Integrand size = 22, antiderivative size = 321 \[ \int x^2 \left (c+a^2 c x^2\right )^2 \arctan (a x)^3 \, dx=-\frac {11 c^2 x^2}{420 a}-\frac {1}{140} a c^2 x^4-\frac {c^2 x \arctan (a x)}{70 a^2}+\frac {17}{210} c^2 x^3 \arctan (a x)+\frac {1}{35} a^2 c^2 x^5 \arctan (a x)+\frac {c^2 \arctan (a x)^2}{140 a^3}-\frac {4 c^2 x^2 \arctan (a x)^2}{35 a}-\frac {27}{140} a c^2 x^4 \arctan (a x)^2-\frac {1}{14} a^3 c^2 x^6 \arctan (a x)^2-\frac {8 i c^2 \arctan (a x)^3}{105 a^3}+\frac {1}{3} c^2 x^3 \arctan (a x)^3+\frac {2}{5} a^2 c^2 x^5 \arctan (a x)^3+\frac {1}{7} a^4 c^2 x^7 \arctan (a x)^3-\frac {8 c^2 \arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )}{35 a^3}+\frac {c^2 \log \left (1+a^2 x^2\right )}{30 a^3}-\frac {8 i c^2 \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{35 a^3}-\frac {4 c^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )}{35 a^3} \]
-11/420*c^2*x^2/a-1/140*a*c^2*x^4-1/70*c^2*x*arctan(a*x)/a^2+17/210*c^2*x^ 3*arctan(a*x)+1/35*a^2*c^2*x^5*arctan(a*x)+1/140*c^2*arctan(a*x)^2/a^3-4/3 5*c^2*x^2*arctan(a*x)^2/a-27/140*a*c^2*x^4*arctan(a*x)^2-1/14*a^3*c^2*x^6* arctan(a*x)^2-8/35*I*c^2*arctan(a*x)*polylog(2,1-2/(1+I*a*x))/a^3+1/3*c^2* x^3*arctan(a*x)^3+2/5*a^2*c^2*x^5*arctan(a*x)^3+1/7*a^4*c^2*x^7*arctan(a*x )^3-8/35*c^2*arctan(a*x)^2*ln(2/(1+I*a*x))/a^3+1/30*c^2*ln(a^2*x^2+1)/a^3- 8/105*I*c^2*arctan(a*x)^3/a^3-4/35*c^2*polylog(3,1-2/(1+I*a*x))/a^3
Time = 0.96 (sec) , antiderivative size = 233, normalized size of antiderivative = 0.73 \[ \int x^2 \left (c+a^2 c x^2\right )^2 \arctan (a x)^3 \, dx=\frac {c^2 \left (-8-11 a^2 x^2-3 a^4 x^4-6 a x \arctan (a x)+34 a^3 x^3 \arctan (a x)+12 a^5 x^5 \arctan (a x)+3 \arctan (a x)^2-48 a^2 x^2 \arctan (a x)^2-81 a^4 x^4 \arctan (a x)^2-30 a^6 x^6 \arctan (a x)^2+32 i \arctan (a x)^3+140 a^3 x^3 \arctan (a x)^3+168 a^5 x^5 \arctan (a x)^3+60 a^7 x^7 \arctan (a x)^3-96 \arctan (a x)^2 \log \left (1+e^{2 i \arctan (a x)}\right )+14 \log \left (1+a^2 x^2\right )+96 i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{2 i \arctan (a x)}\right )-48 \operatorname {PolyLog}\left (3,-e^{2 i \arctan (a x)}\right )\right )}{420 a^3} \]
(c^2*(-8 - 11*a^2*x^2 - 3*a^4*x^4 - 6*a*x*ArcTan[a*x] + 34*a^3*x^3*ArcTan[ a*x] + 12*a^5*x^5*ArcTan[a*x] + 3*ArcTan[a*x]^2 - 48*a^2*x^2*ArcTan[a*x]^2 - 81*a^4*x^4*ArcTan[a*x]^2 - 30*a^6*x^6*ArcTan[a*x]^2 + (32*I)*ArcTan[a*x ]^3 + 140*a^3*x^3*ArcTan[a*x]^3 + 168*a^5*x^5*ArcTan[a*x]^3 + 60*a^7*x^7*A rcTan[a*x]^3 - 96*ArcTan[a*x]^2*Log[1 + E^((2*I)*ArcTan[a*x])] + 14*Log[1 + a^2*x^2] + (96*I)*ArcTan[a*x]*PolyLog[2, -E^((2*I)*ArcTan[a*x])] - 48*Po lyLog[3, -E^((2*I)*ArcTan[a*x])]))/(420*a^3)
Time = 1.78 (sec) , antiderivative size = 321, 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.
| \(\displaystyle \int x^2 \arctan (a x)^3 \left (a^2 c x^2+c\right )^2 \, dx\) |
| \(\Big \downarrow \) 5483 |
| \(\displaystyle \int \left (a^4 c^2 x^6 \arctan (a x)^3+2 a^2 c^2 x^4 \arctan (a x)^3+c^2 x^2 \arctan (a x)^3\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{7} a^4 c^2 x^7 \arctan (a x)^3-\frac {8 i c^2 \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )}{35 a^3}-\frac {1}{14} a^3 c^2 x^6 \arctan (a x)^2-\frac {8 i c^2 \arctan (a x)^3}{105 a^3}+\frac {c^2 \arctan (a x)^2}{140 a^3}-\frac {8 c^2 \arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )}{35 a^3}-\frac {4 c^2 \operatorname {PolyLog}\left (3,1-\frac {2}{i a x+1}\right )}{35 a^3}+\frac {2}{5} a^2 c^2 x^5 \arctan (a x)^3+\frac {1}{35} a^2 c^2 x^5 \arctan (a x)-\frac {c^2 x \arctan (a x)}{70 a^2}+\frac {c^2 \log \left (a^2 x^2+1\right )}{30 a^3}-\frac {27}{140} a c^2 x^4 \arctan (a x)^2+\frac {1}{3} c^2 x^3 \arctan (a x)^3+\frac {17}{210} c^2 x^3 \arctan (a x)-\frac {4 c^2 x^2 \arctan (a x)^2}{35 a}-\frac {1}{140} a c^2 x^4-\frac {11 c^2 x^2}{420 a}\) |
(-11*c^2*x^2)/(420*a) - (a*c^2*x^4)/140 - (c^2*x*ArcTan[a*x])/(70*a^2) + ( 17*c^2*x^3*ArcTan[a*x])/210 + (a^2*c^2*x^5*ArcTan[a*x])/35 + (c^2*ArcTan[a *x]^2)/(140*a^3) - (4*c^2*x^2*ArcTan[a*x]^2)/(35*a) - (27*a*c^2*x^4*ArcTan [a*x]^2)/140 - (a^3*c^2*x^6*ArcTan[a*x]^2)/14 - (((8*I)/105)*c^2*ArcTan[a* x]^3)/a^3 + (c^2*x^3*ArcTan[a*x]^3)/3 + (2*a^2*c^2*x^5*ArcTan[a*x]^3)/5 + (a^4*c^2*x^7*ArcTan[a*x]^3)/7 - (8*c^2*ArcTan[a*x]^2*Log[2/(1 + I*a*x)])/( 35*a^3) + (c^2*Log[1 + a^2*x^2])/(30*a^3) - (((8*I)/35)*c^2*ArcTan[a*x]*Po lyLog[2, 1 - 2/(1 + I*a*x)])/a^3 - (4*c^2*PolyLog[3, 1 - 2/(1 + I*a*x)])/( 35*a^3)
3.4.72.3.1 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 = 72.00 (sec) , antiderivative size = 1256, normalized size of antiderivative = 3.91
| method | result | size |
| derivativedivides | \(\text {Expression too large to display}\) | \(1256\) |
| default | \(\text {Expression too large to display}\) | \(1256\) |
| parts | \(\text {Expression too large to display}\) | \(1256\) |
1/a^3*(1/7*c^2*arctan(a*x)^3*a^7*x^7+2/5*c^2*arctan(a*x)^3*a^5*x^5+1/3*c^2 *arctan(a*x)^3*a^3*x^3-1/35*c^2*(-2*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)*arctan(a*x)^2-1/4*arctan(a*x)^2+4*x^2*arctan(a*x) ^2*a^2+8*arctan(a*x)^2*ln(2)+1/4*(I+a*x)^4+5/2*a^6*x^6*arctan(a*x)^2+27/4* a^4*arctan(a*x)^2*x^4+4*polylog(3,-(1+I*a*x)^2/(a^2*x^2+1))+7/3*ln((1+I*a* x)^2/(a^2*x^2+1)+1)-4*arctan(a*x)^2*ln(a^2*x^2+1)-5*I*arctan(a*x)*(a*x-I)^ 4-8*I*arctan(a*x)*polylog(2,-(1+I*a*x)^2/(a^2*x^2+1))+5*arctan(a*x)*(a*x-I )^4*(I+a*x)-5*arctan(a*x)*(a*x-I)*(I+a*x)^4+10*arctan(a*x)*(a*x-I)^2*(I+a* x)^3-2*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*arctan(a*x)^2+2*I*Pi*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1)^2)^3*arctan(a*x) ^2-2*I*Pi*csgn(I*(1+I*a*x)^2/(a^2*x^2+1))^3*arctan(a*x)^2-30*I*arctan(a*x) *(a*x-I)^2*(I+a*x)^2+20*I*arctan(a*x)*(a*x-I)*(I+a*x)^3-7/12*(I+a*x)^2-8/3 *I*arctan(a*x)^3-5/6*I*(I+a*x)-I*(I+a*x)^3+43/6*arctan(a*x)*(a*x-I)^3-arct an(a*x)*(a*x-I)^5+20*I*arctan(a*x)*(a*x-I)^3*(I+a*x)-3*I*arctan(a*x)*(a*x- I)*(I+a*x)+3/2*I*arctan(a*x)*(a*x-I)^2-10*arctan(a*x)*(a*x-I)^3*(I+a*x)^2+ 43/2*arctan(a*x)*(a*x-I)*(I+a*x)^2-43/2*arctan(a*x)*(a*x-I)^2*(I+a*x)+8*ar ctan(a*x)^2*ln((1+I*a*x)/(a^2*x^2+1)^(1/2))+4*arctan(a*x)*(a*x-I)-2*I*Pi*c sgn(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 n(a*x)^2+2*I*Pi*csgn(I/((1+I*a*x)^2/(a^2*x^2+1)+1)^2)*csgn(I*(1+I*a*x)^...
\[ \int x^2 \left (c+a^2 c x^2\right )^2 \arctan (a x)^3 \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{2} x^{2} \arctan \left (a x\right )^{3} \,d x } \]
\[ \int x^2 \left (c+a^2 c x^2\right )^2 \arctan (a x)^3 \, dx=c^{2} \left (\int x^{2} \operatorname {atan}^{3}{\left (a x \right )}\, dx + \int 2 a^{2} x^{4} \operatorname {atan}^{3}{\left (a x \right )}\, dx + \int a^{4} x^{6} \operatorname {atan}^{3}{\left (a x \right )}\, dx\right ) \]
c**2*(Integral(x**2*atan(a*x)**3, x) + Integral(2*a**2*x**4*atan(a*x)**3, x) + Integral(a**4*x**6*atan(a*x)**3, x))
\[ \int x^2 \left (c+a^2 c x^2\right )^2 \arctan (a x)^3 \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{2} x^{2} \arctan \left (a x\right )^{3} \,d x } \]
1/840*(15*a^4*c^2*x^7 + 42*a^2*c^2*x^5 + 35*c^2*x^3)*arctan(a*x)^3 - 1/112 0*(15*a^4*c^2*x^7 + 42*a^2*c^2*x^5 + 35*c^2*x^3)*arctan(a*x)*log(a^2*x^2 + 1)^2 + integrate(1/1120*(980*(a^6*c^2*x^8 + 3*a^4*c^2*x^6 + 3*a^2*c^2*x^4 + c^2*x^2)*arctan(a*x)^3 - 4*(15*a^5*c^2*x^7 + 42*a^3*c^2*x^5 + 35*a*c^2* x^3)*arctan(a*x)^2 + 4*(15*a^6*c^2*x^8 + 42*a^4*c^2*x^6 + 35*a^2*c^2*x^4)* arctan(a*x)*log(a^2*x^2 + 1) + (15*a^5*c^2*x^7 + 42*a^3*c^2*x^5 + 35*a*c^2 *x^3 + 105*(a^6*c^2*x^8 + 3*a^4*c^2*x^6 + 3*a^2*c^2*x^4 + c^2*x^2)*arctan( a*x))*log(a^2*x^2 + 1)^2)/(a^2*x^2 + 1), x)
\[ \int x^2 \left (c+a^2 c x^2\right )^2 \arctan (a x)^3 \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{2} x^{2} \arctan \left (a x\right )^{3} \,d x } \]
Timed out. \[ \int x^2 \left (c+a^2 c x^2\right )^2 \arctan (a x)^3 \, dx=\int x^2\,{\mathrm {atan}\left (a\,x\right )}^3\,{\left (c\,a^2\,x^2+c\right )}^2 \,d x \]