Integrand size = 23, antiderivative size = 287 \[ \int x^3 (d+i c d x) (a+b \arctan (c x))^2 \, dx=\frac {a b d x}{2 c^3}-\frac {3 i b^2 d x}{10 c^3}+\frac {b^2 d x^2}{12 c^2}+\frac {i b^2 d x^3}{30 c}+\frac {3 i b^2 d \arctan (c x)}{10 c^4}+\frac {b^2 d x \arctan (c x)}{2 c^3}+\frac {i b d x^2 (a+b \arctan (c x))}{5 c^2}-\frac {b d x^3 (a+b \arctan (c x))}{6 c}-\frac {1}{10} i b d x^4 (a+b \arctan (c x))-\frac {9 d (a+b \arctan (c x))^2}{20 c^4}+\frac {1}{4} d x^4 (a+b \arctan (c x))^2+\frac {1}{5} i c d x^5 (a+b \arctan (c x))^2+\frac {2 i b d (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{5 c^4}-\frac {b^2 d \log \left (1+c^2 x^2\right )}{3 c^4}-\frac {b^2 d \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{5 c^4} \]
1/2*a*b*d*x/c^3-3/10*I*b^2*d*x/c^3+1/12*b^2*d*x^2/c^2+1/30*I*b^2*d*x^3/c+3 /10*I*b^2*d*arctan(c*x)/c^4+1/2*b^2*d*x*arctan(c*x)/c^3+1/5*I*b*d*x^2*(a+b *arctan(c*x))/c^2-1/6*b*d*x^3*(a+b*arctan(c*x))/c-1/10*I*b*d*x^4*(a+b*arct an(c*x))-9/20*d*(a+b*arctan(c*x))^2/c^4+1/4*d*x^4*(a+b*arctan(c*x))^2+1/5* I*c*d*x^5*(a+b*arctan(c*x))^2+2/5*I*b*d*(a+b*arctan(c*x))*ln(2/(1+I*c*x))/ c^4-1/3*b^2*d*ln(c^2*x^2+1)/c^4-1/5*b^2*d*polylog(2,1-2/(1+I*c*x))/c^4
Time = 0.65 (sec) , antiderivative size = 285, normalized size of antiderivative = 0.99 \[ \int x^3 (d+i c d x) (a+b \arctan (c x))^2 \, dx=\frac {d \left (18 i a b+5 b^2+30 a b c x-18 i b^2 c x+12 i a b c^2 x^2+5 b^2 c^2 x^2-10 a b c^3 x^3+2 i b^2 c^3 x^3+15 a^2 c^4 x^4-6 i a b c^4 x^4+12 i a^2 c^5 x^5+3 b^2 \left (-1+5 c^4 x^4+4 i c^5 x^5\right ) \arctan (c x)^2+2 b \arctan (c x) \left (b \left (9 i+15 c x+6 i c^2 x^2-5 c^3 x^3-3 i c^4 x^4\right )+3 a \left (-5+5 c^4 x^4+4 i c^5 x^5\right )+12 i b \log \left (1+e^{2 i \arctan (c x)}\right )\right )-12 i a b \log \left (1+c^2 x^2\right )-20 b^2 \log \left (1+c^2 x^2\right )+12 b^2 \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )\right )}{60 c^4} \]
(d*((18*I)*a*b + 5*b^2 + 30*a*b*c*x - (18*I)*b^2*c*x + (12*I)*a*b*c^2*x^2 + 5*b^2*c^2*x^2 - 10*a*b*c^3*x^3 + (2*I)*b^2*c^3*x^3 + 15*a^2*c^4*x^4 - (6 *I)*a*b*c^4*x^4 + (12*I)*a^2*c^5*x^5 + 3*b^2*(-1 + 5*c^4*x^4 + (4*I)*c^5*x ^5)*ArcTan[c*x]^2 + 2*b*ArcTan[c*x]*(b*(9*I + 15*c*x + (6*I)*c^2*x^2 - 5*c ^3*x^3 - (3*I)*c^4*x^4) + 3*a*(-5 + 5*c^4*x^4 + (4*I)*c^5*x^5) + (12*I)*b* Log[1 + E^((2*I)*ArcTan[c*x])]) - (12*I)*a*b*Log[1 + c^2*x^2] - 20*b^2*Log [1 + c^2*x^2] + 12*b^2*PolyLog[2, -E^((2*I)*ArcTan[c*x])]))/(60*c^4)
Time = 0.77 (sec) , antiderivative size = 287, 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 = {5411, 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.
| \(\displaystyle \int x^3 (d+i c d x) (a+b \arctan (c x))^2 \, dx\) |
| \(\Big \downarrow \) 5411 |
| \(\displaystyle \int \left (d x^3 (a+b \arctan (c x))^2+i c d x^4 (a+b \arctan (c x))^2\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {9 d (a+b \arctan (c x))^2}{20 c^4}+\frac {2 i b d \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{5 c^4}+\frac {i b d x^2 (a+b \arctan (c x))}{5 c^2}+\frac {1}{5} i c d x^5 (a+b \arctan (c x))^2+\frac {1}{4} d x^4 (a+b \arctan (c x))^2-\frac {1}{10} i b d x^4 (a+b \arctan (c x))-\frac {b d x^3 (a+b \arctan (c x))}{6 c}+\frac {a b d x}{2 c^3}+\frac {3 i b^2 d \arctan (c x)}{10 c^4}+\frac {b^2 d x \arctan (c x)}{2 c^3}-\frac {b^2 d \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right )}{5 c^4}-\frac {3 i b^2 d x}{10 c^3}+\frac {b^2 d x^2}{12 c^2}-\frac {b^2 d \log \left (c^2 x^2+1\right )}{3 c^4}+\frac {i b^2 d x^3}{30 c}\) |
(a*b*d*x)/(2*c^3) - (((3*I)/10)*b^2*d*x)/c^3 + (b^2*d*x^2)/(12*c^2) + ((I/ 30)*b^2*d*x^3)/c + (((3*I)/10)*b^2*d*ArcTan[c*x])/c^4 + (b^2*d*x*ArcTan[c* x])/(2*c^3) + ((I/5)*b*d*x^2*(a + b*ArcTan[c*x]))/c^2 - (b*d*x^3*(a + b*Ar cTan[c*x]))/(6*c) - (I/10)*b*d*x^4*(a + b*ArcTan[c*x]) - (9*d*(a + b*ArcTa n[c*x])^2)/(20*c^4) + (d*x^4*(a + b*ArcTan[c*x])^2)/4 + (I/5)*c*d*x^5*(a + b*ArcTan[c*x])^2 + (((2*I)/5)*b*d*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)]) /c^4 - (b^2*d*Log[1 + c^2*x^2])/(3*c^4) - (b^2*d*PolyLog[2, 1 - 2/(1 + I*c *x)])/(5*c^4)
3.1.68.3.1 Defintions of rubi rules used
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_ .)*(x_))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*ArcTan[c*x])^p, (f* x)^m*(d + e*x)^q, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[p, 0] & & IntegerQ[q] && (GtQ[q, 0] || NeQ[a, 0] || IntegerQ[m])
Time = 1.48 (sec) , antiderivative size = 375, normalized size of antiderivative = 1.31
| method | result | size |
| parts | \(a^{2} d \left (\frac {1}{5} i c \,x^{5}+\frac {1}{4} x^{4}\right )+\frac {d \,b^{2} \left (-\frac {i \arctan \left (c x \right ) c^{4} x^{4}}{10}+\frac {c^{4} x^{4} \arctan \left (c x \right )^{2}}{4}+\frac {c x \arctan \left (c x \right )}{2}-\frac {i \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right )}{5}-\frac {c^{3} x^{3} \arctan \left (c x \right )}{6}+\frac {i c^{3} x^{3}}{30}+\frac {i \arctan \left (c x \right ) c^{2} x^{2}}{5}-\frac {\arctan \left (c x \right )^{2}}{4}+\frac {\ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )}{10}-\frac {\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )}{10}-\frac {\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )}{10}-\frac {\ln \left (c x -i\right )^{2}}{20}-\frac {\ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )}{10}+\frac {\operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )}{10}+\frac {\ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )}{10}+\frac {\ln \left (c x +i\right )^{2}}{20}+\frac {3 i \arctan \left (c x \right )}{10}-\frac {3 i c x}{10}+\frac {c^{2} x^{2}}{12}-\frac {\ln \left (c^{2} x^{2}+1\right )}{3}+\frac {i \arctan \left (c x \right )^{2} c^{5} x^{5}}{5}\right )}{c^{4}}+\frac {2 a b d \left (\frac {i \arctan \left (c x \right ) c^{5} x^{5}}{5}+\frac {c^{4} x^{4} \arctan \left (c x \right )}{4}+\frac {c x}{4}-\frac {i c^{4} x^{4}}{20}-\frac {c^{3} x^{3}}{12}+\frac {i c^{2} x^{2}}{10}-\frac {i \ln \left (c^{2} x^{2}+1\right )}{10}-\frac {\arctan \left (c x \right )}{4}\right )}{c^{4}}\) | \(375\) |
| derivativedivides | \(\frac {a^{2} d \left (\frac {1}{5} i c^{5} x^{5}+\frac {1}{4} c^{4} x^{4}\right )+d \,b^{2} \left (-\frac {i \arctan \left (c x \right ) c^{4} x^{4}}{10}+\frac {c^{4} x^{4} \arctan \left (c x \right )^{2}}{4}+\frac {c x \arctan \left (c x \right )}{2}-\frac {i \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right )}{5}-\frac {c^{3} x^{3} \arctan \left (c x \right )}{6}+\frac {i c^{3} x^{3}}{30}+\frac {i \arctan \left (c x \right ) c^{2} x^{2}}{5}-\frac {\arctan \left (c x \right )^{2}}{4}+\frac {\ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )}{10}-\frac {\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )}{10}-\frac {\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )}{10}-\frac {\ln \left (c x -i\right )^{2}}{20}-\frac {\ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )}{10}+\frac {\operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )}{10}+\frac {\ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )}{10}+\frac {\ln \left (c x +i\right )^{2}}{20}+\frac {3 i \arctan \left (c x \right )}{10}-\frac {3 i c x}{10}+\frac {c^{2} x^{2}}{12}-\frac {\ln \left (c^{2} x^{2}+1\right )}{3}+\frac {i \arctan \left (c x \right )^{2} c^{5} x^{5}}{5}\right )+2 a b d \left (\frac {i \arctan \left (c x \right ) c^{5} x^{5}}{5}+\frac {c^{4} x^{4} \arctan \left (c x \right )}{4}+\frac {c x}{4}-\frac {i c^{4} x^{4}}{20}-\frac {c^{3} x^{3}}{12}+\frac {i c^{2} x^{2}}{10}-\frac {i \ln \left (c^{2} x^{2}+1\right )}{10}-\frac {\arctan \left (c x \right )}{4}\right )}{c^{4}}\) | \(378\) |
| default | \(\frac {a^{2} d \left (\frac {1}{5} i c^{5} x^{5}+\frac {1}{4} c^{4} x^{4}\right )+d \,b^{2} \left (-\frac {i \arctan \left (c x \right ) c^{4} x^{4}}{10}+\frac {c^{4} x^{4} \arctan \left (c x \right )^{2}}{4}+\frac {c x \arctan \left (c x \right )}{2}-\frac {i \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right )}{5}-\frac {c^{3} x^{3} \arctan \left (c x \right )}{6}+\frac {i c^{3} x^{3}}{30}+\frac {i \arctan \left (c x \right ) c^{2} x^{2}}{5}-\frac {\arctan \left (c x \right )^{2}}{4}+\frac {\ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )}{10}-\frac {\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )}{10}-\frac {\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )}{10}-\frac {\ln \left (c x -i\right )^{2}}{20}-\frac {\ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )}{10}+\frac {\operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )}{10}+\frac {\ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )}{10}+\frac {\ln \left (c x +i\right )^{2}}{20}+\frac {3 i \arctan \left (c x \right )}{10}-\frac {3 i c x}{10}+\frac {c^{2} x^{2}}{12}-\frac {\ln \left (c^{2} x^{2}+1\right )}{3}+\frac {i \arctan \left (c x \right )^{2} c^{5} x^{5}}{5}\right )+2 a b d \left (\frac {i \arctan \left (c x \right ) c^{5} x^{5}}{5}+\frac {c^{4} x^{4} \arctan \left (c x \right )}{4}+\frac {c x}{4}-\frac {i c^{4} x^{4}}{20}-\frac {c^{3} x^{3}}{12}+\frac {i c^{2} x^{2}}{10}-\frac {i \ln \left (c^{2} x^{2}+1\right )}{10}-\frac {\arctan \left (c x \right )}{4}\right )}{c^{4}}\) | \(378\) |
| risch | \(\frac {b^{2} d \,x^{2}}{12 c^{2}}-\frac {b^{2} d \ln \left (c^{2} x^{2}+1\right )}{3 c^{4}}+\frac {a b d x}{2 c^{3}}+\frac {d \,x^{4} a^{2}}{4}-\frac {9 d \,a^{2}}{20 c^{4}}-\frac {b d a \arctan \left (c x \right )}{2 c^{4}}-\frac {d a b \,x^{3}}{6 c}+\frac {5 b^{2} d}{12 c^{4}}+\left (\frac {i d \,b^{2} \left (4 x^{5} c -5 i x^{4}\right ) \ln \left (-i c x +1\right )}{40}+\frac {b d \left (24 a \,c^{5} x^{5}-30 i a \,c^{4} x^{4}-6 b \,c^{4} x^{4}+10 i b \,c^{3} x^{3}+12 b \,c^{2} x^{2}-30 i b c x -27 b \ln \left (-i c x +1\right )\right )}{120 c^{4}}\right ) \ln \left (i c x +1\right )+\frac {i b^{2} d \,x^{3}}{30 c}+\frac {3 i b^{2} d \arctan \left (c x \right )}{10 c^{4}}+\frac {29 i b d a}{30 c^{4}}-\frac {i b d \,x^{4} a}{10}-\frac {d \,b^{2} \ln \left (-i c x +1\right ) x^{2}}{10 c^{2}}+\frac {d \,b^{2} \ln \left (\frac {1}{2}+\frac {i c x}{2}\right ) \ln \left (-i c x +1\right )}{5 c^{4}}-\frac {d \,b^{2} \ln \left (\frac {1}{2}+\frac {i c x}{2}\right ) \ln \left (\frac {1}{2}-\frac {i c x}{2}\right )}{5 c^{4}}+\frac {i d c \,a^{2} x^{5}}{5}-\frac {3 i b^{2} d x}{10 c^{3}}-\frac {d \,b^{2} \ln \left (-i c x +1\right )^{2} x^{4}}{16}+\frac {d \,b^{2} \ln \left (-i c x +1\right ) x^{4}}{20}+\frac {9 d \,b^{2} \ln \left (-i c x +1\right )^{2}}{80 c^{4}}-\frac {d \,b^{2} \operatorname {dilog}\left (\frac {1}{2}-\frac {i c x}{2}\right )}{5 c^{4}}-\frac {i d c \,b^{2} \ln \left (-i c x +1\right )^{2} x^{5}}{20}+\frac {i d a b \ln \left (-i c x +1\right ) x^{4}}{4}-\frac {i d \,b^{2} \left (4 c^{5} x^{5}-5 i c^{4} x^{4}+i\right ) \ln \left (i c x +1\right )^{2}}{80 c^{4}}-\frac {i b d a \ln \left (c^{2} x^{2}+1\right )}{5 c^{4}}-\frac {d c a b \ln \left (-i c x +1\right ) x^{5}}{5}+\frac {i b d \,x^{2} a}{5 c^{2}}+\frac {i d \,b^{2} \ln \left (-i c x +1\right ) x}{4 c^{3}}-\frac {i d \,b^{2} \ln \left (-i c x +1\right ) x^{3}}{12 c}\) | \(577\) |
a^2*d*(1/5*I*c*x^5+1/4*x^4)+d*b^2/c^4*(-1/10*I*arctan(c*x)*c^4*x^4+1/4*c^4 *x^4*arctan(c*x)^2+1/2*c*x*arctan(c*x)-1/5*I*arctan(c*x)*ln(c^2*x^2+1)-1/6 *c^3*x^3*arctan(c*x)+1/30*I*c^3*x^3+1/5*I*arctan(c*x)*c^2*x^2-1/4*arctan(c *x)^2+1/10*ln(c*x-I)*ln(c^2*x^2+1)-1/10*dilog(-1/2*I*(c*x+I))-1/10*ln(c*x- I)*ln(-1/2*I*(c*x+I))-1/20*ln(c*x-I)^2-1/10*ln(c*x+I)*ln(c^2*x^2+1)+1/10*d ilog(1/2*I*(c*x-I))+1/10*ln(c*x+I)*ln(1/2*I*(c*x-I))+1/20*ln(c*x+I)^2+3/10 *I*arctan(c*x)-3/10*I*c*x+1/12*c^2*x^2-1/3*ln(c^2*x^2+1)+1/5*I*arctan(c*x) ^2*c^5*x^5)+2*a*b*d/c^4*(1/5*I*arctan(c*x)*c^5*x^5+1/4*c^4*x^4*arctan(c*x) +1/4*c*x-1/20*I*c^4*x^4-1/12*c^3*x^3+1/10*I*c^2*x^2-1/10*I*ln(c^2*x^2+1)-1 /4*arctan(c*x))
\[ \int x^3 (d+i c d x) (a+b \arctan (c x))^2 \, dx=\int { {\left (i \, c d x + d\right )} {\left (b \arctan \left (c x\right ) + a\right )}^{2} x^{3} \,d x } \]
1/80*(-4*I*b^2*c*d*x^5 - 5*b^2*d*x^4)*log(-(c*x + I)/(c*x - I))^2 + integr al(1/20*(20*I*a^2*c^3*d*x^6 + 20*a^2*c^2*d*x^5 + 20*I*a^2*c*d*x^4 + 20*a^2 *d*x^3 - (20*a*b*c^3*d*x^6 + 4*(-5*I*a*b - b^2)*c^2*d*x^5 + 5*(4*a*b + I*b ^2)*c*d*x^4 - 20*I*a*b*d*x^3)*log(-(c*x + I)/(c*x - I)))/(c^2*x^2 + 1), x)
Timed out. \[ \int x^3 (d+i c d x) (a+b \arctan (c x))^2 \, dx=\text {Timed out} \]
\[ \int x^3 (d+i c d x) (a+b \arctan (c x))^2 \, dx=\int { {\left (i \, c d x + d\right )} {\left (b \arctan \left (c x\right ) + a\right )}^{2} x^{3} \,d x } \]
1/5*I*a^2*c*d*x^5 + 1/4*b^2*d*x^4*arctan(c*x)^2 + 1/4*a^2*d*x^4 + 1/10*I*( 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*c*d + 1/80*I*(4*x^5*arctan(c*x)^2 - x^5*log(c^2*x^2 + 1)^2 + 80*integrat e(1/80*(4*c^2*x^6*log(c^2*x^2 + 1) - 8*c*x^5*arctan(c*x) + 60*(c^2*x^6 + x ^4)*arctan(c*x)^2 + 5*(c^2*x^6 + x^4)*log(c^2*x^2 + 1)^2)/(c^2*x^2 + 1), x ))*b^2*c*d + 1/6*(3*x^4*arctan(c*x) - c*((c^2*x^3 - 3*x)/c^4 + 3*arctan(c* x)/c^5))*a*b*d - 1/12*(2*c*((c^2*x^3 - 3*x)/c^4 + 3*arctan(c*x)/c^5)*arcta n(c*x) - (c^2*x^2 + 3*arctan(c*x)^2 - 4*log(c^2*x^2 + 1))/c^4)*b^2*d
\[ \int x^3 (d+i c d x) (a+b \arctan (c x))^2 \, dx=\int { {\left (i \, c d x + d\right )} {\left (b \arctan \left (c x\right ) + a\right )}^{2} x^{3} \,d x } \]
Timed out. \[ \int x^3 (d+i c d x) (a+b \arctan (c x))^2 \, dx=\int x^3\,{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^2\,\left (d+c\,d\,x\,1{}\mathrm {i}\right ) \,d x \]