Integrand size = 25, antiderivative size = 433 \[ \int \frac {x^4 (a+b \arctan (c x))^2}{(d+i c d x)^2} \, dx=\frac {2 i a b x}{c^4 d^2}-\frac {b^2 x}{3 c^4 d^2}+\frac {b^2}{2 c^5 d^2 (i-c x)}-\frac {b^2 \arctan (c x)}{6 c^5 d^2}+\frac {2 i b^2 x \arctan (c x)}{c^4 d^2}+\frac {b x^2 (a+b \arctan (c x))}{3 c^3 d^2}+\frac {i b (a+b \arctan (c x))}{c^5 d^2 (i-c x)}+\frac {11 i (a+b \arctan (c x))^2}{6 c^5 d^2}+\frac {3 x (a+b \arctan (c x))^2}{c^4 d^2}-\frac {i x^2 (a+b \arctan (c x))^2}{c^3 d^2}-\frac {x^3 (a+b \arctan (c x))^2}{3 c^2 d^2}-\frac {(a+b \arctan (c x))^2}{c^5 d^2 (i-c x)}+\frac {20 b (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{3 c^5 d^2}-\frac {4 i (a+b \arctan (c x))^2 \log \left (\frac {2}{1+i c x}\right )}{c^5 d^2}-\frac {i b^2 \log \left (1+c^2 x^2\right )}{c^5 d^2}+\frac {10 i b^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{3 c^5 d^2}+\frac {4 b (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{c^5 d^2}-\frac {2 i b^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i c x}\right )}{c^5 d^2} \]
2*I*b^2*x*arctan(c*x)/c^4/d^2-1/3*b^2*x/c^4/d^2+1/2*b^2/c^5/d^2/(I-c*x)-1/ 6*b^2*arctan(c*x)/c^5/d^2+11/6*I*(a+b*arctan(c*x))^2/c^5/d^2+1/3*b*x^2*(a+ b*arctan(c*x))/c^3/d^2-4*I*(a+b*arctan(c*x))^2*ln(2/(1+I*c*x))/c^5/d^2+10/ 3*I*b^2*polylog(2,1-2/(1+I*c*x))/c^5/d^2+3*x*(a+b*arctan(c*x))^2/c^4/d^2+2 *I*a*b*x/c^4/d^2-1/3*x^3*(a+b*arctan(c*x))^2/c^2/d^2-(a+b*arctan(c*x))^2/c ^5/d^2/(I-c*x)+20/3*b*(a+b*arctan(c*x))*ln(2/(1+I*c*x))/c^5/d^2-I*b^2*ln(c ^2*x^2+1)/c^5/d^2-2*I*b^2*polylog(3,1-2/(1+I*c*x))/c^5/d^2-I*x^2*(a+b*arct an(c*x))^2/c^3/d^2+4*b*(a+b*arctan(c*x))*polylog(2,1-2/(1+I*c*x))/c^5/d^2+ I*b*(a+b*arctan(c*x))/c^5/d^2/(I-c*x)
Time = 1.97 (sec) , antiderivative size = 502, normalized size of antiderivative = 1.16 \[ \int \frac {x^4 (a+b \arctan (c x))^2}{(d+i c d x)^2} \, dx=-\frac {-36 a^2 c x+12 i a^2 c^2 x^2+4 a^2 c^3 x^3-\frac {12 a^2}{-i+c x}+48 a^2 \arctan (c x)-24 i a^2 \log \left (1+c^2 x^2\right )+2 a b \left (-2-12 i c x-2 c^2 x^2+48 \arctan (c x)^2-3 \cos (2 \arctan (c x))+20 \log \left (1+c^2 x^2\right )+24 \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )+2 \arctan (c x) \left (6 i-18 c x+6 i c^2 x^2+2 c^3 x^3-3 i \cos (2 \arctan (c x))+24 i \log \left (1+e^{2 i \arctan (c x)}\right )-3 \sin (2 \arctan (c x))\right )+3 i \sin (2 \arctan (c x))\right )+b^2 \left (4 c x-4 \arctan (c x)-24 i c x \arctan (c x)-4 c^2 x^2 \arctan (c x)+52 i \arctan (c x)^2-36 c x \arctan (c x)^2+12 i c^2 x^2 \arctan (c x)^2+4 c^3 x^3 \arctan (c x)^2+32 \arctan (c x)^3+3 i \cos (2 \arctan (c x))-6 \arctan (c x) \cos (2 \arctan (c x))-6 i \arctan (c x)^2 \cos (2 \arctan (c x))-80 \arctan (c x) \log \left (1+e^{2 i \arctan (c x)}\right )+48 i \arctan (c x)^2 \log \left (1+e^{2 i \arctan (c x)}\right )+12 i \log \left (1+c^2 x^2\right )+8 (5 i+6 \arctan (c x)) \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )+24 i \operatorname {PolyLog}\left (3,-e^{2 i \arctan (c x)}\right )+3 \sin (2 \arctan (c x))+6 i \arctan (c x) \sin (2 \arctan (c x))-6 \arctan (c x)^2 \sin (2 \arctan (c x))\right )}{12 c^5 d^2} \]
-1/12*(-36*a^2*c*x + (12*I)*a^2*c^2*x^2 + 4*a^2*c^3*x^3 - (12*a^2)/(-I + c *x) + 48*a^2*ArcTan[c*x] - (24*I)*a^2*Log[1 + c^2*x^2] + 2*a*b*(-2 - (12*I )*c*x - 2*c^2*x^2 + 48*ArcTan[c*x]^2 - 3*Cos[2*ArcTan[c*x]] + 20*Log[1 + c ^2*x^2] + 24*PolyLog[2, -E^((2*I)*ArcTan[c*x])] + 2*ArcTan[c*x]*(6*I - 18* c*x + (6*I)*c^2*x^2 + 2*c^3*x^3 - (3*I)*Cos[2*ArcTan[c*x]] + (24*I)*Log[1 + E^((2*I)*ArcTan[c*x])] - 3*Sin[2*ArcTan[c*x]]) + (3*I)*Sin[2*ArcTan[c*x] ]) + b^2*(4*c*x - 4*ArcTan[c*x] - (24*I)*c*x*ArcTan[c*x] - 4*c^2*x^2*ArcTa n[c*x] + (52*I)*ArcTan[c*x]^2 - 36*c*x*ArcTan[c*x]^2 + (12*I)*c^2*x^2*ArcT an[c*x]^2 + 4*c^3*x^3*ArcTan[c*x]^2 + 32*ArcTan[c*x]^3 + (3*I)*Cos[2*ArcTa n[c*x]] - 6*ArcTan[c*x]*Cos[2*ArcTan[c*x]] - (6*I)*ArcTan[c*x]^2*Cos[2*Arc Tan[c*x]] - 80*ArcTan[c*x]*Log[1 + E^((2*I)*ArcTan[c*x])] + (48*I)*ArcTan[ c*x]^2*Log[1 + E^((2*I)*ArcTan[c*x])] + (12*I)*Log[1 + c^2*x^2] + 8*(5*I + 6*ArcTan[c*x])*PolyLog[2, -E^((2*I)*ArcTan[c*x])] + (24*I)*PolyLog[3, -E^ ((2*I)*ArcTan[c*x])] + 3*Sin[2*ArcTan[c*x]] + (6*I)*ArcTan[c*x]*Sin[2*ArcT an[c*x]] - 6*ArcTan[c*x]^2*Sin[2*ArcTan[c*x]]))/(c^5*d^2)
Time = 1.03 (sec) , antiderivative size = 433, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, 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.
| \(\displaystyle \int \frac {x^4 (a+b \arctan (c x))^2}{(d+i c d x)^2} \, dx\) |
| \(\Big \downarrow \) 5411 |
| \(\displaystyle \int \left (\frac {4 i (a+b \arctan (c x))^2}{c^4 d^2 (c x-i)}+\frac {3 (a+b \arctan (c x))^2}{c^4 d^2}-\frac {(a+b \arctan (c x))^2}{c^4 d^2 (c x-i)^2}-\frac {2 i x (a+b \arctan (c x))^2}{c^3 d^2}-\frac {x^2 (a+b \arctan (c x))^2}{c^2 d^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {4 b \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right ) (a+b \arctan (c x))}{c^5 d^2}-\frac {(a+b \arctan (c x))^2}{c^5 d^2 (-c x+i)}+\frac {11 i (a+b \arctan (c x))^2}{6 c^5 d^2}+\frac {i b (a+b \arctan (c x))}{c^5 d^2 (-c x+i)}-\frac {4 i \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))^2}{c^5 d^2}+\frac {20 b \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{3 c^5 d^2}+\frac {3 x (a+b \arctan (c x))^2}{c^4 d^2}-\frac {i x^2 (a+b \arctan (c x))^2}{c^3 d^2}+\frac {b x^2 (a+b \arctan (c x))}{3 c^3 d^2}-\frac {x^3 (a+b \arctan (c x))^2}{3 c^2 d^2}+\frac {2 i a b x}{c^4 d^2}-\frac {b^2 \arctan (c x)}{6 c^5 d^2}+\frac {2 i b^2 x \arctan (c x)}{c^4 d^2}+\frac {10 i b^2 \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right )}{3 c^5 d^2}-\frac {2 i b^2 \operatorname {PolyLog}\left (3,1-\frac {2}{i c x+1}\right )}{c^5 d^2}+\frac {b^2}{2 c^5 d^2 (-c x+i)}-\frac {b^2 x}{3 c^4 d^2}-\frac {i b^2 \log \left (c^2 x^2+1\right )}{c^5 d^2}\) |
((2*I)*a*b*x)/(c^4*d^2) - (b^2*x)/(3*c^4*d^2) + b^2/(2*c^5*d^2*(I - c*x)) - (b^2*ArcTan[c*x])/(6*c^5*d^2) + ((2*I)*b^2*x*ArcTan[c*x])/(c^4*d^2) + (b *x^2*(a + b*ArcTan[c*x]))/(3*c^3*d^2) + (I*b*(a + b*ArcTan[c*x]))/(c^5*d^2 *(I - c*x)) + (((11*I)/6)*(a + b*ArcTan[c*x])^2)/(c^5*d^2) + (3*x*(a + b*A rcTan[c*x])^2)/(c^4*d^2) - (I*x^2*(a + b*ArcTan[c*x])^2)/(c^3*d^2) - (x^3* (a + b*ArcTan[c*x])^2)/(3*c^2*d^2) - (a + b*ArcTan[c*x])^2/(c^5*d^2*(I - c *x)) + (20*b*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(3*c^5*d^2) - ((4*I)* (a + b*ArcTan[c*x])^2*Log[2/(1 + I*c*x)])/(c^5*d^2) - (I*b^2*Log[1 + c^2*x ^2])/(c^5*d^2) + (((10*I)/3)*b^2*PolyLog[2, 1 - 2/(1 + I*c*x)])/(c^5*d^2) + (4*b*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 + I*c*x)])/(c^5*d^2) - ((2* I)*b^2*PolyLog[3, 1 - 2/(1 + I*c*x)])/(c^5*d^2)
3.2.3.3.1 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])
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 36.24 (sec) , antiderivative size = 1199, normalized size of antiderivative = 2.77
| method | result | size |
| derivativedivides | \(\text {Expression too large to display}\) | \(1199\) |
| default | \(\text {Expression too large to display}\) | \(1199\) |
| parts | \(\text {Expression too large to display}\) | \(1254\) |
1/c^5*(2*I*a*b/d^2*c*x+2*a*b/d^2*arctan(c*x)/(c*x-I)+4*a*b/d^2*ln(c*x-I)*l n(-1/2*I*(c*x+I))-11/12*I*a*b/d^2*arctan(1/2*c*x)+11/6*I*a*b/d^2*arctan(1/ 2*c*x-1/2*I)-I*a*b/d^2/(c*x-I)+11/12*I*a*b/d^2*arctan(1/6*c^3*x^3+7/6*c*x) -29/6*I*a*b/d^2*arctan(c*x)+1/3*a*b/d^2*c^2*x^2-I*a^2/d^2*c^2*x^2+2*I*a^2/ d^2*ln(c^2*x^2+1)+3*a^2/d^2*c*x-1/3*a^2/d^2*c^3*x^3-2*a*b/d^2*ln(c*x-I)^2+ 4*a*b/d^2*dilog(-1/2*I*(c*x+I))-11/24*a*b/d^2*ln(c^4*x^4+10*c^2*x^2+9)-29/ 12*a*b/d^2*ln(c^2*x^2+1)-2*I*a*b/d^2*arctan(c*x)*c^2*x^2+6*a*b/d^2*arctan( c*x)*c*x-2/3*a*b/d^2*arctan(c*x)*c^3*x^3+8*I*a*b/d^2*arctan(c*x)*ln(c*x-I) +7/3*a*b/d^2+b^2/d^2*(-1/3*I-1/3*c*x-1/3*c^3*x^3*arctan(c*x)^2-8/3*arctan( c*x)^3+3*arctan(c*x)^2*c*x-4*arctan(c*x)*polylog(2,-(1+I*c*x)^2/(c^2*x^2+1 ))+2/3*arctan(c*x)*(c*x-I)*(c*x+I)-1/3*arctan(c*x)*(c*x-I)^2+20/3*arctan(c *x)*ln(1+I*(1+I*c*x)/(c^2*x^2+1)^(1/2))+20/3*arctan(c*x)*ln(1-I*(1+I*c*x)/ (c^2*x^2+1)^(1/2))+4*Pi*arctan(c*x)^2-2*I*polylog(3,-(1+I*c*x)^2/(c^2*x^2+ 1))-4*Pi*csgn((1+I*c*x)^2/(c^2*x^2+1)/(1+(1+I*c*x)^2/(c^2*x^2+1)))^2*arcta n(c*x)^2-2*Pi*csgn((1+I*c*x)^2/(c^2*x^2+1)/(1+(1+I*c*x)^2/(c^2*x^2+1)))^3* arctan(c*x)^2-2*Pi*csgn((1+I*c*x)^2/(c^2*x^2+1))*csgn((1+I*c*x)^2/(c^2*x^2 +1)/(1+(1+I*c*x)^2/(c^2*x^2+1)))^2*arctan(c*x)^2+2*Pi*csgn(I/(1+(1+I*c*x)^ 2/(c^2*x^2+1)))*csgn((1+I*c*x)^2/(c^2*x^2+1)/(1+(1+I*c*x)^2/(c^2*x^2+1)))^ 2*arctan(c*x)^2-2*Pi*csgn(I/(1+(1+I*c*x)^2/(c^2*x^2+1)))*csgn((1+I*c*x)^2/ (c^2*x^2+1))*csgn((1+I*c*x)^2/(c^2*x^2+1)/(1+(1+I*c*x)^2/(c^2*x^2+1)))*...
\[ \int \frac {x^4 (a+b \arctan (c x))^2}{(d+i c d x)^2} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )}^{2} x^{4}}{{\left (i \, c d x + d\right )}^{2}} \,d x } \]
integral(1/4*(b^2*x^4*log(-(c*x + I)/(c*x - I))^2 - 4*I*a*b*x^4*log(-(c*x + I)/(c*x - I)) - 4*a^2*x^4)/(c^2*d^2*x^2 - 2*I*c*d^2*x - d^2), x)
Timed out. \[ \int \frac {x^4 (a+b \arctan (c x))^2}{(d+i c d x)^2} \, dx=\text {Timed out} \]
\[ \int \frac {x^4 (a+b \arctan (c x))^2}{(d+i c d x)^2} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )}^{2} x^{4}}{{\left (i \, c d x + d\right )}^{2}} \,d x } \]
1/3*a^2*(3/(c^6*d^2*x - I*c^5*d^2) - (c^2*x^3 + 3*I*c*x^2 - 9*x)/(c^4*d^2) + 12*I*log(c*x - I)/(c^5*d^2)) - 1/48*(48*(b^2*c*x - I*b^2)*arctan(c*x)^3 - 6*(-I*b^2*c*x - b^2)*log(c^2*x^2 + 1)^3 + 4*(b^2*c^4*x^4 + 2*I*b^2*c^3* x^3 - 6*b^2*c^2*x^2 + 9*I*b^2*c*x - 3*b^2)*arctan(c*x)^2 - (b^2*c^4*x^4 + 2*I*b^2*c^3*x^3 - 6*b^2*c^2*x^2 + 9*I*b^2*c*x - 3*b^2 - 12*(b^2*c*x - I*b^ 2)*arctan(c*x))*log(c^2*x^2 + 1)^2 + 6*(c^6*d^2*x - I*c^5*d^2)*(288*b^2*c^ 6*integrate(1/48*x^6*arctan(c*x)^2/(c^8*d^2*x^4 + 2*c^6*d^2*x^2 + c^4*d^2) , x) + 24*b^2*c^6*integrate(1/48*x^6*log(c^2*x^2 + 1)^2/(c^8*d^2*x^4 + 2*c ^6*d^2*x^2 + c^4*d^2), x) + 768*a*b*c^6*integrate(1/48*x^6*arctan(c*x)/(c^ 8*d^2*x^4 + 2*c^6*d^2*x^2 + c^4*d^2), x) + 32*b^2*c^6*integrate(1/48*x^6*l og(c^2*x^2 + 1)/(c^8*d^2*x^4 + 2*c^6*d^2*x^2 + c^4*d^2), x) + 192*b^2*c^5* integrate(1/48*x^5*arctan(c*x)*log(c^2*x^2 + 1)/(c^8*d^2*x^4 + 2*c^6*d^2*x ^2 + c^4*d^2), x) + 128*b^2*c^5*integrate(1/48*x^5*arctan(c*x)/(c^8*d^2*x^ 4 + 2*c^6*d^2*x^2 + c^4*d^2), x) - 288*b^2*c^4*integrate(1/48*x^4*arctan(c *x)^2/(c^8*d^2*x^4 + 2*c^6*d^2*x^2 + c^4*d^2), x) - 24*b^2*c^4*integrate(1 /48*x^4*log(c^2*x^2 + 1)^2/(c^8*d^2*x^4 + 2*c^6*d^2*x^2 + c^4*d^2), x) - 7 68*a*b*c^4*integrate(1/48*x^4*arctan(c*x)/(c^8*d^2*x^4 + 2*c^6*d^2*x^2 + c ^4*d^2), x) - 160*b^2*c^4*integrate(1/48*x^4*log(c^2*x^2 + 1)/(c^8*d^2*x^4 + 2*c^6*d^2*x^2 + c^4*d^2), x) + 704*b^2*c^3*integrate(1/48*x^3*arctan(c* x)/(c^8*d^2*x^4 + 2*c^6*d^2*x^2 + c^4*d^2), x) - 768*b^2*c^2*integrate(...
\[ \int \frac {x^4 (a+b \arctan (c x))^2}{(d+i c d x)^2} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )}^{2} x^{4}}{{\left (i \, c d x + d\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {x^4 (a+b \arctan (c x))^2}{(d+i c d x)^2} \, dx=\int \frac {x^4\,{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^2}{{\left (d+c\,d\,x\,1{}\mathrm {i}\right )}^2} \,d x \]