Integrand size = 22, antiderivative size = 381 \[ \int x^3 \left (c+a^2 c x^2\right )^3 \arctan (a x)^3 \, dx=\frac {389 c^3 x}{12600 a^3}-\frac {17 c^3 x^3}{9450 a}-\frac {1}{252} a c^3 x^5-\frac {1}{840} a^3 c^3 x^7-\frac {389 c^3 \arctan (a x)}{12600 a^4}-\frac {107 c^3 x^2 \arctan (a x)}{4200 a^2}+\frac {53 c^3 x^4 \arctan (a x)}{2100}+\frac {71 a^2 c^3 x^6 \arctan (a x)}{2520}+\frac {1}{120} a^4 c^3 x^8 \arctan (a x)+\frac {26 i c^3 \arctan (a x)^2}{525 a^4}+\frac {3 c^3 x \arctan (a x)^2}{40 a^3}-\frac {c^3 x^3 \arctan (a x)^2}{40 a}-\frac {27}{200} a c^3 x^5 \arctan (a x)^2-\frac {33}{280} a^3 c^3 x^7 \arctan (a x)^2-\frac {1}{30} a^5 c^3 x^9 \arctan (a x)^2-\frac {c^3 \arctan (a x)^3}{40 a^4}+\frac {1}{4} c^3 x^4 \arctan (a x)^3+\frac {1}{2} a^2 c^3 x^6 \arctan (a x)^3+\frac {3}{8} a^4 c^3 x^8 \arctan (a x)^3+\frac {1}{10} a^6 c^3 x^{10} \arctan (a x)^3+\frac {52 c^3 \arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{525 a^4}+\frac {26 i c^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{525 a^4} \]
389/12600*c^3*x/a^3-17/9450*c^3*x^3/a-1/252*a*c^3*x^5-1/840*a^3*c^3*x^7-38 9/12600*c^3*arctan(a*x)/a^4-107/4200*c^3*x^2*arctan(a*x)/a^2+53/2100*c^3*x ^4*arctan(a*x)+71/2520*a^2*c^3*x^6*arctan(a*x)+1/120*a^4*c^3*x^8*arctan(a* x)+26/525*I*c^3*polylog(2,1-2/(1+I*a*x))/a^4+3/40*c^3*x*arctan(a*x)^2/a^3- 1/40*c^3*x^3*arctan(a*x)^2/a-27/200*a*c^3*x^5*arctan(a*x)^2-33/280*a^3*c^3 *x^7*arctan(a*x)^2-1/30*a^5*c^3*x^9*arctan(a*x)^2-1/40*c^3*arctan(a*x)^3/a ^4+1/4*c^3*x^4*arctan(a*x)^3+1/2*a^2*c^3*x^6*arctan(a*x)^3+3/8*a^4*c^3*x^8 *arctan(a*x)^3+1/10*a^6*c^3*x^10*arctan(a*x)^3+52/525*c^3*arctan(a*x)*ln(2 /(1+I*a*x))/a^4+26/525*I*c^3*arctan(a*x)^2/a^4
Time = 1.70 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.50 \[ \int x^3 \left (c+a^2 c x^2\right )^3 \arctan (a x)^3 \, dx=\frac {c^3 \left (-a x \left (-1167+68 a^2 x^2+150 a^4 x^4+45 a^6 x^6\right )-9 \left (208 i-315 a x+105 a^3 x^3+567 a^5 x^5+495 a^7 x^7+140 a^9 x^9\right ) \arctan (a x)^2+945 \left (1+a^2 x^2\right )^4 \left (-1+4 a^2 x^2\right ) \arctan (a x)^3+3 \arctan (a x) \left (-389-321 a^2 x^2+318 a^4 x^4+355 a^6 x^6+105 a^8 x^8+1248 \log \left (1+e^{2 i \arctan (a x)}\right )\right )-1872 i \operatorname {PolyLog}\left (2,-e^{2 i \arctan (a x)}\right )\right )}{37800 a^4} \]
(c^3*(-(a*x*(-1167 + 68*a^2*x^2 + 150*a^4*x^4 + 45*a^6*x^6)) - 9*(208*I - 315*a*x + 105*a^3*x^3 + 567*a^5*x^5 + 495*a^7*x^7 + 140*a^9*x^9)*ArcTan[a* x]^2 + 945*(1 + a^2*x^2)^4*(-1 + 4*a^2*x^2)*ArcTan[a*x]^3 + 3*ArcTan[a*x]* (-389 - 321*a^2*x^2 + 318*a^4*x^4 + 355*a^6*x^6 + 105*a^8*x^8 + 1248*Log[1 + E^((2*I)*ArcTan[a*x])]) - (1872*I)*PolyLog[2, -E^((2*I)*ArcTan[a*x])])) /(37800*a^4)
Time = 3.48 (sec) , antiderivative size = 381, 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.
| \(\displaystyle \int x^3 \arctan (a x)^3 \left (a^2 c x^2+c\right )^3 \, dx\) |
| \(\Big \downarrow \) 5483 |
| \(\displaystyle \int \left (a^6 c^3 x^9 \arctan (a x)^3+3 a^4 c^3 x^7 \arctan (a x)^3+3 a^2 c^3 x^5 \arctan (a x)^3+c^3 x^3 \arctan (a x)^3\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{10} a^6 c^3 x^{10} \arctan (a x)^3-\frac {1}{30} a^5 c^3 x^9 \arctan (a x)^2+\frac {3}{8} a^4 c^3 x^8 \arctan (a x)^3+\frac {1}{120} a^4 c^3 x^8 \arctan (a x)-\frac {c^3 \arctan (a x)^3}{40 a^4}+\frac {26 i c^3 \arctan (a x)^2}{525 a^4}-\frac {389 c^3 \arctan (a x)}{12600 a^4}+\frac {52 c^3 \arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{525 a^4}+\frac {26 i c^3 \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )}{525 a^4}-\frac {33}{280} a^3 c^3 x^7 \arctan (a x)^2+\frac {3 c^3 x \arctan (a x)^2}{40 a^3}-\frac {1}{840} a^3 c^3 x^7+\frac {389 c^3 x}{12600 a^3}+\frac {1}{2} a^2 c^3 x^6 \arctan (a x)^3+\frac {71 a^2 c^3 x^6 \arctan (a x)}{2520}-\frac {107 c^3 x^2 \arctan (a x)}{4200 a^2}-\frac {27}{200} a c^3 x^5 \arctan (a x)^2+\frac {1}{4} c^3 x^4 \arctan (a x)^3+\frac {53 c^3 x^4 \arctan (a x)}{2100}-\frac {c^3 x^3 \arctan (a x)^2}{40 a}-\frac {1}{252} a c^3 x^5-\frac {17 c^3 x^3}{9450 a}\) |
(389*c^3*x)/(12600*a^3) - (17*c^3*x^3)/(9450*a) - (a*c^3*x^5)/252 - (a^3*c ^3*x^7)/840 - (389*c^3*ArcTan[a*x])/(12600*a^4) - (107*c^3*x^2*ArcTan[a*x] )/(4200*a^2) + (53*c^3*x^4*ArcTan[a*x])/2100 + (71*a^2*c^3*x^6*ArcTan[a*x] )/2520 + (a^4*c^3*x^8*ArcTan[a*x])/120 + (((26*I)/525)*c^3*ArcTan[a*x]^2)/ a^4 + (3*c^3*x*ArcTan[a*x]^2)/(40*a^3) - (c^3*x^3*ArcTan[a*x]^2)/(40*a) - (27*a*c^3*x^5*ArcTan[a*x]^2)/200 - (33*a^3*c^3*x^7*ArcTan[a*x]^2)/280 - (a ^5*c^3*x^9*ArcTan[a*x]^2)/30 - (c^3*ArcTan[a*x]^3)/(40*a^4) + (c^3*x^4*Arc Tan[a*x]^3)/4 + (a^2*c^3*x^6*ArcTan[a*x]^3)/2 + (3*a^4*c^3*x^8*ArcTan[a*x] ^3)/8 + (a^6*c^3*x^10*ArcTan[a*x]^3)/10 + (52*c^3*ArcTan[a*x]*Log[2/(1 + I *a*x)])/(525*a^4) + (((26*I)/525)*c^3*PolyLog[2, 1 - 2/(1 + I*a*x)])/a^4
3.4.79.3.1 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 = 4.95 (sec) , antiderivative size = 383, normalized size of antiderivative = 1.01
| method | result | size |
| derivativedivides | \(\frac {\frac {c^{3} \arctan \left (a x \right )^{3} a^{10} x^{10}}{10}+\frac {3 c^{3} \arctan \left (a x \right )^{3} a^{8} x^{8}}{8}+\frac {a^{6} c^{3} x^{6} \arctan \left (a x \right )^{3}}{2}+\frac {a^{4} c^{3} x^{4} \arctan \left (a x \right )^{3}}{4}-\frac {c^{3} \arctan \left (a x \right )^{3}}{40}-\frac {3 c^{3} \left (\frac {4 \arctan \left (a x \right )^{2} a^{9} x^{9}}{9}+\frac {11 \arctan \left (a x \right )^{2} a^{7} x^{7}}{7}+\frac {9 a^{5} \arctan \left (a x \right )^{2} x^{5}}{5}+\frac {a^{3} \arctan \left (a x \right )^{2} x^{3}}{3}-a \arctan \left (a x \right )^{2} x -\frac {\arctan \left (a x \right ) a^{8} x^{8}}{9}-\frac {71 a^{6} \arctan \left (a x \right ) x^{6}}{189}-\frac {106 \arctan \left (a x \right ) a^{4} x^{4}}{315}+\frac {107 a^{2} \arctan \left (a x \right ) x^{2}}{315}+\frac {208 \arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )}{315}+\frac {a^{7} x^{7}}{63}+\frac {10 a^{5} x^{5}}{189}+\frac {68 a^{3} x^{3}}{2835}-\frac {389 a x}{945}+\frac {389 \arctan \left (a x \right )}{945}+\frac {104 i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )-\frac {\ln \left (a x -i\right )^{2}}{2}\right )}{315}-\frac {104 i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )-\frac {\ln \left (a x +i\right )^{2}}{2}\right )}{315}\right )}{40}}{a^{4}}\) | \(383\) |
| default | \(\frac {\frac {c^{3} \arctan \left (a x \right )^{3} a^{10} x^{10}}{10}+\frac {3 c^{3} \arctan \left (a x \right )^{3} a^{8} x^{8}}{8}+\frac {a^{6} c^{3} x^{6} \arctan \left (a x \right )^{3}}{2}+\frac {a^{4} c^{3} x^{4} \arctan \left (a x \right )^{3}}{4}-\frac {c^{3} \arctan \left (a x \right )^{3}}{40}-\frac {3 c^{3} \left (\frac {4 \arctan \left (a x \right )^{2} a^{9} x^{9}}{9}+\frac {11 \arctan \left (a x \right )^{2} a^{7} x^{7}}{7}+\frac {9 a^{5} \arctan \left (a x \right )^{2} x^{5}}{5}+\frac {a^{3} \arctan \left (a x \right )^{2} x^{3}}{3}-a \arctan \left (a x \right )^{2} x -\frac {\arctan \left (a x \right ) a^{8} x^{8}}{9}-\frac {71 a^{6} \arctan \left (a x \right ) x^{6}}{189}-\frac {106 \arctan \left (a x \right ) a^{4} x^{4}}{315}+\frac {107 a^{2} \arctan \left (a x \right ) x^{2}}{315}+\frac {208 \arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )}{315}+\frac {a^{7} x^{7}}{63}+\frac {10 a^{5} x^{5}}{189}+\frac {68 a^{3} x^{3}}{2835}-\frac {389 a x}{945}+\frac {389 \arctan \left (a x \right )}{945}+\frac {104 i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )-\frac {\ln \left (a x -i\right )^{2}}{2}\right )}{315}-\frac {104 i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )-\frac {\ln \left (a x +i\right )^{2}}{2}\right )}{315}\right )}{40}}{a^{4}}\) | \(383\) |
| parts | \(\frac {a^{6} c^{3} x^{10} \arctan \left (a x \right )^{3}}{10}+\frac {3 a^{4} c^{3} x^{8} \arctan \left (a x \right )^{3}}{8}+\frac {a^{2} c^{3} x^{6} \arctan \left (a x \right )^{3}}{2}+\frac {c^{3} x^{4} \arctan \left (a x \right )^{3}}{4}-\frac {3 c^{3} \left (\frac {4 a^{5} \arctan \left (a x \right )^{2} x^{9}}{9}+\frac {11 a^{3} \arctan \left (a x \right )^{2} x^{7}}{7}+\frac {9 a \arctan \left (a x \right )^{2} x^{5}}{5}+\frac {\arctan \left (a x \right )^{2} x^{3}}{3 a}-\frac {\arctan \left (a x \right )^{2} x}{a^{3}}+\frac {\arctan \left (a x \right )^{3}}{a^{4}}-\frac {2 \left (\frac {35 \arctan \left (a x \right ) a^{8} x^{8}}{2}+\frac {355 a^{6} \arctan \left (a x \right ) x^{6}}{6}+53 \arctan \left (a x \right ) a^{4} x^{4}-\frac {107 a^{2} \arctan \left (a x \right ) x^{2}}{2}-104 \arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )-\frac {5 a^{7} x^{7}}{2}-\frac {25 a^{5} x^{5}}{3}-\frac {34 a^{3} x^{3}}{9}+\frac {389 a x}{6}-\frac {389 \arctan \left (a x \right )}{6}-52 i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )-\frac {\ln \left (a x -i\right )^{2}}{2}\right )+52 i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )-\frac {\ln \left (a x +i\right )^{2}}{2}\right )+105 \arctan \left (a x \right )^{3}\right )}{315 a^{4}}\right )}{40}\) | \(389\) |
1/a^4*(1/10*c^3*arctan(a*x)^3*a^10*x^10+3/8*c^3*arctan(a*x)^3*a^8*x^8+1/2* a^6*c^3*x^6*arctan(a*x)^3+1/4*a^4*c^3*x^4*arctan(a*x)^3-1/40*c^3*arctan(a* x)^3-3/40*c^3*(4/9*arctan(a*x)^2*a^9*x^9+11/7*arctan(a*x)^2*a^7*x^7+9/5*a^ 5*arctan(a*x)^2*x^5+1/3*a^3*arctan(a*x)^2*x^3-a*arctan(a*x)^2*x-1/9*arctan (a*x)*a^8*x^8-71/189*a^6*arctan(a*x)*x^6-106/315*arctan(a*x)*a^4*x^4+107/3 15*a^2*arctan(a*x)*x^2+208/315*arctan(a*x)*ln(a^2*x^2+1)+1/63*a^7*x^7+10/1 89*a^5*x^5+68/2835*a^3*x^3-389/945*a*x+389/945*arctan(a*x)+104/315*I*(ln(a *x-I)*ln(a^2*x^2+1)-dilog(-1/2*I*(I+a*x))-ln(a*x-I)*ln(-1/2*I*(I+a*x))-1/2 *ln(a*x-I)^2)-104/315*I*(ln(I+a*x)*ln(a^2*x^2+1)-dilog(1/2*I*(a*x-I))-ln(I +a*x)*ln(1/2*I*(a*x-I))-1/2*ln(I+a*x)^2)))
\[ \int x^3 \left (c+a^2 c x^2\right )^3 \arctan (a x)^3 \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{3} x^{3} \arctan \left (a x\right )^{3} \,d x } \]
\[ \int x^3 \left (c+a^2 c x^2\right )^3 \arctan (a x)^3 \, dx=c^{3} \left (\int x^{3} \operatorname {atan}^{3}{\left (a x \right )}\, dx + \int 3 a^{2} x^{5} \operatorname {atan}^{3}{\left (a x \right )}\, dx + \int 3 a^{4} x^{7} \operatorname {atan}^{3}{\left (a x \right )}\, dx + \int a^{6} x^{9} \operatorname {atan}^{3}{\left (a x \right )}\, dx\right ) \]
c**3*(Integral(x**3*atan(a*x)**3, x) + Integral(3*a**2*x**5*atan(a*x)**3, x) + Integral(3*a**4*x**7*atan(a*x)**3, x) + Integral(a**6*x**9*atan(a*x)* *3, x))
\[ \int x^3 \left (c+a^2 c x^2\right )^3 \arctan (a x)^3 \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{3} x^{3} \arctan \left (a x\right )^{3} \,d x } \]
1/67200*(420*(5376000*a^11*c^3*integrate(1/67200*x^11*arctan(a*x)^3/(a^5*x ^2 + a^3), x) - 806400*a^10*c^3*integrate(1/67200*x^10*arctan(a*x)^2/(a^5* x^2 + a^3), x) - 201600*a^10*c^3*integrate(1/67200*x^10*log(a^2*x^2 + 1)^2 /(a^5*x^2 + a^3), x) - 89600*a^10*c^3*integrate(1/67200*x^10*log(a^2*x^2 + 1)/(a^5*x^2 + a^3), x) + 21504000*a^9*c^3*integrate(1/67200*x^9*arctan(a* x)^3/(a^5*x^2 + a^3), x) + 179200*a^9*c^3*integrate(1/67200*x^9*arctan(a*x )/(a^5*x^2 + a^3), x) - 3024000*a^8*c^3*integrate(1/67200*x^8*arctan(a*x)^ 2/(a^5*x^2 + a^3), x) - 756000*a^8*c^3*integrate(1/67200*x^8*log(a^2*x^2 + 1)^2/(a^5*x^2 + a^3), x) - 316800*a^8*c^3*integrate(1/67200*x^8*log(a^2*x ^2 + 1)/(a^5*x^2 + a^3), x) + 32256000*a^7*c^3*integrate(1/67200*x^7*arcta n(a*x)^3/(a^5*x^2 + a^3), x) + 633600*a^7*c^3*integrate(1/67200*x^7*arctan (a*x)/(a^5*x^2 + a^3), x) - 4032000*a^6*c^3*integrate(1/67200*x^6*arctan(a *x)^2/(a^5*x^2 + a^3), x) - 1008000*a^6*c^3*integrate(1/67200*x^6*log(a^2* x^2 + 1)^2/(a^5*x^2 + a^3), x) - 362880*a^6*c^3*integrate(1/67200*x^6*log( a^2*x^2 + 1)/(a^5*x^2 + a^3), x) + 21504000*a^5*c^3*integrate(1/67200*x^5* arctan(a*x)^3/(a^5*x^2 + a^3), x) + 725760*a^5*c^3*integrate(1/67200*x^5*a rctan(a*x)/(a^5*x^2 + a^3), x) - 2016000*a^4*c^3*integrate(1/67200*x^4*arc tan(a*x)^2/(a^5*x^2 + a^3), x) - 504000*a^4*c^3*integrate(1/67200*x^4*log( a^2*x^2 + 1)^2/(a^5*x^2 + a^3), x) - 67200*a^4*c^3*integrate(1/67200*x^4*l og(a^2*x^2 + 1)/(a^5*x^2 + a^3), x) + 5376000*a^3*c^3*integrate(1/67200...
\[ \int x^3 \left (c+a^2 c x^2\right )^3 \arctan (a x)^3 \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{3} x^{3} \arctan \left (a x\right )^{3} \,d x } \]
Timed out. \[ \int x^3 \left (c+a^2 c x^2\right )^3 \arctan (a x)^3 \, dx=\int x^3\,{\mathrm {atan}\left (a\,x\right )}^3\,{\left (c\,a^2\,x^2+c\right )}^3 \,d x \]