Integrand size = 23, antiderivative size = 580 \[ \int x^2 \left (d+e x^2\right )^2 (a+b \arctan (c x))^2 \, dx=\frac {b^2 d^2 x}{3 c^2}-\frac {3 b^2 d e x}{5 c^4}+\frac {11 b^2 e^2 x}{42 c^6}+\frac {b^2 d e x^3}{15 c^2}-\frac {5 b^2 e^2 x^3}{126 c^4}+\frac {b^2 e^2 x^5}{105 c^2}-\frac {b^2 d^2 \arctan (c x)}{3 c^3}+\frac {3 b^2 d e \arctan (c x)}{5 c^5}-\frac {11 b^2 e^2 \arctan (c x)}{42 c^7}-\frac {b d^2 x^2 (a+b \arctan (c x))}{3 c}+\frac {2 b d e x^2 (a+b \arctan (c x))}{5 c^3}-\frac {b e^2 x^2 (a+b \arctan (c x))}{7 c^5}-\frac {b d e x^4 (a+b \arctan (c x))}{5 c}+\frac {b e^2 x^4 (a+b \arctan (c x))}{14 c^3}-\frac {b e^2 x^6 (a+b \arctan (c x))}{21 c}-\frac {i d^2 (a+b \arctan (c x))^2}{3 c^3}+\frac {2 i d e (a+b \arctan (c x))^2}{5 c^5}-\frac {i e^2 (a+b \arctan (c x))^2}{7 c^7}+\frac {1}{3} d^2 x^3 (a+b \arctan (c x))^2+\frac {2}{5} d e x^5 (a+b \arctan (c x))^2+\frac {1}{7} e^2 x^7 (a+b \arctan (c x))^2-\frac {2 b d^2 (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{3 c^3}+\frac {4 b d e (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{5 c^5}-\frac {2 b e^2 (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{7 c^7}-\frac {i b^2 d^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{3 c^3}+\frac {2 i b^2 d e \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{5 c^5}-\frac {i b^2 e^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{7 c^7} \]
2/5*I*d*e*(a+b*arctan(c*x))^2/c^5+2/5*I*b^2*d*e*polylog(2,1-2/(1+I*c*x))/c ^5-1/3*b*d^2*x^2*(a+b*arctan(c*x))/c-2/3*b*d^2*(a+b*arctan(c*x))*ln(2/(1+I *c*x))/c^3+1/7*e^2*x^7*(a+b*arctan(c*x))^2-2/7*b*e^2*(a+b*arctan(c*x))*ln( 2/(1+I*c*x))/c^7-3/5*b^2*d*e*x/c^4+1/15*b^2*d*e*x^3/c^2+3/5*b^2*d*e*arctan (c*x)/c^5-1/7*b*e^2*x^2*(a+b*arctan(c*x))/c^5+1/14*b*e^2*x^4*(a+b*arctan(c *x))/c^3-1/21*b*e^2*x^6*(a+b*arctan(c*x))/c-1/3*I*b^2*d^2*polylog(2,1-2/(1 +I*c*x))/c^3-1/7*I*b^2*e^2*polylog(2,1-2/(1+I*c*x))/c^7+1/3*d^2*x^3*(a+b*a rctan(c*x))^2+1/3*b^2*d^2*x/c^2-1/3*b^2*d^2*arctan(c*x)/c^3+11/42*b^2*e^2* x/c^6-5/126*b^2*e^2*x^3/c^4+1/105*b^2*e^2*x^5/c^2-11/42*b^2*e^2*arctan(c*x )/c^7+2/5*d*e*x^5*(a+b*arctan(c*x))^2-1/3*I*d^2*(a+b*arctan(c*x))^2/c^3-1/ 7*I*e^2*(a+b*arctan(c*x))^2/c^7+2/5*b*d*e*x^2*(a+b*arctan(c*x))/c^3-1/5*b* d*e*x^4*(a+b*arctan(c*x))/c+4/5*b*d*e*(a+b*arctan(c*x))*ln(2/(1+I*c*x))/c^ 5
Time = 1.18 (sec) , antiderivative size = 513, normalized size of antiderivative = 0.88 \[ \int x^2 \left (d+e x^2\right )^2 (a+b \arctan (c x))^2 \, dx=\frac {378 a b c^2 d e-165 a b e^2+210 b^2 c^5 d^2 x-378 b^2 c^3 d e x+165 b^2 c e^2 x-210 a b c^6 d^2 x^2+252 a b c^4 d e x^2-90 a b c^2 e^2 x^2+210 a^2 c^7 d^2 x^3+42 b^2 c^5 d e x^3-25 b^2 c^3 e^2 x^3-126 a b c^6 d e x^4+45 a b c^4 e^2 x^4+252 a^2 c^7 d e x^5+6 b^2 c^5 e^2 x^5-30 a b c^6 e^2 x^6+90 a^2 c^7 e^2 x^7+6 b^2 \left (35 i c^4 d^2-42 i c^2 d e+15 i e^2+c^7 \left (35 d^2 x^3+42 d e x^5+15 e^2 x^7\right )\right ) \arctan (c x)^2-3 b \arctan (c x) \left (-4 a c^7 x^3 \left (35 d^2+42 d e x^2+15 e^2 x^4\right )+b \left (1+c^2 x^2\right ) \left (55 e^2-c^2 e \left (126 d+25 e x^2\right )+2 c^4 \left (35 d^2+21 d e x^2+5 e^2 x^4\right )\right )+4 b \left (35 c^4 d^2-42 c^2 d e+15 e^2\right ) \log \left (1+e^{2 i \arctan (c x)}\right )\right )+210 a b c^4 d^2 \log \left (1+c^2 x^2\right )-252 a b c^2 d e \log \left (1+c^2 x^2\right )+90 a b e^2 \log \left (1+c^2 x^2\right )+6 i b^2 \left (35 c^4 d^2-42 c^2 d e+15 e^2\right ) \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )}{630 c^7} \]
(378*a*b*c^2*d*e - 165*a*b*e^2 + 210*b^2*c^5*d^2*x - 378*b^2*c^3*d*e*x + 1 65*b^2*c*e^2*x - 210*a*b*c^6*d^2*x^2 + 252*a*b*c^4*d*e*x^2 - 90*a*b*c^2*e^ 2*x^2 + 210*a^2*c^7*d^2*x^3 + 42*b^2*c^5*d*e*x^3 - 25*b^2*c^3*e^2*x^3 - 12 6*a*b*c^6*d*e*x^4 + 45*a*b*c^4*e^2*x^4 + 252*a^2*c^7*d*e*x^5 + 6*b^2*c^5*e ^2*x^5 - 30*a*b*c^6*e^2*x^6 + 90*a^2*c^7*e^2*x^7 + 6*b^2*((35*I)*c^4*d^2 - (42*I)*c^2*d*e + (15*I)*e^2 + c^7*(35*d^2*x^3 + 42*d*e*x^5 + 15*e^2*x^7)) *ArcTan[c*x]^2 - 3*b*ArcTan[c*x]*(-4*a*c^7*x^3*(35*d^2 + 42*d*e*x^2 + 15*e ^2*x^4) + b*(1 + c^2*x^2)*(55*e^2 - c^2*e*(126*d + 25*e*x^2) + 2*c^4*(35*d ^2 + 21*d*e*x^2 + 5*e^2*x^4)) + 4*b*(35*c^4*d^2 - 42*c^2*d*e + 15*e^2)*Log [1 + E^((2*I)*ArcTan[c*x])]) + 210*a*b*c^4*d^2*Log[1 + c^2*x^2] - 252*a*b* c^2*d*e*Log[1 + c^2*x^2] + 90*a*b*e^2*Log[1 + c^2*x^2] + (6*I)*b^2*(35*c^4 *d^2 - 42*c^2*d*e + 15*e^2)*PolyLog[2, -E^((2*I)*ArcTan[c*x])])/(630*c^7)
Time = 1.27 (sec) , antiderivative size = 580, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {5515, 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.
| \(\displaystyle \int x^2 \left (d+e x^2\right )^2 (a+b \arctan (c x))^2 \, dx\) |
| \(\Big \downarrow \) 5515 |
| \(\displaystyle \int \left (d^2 x^2 (a+b \arctan (c x))^2+2 d e x^4 (a+b \arctan (c x))^2+e^2 x^6 (a+b \arctan (c x))^2\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {i e^2 (a+b \arctan (c x))^2}{7 c^7}-\frac {2 b e^2 \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{7 c^7}+\frac {2 i d e (a+b \arctan (c x))^2}{5 c^5}+\frac {4 b d e \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{5 c^5}-\frac {b e^2 x^2 (a+b \arctan (c x))}{7 c^5}-\frac {i d^2 (a+b \arctan (c x))^2}{3 c^3}-\frac {2 b d^2 \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{3 c^3}+\frac {2 b d e x^2 (a+b \arctan (c x))}{5 c^3}+\frac {b e^2 x^4 (a+b \arctan (c x))}{14 c^3}+\frac {1}{3} d^2 x^3 (a+b \arctan (c x))^2-\frac {b d^2 x^2 (a+b \arctan (c x))}{3 c}+\frac {2}{5} d e x^5 (a+b \arctan (c x))^2-\frac {b d e x^4 (a+b \arctan (c x))}{5 c}+\frac {1}{7} e^2 x^7 (a+b \arctan (c x))^2-\frac {b e^2 x^6 (a+b \arctan (c x))}{21 c}-\frac {11 b^2 e^2 \arctan (c x)}{42 c^7}+\frac {3 b^2 d e \arctan (c x)}{5 c^5}-\frac {b^2 d^2 \arctan (c x)}{3 c^3}-\frac {i b^2 e^2 \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right )}{7 c^7}+\frac {11 b^2 e^2 x}{42 c^6}+\frac {2 i b^2 d e \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right )}{5 c^5}-\frac {3 b^2 d e x}{5 c^4}-\frac {5 b^2 e^2 x^3}{126 c^4}-\frac {i b^2 d^2 \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right )}{3 c^3}+\frac {b^2 d^2 x}{3 c^2}+\frac {b^2 d e x^3}{15 c^2}+\frac {b^2 e^2 x^5}{105 c^2}\) |
(b^2*d^2*x)/(3*c^2) - (3*b^2*d*e*x)/(5*c^4) + (11*b^2*e^2*x)/(42*c^6) + (b ^2*d*e*x^3)/(15*c^2) - (5*b^2*e^2*x^3)/(126*c^4) + (b^2*e^2*x^5)/(105*c^2) - (b^2*d^2*ArcTan[c*x])/(3*c^3) + (3*b^2*d*e*ArcTan[c*x])/(5*c^5) - (11*b ^2*e^2*ArcTan[c*x])/(42*c^7) - (b*d^2*x^2*(a + b*ArcTan[c*x]))/(3*c) + (2* b*d*e*x^2*(a + b*ArcTan[c*x]))/(5*c^3) - (b*e^2*x^2*(a + b*ArcTan[c*x]))/( 7*c^5) - (b*d*e*x^4*(a + b*ArcTan[c*x]))/(5*c) + (b*e^2*x^4*(a + b*ArcTan[ c*x]))/(14*c^3) - (b*e^2*x^6*(a + b*ArcTan[c*x]))/(21*c) - ((I/3)*d^2*(a + b*ArcTan[c*x])^2)/c^3 + (((2*I)/5)*d*e*(a + b*ArcTan[c*x])^2)/c^5 - ((I/7 )*e^2*(a + b*ArcTan[c*x])^2)/c^7 + (d^2*x^3*(a + b*ArcTan[c*x])^2)/3 + (2* d*e*x^5*(a + b*ArcTan[c*x])^2)/5 + (e^2*x^7*(a + b*ArcTan[c*x])^2)/7 - (2* b*d^2*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(3*c^3) + (4*b*d*e*(a + b*Ar cTan[c*x])*Log[2/(1 + I*c*x)])/(5*c^5) - (2*b*e^2*(a + b*ArcTan[c*x])*Log[ 2/(1 + I*c*x)])/(7*c^7) - ((I/3)*b^2*d^2*PolyLog[2, 1 - 2/(1 + I*c*x)])/c^ 3 + (((2*I)/5)*b^2*d*e*PolyLog[2, 1 - 2/(1 + I*c*x)])/c^5 - ((I/7)*b^2*e^2 *PolyLog[2, 1 - 2/(1 + I*c*x)])/c^7
3.13.55.3.1 Defintions of rubi rules used
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_ .)*(x_)^2)^(q_.), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*ArcTan[c*x] )^p, (f*x)^m*(d + e*x^2)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d , e, f, m}, x] && IntegerQ[q] && IGtQ[p, 0] && ((EqQ[p, 1] && GtQ[q, 0]) || IntegerQ[m])
Time = 0.87 (sec) , antiderivative size = 634, normalized size of antiderivative = 1.09
| method | result | size |
| parts | \(a^{2} \left (\frac {1}{7} e^{2} x^{7}+\frac {2}{5} e d \,x^{5}+\frac {1}{3} d^{2} x^{3}\right )+\frac {b^{2} \left (\frac {\arctan \left (c x \right )^{2} c^{3} e^{2} x^{7}}{7}+\frac {2 \arctan \left (c x \right )^{2} c^{3} d e \,x^{5}}{5}+\frac {\arctan \left (c x \right )^{2} d^{2} c^{3} x^{3}}{3}-\frac {2 \left (\frac {35 \arctan \left (c x \right ) c^{6} d^{2} x^{2}}{2}+\frac {21 \arctan \left (c x \right ) e \,c^{6} d \,x^{4}}{2}+\frac {5 \arctan \left (c x \right ) e^{2} c^{6} x^{6}}{2}-21 \arctan \left (c x \right ) d \,c^{4} e \,x^{2}-\frac {15 \arctan \left (c x \right ) e^{2} c^{4} x^{4}}{4}+\frac {15 \arctan \left (c x \right ) e^{2} c^{2} x^{2}}{2}-\frac {35 \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right ) c^{4} d^{2}}{2}+21 \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right ) c^{2} d e -\frac {15 \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right ) e^{2}}{2}-\frac {e^{2} c^{5} x^{5}}{2}-\frac {7 d \,c^{5} e \,x^{3}}{2}-\frac {35 c^{5} x \,d^{2}}{2}+\frac {25 e^{2} c^{3} x^{3}}{12}+\frac {63 c^{3} d e x}{2}-\frac {55 c x \,e^{2}}{4}-\frac {\left (-70 c^{4} d^{2}+126 c^{2} d e -55 e^{2}\right ) \arctan \left (c x \right )}{4}-\frac {\left (-70 c^{4} d^{2}+84 c^{2} d e -30 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 )}{105 c^{4}}\right )}{c^{3}}+\frac {2 a b \left (\frac {\arctan \left (c x \right ) c^{3} e^{2} x^{7}}{7}+\frac {2 \arctan \left (c x \right ) c^{3} d e \,x^{5}}{5}+\frac {\arctan \left (c x \right ) d^{2} c^{3} x^{3}}{3}-\frac {\frac {35 d^{2} c^{6} x^{2}}{2}+\frac {21 d \,c^{6} e \,x^{4}}{2}+\frac {5 e^{2} c^{6} x^{6}}{2}-21 d \,c^{4} e \,x^{2}-\frac {15 e^{2} c^{4} x^{4}}{4}+\frac {15 e^{2} c^{2} x^{2}}{2}+\frac {\left (-35 c^{4} d^{2}+42 c^{2} d e -15 e^{2}\right ) \ln \left (c^{2} x^{2}+1\right )}{2}}{105 c^{4}}\right )}{c^{3}}\) | \(634\) |
| derivativedivides | \(\frac {\frac {a^{2} \left (\frac {1}{3} d^{2} c^{7} x^{3}+\frac {2}{5} d \,c^{7} e \,x^{5}+\frac {1}{7} e^{2} c^{7} x^{7}\right )}{c^{4}}+\frac {b^{2} \left (\frac {\arctan \left (c x \right )^{2} d^{2} c^{7} x^{3}}{3}+\frac {2 \arctan \left (c x \right )^{2} d \,c^{7} e \,x^{5}}{5}+\frac {\arctan \left (c x \right )^{2} e^{2} c^{7} x^{7}}{7}-\frac {\arctan \left (c x \right ) c^{6} d^{2} x^{2}}{3}-\frac {\arctan \left (c x \right ) e \,c^{6} d \,x^{4}}{5}+\frac {2 \arctan \left (c x \right ) d \,c^{4} e \,x^{2}}{5}-\frac {\arctan \left (c x \right ) e^{2} c^{6} x^{6}}{21}+\frac {\arctan \left (c x \right ) e^{2} c^{4} x^{4}}{14}-\frac {\arctan \left (c x \right ) e^{2} c^{2} x^{2}}{7}+\frac {\arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right ) c^{4} d^{2}}{3}-\frac {2 \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right ) c^{2} d e}{5}+\frac {\arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right ) e^{2}}{7}+\frac {c^{5} x \,d^{2}}{3}+\frac {d \,c^{5} e \,x^{3}}{15}+\frac {e^{2} c^{5} x^{5}}{105}-\frac {3 c^{3} d e x}{5}-\frac {5 e^{2} c^{3} x^{3}}{126}+\frac {11 c x \,e^{2}}{42}+\frac {\left (-70 c^{4} d^{2}+126 c^{2} d e -55 e^{2}\right ) \arctan \left (c x \right )}{210}+\frac {\left (-70 c^{4} d^{2}+84 c^{2} d e -30 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 )}{210}\right )}{c^{4}}+\frac {2 a b \left (\frac {\arctan \left (c x \right ) d^{2} c^{7} x^{3}}{3}+\frac {2 \arctan \left (c x \right ) d \,c^{7} e \,x^{5}}{5}+\frac {\arctan \left (c x \right ) e^{2} c^{7} x^{7}}{7}-\frac {d^{2} c^{6} x^{2}}{6}-\frac {d \,c^{6} e \,x^{4}}{10}+\frac {d \,c^{4} e \,x^{2}}{5}-\frac {e^{2} c^{6} x^{6}}{42}+\frac {e^{2} c^{4} x^{4}}{28}-\frac {e^{2} c^{2} x^{2}}{14}-\frac {\left (-35 c^{4} d^{2}+42 c^{2} d e -15 e^{2}\right ) \ln \left (c^{2} x^{2}+1\right )}{210}\right )}{c^{4}}}{c^{3}}\) | \(638\) |
| default | \(\frac {\frac {a^{2} \left (\frac {1}{3} d^{2} c^{7} x^{3}+\frac {2}{5} d \,c^{7} e \,x^{5}+\frac {1}{7} e^{2} c^{7} x^{7}\right )}{c^{4}}+\frac {b^{2} \left (\frac {\arctan \left (c x \right )^{2} d^{2} c^{7} x^{3}}{3}+\frac {2 \arctan \left (c x \right )^{2} d \,c^{7} e \,x^{5}}{5}+\frac {\arctan \left (c x \right )^{2} e^{2} c^{7} x^{7}}{7}-\frac {\arctan \left (c x \right ) c^{6} d^{2} x^{2}}{3}-\frac {\arctan \left (c x \right ) e \,c^{6} d \,x^{4}}{5}+\frac {2 \arctan \left (c x \right ) d \,c^{4} e \,x^{2}}{5}-\frac {\arctan \left (c x \right ) e^{2} c^{6} x^{6}}{21}+\frac {\arctan \left (c x \right ) e^{2} c^{4} x^{4}}{14}-\frac {\arctan \left (c x \right ) e^{2} c^{2} x^{2}}{7}+\frac {\arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right ) c^{4} d^{2}}{3}-\frac {2 \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right ) c^{2} d e}{5}+\frac {\arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right ) e^{2}}{7}+\frac {c^{5} x \,d^{2}}{3}+\frac {d \,c^{5} e \,x^{3}}{15}+\frac {e^{2} c^{5} x^{5}}{105}-\frac {3 c^{3} d e x}{5}-\frac {5 e^{2} c^{3} x^{3}}{126}+\frac {11 c x \,e^{2}}{42}+\frac {\left (-70 c^{4} d^{2}+126 c^{2} d e -55 e^{2}\right ) \arctan \left (c x \right )}{210}+\frac {\left (-70 c^{4} d^{2}+84 c^{2} d e -30 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 )}{210}\right )}{c^{4}}+\frac {2 a b \left (\frac {\arctan \left (c x \right ) d^{2} c^{7} x^{3}}{3}+\frac {2 \arctan \left (c x \right ) d \,c^{7} e \,x^{5}}{5}+\frac {\arctan \left (c x \right ) e^{2} c^{7} x^{7}}{7}-\frac {d^{2} c^{6} x^{2}}{6}-\frac {d \,c^{6} e \,x^{4}}{10}+\frac {d \,c^{4} e \,x^{2}}{5}-\frac {e^{2} c^{6} x^{6}}{42}+\frac {e^{2} c^{4} x^{4}}{28}-\frac {e^{2} c^{2} x^{2}}{14}-\frac {\left (-35 c^{4} d^{2}+42 c^{2} d e -15 e^{2}\right ) \ln \left (c^{2} x^{2}+1\right )}{210}\right )}{c^{4}}}{c^{3}}\) | \(638\) |
| risch | \(\text {Expression too large to display}\) | \(1597\) |
a^2*(1/7*e^2*x^7+2/5*e*d*x^5+1/3*d^2*x^3)+b^2/c^3*(1/7*arctan(c*x)^2*c^3*e ^2*x^7+2/5*arctan(c*x)^2*c^3*d*e*x^5+1/3*arctan(c*x)^2*d^2*c^3*x^3-2/105/c ^4*(35/2*arctan(c*x)*c^6*d^2*x^2+21/2*arctan(c*x)*e*c^6*d*x^4+5/2*arctan(c *x)*e^2*c^6*x^6-21*arctan(c*x)*d*c^4*e*x^2-15/4*arctan(c*x)*e^2*c^4*x^4+15 /2*arctan(c*x)*e^2*c^2*x^2-35/2*arctan(c*x)*ln(c^2*x^2+1)*c^4*d^2+21*arcta n(c*x)*ln(c^2*x^2+1)*c^2*d*e-15/2*arctan(c*x)*ln(c^2*x^2+1)*e^2-1/2*e^2*c^ 5*x^5-7/2*d*c^5*e*x^3-35/2*c^5*x*d^2+25/12*e^2*c^3*x^3+63/2*c^3*d*e*x-55/4 *c*x*e^2-1/4*(-70*c^4*d^2+126*c^2*d*e-55*e^2)*arctan(c*x)-1/4*(-70*c^4*d^2 +84*c^2*d*e-30*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^3*(1/7*arctan(c*x)*c^3*e^2*x^7+2/5*arctan(c*x)*c^3*d*e*x^5+1/3*arct an(c*x)*d^2*c^3*x^3-1/105/c^4*(35/2*d^2*c^6*x^2+21/2*d*c^6*e*x^4+5/2*e^2*c ^6*x^6-21*d*c^4*e*x^2-15/4*e^2*c^4*x^4+15/2*e^2*c^2*x^2+1/2*(-35*c^4*d^2+4 2*c^2*d*e-15*e^2)*ln(c^2*x^2+1)))
\[ \int x^2 \left (d+e x^2\right )^2 (a+b \arctan (c x))^2 \, dx=\int { {\left (e x^{2} + d\right )}^{2} {\left (b \arctan \left (c x\right ) + a\right )}^{2} x^{2} \,d x } \]
integral(a^2*e^2*x^6 + 2*a^2*d*e*x^4 + a^2*d^2*x^2 + (b^2*e^2*x^6 + 2*b^2* d*e*x^4 + b^2*d^2*x^2)*arctan(c*x)^2 + 2*(a*b*e^2*x^6 + 2*a*b*d*e*x^4 + a* b*d^2*x^2)*arctan(c*x), x)
\[ \int x^2 \left (d+e x^2\right )^2 (a+b \arctan (c x))^2 \, dx=\int x^{2} \left (a + b \operatorname {atan}{\left (c x \right )}\right )^{2} \left (d + e x^{2}\right )^{2}\, dx \]
\[ \int x^2 \left (d+e x^2\right )^2 (a+b \arctan (c x))^2 \, dx=\int { {\left (e x^{2} + d\right )}^{2} {\left (b \arctan \left (c x\right ) + a\right )}^{2} x^{2} \,d x } \]
1/7*a^2*e^2*x^7 + 2/5*a^2*d*e*x^5 + 1/3*a^2*d^2*x^3 + 1/3*(2*x^3*arctan(c* x) - c*(x^2/c^2 - log(c^2*x^2 + 1)/c^4))*a*b*d^2 + 1/5*(4*x^5*arctan(c*x) - c*((c^2*x^4 - 2*x^2)/c^4 + 2*log(c^2*x^2 + 1)/c^6))*a*b*d*e + 1/42*(12*x ^7*arctan(c*x) - c*((2*c^4*x^6 - 3*c^2*x^4 + 6*x^2)/c^6 - 6*log(c^2*x^2 + 1)/c^8))*a*b*e^2 + 1/420*(15*b^2*e^2*x^7 + 42*b^2*d*e*x^5 + 35*b^2*d^2*x^3 )*arctan(c*x)^2 - 1/1680*(15*b^2*e^2*x^7 + 42*b^2*d*e*x^5 + 35*b^2*d^2*x^3 )*log(c^2*x^2 + 1)^2 + integrate(1/1680*(1260*(b^2*c^2*e^2*x^8 + (2*b^2*c^ 2*d*e + b^2*e^2)*x^6 + b^2*d^2*x^2 + (b^2*c^2*d^2 + 2*b^2*d*e)*x^4)*arctan (c*x)^2 + 105*(b^2*c^2*e^2*x^8 + (2*b^2*c^2*d*e + b^2*e^2)*x^6 + b^2*d^2*x ^2 + (b^2*c^2*d^2 + 2*b^2*d*e)*x^4)*log(c^2*x^2 + 1)^2 - 8*(15*b^2*c*e^2*x ^7 + 42*b^2*c*d*e*x^5 + 35*b^2*c*d^2*x^3)*arctan(c*x) + 4*(15*b^2*c^2*e^2* x^8 + 42*b^2*c^2*d*e*x^6 + 35*b^2*c^2*d^2*x^4)*log(c^2*x^2 + 1))/(c^2*x^2 + 1), x)
\[ \int x^2 \left (d+e x^2\right )^2 (a+b \arctan (c x))^2 \, dx=\int { {\left (e x^{2} + d\right )}^{2} {\left (b \arctan \left (c x\right ) + a\right )}^{2} x^{2} \,d x } \]
Timed out. \[ \int x^2 \left (d+e x^2\right )^2 (a+b \arctan (c x))^2 \, dx=\int x^2\,{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^2\,{\left (e\,x^2+d\right )}^2 \,d x \]