTan[a*x]*PolyLog[3, E^((-2*I)* ArcTan[a*x])] - 1440*ArcTan[a*x]*PolyLog[3, -E^((2*I)*ArcTan[a*x])] - (720 *I)*PolyLog[4, E^((-2*I)*ArcTan[a*x])] - (720*I)*PolyLog[4, -E^((2*I)*ArcT an[a*x])]))/960
Time = 1.80 (sec) , antiderivative size = 447, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {5483, 2009}
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.
| \(\displaystyle \int \frac {\arctan (a x)^3 \left (a^2 c x^2+c\right )^3}{x} \, dx\) |
| \(\Big \downarrow \) 5483 |
| \(\displaystyle \int \left (a^6 c^3 x^5 \arctan (a x)^3+3 a^4 c^3 x^3 \arctan (a x)^3+3 a^2 c^3 x \arctan (a x)^3+\frac {c^3 \arctan (a x)^3}{x}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{6} a^6 c^3 x^6 \arctan (a x)^3-\frac {1}{10} a^5 c^3 x^5 \arctan (a x)^2+\frac {3}{4} a^4 c^3 x^4 \arctan (a x)^3+\frac {1}{20} a^4 c^3 x^4 \arctan (a x)-\frac {7}{12} a^3 c^3 x^3 \arctan (a x)^2-\frac {1}{60} a^3 c^3 x^3+\frac {3}{2} a^2 c^3 x^2 \arctan (a x)^3+\frac {29}{60} a^2 c^3 x^2 \arctan (a x)+2 c^3 \arctan (a x)^3 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )-\frac {3}{2} i c^3 \arctan (a x)^2 \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )+\frac {3}{2} i c^3 \arctan (a x)^2 \operatorname {PolyLog}\left (2,\frac {2}{i a x+1}-1\right )-\frac {3}{2} c^3 \arctan (a x) \operatorname {PolyLog}\left (3,1-\frac {2}{i a x+1}\right )+\frac {3}{2} c^3 \arctan (a x) \operatorname {PolyLog}\left (3,\frac {2}{i a x+1}-1\right )-\frac {11}{4} a c^3 x \arctan (a x)^2+\frac {11}{12} c^3 \arctan (a x)^3-\frac {34}{15} i c^3 \arctan (a x)^2+\frac {13}{30} c^3 \arctan (a x)-\frac {68}{15} c^3 \arctan (a x) \log \left (\frac {2}{1+i a x}\right )-\frac {34}{15} i c^3 \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )+\frac {3}{4} i c^3 \operatorname {PolyLog}\left (4,1-\frac {2}{i a x+1}\right )-\frac {3}{4} i c^3 \operatorname {PolyLog}\left (4,\frac {2}{i a x+1}-1\right )-\frac {13}{30} a c^3 x\) |
(-13*a*c^3*x)/30 - (a^3*c^3*x^3)/60 + (13*c^3*ArcTan[a*x])/30 + (29*a^2*c^ 3*x^2*ArcTan[a*x])/60 + (a^4*c^3*x^4*ArcTan[a*x])/20 - ((34*I)/15)*c^3*Arc Tan[a*x]^2 - (11*a*c^3*x*ArcTan[a*x]^2)/4 - (7*a^3*c^3*x^3*ArcTan[a*x]^2)/ 12 - (a^5*c^3*x^5*ArcTan[a*x]^2)/10 + (11*c^3*ArcTan[a*x]^3)/12 + (3*a^2*c ^3*x^2*ArcTan[a*x]^3)/2 + (3*a^4*c^3*x^4*ArcTan[a*x]^3)/4 + (a^6*c^3*x^6*A rcTan[a*x]^3)/6 + 2*c^3*ArcTan[a*x]^3*ArcTanh[1 - 2/(1 + I*a*x)] - (68*c^3 *ArcTan[a*x]*Log[2/(1 + I*a*x)])/15 - ((34*I)/15)*c^3*PolyLog[2, 1 - 2/(1 + I*a*x)] - ((3*I)/2)*c^3*ArcTan[a*x]^2*PolyLog[2, 1 - 2/(1 + I*a*x)] + (( 3*I)/2)*c^3*ArcTan[a*x]^2*PolyLog[2, -1 + 2/(1 + I*a*x)] - (3*c^3*ArcTan[a *x]*PolyLog[3, 1 - 2/(1 + I*a*x)])/2 + (3*c^3*ArcTan[a*x]*PolyLog[3, -1 + 2/(1 + I*a*x)])/2 + ((3*I)/4)*c^3*PolyLog[4, 1 - 2/(1 + I*a*x)] - ((3*I)/4 )*c^3*PolyLog[4, -1 + 2/(1 + I*a*x)]
3.4.83.3.1 Defintions of rubi rules used
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_)*((d_) + (e_. )*(x_)^2)^(q_), x_Symbol] :> Int[ExpandIntegrand[(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])
Time = 56.76 (sec) , antiderivative size = 664, normalized size of antiderivative = 1.49
| method | result | size |
| derivativedivides | \(\frac {c^{3} \left (-136 \arctan \left (a x \right )^{2}-55 i \arctan \left (a x \right )^{3}+i a x +55 \arctan \left (a x \right )^{3} a x -29 x^{2} \arctan \left (a x \right )^{2} a^{2}-26 i \arctan \left (a x \right )-3 i \arctan \left (a x \right ) a^{2} x^{2}+35 \arctan \left (a x \right )^{3} a^{3} x^{3}-6 a^{4} \arctan \left (a x \right )^{2} x^{4}-35 i \arctan \left (a x \right )^{3} a^{2} x^{2}+10 \arctan \left (a x \right )^{3} a^{5} x^{5}-25+29 i \arctan \left (a x \right )^{2} a x -10 i \arctan \left (a x \right )^{3} a^{4} x^{4}+26 x \arctan \left (a x \right ) a -a^{2} x^{2}+6 i \arctan \left (a x \right )^{2} a^{3} x^{3}+3 \arctan \left (a x \right ) x^{3} a^{3}\right ) \left (a x +i\right )}{60}+\frac {3 i c^{3} \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{2}-\frac {68 c^{3} \arctan \left (a x \right ) \ln \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )}{15}+6 i c^{3} \operatorname {polylog}\left (4, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+c^{3} \arctan \left (a x \right )^{3} \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+1\right )+\frac {34 i c^{3} \operatorname {polylog}\left (2, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{15}+6 c^{3} \arctan \left (a x \right ) \operatorname {polylog}\left (3, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-3 i c^{3} \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-c^{3} \arctan \left (a x \right )^{3} \ln \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )+\frac {68 i c^{3} \arctan \left (a x \right )^{2}}{15}-\frac {3 c^{3} \arctan \left (a x \right ) \operatorname {polylog}\left (3, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{2}-\frac {3 i c^{3} \operatorname {polylog}\left (4, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{4}+c^{3} \arctan \left (a x \right )^{3} \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-3 i c^{3} \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+6 c^{3} \arctan \left (a x \right ) \operatorname {polylog}\left (3, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+6 i c^{3} \operatorname {polylog}\left (4, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )\) | \(664\) |
| default | \(\frac {c^{3} \left (-136 \arctan \left (a x \right )^{2}-55 i \arctan \left (a x \right )^{3}+i a x +55 \arctan \left (a x \right )^{3} a x -29 x^{2} \arctan \left (a x \right )^{2} a^{2}-26 i \arctan \left (a x \right )-3 i \arctan \left (a x \right ) a^{2} x^{2}+35 \arctan \left (a x \right )^{3} a^{3} x^{3}-6 a^{4} \arctan \left (a x \right )^{2} x^{4}-35 i \arctan \left (a x \right )^{3} a^{2} x^{2}+10 \arctan \left (a x \right )^{3} a^{5} x^{5}-25+29 i \arctan \left (a x \right )^{2} a x -10 i \arctan \left (a x \right )^{3} a^{4} x^{4}+26 x \arctan \left (a x \right ) a -a^{2} x^{2}+6 i \arctan \left (a x \right )^{2} a^{3} x^{3}+3 \arctan \left (a x \right ) x^{3} a^{3}\right ) \left (a x +i\right )}{60}+\frac {3 i c^{3} \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{2}-\frac {68 c^{3} \arctan \left (a x \right ) \ln \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )}{15}+6 i c^{3} \operatorname {polylog}\left (4, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+c^{3} \arctan \left (a x \right )^{3} \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+1\right )+\frac {34 i c^{3} \operatorname {polylog}\left (2, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{15}+6 c^{3} \arctan \left (a x \right ) \operatorname {polylog}\left (3, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-3 i c^{3} \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-c^{3} \arctan \left (a x \right )^{3} \ln \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )+\frac {68 i c^{3} \arctan \left (a x \right )^{2}}{15}-\frac {3 c^{3} \arctan \left (a x \right ) \operatorname {polylog}\left (3, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{2}-\frac {3 i c^{3} \operatorname {polylog}\left (4, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{4}+c^{3} \arctan \left (a x \right )^{3} \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-3 i c^{3} \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+6 c^{3} \arctan \left (a x \right ) \operatorname {polylog}\left (3, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+6 i c^{3} \operatorname {polylog}\left (4, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )\) | \(664\) |
1/60*c^3*(-136*arctan(a*x)^2-55*I*arctan(a*x)^3+I*a*x+55*arctan(a*x)^3*a*x -29*x^2*arctan(a*x)^2*a^2-26*I*arctan(a*x)-3*I*arctan(a*x)*a^2*x^2+35*arct an(a*x)^3*a^3*x^3-6*a^4*arctan(a*x)^2*x^4-35*I*arctan(a*x)^3*a^2*x^2+10*ar ctan(a*x)^3*a^5*x^5-25+29*I*arctan(a*x)^2*a*x-10*I*arctan(a*x)^3*a^4*x^4+2 6*x*arctan(a*x)*a-a^2*x^2+6*I*arctan(a*x)^2*a^3*x^3+3*arctan(a*x)*x^3*a^3) *(I+a*x)+3/2*I*c^3*arctan(a*x)^2*polylog(2,-(1+I*a*x)^2/(a^2*x^2+1))-68/15 *c^3*arctan(a*x)*ln((1+I*a*x)^2/(a^2*x^2+1)+1)+6*I*c^3*polylog(4,-(1+I*a*x )/(a^2*x^2+1)^(1/2))+c^3*arctan(a*x)^3*ln((1+I*a*x)/(a^2*x^2+1)^(1/2)+1)+3 4/15*I*c^3*polylog(2,-(1+I*a*x)^2/(a^2*x^2+1))+6*c^3*arctan(a*x)*polylog(3 ,-(1+I*a*x)/(a^2*x^2+1)^(1/2))-3*I*c^3*arctan(a*x)^2*polylog(2,(1+I*a*x)/( a^2*x^2+1)^(1/2))-c^3*arctan(a*x)^3*ln((1+I*a*x)^2/(a^2*x^2+1)+1)+68/15*I* c^3*arctan(a*x)^2-3/2*c^3*arctan(a*x)*polylog(3,-(1+I*a*x)^2/(a^2*x^2+1))- 3/4*I*c^3*polylog(4,-(1+I*a*x)^2/(a^2*x^2+1))+c^3*arctan(a*x)^3*ln(1-(1+I* a*x)/(a^2*x^2+1)^(1/2))-3*I*c^3*arctan(a*x)^2*polylog(2,-(1+I*a*x)/(a^2*x^ 2+1)^(1/2))+6*c^3*arctan(a*x)*polylog(3,(1+I*a*x)/(a^2*x^2+1)^(1/2))+6*I*c ^3*polylog(4,(1+I*a*x)/(a^2*x^2+1)^(1/2))
\[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^3}{x} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{3} \arctan \left (a x\right )^{3}}{x} \,d x } \]
\[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^3}{x} \, dx=c^{3} \left (\int \frac {\operatorname {atan}^{3}{\left (a x \right )}}{x}\, dx + \int 3 a^{2} x \operatorname {atan}^{3}{\left (a x \right )}\, dx + \int 3 a^{4} x^{3} \operatorname {atan}^{3}{\left (a x \right )}\, dx + \int a^{6} x^{5} \operatorname {atan}^{3}{\left (a x \right )}\, dx\right ) \]
c**3*(Integral(atan(a*x)**3/x, x) + Integral(3*a**2*x*atan(a*x)**3, x) + I ntegral(3*a**4*x**3*atan(a*x)**3, x) + Integral(a**6*x**5*atan(a*x)**3, x) )
\[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^3}{x} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{3} \arctan \left (a x\right )^{3}}{x} \,d x } \]
1/96*(2*a^6*c^3*x^6 + 9*a^4*c^3*x^4 + 18*a^2*c^3*x^2)*arctan(a*x)^3 - 1/12 8*(2*a^6*c^3*x^6 + 9*a^4*c^3*x^4 + 18*a^2*c^3*x^2)*arctan(a*x)*log(a^2*x^2 + 1)^2 + integrate(1/128*(112*(a^8*c^3*x^8 + 4*a^6*c^3*x^6 + 6*a^4*c^3*x^ 4 + 4*a^2*c^3*x^2 + c^3)*arctan(a*x)^3 - 4*(2*a^7*c^3*x^7 + 9*a^5*c^3*x^5 + 18*a^3*c^3*x^3)*arctan(a*x)^2 + 4*(2*a^8*c^3*x^8 + 9*a^6*c^3*x^6 + 18*a^ 4*c^3*x^4)*arctan(a*x)*log(a^2*x^2 + 1) + (2*a^7*c^3*x^7 + 9*a^5*c^3*x^5 + 18*a^3*c^3*x^3 + 12*(a^8*c^3*x^8 + 4*a^6*c^3*x^6 + 6*a^4*c^3*x^4 + 4*a^2* c^3*x^2 + c^3)*arctan(a*x))*log(a^2*x^2 + 1)^2)/(a^2*x^3 + x), x)
Timed out. \[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^3}{x} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^3}{x} \, dx=\int \frac {{\mathrm {atan}\left (a\,x\right )}^3\,{\left (c\,a^2\,x^2+c\right )}^3}{x} \,d x \]