Integrand size = 22, antiderivative size = 389 \[ \int x^2 \left (c+a^2 c x^2\right )^3 \arctan (a x)^3 \, dx=-\frac {107 c^3 x^2}{7560 a}-\frac {11 a c^3 x^4}{1260}-\frac {1}{504} a^3 c^3 x^6-\frac {47 c^3 x \arctan (a x)}{1260 a^2}+\frac {239 c^3 x^3 \arctan (a x)}{3780}+\frac {59 a^2 c^3 x^5 \arctan (a x)}{1260}+\frac {1}{84} a^4 c^3 x^7 \arctan (a x)+\frac {47 c^3 \arctan (a x)^2}{2520 a^3}-\frac {8 c^3 x^2 \arctan (a x)^2}{105 a}-\frac {89}{420} a c^3 x^4 \arctan (a x)^2-\frac {10}{63} a^3 c^3 x^6 \arctan (a x)^2-\frac {1}{24} a^5 c^3 x^8 \arctan (a x)^2-\frac {16 i c^3 \arctan (a x)^3}{315 a^3}+\frac {1}{3} c^3 x^3 \arctan (a x)^3+\frac {3}{5} a^2 c^3 x^5 \arctan (a x)^3+\frac {3}{7} a^4 c^3 x^7 \arctan (a x)^3+\frac {1}{9} a^6 c^3 x^9 \arctan (a x)^3-\frac {16 c^3 \arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )}{105 a^3}+\frac {31 c^3 \log \left (1+a^2 x^2\right )}{945 a^3}-\frac {16 i c^3 \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{105 a^3}-\frac {8 c^3 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )}{105 a^3} \]
-107/7560*c^3*x^2/a-11/1260*a*c^3*x^4-1/504*a^3*c^3*x^6-47/1260*c^3*x*arct an(a*x)/a^2+239/3780*c^3*x^3*arctan(a*x)+59/1260*a^2*c^3*x^5*arctan(a*x)+1 /84*a^4*c^3*x^7*arctan(a*x)+47/2520*c^3*arctan(a*x)^2/a^3-8/105*c^3*x^2*ar ctan(a*x)^2/a-89/420*a*c^3*x^4*arctan(a*x)^2-10/63*a^3*c^3*x^6*arctan(a*x) ^2-1/24*a^5*c^3*x^8*arctan(a*x)^2-16/105*I*c^3*arctan(a*x)*polylog(2,1-2/( 1+I*a*x))/a^3+1/3*c^3*x^3*arctan(a*x)^3+3/5*a^2*c^3*x^5*arctan(a*x)^3+3/7* a^4*c^3*x^7*arctan(a*x)^3+1/9*a^6*c^3*x^9*arctan(a*x)^3-16/105*c^3*arctan( a*x)^2*ln(2/(1+I*a*x))/a^3+31/945*c^3*ln(a^2*x^2+1)/a^3-16/315*I*c^3*arcta n(a*x)^3/a^3-8/105*c^3*polylog(3,1-2/(1+I*a*x))/a^3
Time = 1.60 (sec) , antiderivative size = 281, normalized size of antiderivative = 0.72 \[ \int x^2 \left (c+a^2 c x^2\right )^3 \arctan (a x)^3 \, dx=\frac {c^3 \left (-56-107 a^2 x^2-66 a^4 x^4-15 a^6 x^6-282 a x \arctan (a x)+478 a^3 x^3 \arctan (a x)+354 a^5 x^5 \arctan (a x)+90 a^7 x^7 \arctan (a x)+141 \arctan (a x)^2-576 a^2 x^2 \arctan (a x)^2-1602 a^4 x^4 \arctan (a x)^2-1200 a^6 x^6 \arctan (a x)^2-315 a^8 x^8 \arctan (a x)^2+384 i \arctan (a x)^3+2520 a^3 x^3 \arctan (a x)^3+4536 a^5 x^5 \arctan (a x)^3+3240 a^7 x^7 \arctan (a x)^3+840 a^9 x^9 \arctan (a x)^3-1152 \arctan (a x)^2 \log \left (1+e^{2 i \arctan (a x)}\right )+248 \log \left (1+a^2 x^2\right )+1152 i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{2 i \arctan (a x)}\right )-576 \operatorname {PolyLog}\left (3,-e^{2 i \arctan (a x)}\right )\right )}{7560 a^3} \]
(c^3*(-56 - 107*a^2*x^2 - 66*a^4*x^4 - 15*a^6*x^6 - 282*a*x*ArcTan[a*x] + 478*a^3*x^3*ArcTan[a*x] + 354*a^5*x^5*ArcTan[a*x] + 90*a^7*x^7*ArcTan[a*x] + 141*ArcTan[a*x]^2 - 576*a^2*x^2*ArcTan[a*x]^2 - 1602*a^4*x^4*ArcTan[a*x ]^2 - 1200*a^6*x^6*ArcTan[a*x]^2 - 315*a^8*x^8*ArcTan[a*x]^2 + (384*I)*Arc Tan[a*x]^3 + 2520*a^3*x^3*ArcTan[a*x]^3 + 4536*a^5*x^5*ArcTan[a*x]^3 + 324 0*a^7*x^7*ArcTan[a*x]^3 + 840*a^9*x^9*ArcTan[a*x]^3 - 1152*ArcTan[a*x]^2*L og[1 + E^((2*I)*ArcTan[a*x])] + 248*Log[1 + a^2*x^2] + (1152*I)*ArcTan[a*x ]*PolyLog[2, -E^((2*I)*ArcTan[a*x])] - 576*PolyLog[3, -E^((2*I)*ArcTan[a*x ])]))/(7560*a^3)
Time = 2.84 (sec) , antiderivative size = 389, 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.
| \(\displaystyle \int x^2 \arctan (a x)^3 \left (a^2 c x^2+c\right )^3 \, dx\) |
| \(\Big \downarrow \) 5483 |
| \(\displaystyle \int \left (a^6 c^3 x^8 \arctan (a x)^3+3 a^4 c^3 x^6 \arctan (a x)^3+3 a^2 c^3 x^4 \arctan (a x)^3+c^3 x^2 \arctan (a x)^3\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{9} a^6 c^3 x^9 \arctan (a x)^3-\frac {1}{24} a^5 c^3 x^8 \arctan (a x)^2+\frac {3}{7} a^4 c^3 x^7 \arctan (a x)^3+\frac {1}{84} a^4 c^3 x^7 \arctan (a x)-\frac {16 i c^3 \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )}{105 a^3}-\frac {10}{63} a^3 c^3 x^6 \arctan (a x)^2-\frac {16 i c^3 \arctan (a x)^3}{315 a^3}+\frac {47 c^3 \arctan (a x)^2}{2520 a^3}-\frac {16 c^3 \arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )}{105 a^3}-\frac {8 c^3 \operatorname {PolyLog}\left (3,1-\frac {2}{i a x+1}\right )}{105 a^3}-\frac {1}{504} a^3 c^3 x^6+\frac {3}{5} a^2 c^3 x^5 \arctan (a x)^3+\frac {59 a^2 c^3 x^5 \arctan (a x)}{1260}-\frac {47 c^3 x \arctan (a x)}{1260 a^2}+\frac {31 c^3 \log \left (a^2 x^2+1\right )}{945 a^3}-\frac {89}{420} a c^3 x^4 \arctan (a x)^2+\frac {1}{3} c^3 x^3 \arctan (a x)^3+\frac {239 c^3 x^3 \arctan (a x)}{3780}-\frac {8 c^3 x^2 \arctan (a x)^2}{105 a}-\frac {11 a c^3 x^4}{1260}-\frac {107 c^3 x^2}{7560 a}\) |
(-107*c^3*x^2)/(7560*a) - (11*a*c^3*x^4)/1260 - (a^3*c^3*x^6)/504 - (47*c^ 3*x*ArcTan[a*x])/(1260*a^2) + (239*c^3*x^3*ArcTan[a*x])/3780 + (59*a^2*c^3 *x^5*ArcTan[a*x])/1260 + (a^4*c^3*x^7*ArcTan[a*x])/84 + (47*c^3*ArcTan[a*x ]^2)/(2520*a^3) - (8*c^3*x^2*ArcTan[a*x]^2)/(105*a) - (89*a*c^3*x^4*ArcTan [a*x]^2)/420 - (10*a^3*c^3*x^6*ArcTan[a*x]^2)/63 - (a^5*c^3*x^8*ArcTan[a*x ]^2)/24 - (((16*I)/315)*c^3*ArcTan[a*x]^3)/a^3 + (c^3*x^3*ArcTan[a*x]^3)/3 + (3*a^2*c^3*x^5*ArcTan[a*x]^3)/5 + (3*a^4*c^3*x^7*ArcTan[a*x]^3)/7 + (a^ 6*c^3*x^9*ArcTan[a*x]^3)/9 - (16*c^3*ArcTan[a*x]^2*Log[2/(1 + I*a*x)])/(10 5*a^3) + (31*c^3*Log[1 + a^2*x^2])/(945*a^3) - (((16*I)/105)*c^3*ArcTan[a* x]*PolyLog[2, 1 - 2/(1 + I*a*x)])/a^3 - (8*c^3*PolyLog[3, 1 - 2/(1 + I*a*x )])/(105*a^3)
3.4.80.3.1 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])
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 121.68 (sec) , antiderivative size = 1576, normalized size of antiderivative = 4.05
| method | result | size |
| derivativedivides | \(\text {Expression too large to display}\) | \(1576\) |
| default | \(\text {Expression too large to display}\) | \(1576\) |
| parts | \(\text {Expression too large to display}\) | \(1576\) |
1/a^3*(1/9*c^3*arctan(a*x)^3*a^9*x^9+3/7*c^3*arctan(a*x)^3*a^7*x^7+3/5*c^3 *arctan(a*x)^3*a^5*x^5+1/3*c^3*arctan(a*x)^3*a^3*x^3-1/105*c^3*(-4*I*Pi*cs gn(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)^2-47/2 4*arctan(a*x)^2+35/8*arctan(a*x)^2*a^8*x^8+8*x^2*arctan(a*x)^2*a^2+175/4*a rctan(a*x)*(a*x-I)^4*(I+a*x)^3-35/4*I*arctan(a*x)*(a*x-I)^6-105/4*arctan(a *x)*(a*x-I)^5*(I+a*x)^2+35/4*arctan(a*x)*(a*x-I)^6*(I+a*x)+115/6*I*arctan( a*x)*(a*x-I)^4+3*I*arctan(a*x)*(a*x-I)^2-16*I*arctan(a*x)*polylog(2,-(1+I* a*x)^2/(a^2*x^2+1))-175/4*arctan(a*x)*(a*x-I)^3*(I+a*x)^4+16*arctan(a*x)^2 *ln(2)+4*I*Pi*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1)^2)^3*arctan(a*x)^2-230/3* I*arctan(a*x)*(a*x-I)^3*(I+a*x)+115*I*arctan(a*x)*(a*x-I)^2*(I+a*x)^2+105/ 2*I*arctan(a*x)*(a*x-I)^5*(I+a*x)-525/4*I*arctan(a*x)*(a*x-I)^4*(I+a*x)^2- 4*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*arc tan(a*x)^2-4*I*Pi*csgn(I*(1+I*a*x)^2/(a^2*x^2+1))^3*arctan(a*x)^2-525/4*I* arctan(a*x)*(a*x-I)^2*(I+a*x)^4-6*I*arctan(a*x)*(a*x-I)*(I+a*x)+105/2*I*ar ctan(a*x)*(a*x-I)*(I+a*x)^5-230/3*I*arctan(a*x)*(a*x-I)*(I+a*x)^3+175*I*ar ctan(a*x)*(a*x-I)^3*(I+a*x)^3-53/24*(I+a*x)^4+50/3*a^6*x^6*arctan(a*x)^2+5 /24*(I+a*x)^6+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)/((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))^2*csgn(I*(1+I*a*x)^2/(a^2*x^2+1))*arcta...
\[ \int x^2 \left (c+a^2 c x^2\right )^3 \arctan (a x)^3 \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{3} x^{2} \arctan \left (a x\right )^{3} \,d x } \]
\[ \int x^2 \left (c+a^2 c x^2\right )^3 \arctan (a x)^3 \, dx=c^{3} \left (\int x^{2} \operatorname {atan}^{3}{\left (a x \right )}\, dx + \int 3 a^{2} x^{4} \operatorname {atan}^{3}{\left (a x \right )}\, dx + \int 3 a^{4} x^{6} \operatorname {atan}^{3}{\left (a x \right )}\, dx + \int a^{6} x^{8} \operatorname {atan}^{3}{\left (a x \right )}\, dx\right ) \]
c**3*(Integral(x**2*atan(a*x)**3, x) + Integral(3*a**2*x**4*atan(a*x)**3, x) + Integral(3*a**4*x**6*atan(a*x)**3, x) + Integral(a**6*x**8*atan(a*x)* *3, x))
\[ \int x^2 \left (c+a^2 c x^2\right )^3 \arctan (a x)^3 \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{3} x^{2} \arctan \left (a x\right )^{3} \,d x } \]
1/2520*(35*a^6*c^3*x^9 + 135*a^4*c^3*x^7 + 189*a^2*c^3*x^5 + 105*c^3*x^3)* arctan(a*x)^3 - 1/3360*(35*a^6*c^3*x^9 + 135*a^4*c^3*x^7 + 189*a^2*c^3*x^5 + 105*c^3*x^3)*arctan(a*x)*log(a^2*x^2 + 1)^2 + integrate(1/3360*(2940*(a ^8*c^3*x^10 + 4*a^6*c^3*x^8 + 6*a^4*c^3*x^6 + 4*a^2*c^3*x^4 + c^3*x^2)*arc tan(a*x)^3 - 4*(35*a^7*c^3*x^9 + 135*a^5*c^3*x^7 + 189*a^3*c^3*x^5 + 105*a *c^3*x^3)*arctan(a*x)^2 + 4*(35*a^8*c^3*x^10 + 135*a^6*c^3*x^8 + 189*a^4*c ^3*x^6 + 105*a^2*c^3*x^4)*arctan(a*x)*log(a^2*x^2 + 1) + (35*a^7*c^3*x^9 + 135*a^5*c^3*x^7 + 189*a^3*c^3*x^5 + 105*a*c^3*x^3 + 315*(a^8*c^3*x^10 + 4 *a^6*c^3*x^8 + 6*a^4*c^3*x^6 + 4*a^2*c^3*x^4 + c^3*x^2)*arctan(a*x))*log(a ^2*x^2 + 1)^2)/(a^2*x^2 + 1), x)
\[ \int x^2 \left (c+a^2 c x^2\right )^3 \arctan (a x)^3 \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{3} x^{2} \arctan \left (a x\right )^{3} \,d x } \]
Timed out. \[ \int x^2 \left (c+a^2 c x^2\right )^3 \arctan (a x)^3 \, dx=\int x^2\,{\mathrm {atan}\left (a\,x\right )}^3\,{\left (c\,a^2\,x^2+c\right )}^3 \,d x \]