Integrand size = 20, antiderivative size = 442 \[ \int \left (d+e x^2\right )^2 (a+b \arctan (c x))^2 \, dx=\frac {2 b^2 d e x}{3 c^2}-\frac {3 b^2 e^2 x}{10 c^4}+\frac {b^2 e^2 x^3}{30 c^2}-\frac {2 b^2 d e \arctan (c x)}{3 c^3}+\frac {3 b^2 e^2 \arctan (c x)}{10 c^5}-\frac {2 b d e x^2 (a+b \arctan (c x))}{3 c}+\frac {b e^2 x^2 (a+b \arctan (c x))}{5 c^3}-\frac {b e^2 x^4 (a+b \arctan (c x))}{10 c}+\frac {i d^2 (a+b \arctan (c x))^2}{c}-\frac {2 i d e (a+b \arctan (c x))^2}{3 c^3}+\frac {i e^2 (a+b \arctan (c x))^2}{5 c^5}+d^2 x (a+b \arctan (c x))^2+\frac {2}{3} d e x^3 (a+b \arctan (c x))^2+\frac {1}{5} e^2 x^5 (a+b \arctan (c x))^2+\frac {2 b d^2 (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{c}-\frac {4 b d e (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{3 c^3}+\frac {2 b e^2 (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{5 c^5}+\frac {i b^2 d^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{c}-\frac {2 i b^2 d e \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{3 c^3}+\frac {i b^2 e^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{5 c^5} \]
2/3*b^2*d*e*x/c^2-3/10*b^2*e^2*x/c^4+1/30*b^2*e^2*x^3/c^2-2/3*b^2*d*e*arct an(c*x)/c^3+3/10*b^2*e^2*arctan(c*x)/c^5-2/3*b*d*e*x^2*(a+b*arctan(c*x))/c +1/5*b*e^2*x^2*(a+b*arctan(c*x))/c^3-1/10*b*e^2*x^4*(a+b*arctan(c*x))/c+I* b^2*d^2*polylog(2,1-2/(1+I*c*x))/c-2/3*I*b^2*d*e*polylog(2,1-2/(1+I*c*x))/ c^3-2/3*I*d*e*(a+b*arctan(c*x))^2/c^3+d^2*x*(a+b*arctan(c*x))^2+2/3*d*e*x^ 3*(a+b*arctan(c*x))^2+1/5*e^2*x^5*(a+b*arctan(c*x))^2+2*b*d^2*(a+b*arctan( c*x))*ln(2/(1+I*c*x))/c-4/3*b*d*e*(a+b*arctan(c*x))*ln(2/(1+I*c*x))/c^3+2/ 5*b*e^2*(a+b*arctan(c*x))*ln(2/(1+I*c*x))/c^5+I*d^2*(a+b*arctan(c*x))^2/c+ 1/5*I*e^2*(a+b*arctan(c*x))^2/c^5+1/5*I*b^2*e^2*polylog(2,1-2/(1+I*c*x))/c ^5
Time = 0.69 (sec) , antiderivative size = 391, normalized size of antiderivative = 0.88 \[ \int \left (d+e x^2\right )^2 (a+b \arctan (c x))^2 \, dx=\frac {9 a b e^2+30 a^2 c^5 d^2 x+20 b^2 c^3 d e x-9 b^2 c e^2 x-20 a b c^4 d e x^2+6 a b c^2 e^2 x^2+20 a^2 c^5 d e x^3+b^2 c^3 e^2 x^3-3 a b c^4 e^2 x^4+6 a^2 c^5 e^2 x^5+2 b^2 \left (-15 i c^4 d^2+10 i c^2 d e-3 i e^2+c^5 \left (15 d^2 x+10 d e x^3+3 e^2 x^5\right )\right ) \arctan (c x)^2+b \arctan (c x) \left (4 a c^5 x \left (15 d^2+10 d e x^2+3 e^2 x^4\right )-b e \left (1+c^2 x^2\right ) \left (-9 e+c^2 \left (20 d+3 e x^2\right )\right )+4 b \left (15 c^4 d^2-10 c^2 d e+3 e^2\right ) \log \left (1+e^{2 i \arctan (c x)}\right )\right )-30 a b c^4 d^2 \log \left (1+c^2 x^2\right )+20 a b c^2 d e \log \left (1+c^2 x^2\right )-6 a b e^2 \log \left (1+c^2 x^2\right )-2 i b^2 \left (15 c^4 d^2-10 c^2 d e+3 e^2\right ) \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )}{30 c^5} \]
(9*a*b*e^2 + 30*a^2*c^5*d^2*x + 20*b^2*c^3*d*e*x - 9*b^2*c*e^2*x - 20*a*b* c^4*d*e*x^2 + 6*a*b*c^2*e^2*x^2 + 20*a^2*c^5*d*e*x^3 + b^2*c^3*e^2*x^3 - 3 *a*b*c^4*e^2*x^4 + 6*a^2*c^5*e^2*x^5 + 2*b^2*((-15*I)*c^4*d^2 + (10*I)*c^2 *d*e - (3*I)*e^2 + c^5*(15*d^2*x + 10*d*e*x^3 + 3*e^2*x^5))*ArcTan[c*x]^2 + b*ArcTan[c*x]*(4*a*c^5*x*(15*d^2 + 10*d*e*x^2 + 3*e^2*x^4) - b*e*(1 + c^ 2*x^2)*(-9*e + c^2*(20*d + 3*e*x^2)) + 4*b*(15*c^4*d^2 - 10*c^2*d*e + 3*e^ 2)*Log[1 + E^((2*I)*ArcTan[c*x])]) - 30*a*b*c^4*d^2*Log[1 + c^2*x^2] + 20* a*b*c^2*d*e*Log[1 + c^2*x^2] - 6*a*b*e^2*Log[1 + c^2*x^2] - (2*I)*b^2*(15* c^4*d^2 - 10*c^2*d*e + 3*e^2)*PolyLog[2, -E^((2*I)*ArcTan[c*x])])/(30*c^5)
Time = 0.92 (sec) , antiderivative size = 442, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {5449, 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.
| \(\displaystyle \int \left (d+e x^2\right )^2 (a+b \arctan (c x))^2 \, dx\) |
| \(\Big \downarrow \) 5449 |
| \(\displaystyle \int \left (d^2 (a+b \arctan (c x))^2+2 d e x^2 (a+b \arctan (c x))^2+e^2 x^4 (a+b \arctan (c x))^2\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {i e^2 (a+b \arctan (c x))^2}{5 c^5}+\frac {2 b e^2 \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{5 c^5}-\frac {2 i d e (a+b \arctan (c x))^2}{3 c^3}-\frac {4 b d e \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{3 c^3}+\frac {b e^2 x^2 (a+b \arctan (c x))}{5 c^3}+d^2 x (a+b \arctan (c x))^2+\frac {i d^2 (a+b \arctan (c x))^2}{c}+\frac {2 b d^2 \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{c}+\frac {2}{3} d e x^3 (a+b \arctan (c x))^2-\frac {2 b d e x^2 (a+b \arctan (c x))}{3 c}+\frac {1}{5} e^2 x^5 (a+b \arctan (c x))^2-\frac {b e^2 x^4 (a+b \arctan (c x))}{10 c}+\frac {3 b^2 e^2 \arctan (c x)}{10 c^5}-\frac {2 b^2 d e \arctan (c x)}{3 c^3}+\frac {i b^2 e^2 \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right )}{5 c^5}-\frac {3 b^2 e^2 x}{10 c^4}-\frac {2 i b^2 d e \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right )}{3 c^3}+\frac {2 b^2 d e x}{3 c^2}+\frac {b^2 e^2 x^3}{30 c^2}+\frac {i b^2 d^2 \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right )}{c}\) |
(2*b^2*d*e*x)/(3*c^2) - (3*b^2*e^2*x)/(10*c^4) + (b^2*e^2*x^3)/(30*c^2) - (2*b^2*d*e*ArcTan[c*x])/(3*c^3) + (3*b^2*e^2*ArcTan[c*x])/(10*c^5) - (2*b* d*e*x^2*(a + b*ArcTan[c*x]))/(3*c) + (b*e^2*x^2*(a + b*ArcTan[c*x]))/(5*c^ 3) - (b*e^2*x^4*(a + b*ArcTan[c*x]))/(10*c) + (I*d^2*(a + b*ArcTan[c*x])^2 )/c - (((2*I)/3)*d*e*(a + b*ArcTan[c*x])^2)/c^3 + ((I/5)*e^2*(a + b*ArcTan [c*x])^2)/c^5 + d^2*x*(a + b*ArcTan[c*x])^2 + (2*d*e*x^3*(a + b*ArcTan[c*x ])^2)/3 + (e^2*x^5*(a + b*ArcTan[c*x])^2)/5 + (2*b*d^2*(a + b*ArcTan[c*x]) *Log[2/(1 + I*c*x)])/c - (4*b*d*e*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/ (3*c^3) + (2*b*e^2*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(5*c^5) + (I*b^ 2*d^2*PolyLog[2, 1 - 2/(1 + I*c*x)])/c - (((2*I)/3)*b^2*d*e*PolyLog[2, 1 - 2/(1 + I*c*x)])/c^3 + ((I/5)*b^2*e^2*PolyLog[2, 1 - 2/(1 + I*c*x)])/c^5
3.13.57.3.1 Defintions of rubi rules used
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((d_) + (e_.)*(x_)^2)^(q_.), x _Symbol] :> Int[ExpandIntegrand[(a + b*ArcTan[c*x])^p, (d + e*x^2)^q, x], x ] /; FreeQ[{a, b, c, d, e}, x] && IntegerQ[q] && IGtQ[p, 0]
Time = 0.62 (sec) , antiderivative size = 493, normalized size of antiderivative = 1.12
| method | result | size |
| parts | \(a^{2} \left (\frac {1}{5} x^{5} e^{2}+\frac {2}{3} x^{3} e d +x \,d^{2}\right )+\frac {b^{2} \left (\frac {\arctan \left (c x \right )^{2} c \,e^{2} x^{5}}{5}+\frac {2 \arctan \left (c x \right )^{2} c d e \,x^{3}}{3}+\arctan \left (c x \right )^{2} c x \,d^{2}-\frac {2 \left (5 \arctan \left (c x \right ) d \,c^{4} e \,x^{2}+\frac {3 \arctan \left (c x \right ) e^{2} c^{4} x^{4}}{4}-\frac {3 \arctan \left (c x \right ) e^{2} c^{2} x^{2}}{2}+\frac {15 \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right ) c^{4} d^{2}}{2}-5 \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right ) c^{2} d e +\frac {3 \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right ) e^{2}}{2}-\frac {e \left (20 c^{3} x d +e \,c^{3} x^{3}-9 e c x +\left (-20 c^{2} d +9 e \right ) \arctan \left (c x \right )\right )}{4}-\frac {\left (30 c^{4} d^{2}-20 c^{2} d e +6 e^{2}\right ) \left (-\frac {i \left (\ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )-\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right )}{2}+\frac {i \left (\ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )-\ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )\right )}{2}\right )}{4}\right )}{15 c^{4}}\right )}{c}+\frac {2 a b \left (\frac {\arctan \left (c x \right ) c \,e^{2} x^{5}}{5}+\frac {2 \arctan \left (c x \right ) c d e \,x^{3}}{3}+\arctan \left (c x \right ) c x \,d^{2}-\frac {5 d \,c^{4} e \,x^{2}+\frac {3 e^{2} c^{4} x^{4}}{4}-\frac {3 e^{2} c^{2} x^{2}}{2}+\frac {\left (15 c^{4} d^{2}-10 c^{2} d e +3 e^{2}\right ) \ln \left (c^{2} x^{2}+1\right )}{2}}{15 c^{4}}\right )}{c}\) | \(493\) |
| derivativedivides | \(\frac {\frac {a^{2} \left (c^{5} x \,d^{2}+\frac {2}{3} d \,c^{5} e \,x^{3}+\frac {1}{5} e^{2} c^{5} x^{5}\right )}{c^{4}}+\frac {b^{2} \left (\arctan \left (c x \right )^{2} c^{5} x \,d^{2}+\frac {2 \arctan \left (c x \right )^{2} d \,c^{5} e \,x^{3}}{3}+\frac {\arctan \left (c x \right )^{2} e^{2} c^{5} x^{5}}{5}-\frac {2 \arctan \left (c x \right ) d \,c^{4} e \,x^{2}}{3}-\frac {\arctan \left (c x \right ) e^{2} c^{4} x^{4}}{10}+\frac {\arctan \left (c x \right ) e^{2} c^{2} x^{2}}{5}-\arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right ) c^{4} d^{2}+\frac {2 \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right ) c^{2} d e}{3}-\frac {\arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right ) e^{2}}{5}+\frac {e \left (20 c^{3} x d +e \,c^{3} x^{3}-9 e c x +\left (-20 c^{2} d +9 e \right ) \arctan \left (c x \right )\right )}{30}+\frac {\left (30 c^{4} d^{2}-20 c^{2} d e +6 e^{2}\right ) \left (-\frac {i \left (\ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )-\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right )}{2}+\frac {i \left (\ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )-\ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )\right )}{2}\right )}{30}\right )}{c^{4}}+\frac {2 a b \left (\arctan \left (c x \right ) c^{5} x \,d^{2}+\frac {2 \arctan \left (c x \right ) d \,c^{5} e \,x^{3}}{3}+\frac {\arctan \left (c x \right ) e^{2} c^{5} x^{5}}{5}-\frac {d \,c^{4} e \,x^{2}}{3}-\frac {e^{2} c^{4} x^{4}}{20}+\frac {e^{2} c^{2} x^{2}}{10}-\frac {\left (15 c^{4} d^{2}-10 c^{2} d e +3 e^{2}\right ) \ln \left (c^{2} x^{2}+1\right )}{30}\right )}{c^{4}}}{c}\) | \(509\) |
| default | \(\frac {\frac {a^{2} \left (c^{5} x \,d^{2}+\frac {2}{3} d \,c^{5} e \,x^{3}+\frac {1}{5} e^{2} c^{5} x^{5}\right )}{c^{4}}+\frac {b^{2} \left (\arctan \left (c x \right )^{2} c^{5} x \,d^{2}+\frac {2 \arctan \left (c x \right )^{2} d \,c^{5} e \,x^{3}}{3}+\frac {\arctan \left (c x \right )^{2} e^{2} c^{5} x^{5}}{5}-\frac {2 \arctan \left (c x \right ) d \,c^{4} e \,x^{2}}{3}-\frac {\arctan \left (c x \right ) e^{2} c^{4} x^{4}}{10}+\frac {\arctan \left (c x \right ) e^{2} c^{2} x^{2}}{5}-\arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right ) c^{4} d^{2}+\frac {2 \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right ) c^{2} d e}{3}-\frac {\arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right ) e^{2}}{5}+\frac {e \left (20 c^{3} x d +e \,c^{3} x^{3}-9 e c x +\left (-20 c^{2} d +9 e \right ) \arctan \left (c x \right )\right )}{30}+\frac {\left (30 c^{4} d^{2}-20 c^{2} d e +6 e^{2}\right ) \left (-\frac {i \left (\ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )-\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right )}{2}+\frac {i \left (\ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )-\ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )\right )}{2}\right )}{30}\right )}{c^{4}}+\frac {2 a b \left (\arctan \left (c x \right ) c^{5} x \,d^{2}+\frac {2 \arctan \left (c x \right ) d \,c^{5} e \,x^{3}}{3}+\frac {\arctan \left (c x \right ) e^{2} c^{5} x^{5}}{5}-\frac {d \,c^{4} e \,x^{2}}{3}-\frac {e^{2} c^{4} x^{4}}{20}+\frac {e^{2} c^{2} x^{2}}{10}-\frac {\left (15 c^{4} d^{2}-10 c^{2} d e +3 e^{2}\right ) \ln \left (c^{2} x^{2}+1\right )}{30}\right )}{c^{4}}}{c}\) | \(509\) |
| risch | \(\text {Expression too large to display}\) | \(1389\) |
a^2*(1/5*x^5*e^2+2/3*x^3*e*d+x*d^2)+b^2/c*(1/5*arctan(c*x)^2*c*e^2*x^5+2/3 *arctan(c*x)^2*c*d*e*x^3+arctan(c*x)^2*c*x*d^2-2/15/c^4*(5*arctan(c*x)*d*c ^4*e*x^2+3/4*arctan(c*x)*e^2*c^4*x^4-3/2*arctan(c*x)*e^2*c^2*x^2+15/2*arct an(c*x)*ln(c^2*x^2+1)*c^4*d^2-5*arctan(c*x)*ln(c^2*x^2+1)*c^2*d*e+3/2*arct an(c*x)*ln(c^2*x^2+1)*e^2-1/4*e*(20*c^3*x*d+e*c^3*x^3-9*e*c*x+(-20*c^2*d+9 *e)*arctan(c*x))-1/4*(30*c^4*d^2-20*c^2*d*e+6*e^2)*(-1/2*I*(ln(c*x-I)*ln(c ^2*x^2+1)-1/2*ln(c*x-I)^2-dilog(-1/2*I*(I+c*x))-ln(c*x-I)*ln(-1/2*I*(I+c*x )))+1/2*I*(ln(I+c*x)*ln(c^2*x^2+1)-1/2*ln(I+c*x)^2-dilog(1/2*I*(c*x-I))-ln (I+c*x)*ln(1/2*I*(c*x-I))))))+2*a*b/c*(1/5*arctan(c*x)*c*e^2*x^5+2/3*arcta n(c*x)*c*d*e*x^3+arctan(c*x)*c*x*d^2-1/15/c^4*(5*d*c^4*e*x^2+3/4*e^2*c^4*x ^4-3/2*e^2*c^2*x^2+1/2*(15*c^4*d^2-10*c^2*d*e+3*e^2)*ln(c^2*x^2+1)))
\[ \int \left (d+e x^2\right )^2 (a+b \arctan (c x))^2 \, dx=\int { {\left (e x^{2} + d\right )}^{2} {\left (b \arctan \left (c x\right ) + a\right )}^{2} \,d x } \]
integral(a^2*e^2*x^4 + 2*a^2*d*e*x^2 + a^2*d^2 + (b^2*e^2*x^4 + 2*b^2*d*e* x^2 + b^2*d^2)*arctan(c*x)^2 + 2*(a*b*e^2*x^4 + 2*a*b*d*e*x^2 + a*b*d^2)*a rctan(c*x), x)
\[ \int \left (d+e x^2\right )^2 (a+b \arctan (c x))^2 \, dx=\int \left (a + b \operatorname {atan}{\left (c x \right )}\right )^{2} \left (d + e x^{2}\right )^{2}\, dx \]
\[ \int \left (d+e x^2\right )^2 (a+b \arctan (c x))^2 \, dx=\int { {\left (e x^{2} + d\right )}^{2} {\left (b \arctan \left (c x\right ) + a\right )}^{2} \,d x } \]
1/5*a^2*e^2*x^5 + 2/3*a^2*d*e*x^3 + 180*b^2*c^2*e^2*integrate(1/240*x^6*ar ctan(c*x)^2/(c^2*x^2 + 1), x) + 15*b^2*c^2*e^2*integrate(1/240*x^6*log(c^2 *x^2 + 1)^2/(c^2*x^2 + 1), x) + 12*b^2*c^2*e^2*integrate(1/240*x^6*log(c^2 *x^2 + 1)/(c^2*x^2 + 1), x) + 360*b^2*c^2*d*e*integrate(1/240*x^4*arctan(c *x)^2/(c^2*x^2 + 1), x) + 30*b^2*c^2*d*e*integrate(1/240*x^4*log(c^2*x^2 + 1)^2/(c^2*x^2 + 1), x) + 40*b^2*c^2*d*e*integrate(1/240*x^4*log(c^2*x^2 + 1)/(c^2*x^2 + 1), x) + 180*b^2*c^2*d^2*integrate(1/240*x^2*arctan(c*x)^2/ (c^2*x^2 + 1), x) + 15*b^2*c^2*d^2*integrate(1/240*x^2*log(c^2*x^2 + 1)^2/ (c^2*x^2 + 1), x) + 60*b^2*c^2*d^2*integrate(1/240*x^2*log(c^2*x^2 + 1)/(c ^2*x^2 + 1), x) + 1/4*b^2*d^2*arctan(c*x)^3/c - 24*b^2*c*e^2*integrate(1/2 40*x^5*arctan(c*x)/(c^2*x^2 + 1), x) - 80*b^2*c*d*e*integrate(1/240*x^3*ar ctan(c*x)/(c^2*x^2 + 1), x) - 120*b^2*c*d^2*integrate(1/240*x*arctan(c*x)/ (c^2*x^2 + 1), x) + 2/3*(2*x^3*arctan(c*x) - c*(x^2/c^2 - log(c^2*x^2 + 1) /c^4))*a*b*d*e + 1/10*(4*x^5*arctan(c*x) - c*((c^2*x^4 - 2*x^2)/c^4 + 2*lo g(c^2*x^2 + 1)/c^6))*a*b*e^2 + a^2*d^2*x + 180*b^2*e^2*integrate(1/240*x^4 *arctan(c*x)^2/(c^2*x^2 + 1), x) + 15*b^2*e^2*integrate(1/240*x^4*log(c^2* x^2 + 1)^2/(c^2*x^2 + 1), x) + 360*b^2*d*e*integrate(1/240*x^2*arctan(c*x) ^2/(c^2*x^2 + 1), x) + 30*b^2*d*e*integrate(1/240*x^2*log(c^2*x^2 + 1)^2/( c^2*x^2 + 1), x) + 15*b^2*d^2*integrate(1/240*log(c^2*x^2 + 1)^2/(c^2*x^2 + 1), x) + (2*c*x*arctan(c*x) - log(c^2*x^2 + 1))*a*b*d^2/c + 1/60*(3*b...
\[ \int \left (d+e x^2\right )^2 (a+b \arctan (c x))^2 \, dx=\int { {\left (e x^{2} + d\right )}^{2} {\left (b \arctan \left (c x\right ) + a\right )}^{2} \,d x } \]
Timed out. \[ \int \left (d+e x^2\right )^2 (a+b \arctan (c x))^2 \, dx=\int {\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^2\,{\left (e\,x^2+d\right )}^2 \,d x \]