Integrand size = 22, antiderivative size = 503 \[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^3}{x^3} \, dx=-\frac {1}{4} a^3 c^3 x+\frac {1}{4} a^2 c^3 \arctan (a x)+\frac {1}{4} a^4 c^3 x^2 \arctan (a x)-5 i a^2 c^3 \arctan (a x)^2-\frac {3 a c^3 \arctan (a x)^2}{2 x}-\frac {15}{4} a^3 c^3 x \arctan (a x)^2-\frac {1}{4} a^5 c^3 x^3 \arctan (a x)^2+\frac {3}{4} a^2 c^3 \arctan (a x)^3-\frac {c^3 \arctan (a x)^3}{2 x^2}+\frac {3}{2} a^4 c^3 x^2 \arctan (a x)^3+\frac {1}{4} a^6 c^3 x^4 \arctan (a x)^3+6 a^2 c^3 \arctan (a x)^3 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )-7 a^2 c^3 \arctan (a x) \log \left (\frac {2}{1+i a x}\right )+3 a^2 c^3 \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )-\frac {3}{2} i a^2 c^3 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )-\frac {7}{2} i a^2 c^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )-\frac {9}{2} i a^2 c^3 \arctan (a x)^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )+\frac {9}{2} i a^2 c^3 \arctan (a x)^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i a x}\right )-\frac {9}{2} a^2 c^3 \arctan (a x) \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )+\frac {9}{2} a^2 c^3 \arctan (a x) \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i a x}\right )+\frac {9}{4} i a^2 c^3 \operatorname {PolyLog}\left (4,1-\frac {2}{1+i a x}\right )-\frac {9}{4} i a^2 c^3 \operatorname {PolyLog}\left (4,-1+\frac {2}{1+i a x}\right ) \]
-1/4*a^3*c^3*x+1/4*a^2*c^3*arctan(a*x)+1/4*a^4*c^3*x^2*arctan(a*x)-9/4*I*a ^2*c^3*polylog(4,-1+2/(1+I*a*x))-3/2*a*c^3*arctan(a*x)^2/x-15/4*a^3*c^3*x* arctan(a*x)^2-1/4*a^5*c^3*x^3*arctan(a*x)^2+3/4*a^2*c^3*arctan(a*x)^3-1/2* c^3*arctan(a*x)^3/x^2+3/2*a^4*c^3*x^2*arctan(a*x)^3+1/4*a^6*c^3*x^4*arctan (a*x)^3-6*a^2*c^3*arctan(a*x)^3*arctanh(-1+2/(1+I*a*x))-7*a^2*c^3*arctan(a *x)*ln(2/(1+I*a*x))+3*a^2*c^3*arctan(a*x)*ln(2-2/(1-I*a*x))+9/4*I*a^2*c^3* polylog(4,1-2/(1+I*a*x))-3/2*I*a^2*c^3*polylog(2,-1+2/(1-I*a*x))-5*I*a^2*c ^3*arctan(a*x)^2-7/2*I*a^2*c^3*polylog(2,1-2/(1+I*a*x))-9/2*a^2*c^3*arctan (a*x)*polylog(3,1-2/(1+I*a*x))+9/2*a^2*c^3*arctan(a*x)*polylog(3,-1+2/(1+I *a*x))-9/2*I*a^2*c^3*arctan(a*x)^2*polylog(2,1-2/(1+I*a*x))+9/2*I*a^2*c^3* arctan(a*x)^2*polylog(2,-1+2/(1+I*a*x))
Time = 0.57 (sec) , antiderivative size = 464, normalized size of antiderivative = 0.92 \[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^3}{x^3} \, dx=\frac {c^3 \left (-3 i a^2 \pi ^4 x^2-16 a^3 x^3+16 a^2 x^2 \arctan (a x)+16 a^4 x^4 \arctan (a x)-96 a x \arctan (a x)^2+128 i a^2 x^2 \arctan (a x)^2-240 a^3 x^3 \arctan (a x)^2-16 a^5 x^5 \arctan (a x)^2-32 \arctan (a x)^3+48 a^2 x^2 \arctan (a x)^3+96 a^4 x^4 \arctan (a x)^3+16 a^6 x^6 \arctan (a x)^3+96 i a^2 x^2 \arctan (a x)^4+192 a^2 x^2 \arctan (a x)^3 \log \left (1-e^{-2 i \arctan (a x)}\right )+192 a^2 x^2 \arctan (a x) \log \left (1-e^{2 i \arctan (a x)}\right )-448 a^2 x^2 \arctan (a x) \log \left (1+e^{2 i \arctan (a x)}\right )-192 a^2 x^2 \arctan (a x)^3 \log \left (1+e^{2 i \arctan (a x)}\right )+288 i a^2 x^2 \arctan (a x)^2 \operatorname {PolyLog}\left (2,e^{-2 i \arctan (a x)}\right )+32 i a^2 x^2 \left (7+9 \arctan (a x)^2\right ) \operatorname {PolyLog}\left (2,-e^{2 i \arctan (a x)}\right )-96 i a^2 x^2 \operatorname {PolyLog}\left (2,e^{2 i \arctan (a x)}\right )+288 a^2 x^2 \arctan (a x) \operatorname {PolyLog}\left (3,e^{-2 i \arctan (a x)}\right )-288 a^2 x^2 \arctan (a x) \operatorname {PolyLog}\left (3,-e^{2 i \arctan (a x)}\right )-144 i a^2 x^2 \operatorname {PolyLog}\left (4,e^{-2 i \arctan (a x)}\right )-144 i a^2 x^2 \operatorname {PolyLog}\left (4,-e^{2 i \arctan (a x)}\right )\right )}{64 x^2} \]
(c^3*((-3*I)*a^2*Pi^4*x^2 - 16*a^3*x^3 + 16*a^2*x^2*ArcTan[a*x] + 16*a^4*x ^4*ArcTan[a*x] - 96*a*x*ArcTan[a*x]^2 + (128*I)*a^2*x^2*ArcTan[a*x]^2 - 24 0*a^3*x^3*ArcTan[a*x]^2 - 16*a^5*x^5*ArcTan[a*x]^2 - 32*ArcTan[a*x]^3 + 48 *a^2*x^2*ArcTan[a*x]^3 + 96*a^4*x^4*ArcTan[a*x]^3 + 16*a^6*x^6*ArcTan[a*x] ^3 + (96*I)*a^2*x^2*ArcTan[a*x]^4 + 192*a^2*x^2*ArcTan[a*x]^3*Log[1 - E^(( -2*I)*ArcTan[a*x])] + 192*a^2*x^2*ArcTan[a*x]*Log[1 - E^((2*I)*ArcTan[a*x] )] - 448*a^2*x^2*ArcTan[a*x]*Log[1 + E^((2*I)*ArcTan[a*x])] - 192*a^2*x^2* ArcTan[a*x]^3*Log[1 + E^((2*I)*ArcTan[a*x])] + (288*I)*a^2*x^2*ArcTan[a*x] ^2*PolyLog[2, E^((-2*I)*ArcTan[a*x])] + (32*I)*a^2*x^2*(7 + 9*ArcTan[a*x]^ 2)*PolyLog[2, -E^((2*I)*ArcTan[a*x])] - (96*I)*a^2*x^2*PolyLog[2, E^((2*I) *ArcTan[a*x])] + 288*a^2*x^2*ArcTan[a*x]*PolyLog[3, E^((-2*I)*ArcTan[a*x]) ] - 288*a^2*x^2*ArcTan[a*x]*PolyLog[3, -E^((2*I)*ArcTan[a*x])] - (144*I)*a ^2*x^2*PolyLog[4, E^((-2*I)*ArcTan[a*x])] - (144*I)*a^2*x^2*PolyLog[4, -E^ ((2*I)*ArcTan[a*x])]))/(64*x^2)
Time = 1.43 (sec) , antiderivative size = 503, 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.
| \(\displaystyle \int \frac {\arctan (a x)^3 \left (a^2 c x^2+c\right )^3}{x^3} \, dx\) |
| \(\Big \downarrow \) 5483 |
| \(\displaystyle \int \left (a^6 c^3 x^3 \arctan (a x)^3+3 a^4 c^3 x \arctan (a x)^3+\frac {3 a^2 c^3 \arctan (a x)^3}{x}+\frac {c^3 \arctan (a x)^3}{x^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{4} a^6 c^3 x^4 \arctan (a x)^3-\frac {1}{4} a^5 c^3 x^3 \arctan (a x)^2+\frac {3}{2} a^4 c^3 x^2 \arctan (a x)^3+\frac {1}{4} a^4 c^3 x^2 \arctan (a x)-\frac {15}{4} a^3 c^3 x \arctan (a x)^2-\frac {1}{4} a^3 c^3 x+6 a^2 c^3 \arctan (a x)^3 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )-\frac {9}{2} i a^2 c^3 \arctan (a x)^2 \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )+\frac {9}{2} i a^2 c^3 \arctan (a x)^2 \operatorname {PolyLog}\left (2,\frac {2}{i a x+1}-1\right )-\frac {9}{2} a^2 c^3 \arctan (a x) \operatorname {PolyLog}\left (3,1-\frac {2}{i a x+1}\right )+\frac {9}{2} a^2 c^3 \arctan (a x) \operatorname {PolyLog}\left (3,\frac {2}{i a x+1}-1\right )+\frac {3}{4} a^2 c^3 \arctan (a x)^3-5 i a^2 c^3 \arctan (a x)^2+\frac {1}{4} a^2 c^3 \arctan (a x)-7 a^2 c^3 \arctan (a x) \log \left (\frac {2}{1+i a x}\right )+3 a^2 c^3 \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )-\frac {3}{2} i a^2 c^3 \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )-\frac {7}{2} i a^2 c^3 \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )+\frac {9}{4} i a^2 c^3 \operatorname {PolyLog}\left (4,1-\frac {2}{i a x+1}\right )-\frac {9}{4} i a^2 c^3 \operatorname {PolyLog}\left (4,\frac {2}{i a x+1}-1\right )-\frac {c^3 \arctan (a x)^3}{2 x^2}-\frac {3 a c^3 \arctan (a x)^2}{2 x}\) |
-1/4*(a^3*c^3*x) + (a^2*c^3*ArcTan[a*x])/4 + (a^4*c^3*x^2*ArcTan[a*x])/4 - (5*I)*a^2*c^3*ArcTan[a*x]^2 - (3*a*c^3*ArcTan[a*x]^2)/(2*x) - (15*a^3*c^3 *x*ArcTan[a*x]^2)/4 - (a^5*c^3*x^3*ArcTan[a*x]^2)/4 + (3*a^2*c^3*ArcTan[a* x]^3)/4 - (c^3*ArcTan[a*x]^3)/(2*x^2) + (3*a^4*c^3*x^2*ArcTan[a*x]^3)/2 + (a^6*c^3*x^4*ArcTan[a*x]^3)/4 + 6*a^2*c^3*ArcTan[a*x]^3*ArcTanh[1 - 2/(1 + I*a*x)] - 7*a^2*c^3*ArcTan[a*x]*Log[2/(1 + I*a*x)] + 3*a^2*c^3*ArcTan[a*x ]*Log[2 - 2/(1 - I*a*x)] - ((3*I)/2)*a^2*c^3*PolyLog[2, -1 + 2/(1 - I*a*x) ] - ((7*I)/2)*a^2*c^3*PolyLog[2, 1 - 2/(1 + I*a*x)] - ((9*I)/2)*a^2*c^3*Ar cTan[a*x]^2*PolyLog[2, 1 - 2/(1 + I*a*x)] + ((9*I)/2)*a^2*c^3*ArcTan[a*x]^ 2*PolyLog[2, -1 + 2/(1 + I*a*x)] - (9*a^2*c^3*ArcTan[a*x]*PolyLog[3, 1 - 2 /(1 + I*a*x)])/2 + (9*a^2*c^3*ArcTan[a*x]*PolyLog[3, -1 + 2/(1 + I*a*x)])/ 2 + ((9*I)/4)*a^2*c^3*PolyLog[4, 1 - 2/(1 + I*a*x)] - ((9*I)/4)*a^2*c^3*Po lyLog[4, -1 + 2/(1 + I*a*x)]
3.4.85.3.1 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 = 77.00 (sec) , antiderivative size = 763, normalized size of antiderivative = 1.52
| method | result | size |
| derivativedivides | \(a^{2} \left (\frac {c^{3} \left (2 i \arctan \left (a x \right )^{3}+6 i \arctan \left (a x \right )^{2} a x -2 \arctan \left (a x \right )^{3} a x -14 x^{2} \arctan \left (a x \right )^{2} a^{2}-5 i \arctan \left (a x \right )^{3} a^{2} x^{2}+i \arctan \left (a x \right )^{2} a^{3} x^{3}+5 \arctan \left (a x \right )^{3} a^{3} x^{3}-a^{4} \arctan \left (a x \right )^{2} x^{4}-i \arctan \left (a x \right )^{3} a^{4} x^{4}+\arctan \left (a x \right )^{3} a^{5} x^{5}-a^{2} x^{2}-i \arctan \left (a x \right ) a^{2} x^{2}+\arctan \left (a x \right ) x^{3} a^{3}\right ) \left (a x +i\right )}{4 a^{2} x^{2}}-9 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 )-7 c^{3} \arctan \left (a x \right ) \ln \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )-3 i c^{3} \operatorname {polylog}\left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+3 c^{3} \arctan \left (a x \right ) \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+\frac {7 i c^{3} \operatorname {polylog}\left (2, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{2}+3 c^{3} \arctan \left (a x \right )^{3} \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+1\right )+18 i c^{3} \operatorname {polylog}\left (4, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+18 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} \operatorname {polylog}\left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+3 c^{3} \arctan \left (a x \right )^{3} \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+18 i c^{3} \operatorname {polylog}\left (4, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-\frac {9 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}+4 i c^{3} \arctan \left (a x \right )^{2}+18 c^{3} \arctan \left (a x \right ) \operatorname {polylog}\left (3, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-9 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 )-3 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 {9 i c^{3} \operatorname {polylog}\left (4, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{4}+3 c^{3} \arctan \left (a x \right ) \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+1\right )+\frac {9 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}\right )\) | \(763\) |
| default | \(a^{2} \left (\frac {c^{3} \left (2 i \arctan \left (a x \right )^{3}+6 i \arctan \left (a x \right )^{2} a x -2 \arctan \left (a x \right )^{3} a x -14 x^{2} \arctan \left (a x \right )^{2} a^{2}-5 i \arctan \left (a x \right )^{3} a^{2} x^{2}+i \arctan \left (a x \right )^{2} a^{3} x^{3}+5 \arctan \left (a x \right )^{3} a^{3} x^{3}-a^{4} \arctan \left (a x \right )^{2} x^{4}-i \arctan \left (a x \right )^{3} a^{4} x^{4}+\arctan \left (a x \right )^{3} a^{5} x^{5}-a^{2} x^{2}-i \arctan \left (a x \right ) a^{2} x^{2}+\arctan \left (a x \right ) x^{3} a^{3}\right ) \left (a x +i\right )}{4 a^{2} x^{2}}-9 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 )-7 c^{3} \arctan \left (a x \right ) \ln \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )-3 i c^{3} \operatorname {polylog}\left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+3 c^{3} \arctan \left (a x \right ) \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+\frac {7 i c^{3} \operatorname {polylog}\left (2, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{2}+3 c^{3} \arctan \left (a x \right )^{3} \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+1\right )+18 i c^{3} \operatorname {polylog}\left (4, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+18 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} \operatorname {polylog}\left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+3 c^{3} \arctan \left (a x \right )^{3} \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+18 i c^{3} \operatorname {polylog}\left (4, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-\frac {9 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}+4 i c^{3} \arctan \left (a x \right )^{2}+18 c^{3} \arctan \left (a x \right ) \operatorname {polylog}\left (3, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-9 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 )-3 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 {9 i c^{3} \operatorname {polylog}\left (4, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{4}+3 c^{3} \arctan \left (a x \right ) \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+1\right )+\frac {9 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}\right )\) | \(763\) |
a^2*(1/4*c^3*(2*I*arctan(a*x)^3+6*I*arctan(a*x)^2*a*x-2*arctan(a*x)^3*a*x- 14*x^2*arctan(a*x)^2*a^2-5*I*arctan(a*x)^3*a^2*x^2+I*arctan(a*x)^2*a^3*x^3 +5*arctan(a*x)^3*a^3*x^3-a^4*arctan(a*x)^2*x^4-I*arctan(a*x)^3*a^4*x^4+arc tan(a*x)^3*a^5*x^5-a^2*x^2-I*arctan(a*x)*a^2*x^2+arctan(a*x)*x^3*a^3)*(I+a *x)/a^2/x^2-9*I*c^3*arctan(a*x)^2*polylog(2,(1+I*a*x)/(a^2*x^2+1)^(1/2))-7 *c^3*arctan(a*x)*ln((1+I*a*x)^2/(a^2*x^2+1)+1)-3*I*c^3*polylog(2,(1+I*a*x) /(a^2*x^2+1)^(1/2))+3*c^3*arctan(a*x)*ln(1-(1+I*a*x)/(a^2*x^2+1)^(1/2))+7/ 2*I*c^3*polylog(2,-(1+I*a*x)^2/(a^2*x^2+1))+3*c^3*arctan(a*x)^3*ln((1+I*a* x)/(a^2*x^2+1)^(1/2)+1)+18*I*c^3*polylog(4,-(1+I*a*x)/(a^2*x^2+1)^(1/2))+1 8*c^3*arctan(a*x)*polylog(3,-(1+I*a*x)/(a^2*x^2+1)^(1/2))-3*I*c^3*polylog( 2,-(1+I*a*x)/(a^2*x^2+1)^(1/2))+3*c^3*arctan(a*x)^3*ln(1-(1+I*a*x)/(a^2*x^ 2+1)^(1/2))+18*I*c^3*polylog(4,(1+I*a*x)/(a^2*x^2+1)^(1/2))-9/2*c^3*arctan (a*x)*polylog(3,-(1+I*a*x)^2/(a^2*x^2+1))+4*I*c^3*arctan(a*x)^2+18*c^3*arc tan(a*x)*polylog(3,(1+I*a*x)/(a^2*x^2+1)^(1/2))-9*I*c^3*arctan(a*x)^2*poly log(2,-(1+I*a*x)/(a^2*x^2+1)^(1/2))-3*c^3*arctan(a*x)^3*ln((1+I*a*x)^2/(a^ 2*x^2+1)+1)-9/4*I*c^3*polylog(4,-(1+I*a*x)^2/(a^2*x^2+1))+3*c^3*arctan(a*x )*ln((1+I*a*x)/(a^2*x^2+1)^(1/2)+1)+9/2*I*c^3*arctan(a*x)^2*polylog(2,-(1+ I*a*x)^2/(a^2*x^2+1)))
\[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^3}{x^3} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{3} \arctan \left (a x\right )^{3}}{x^{3}} \,d x } \]
\[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^3}{x^3} \, dx=c^{3} \left (\int \frac {\operatorname {atan}^{3}{\left (a x \right )}}{x^{3}}\, dx + \int \frac {3 a^{2} \operatorname {atan}^{3}{\left (a x \right )}}{x}\, dx + \int 3 a^{4} x \operatorname {atan}^{3}{\left (a x \right )}\, dx + \int a^{6} x^{3} \operatorname {atan}^{3}{\left (a x \right )}\, dx\right ) \]
c**3*(Integral(atan(a*x)**3/x**3, x) + Integral(3*a**2*atan(a*x)**3/x, x) + Integral(3*a**4*x*atan(a*x)**3, x) + Integral(a**6*x**3*atan(a*x)**3, x) )
\[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^3}{x^3} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{3} \arctan \left (a x\right )^{3}}{x^{3}} \,d x } \]
1/128*(4*(a^6*c^3*x^6 + 6*a^4*c^3*x^4 - 2*c^3)*arctan(a*x)^3 - 3*(a^6*c^3* x^6 + 6*a^4*c^3*x^4 - 2*c^3)*arctan(a*x)*log(a^2*x^2 + 1)^2 + 128*x^2*inte grate(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 - 12*(a^7*c^3*x^7 + 6*a^5*c^3*x^5 - 2*a*c^3*x)*ar ctan(a*x)^2 + 12*(a^8*c^3*x^8 + 6*a^6*c^3*x^6 - 2*a^2*c^3*x^2)*arctan(a*x) *log(a^2*x^2 + 1) + 3*(a^7*c^3*x^7 + 6*a^5*c^3*x^5 - 2*a*c^3*x + 4*(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^5 + x^3), x))/x^2
Timed out. \[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^3}{x^3} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^3}{x^3} \, dx=\int \frac {{\mathrm {atan}\left (a\,x\right )}^3\,{\left (c\,a^2\,x^2+c\right )}^3}{x^3} \,d x \]