Integrand size = 23, antiderivative size = 307 \[ \int x (d+i c d x)^3 (a+b \arctan (c x))^2 \, dx=-\frac {5 a b d^3 x}{2 c}+\frac {13 i b^2 d^3 x}{10 c}-\frac {1}{4} b^2 d^3 x^2-\frac {1}{30} i b^2 c d^3 x^3-\frac {13 i b^2 d^3 \arctan (c x)}{10 c^2}-\frac {5 b^2 d^3 x \arctan (c x)}{2 c}-\frac {6}{5} i b d^3 x^2 (a+b \arctan (c x))+\frac {1}{2} b c d^3 x^3 (a+b \arctan (c x))+\frac {1}{10} i b c^2 d^3 x^4 (a+b \arctan (c x))+\frac {d^3 (1+i c x)^4 (a+b \arctan (c x))^2}{4 c^2}-\frac {d^3 (1+i c x)^5 (a+b \arctan (c x))^2}{5 c^2}-\frac {12 i b d^3 (a+b \arctan (c x)) \log \left (\frac {2}{1-i c x}\right )}{5 c^2}+\frac {3 b^2 d^3 \log \left (1+c^2 x^2\right )}{2 c^2}-\frac {6 b^2 d^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{5 c^2} \] Output:
-5/2*a*b*d^3*x/c+13/10*I*b^2*d^3*x/c-1/4*b^2*d^3*x^2-1/30*I*b^2*c*d^3*x^3- 13/10*I*b^2*d^3*arctan(c*x)/c^2-5/2*b^2*d^3*x*arctan(c*x)/c-6/5*I*b*d^3*x^ 2*(a+b*arctan(c*x))+1/2*b*c*d^3*x^3*(a+b*arctan(c*x))+1/10*I*b*c^2*d^3*x^4 *(a+b*arctan(c*x))+1/4*d^3*(1+I*c*x)^4*(a+b*arctan(c*x))^2/c^2-1/5*d^3*(1+ I*c*x)^5*(a+b*arctan(c*x))^2/c^2-12/5*I*b*d^3*(a+b*arctan(c*x))*ln(2/(1-I* c*x))/c^2+3/2*b^2*d^3*ln(c^2*x^2+1)/c^2-6/5*b^2*d^3*polylog(2,1-2/(1-I*c*x ))/c^2
Time = 1.90 (sec) , antiderivative size = 325, normalized size of antiderivative = 1.06 \[ \int x (d+i c d x)^3 (a+b \arctan (c x))^2 \, dx=\frac {d^3 \left (-18 i a b-15 b^2-150 a b c x+78 i b^2 c x+30 a^2 c^2 x^2-72 i a b c^2 x^2-15 b^2 c^2 x^2+60 i a^2 c^3 x^3+30 a b c^3 x^3-2 i b^2 c^3 x^3-45 a^2 c^4 x^4+6 i a b c^4 x^4-12 i a^2 c^5 x^5+3 b^2 (1-4 i c x) (-i+c x)^4 \arctan (c x)^2+6 b \arctan (c x) \left (b \left (-13 i-25 c x-12 i c^2 x^2+5 c^3 x^3+i c^4 x^4\right )+a \left (25+10 c^2 x^2+20 i c^3 x^3-15 c^4 x^4-4 i c^5 x^5\right )-24 i b \log \left (1+e^{2 i \arctan (c x)}\right )\right )+72 i a b \log \left (1+c^2 x^2\right )+90 b^2 \log \left (1+c^2 x^2\right )-72 b^2 \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )\right )}{60 c^2} \] Input:
Integrate[x*(d + I*c*d*x)^3*(a + b*ArcTan[c*x])^2,x]
Output:
(d^3*((-18*I)*a*b - 15*b^2 - 150*a*b*c*x + (78*I)*b^2*c*x + 30*a^2*c^2*x^2 - (72*I)*a*b*c^2*x^2 - 15*b^2*c^2*x^2 + (60*I)*a^2*c^3*x^3 + 30*a*b*c^3*x ^3 - (2*I)*b^2*c^3*x^3 - 45*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 - (4*I)*c*x)*(-I + c*x)^4*ArcTan[c*x]^2 + 6*b*ArcTan[c*x ]*(b*(-13*I - 25*c*x - (12*I)*c^2*x^2 + 5*c^3*x^3 + I*c^4*x^4) + a*(25 + 1 0*c^2*x^2 + (20*I)*c^3*x^3 - 15*c^4*x^4 - (4*I)*c^5*x^5) - (24*I)*b*Log[1 + E^((2*I)*ArcTan[c*x])]) + (72*I)*a*b*Log[1 + c^2*x^2] + 90*b^2*Log[1 + c ^2*x^2] - 72*b^2*PolyLog[2, -E^((2*I)*ArcTan[c*x])]))/(60*c^2)
Time = 0.81 (sec) , antiderivative size = 307, 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.
| \(\displaystyle \int x (d+i c d x)^3 (a+b \arctan (c x))^2 \, dx\) |
| \(\Big \downarrow \) 5411 |
| \(\displaystyle \int \left (\frac {i (d+i c d x)^3 (a+b \arctan (c x))^2}{c}-\frac {i (d+i c d x)^4 (a+b \arctan (c x))^2}{c d}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{10} i b c^2 d^3 x^4 (a+b \arctan (c x))-\frac {d^3 (1+i c x)^5 (a+b \arctan (c x))^2}{5 c^2}+\frac {d^3 (1+i c x)^4 (a+b \arctan (c x))^2}{4 c^2}-\frac {12 i b d^3 \log \left (\frac {2}{1-i c x}\right ) (a+b \arctan (c x))}{5 c^2}+\frac {1}{2} b c d^3 x^3 (a+b \arctan (c x))-\frac {6}{5} i b d^3 x^2 (a+b \arctan (c x))-\frac {5 a b d^3 x}{2 c}-\frac {13 i b^2 d^3 \arctan (c x)}{10 c^2}-\frac {5 b^2 d^3 x \arctan (c x)}{2 c}-\frac {6 b^2 d^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{5 c^2}+\frac {3 b^2 d^3 \log \left (c^2 x^2+1\right )}{2 c^2}-\frac {1}{30} i b^2 c d^3 x^3+\frac {13 i b^2 d^3 x}{10 c}-\frac {1}{4} b^2 d^3 x^2\) |
Input:
Int[x*(d + I*c*d*x)^3*(a + b*ArcTan[c*x])^2,x]
Output:
(-5*a*b*d^3*x)/(2*c) + (((13*I)/10)*b^2*d^3*x)/c - (b^2*d^3*x^2)/4 - (I/30 )*b^2*c*d^3*x^3 - (((13*I)/10)*b^2*d^3*ArcTan[c*x])/c^2 - (5*b^2*d^3*x*Arc Tan[c*x])/(2*c) - ((6*I)/5)*b*d^3*x^2*(a + b*ArcTan[c*x]) + (b*c*d^3*x^3*( a + b*ArcTan[c*x]))/2 + (I/10)*b*c^2*d^3*x^4*(a + b*ArcTan[c*x]) + (d^3*(1 + I*c*x)^4*(a + b*ArcTan[c*x])^2)/(4*c^2) - (d^3*(1 + I*c*x)^5*(a + b*Arc Tan[c*x])^2)/(5*c^2) - (((12*I)/5)*b*d^3*(a + b*ArcTan[c*x])*Log[2/(1 - I* c*x)])/c^2 + (3*b^2*d^3*Log[1 + c^2*x^2])/(2*c^2) - (6*b^2*d^3*PolyLog[2, 1 - 2/(1 - I*c*x)])/(5*c^2)
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 = 1.47 (sec) , antiderivative size = 452, normalized size of antiderivative = 1.47
| method | result | size |
| parts | \(d^{3} a^{2} \left (-\frac {1}{5} i c^{3} x^{5}-\frac {3}{4} c^{2} x^{4}+i c \,x^{3}+\frac {1}{2} x^{2}\right )+\frac {d^{3} b^{2} \left (i \arctan \left (c x \right )^{2} c^{3} x^{3}-\frac {3 c^{4} x^{4} \arctan \left (c x \right )^{2}}{4}-\frac {13 i \arctan \left (c x \right )}{10}+\frac {c^{2} x^{2} \arctan \left (c x \right )^{2}}{2}-\frac {i \arctan \left (c x \right )^{2} c^{5} x^{5}}{5}+\frac {6 i \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right )}{5}+\frac {c^{3} x^{3} \arctan \left (c x \right )}{2}+\frac {i \arctan \left (c x \right ) c^{4} x^{4}}{10}+\frac {5 \arctan \left (c x \right )^{2}}{4}-\frac {5 c x \arctan \left (c x \right )}{2}+\frac {3 \ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )}{5}-\frac {3 \ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )}{5}-\frac {3 \ln \left (c x +i\right )^{2}}{10}-\frac {3 \ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )}{5}+\frac {3 \ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )}{5}+\frac {3 \ln \left (c x -i\right )^{2}}{10}-\frac {3 \operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )}{5}+\frac {3 \operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )}{5}-\frac {i c^{3} x^{3}}{30}-\frac {6 i \arctan \left (c x \right ) c^{2} x^{2}}{5}-\frac {c^{2} x^{2}}{4}+\frac {3 \ln \left (c^{2} x^{2}+1\right )}{2}+\frac {13 i c x}{10}\right )}{c^{2}}+\frac {2 d^{3} a b \left (-\frac {i \arctan \left (c x \right ) c^{5} x^{5}}{5}-\frac {3 c^{4} x^{4} \arctan \left (c x \right )}{4}+i \arctan \left (c x \right ) c^{3} x^{3}+\frac {c^{2} x^{2} \arctan \left (c x \right )}{2}-\frac {5 c x}{4}+\frac {i c^{4} x^{4}}{20}+\frac {c^{3} x^{3}}{4}-\frac {3 i c^{2} x^{2}}{5}+\frac {3 i \ln \left (c^{2} x^{2}+1\right )}{5}+\frac {5 \arctan \left (c x \right )}{4}\right )}{c^{2}}\) | \(452\) |
| derivativedivides | \(\frac {d^{3} a^{2} \left (-\frac {1}{5} i c^{5} x^{5}-\frac {3}{4} c^{4} x^{4}+i c^{3} x^{3}+\frac {1}{2} c^{2} x^{2}\right )+d^{3} b^{2} \left (i \arctan \left (c x \right )^{2} c^{3} x^{3}-\frac {3 c^{4} x^{4} \arctan \left (c x \right )^{2}}{4}-\frac {13 i \arctan \left (c x \right )}{10}+\frac {c^{2} x^{2} \arctan \left (c x \right )^{2}}{2}-\frac {i \arctan \left (c x \right )^{2} c^{5} x^{5}}{5}+\frac {6 i \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right )}{5}+\frac {c^{3} x^{3} \arctan \left (c x \right )}{2}+\frac {i \arctan \left (c x \right ) c^{4} x^{4}}{10}+\frac {5 \arctan \left (c x \right )^{2}}{4}-\frac {5 c x \arctan \left (c x \right )}{2}+\frac {3 \ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )}{5}-\frac {3 \ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )}{5}-\frac {3 \ln \left (c x +i\right )^{2}}{10}-\frac {3 \ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )}{5}+\frac {3 \ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )}{5}+\frac {3 \ln \left (c x -i\right )^{2}}{10}-\frac {3 \operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )}{5}+\frac {3 \operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )}{5}-\frac {i c^{3} x^{3}}{30}-\frac {6 i \arctan \left (c x \right ) c^{2} x^{2}}{5}-\frac {c^{2} x^{2}}{4}+\frac {3 \ln \left (c^{2} x^{2}+1\right )}{2}+\frac {13 i c x}{10}\right )+2 d^{3} a b \left (-\frac {i \arctan \left (c x \right ) c^{5} x^{5}}{5}-\frac {3 c^{4} x^{4} \arctan \left (c x \right )}{4}+i \arctan \left (c x \right ) c^{3} x^{3}+\frac {c^{2} x^{2} \arctan \left (c x \right )}{2}-\frac {5 c x}{4}+\frac {i c^{4} x^{4}}{20}+\frac {c^{3} x^{3}}{4}-\frac {3 i c^{2} x^{2}}{5}+\frac {3 i \ln \left (c^{2} x^{2}+1\right )}{5}+\frac {5 \arctan \left (c x \right )}{4}\right )}{c^{2}}\) | \(455\) |
| default | \(\frac {d^{3} a^{2} \left (-\frac {1}{5} i c^{5} x^{5}-\frac {3}{4} c^{4} x^{4}+i c^{3} x^{3}+\frac {1}{2} c^{2} x^{2}\right )+d^{3} b^{2} \left (i \arctan \left (c x \right )^{2} c^{3} x^{3}-\frac {3 c^{4} x^{4} \arctan \left (c x \right )^{2}}{4}-\frac {13 i \arctan \left (c x \right )}{10}+\frac {c^{2} x^{2} \arctan \left (c x \right )^{2}}{2}-\frac {i \arctan \left (c x \right )^{2} c^{5} x^{5}}{5}+\frac {6 i \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right )}{5}+\frac {c^{3} x^{3} \arctan \left (c x \right )}{2}+\frac {i \arctan \left (c x \right ) c^{4} x^{4}}{10}+\frac {5 \arctan \left (c x \right )^{2}}{4}-\frac {5 c x \arctan \left (c x \right )}{2}+\frac {3 \ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )}{5}-\frac {3 \ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )}{5}-\frac {3 \ln \left (c x +i\right )^{2}}{10}-\frac {3 \ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )}{5}+\frac {3 \ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )}{5}+\frac {3 \ln \left (c x -i\right )^{2}}{10}-\frac {3 \operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )}{5}+\frac {3 \operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )}{5}-\frac {i c^{3} x^{3}}{30}-\frac {6 i \arctan \left (c x \right ) c^{2} x^{2}}{5}-\frac {c^{2} x^{2}}{4}+\frac {3 \ln \left (c^{2} x^{2}+1\right )}{2}+\frac {13 i c x}{10}\right )+2 d^{3} a b \left (-\frac {i \arctan \left (c x \right ) c^{5} x^{5}}{5}-\frac {3 c^{4} x^{4} \arctan \left (c x \right )}{4}+i \arctan \left (c x \right ) c^{3} x^{3}+\frac {c^{2} x^{2} \arctan \left (c x \right )}{2}-\frac {5 c x}{4}+\frac {i c^{4} x^{4}}{20}+\frac {c^{3} x^{3}}{4}-\frac {3 i c^{2} x^{2}}{5}+\frac {3 i \ln \left (c^{2} x^{2}+1\right )}{5}+\frac {5 \arctan \left (c x \right )}{4}\right )}{c^{2}}\) | \(455\) |
| risch | \(\frac {5 d^{3} b a \arctan \left (c x \right )}{2 c^{2}}+\frac {49 d^{3} a^{2}}{20 c^{2}}+\frac {i d^{3} b a \ln \left (-i c x +1\right ) x^{2}}{2}-d^{3} c b a \ln \left (-i c x +1\right ) x^{3}+\frac {d^{3} c^{3} b a \ln \left (-i c x +1\right ) x^{5}}{5}+\frac {6 i d^{3} b a \ln \left (c^{2} x^{2}+1\right )}{5 c^{2}}+\frac {i d^{3} b^{2} \left (4 c^{5} x^{5}-15 i c^{4} x^{4}-20 c^{3} x^{3}+10 i c^{2} x^{2}+i\right ) \ln \left (i c x +1\right )^{2}}{80 c^{2}}+\frac {i d^{3} c^{2} a b \,x^{4}}{10}+\frac {i d^{3} c \,b^{2} \ln \left (-i c x +1\right ) x^{3}}{4}-\frac {5 i d^{3} b^{2} \ln \left (-i c x +1\right ) x}{4 c}-\frac {i d^{3} c \,b^{2} \ln \left (-i c x +1\right )^{2} x^{3}}{4}+\frac {i d^{3} c^{3} b^{2} \ln \left (-i c x +1\right )^{2} x^{5}}{20}+\frac {d^{3} c b a \,x^{3}}{2}-\frac {19 d^{3} b^{2}}{12 c^{2}}+\frac {d^{3} x^{2} a^{2}}{2}-\frac {3 i d^{3} c^{2} a b \ln \left (-i c x +1\right ) x^{4}}{4}-\frac {3 d^{3} c^{2} a^{2} x^{4}}{4}-\frac {b^{2} d^{3} x^{2}}{4}+\frac {3 d^{3} b^{2} \ln \left (-i c x +1\right ) x^{2}}{5}+\frac {6 d^{3} b^{2} \operatorname {dilog}\left (\frac {1}{2}-\frac {i c x}{2}\right )}{5 c^{2}}-\frac {49 d^{3} b^{2} \ln \left (-i c x +1\right )^{2}}{80 c^{2}}-\frac {d^{3} b^{2} \ln \left (-i c x +1\right )^{2} x^{2}}{8}-\frac {43 i d^{3} b a}{10 c^{2}}-\frac {6 i d^{3} a b \,x^{2}}{5}-\frac {i d^{3} c^{3} a^{2} x^{5}}{5}-\frac {6 d^{3} b^{2} \ln \left (\frac {1}{2}+\frac {i c x}{2}\right ) \ln \left (-i c x +1\right )}{5 c^{2}}+\frac {6 d^{3} 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^{2}}-\frac {d^{3} b^{2} c^{2} \ln \left (-i c x +1\right ) x^{4}}{20}+\frac {3 d^{3} c^{2} b^{2} \ln \left (-i c x +1\right )^{2} x^{4}}{16}+i d^{3} c \,x^{3} a^{2}+\frac {13 i b^{2} d^{3} x}{10 c}-\frac {i b^{2} c \,d^{3} x^{3}}{30}-\frac {13 i b^{2} d^{3} \arctan \left (c x \right )}{10 c^{2}}+\left (-\frac {i d^{3} b^{2} \left (4 c^{3} x^{5}-15 i c^{2} x^{4}-20 c \,x^{3}+10 i x^{2}\right ) \ln \left (-i c x +1\right )}{40}-\frac {d^{3} b \left (8 c^{5} x^{5} a -30 i a \,c^{4} x^{4}-2 b \,c^{4} x^{4}+10 i b \,c^{3} x^{3}-40 c^{3} x^{3} a +20 i a \,c^{2} x^{2}+24 b \,c^{2} x^{2}-50 i x b c -49 b \ln \left (-i c x +1\right )\right )}{40 c^{2}}\right ) \ln \left (i c x +1\right )+\frac {3 b^{2} d^{3} \ln \left (c^{2} x^{2}+1\right )}{2 c^{2}}-\frac {5 a b \,d^{3} x}{2 c}\) | \(804\) |
Input:
int(x*(d+I*c*d*x)^3*(a+b*arctan(c*x))^2,x,method=_RETURNVERBOSE)
Output:
d^3*a^2*(-1/5*I*c^3*x^5-3/4*c^2*x^4+I*c*x^3+1/2*x^2)+d^3*b^2/c^2*(I*arctan (c*x)^2*c^3*x^3-3/4*c^4*x^4*arctan(c*x)^2-13/10*I*arctan(c*x)+1/2*c^2*x^2* arctan(c*x)^2-1/5*I*arctan(c*x)^2*c^5*x^5+6/5*I*ln(c^2*x^2+1)*arctan(c*x)+ 1/2*c^3*x^3*arctan(c*x)+1/10*I*arctan(c*x)*c^4*x^4+5/4*arctan(c*x)^2-5/2*c *x*arctan(c*x)+3/5*ln(c*x+I)*ln(c^2*x^2+1)-3/5*ln(c*x-I)*ln(c^2*x^2+1)-3/1 0*ln(c*x+I)^2-3/5*ln(c*x+I)*ln(1/2*I*(c*x-I))+3/5*ln(c*x-I)*ln(-1/2*I*(c*x +I))+3/10*ln(c*x-I)^2-3/5*dilog(1/2*I*(c*x-I))+3/5*dilog(-1/2*I*(c*x+I))-1 /30*I*c^3*x^3-6/5*I*arctan(c*x)*c^2*x^2-1/4*c^2*x^2+3/2*ln(c^2*x^2+1)+13/1 0*I*c*x)+2*d^3*a*b/c^2*(-1/5*I*arctan(c*x)*c^5*x^5-3/4*c^4*x^4*arctan(c*x) +I*arctan(c*x)*c^3*x^3+1/2*c^2*x^2*arctan(c*x)-5/4*c*x+1/20*I*c^4*x^4+1/4* c^3*x^3-3/5*I*c^2*x^2+3/5*I*ln(c^2*x^2+1)+5/4*arctan(c*x))
\[ \int x (d+i c d x)^3 (a+b \arctan (c x))^2 \, dx=\int { {\left (i \, c d x + d\right )}^{3} {\left (b \arctan \left (c x\right ) + a\right )}^{2} x \,d x } \] Input:
integrate(x*(d+I*c*d*x)^3*(a+b*arctan(c*x))^2,x, algorithm="fricas")
Output:
1/80*(4*I*b^2*c^3*d^3*x^5 + 15*b^2*c^2*d^3*x^4 - 20*I*b^2*c*d^3*x^3 - 10*b ^2*d^3*x^2)*log(-(c*x + I)/(c*x - I))^2 + integral(1/20*(-20*I*a^2*c^5*d^3 *x^6 - 60*a^2*c^4*d^3*x^5 + 40*I*a^2*c^3*d^3*x^4 - 40*a^2*c^2*d^3*x^3 + 60 *I*a^2*c*d^3*x^2 + 20*a^2*d^3*x + (20*a*b*c^5*d^3*x^6 - 4*(15*I*a*b + b^2) *c^4*d^3*x^5 - 5*(8*a*b - 3*I*b^2)*c^3*d^3*x^4 - 20*(2*I*a*b - b^2)*c^2*d^ 3*x^3 - 10*(6*a*b + I*b^2)*c*d^3*x^2 + 20*I*a*b*d^3*x)*log(-(c*x + I)/(c*x - I)))/(c^2*x^2 + 1), x)
Timed out. \[ \int x (d+i c d x)^3 (a+b \arctan (c x))^2 \, dx=\text {Timed out} \] Input:
integrate(x*(d+I*c*d*x)**3*(a+b*atan(c*x))**2,x)
Output:
Timed out
\[ \int x (d+i c d x)^3 (a+b \arctan (c x))^2 \, dx=\int { {\left (i \, c d x + d\right )}^{3} {\left (b \arctan \left (c x\right ) + a\right )}^{2} x \,d x } \] Input:
integrate(x*(d+I*c*d*x)^3*(a+b*arctan(c*x))^2,x, algorithm="maxima")
Output:
-1/5*I*a^2*c^3*d^3*x^5 - 3/4*a^2*c^2*d^3*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^3*d^3 + I*a^2*c *d^3*x^3 + 1/2*b^2*d^3*x^2*arctan(c*x)^2 - 1/2*(3*x^4*arctan(c*x) - c*((c^ 2*x^3 - 3*x)/c^4 + 3*arctan(c*x)/c^5))*a*b*c^2*d^3 + I*(2*x^3*arctan(c*x) - c*(x^2/c^2 - log(c^2*x^2 + 1)/c^4))*a*b*c*d^3 + 1/2*a^2*d^3*x^2 + (x^2*a rctan(c*x) - c*(x/c^2 - arctan(c*x)/c^3))*a*b*d^3 - 1/2*(2*c*(x/c^2 - arct an(c*x)/c^3)*arctan(c*x) + (arctan(c*x)^2 - log(c^2*x^2 + 1))/c^2)*b^2*d^3 + 1/80*(-4*I*b^2*c^3*d^3*x^5 - 15*b^2*c^2*d^3*x^4 + 20*I*b^2*c*d^3*x^3)*a rctan(c*x)^2 + 1/80*(4*b^2*c^3*d^3*x^5 - 15*I*b^2*c^2*d^3*x^4 - 20*b^2*c*d ^3*x^3)*arctan(c*x)*log(c^2*x^2 + 1) - 1/320*(-4*I*b^2*c^3*d^3*x^5 - 15*b^ 2*c^2*d^3*x^4 + 20*I*b^2*c*d^3*x^3)*log(c^2*x^2 + 1)^2 - I*integrate(1/80* (60*(b^2*c^5*d^3*x^6 - 2*b^2*c^3*d^3*x^4 - 3*b^2*c*d^3*x^2)*arctan(c*x)^2 + 5*(b^2*c^5*d^3*x^6 - 2*b^2*c^3*d^3*x^4 - 3*b^2*c*d^3*x^2)*log(c^2*x^2 + 1)^2 - 2*(19*b^2*c^4*d^3*x^5 - 20*b^2*c^2*d^3*x^3)*arctan(c*x) + (4*b^2*c^ 5*d^3*x^6 - 35*b^2*c^3*d^3*x^4 - 60*(b^2*c^4*d^3*x^5 + b^2*c^2*d^3*x^3)*ar ctan(c*x))*log(c^2*x^2 + 1))/(c^2*x^2 + 1), x) - integrate(1/80*(180*(b^2* c^4*d^3*x^5 + b^2*c^2*d^3*x^3)*arctan(c*x)^2 + 15*(b^2*c^4*d^3*x^5 + b^2*c ^2*d^3*x^3)*log(c^2*x^2 + 1)^2 + 2*(4*b^2*c^5*d^3*x^6 - 35*b^2*c^3*d^3*x^4 )*arctan(c*x) + (19*b^2*c^4*d^3*x^5 - 20*b^2*c^2*d^3*x^3 + 20*(b^2*c^5*d^3 *x^6 - 2*b^2*c^3*d^3*x^4 - 3*b^2*c*d^3*x^2)*arctan(c*x))*log(c^2*x^2 + ...
\[ \int x (d+i c d x)^3 (a+b \arctan (c x))^2 \, dx=\int { {\left (i \, c d x + d\right )}^{3} {\left (b \arctan \left (c x\right ) + a\right )}^{2} x \,d x } \] Input:
integrate(x*(d+I*c*d*x)^3*(a+b*arctan(c*x))^2,x, algorithm="giac")
Output:
integrate((I*c*d*x + d)^3*(b*arctan(c*x) + a)^2*x, x)
Timed out. \[ \int x (d+i c d x)^3 (a+b \arctan (c x))^2 \, dx=\int x\,{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^2\,{\left (d+c\,d\,x\,1{}\mathrm {i}\right )}^3 \,d x \] Input:
int(x*(a + b*atan(c*x))^2*(d + c*d*x*1i)^3,x)
Output:
int(x*(a + b*atan(c*x))^2*(d + c*d*x*1i)^3, x)
\[ \int x (d+i c d x)^3 (a+b \arctan (c x))^2 \, dx=\frac {d^{3} \left (60 \mathit {atan} \left (c x \right )^{2} b^{2} c^{3} i \,x^{3}-90 \mathit {atan} \left (c x \right ) a b \,c^{4} x^{4}-12 \mathit {atan} \left (c x \right )^{2} b^{2} c^{5} i \,x^{5}+72 \mathit {atan} \left (c x \right )^{2} b^{2} c i x +6 \mathit {atan} \left (c x \right ) b^{2} c^{4} i \,x^{4}-72 \mathit {atan} \left (c x \right ) b^{2} c^{2} i \,x^{2}+6 a b \,c^{4} i \,x^{4}-72 a b \,c^{2} i \,x^{2}-150 \mathit {atan} \left (c x \right ) b^{2} c x +30 a b \,c^{3} x^{3}-150 a b c x -24 \mathit {atan} \left (c x \right ) a b \,c^{5} i \,x^{5}+30 \mathit {atan} \left (c x \right )^{2} b^{2} c^{2} x^{2}+120 \mathit {atan} \left (c x \right ) a b \,c^{3} i \,x^{3}+60 a^{2} c^{3} i \,x^{3}+30 a^{2} c^{2} x^{2}-78 \mathit {atan} \left (c x \right ) b^{2} i +60 \mathit {atan} \left (c x \right ) a b \,c^{2} x^{2}-45 \mathit {atan} \left (c x \right )^{2} b^{2} c^{4} x^{4}+78 b^{2} c i x -72 \left (\int \mathit {atan} \left (c x \right )^{2}d x \right ) b^{2} c i +30 \mathit {atan} \left (c x \right ) b^{2} c^{3} x^{3}+72 \,\mathrm {log}\left (c^{2} x^{2}+1\right ) a b i -12 a^{2} c^{5} i \,x^{5}-2 b^{2} c^{3} i \,x^{3}-45 a^{2} c^{4} x^{4}+75 \mathit {atan} \left (c x \right )^{2} b^{2}+90 \,\mathrm {log}\left (c^{2} x^{2}+1\right ) b^{2}+150 \mathit {atan} \left (c x \right ) a b -15 b^{2} c^{2} x^{2}\right )}{60 c^{2}} \] Input:
int(x*(d+I*c*d*x)^3*(a+b*atan(c*x))^2,x)
Output:
(d**3*( - 12*atan(c*x)**2*b**2*c**5*i*x**5 - 45*atan(c*x)**2*b**2*c**4*x** 4 + 60*atan(c*x)**2*b**2*c**3*i*x**3 + 30*atan(c*x)**2*b**2*c**2*x**2 + 72 *atan(c*x)**2*b**2*c*i*x + 75*atan(c*x)**2*b**2 - 24*atan(c*x)*a*b*c**5*i* x**5 - 90*atan(c*x)*a*b*c**4*x**4 + 120*atan(c*x)*a*b*c**3*i*x**3 + 60*ata n(c*x)*a*b*c**2*x**2 + 150*atan(c*x)*a*b + 6*atan(c*x)*b**2*c**4*i*x**4 + 30*atan(c*x)*b**2*c**3*x**3 - 72*atan(c*x)*b**2*c**2*i*x**2 - 150*atan(c*x )*b**2*c*x - 78*atan(c*x)*b**2*i - 72*int(atan(c*x)**2,x)*b**2*c*i + 72*lo g(c**2*x**2 + 1)*a*b*i + 90*log(c**2*x**2 + 1)*b**2 - 12*a**2*c**5*i*x**5 - 45*a**2*c**4*x**4 + 60*a**2*c**3*i*x**3 + 30*a**2*c**2*x**2 + 6*a*b*c**4 *i*x**4 + 30*a*b*c**3*x**3 - 72*a*b*c**2*i*x**2 - 150*a*b*c*x - 2*b**2*c** 3*i*x**3 - 15*b**2*c**2*x**2 + 78*b**2*c*i*x))/(60*c**2)