Integrand size = 25, antiderivative size = 402 \[ \int \frac {(d+i c d x)^3 (a+b \arctan (c x))^2}{x^2} \, dx=i a b c^2 d^3 x+i b^2 c^2 d^3 x \arctan (c x)-\frac {9}{2} i c d^3 (a+b \arctan (c x))^2-\frac {d^3 (a+b \arctan (c x))^2}{x}-3 c^2 d^3 x (a+b \arctan (c x))^2-\frac {1}{2} i c^3 d^3 x^2 (a+b \arctan (c x))^2+6 i c d^3 (a+b \arctan (c x))^2 \text {arctanh}\left (1-\frac {2}{1+i c x}\right )-6 b c d^3 (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )-\frac {1}{2} i b^2 c d^3 \log \left (1+c^2 x^2\right )+2 b c d^3 (a+b \arctan (c x)) \log \left (2-\frac {2}{1-i c x}\right )-i b^2 c d^3 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i c x}\right )-3 i b^2 c d^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )+3 b c d^3 (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )-3 b c d^3 (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i c x}\right )-\frac {3}{2} i b^2 c d^3 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i c x}\right )+\frac {3}{2} i b^2 c d^3 \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i c x}\right ) \]
-I*b^2*c*d^3*polylog(2,-1+2/(1-I*c*x))+3/2*I*b^2*c*d^3*polylog(3,-1+2/(1+I *c*x))-9/2*I*c*d^3*(a+b*arctan(c*x))^2-d^3*(a+b*arctan(c*x))^2/x-3*c^2*d^3 *x*(a+b*arctan(c*x))^2-3*I*b^2*c*d^3*polylog(2,1-2/(1+I*c*x))-1/2*I*c^3*d^ 3*x^2*(a+b*arctan(c*x))^2-6*b*c*d^3*(a+b*arctan(c*x))*ln(2/(1+I*c*x))-6*I* c*d^3*(a+b*arctan(c*x))^2*arctanh(-1+2/(1+I*c*x))+2*b*c*d^3*(a+b*arctan(c* x))*ln(2-2/(1-I*c*x))-3/2*I*b^2*c*d^3*polylog(3,1-2/(1+I*c*x))-1/2*I*b^2*c *d^3*ln(c^2*x^2+1)+3*b*c*d^3*(a+b*arctan(c*x))*polylog(2,1-2/(1+I*c*x))-3* b*c*d^3*(a+b*arctan(c*x))*polylog(2,-1+2/(1+I*c*x))+I*a*b*c^2*d^3*x+I*b^2* c^2*d^3*x*arctan(c*x)
Time = 0.55 (sec) , antiderivative size = 512, normalized size of antiderivative = 1.27 \[ \int \frac {(d+i c d x)^3 (a+b \arctan (c x))^2}{x^2} \, dx=\frac {d^3 \left (-8 a^2+b^2 c \pi ^3 x-24 a^2 c^2 x^2+8 i a b c^2 x^2-4 i a^2 c^3 x^3-16 a b \arctan (c x)-8 i a b c x \arctan (c x)-48 a b c^2 x^2 \arctan (c x)+8 i b^2 c^2 x^2 \arctan (c x)-8 i a b c^3 x^3 \arctan (c x)-8 b^2 \arctan (c x)^2+12 i b^2 c x \arctan (c x)^2-24 b^2 c^2 x^2 \arctan (c x)^2-4 i b^2 c^3 x^3 \arctan (c x)^2-16 b^2 c x \arctan (c x)^3+24 i b^2 c x \arctan (c x)^2 \log \left (1-e^{-2 i \arctan (c x)}\right )+16 b^2 c x \arctan (c x) \log \left (1-e^{2 i \arctan (c x)}\right )-48 b^2 c x \arctan (c x) \log \left (1+e^{2 i \arctan (c x)}\right )-24 i b^2 c x \arctan (c x)^2 \log \left (1+e^{2 i \arctan (c x)}\right )+24 i a^2 c x \log (x)+16 a b c x \log (c x)+16 a b c x \log \left (1+c^2 x^2\right )-4 i b^2 c x \log \left (1+c^2 x^2\right )-24 b^2 c x \arctan (c x) \operatorname {PolyLog}\left (2,e^{-2 i \arctan (c x)}\right )-24 b^2 c x (-i+\arctan (c x)) \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )-8 i b^2 c x \operatorname {PolyLog}\left (2,e^{2 i \arctan (c x)}\right )-24 a b c x \operatorname {PolyLog}(2,-i c x)+24 a b c x \operatorname {PolyLog}(2,i c x)+12 i b^2 c x \operatorname {PolyLog}\left (3,e^{-2 i \arctan (c x)}\right )-12 i b^2 c x \operatorname {PolyLog}\left (3,-e^{2 i \arctan (c x)}\right )\right )}{8 x} \]
(d^3*(-8*a^2 + b^2*c*Pi^3*x - 24*a^2*c^2*x^2 + (8*I)*a*b*c^2*x^2 - (4*I)*a ^2*c^3*x^3 - 16*a*b*ArcTan[c*x] - (8*I)*a*b*c*x*ArcTan[c*x] - 48*a*b*c^2*x ^2*ArcTan[c*x] + (8*I)*b^2*c^2*x^2*ArcTan[c*x] - (8*I)*a*b*c^3*x^3*ArcTan[ c*x] - 8*b^2*ArcTan[c*x]^2 + (12*I)*b^2*c*x*ArcTan[c*x]^2 - 24*b^2*c^2*x^2 *ArcTan[c*x]^2 - (4*I)*b^2*c^3*x^3*ArcTan[c*x]^2 - 16*b^2*c*x*ArcTan[c*x]^ 3 + (24*I)*b^2*c*x*ArcTan[c*x]^2*Log[1 - E^((-2*I)*ArcTan[c*x])] + 16*b^2* c*x*ArcTan[c*x]*Log[1 - E^((2*I)*ArcTan[c*x])] - 48*b^2*c*x*ArcTan[c*x]*Lo g[1 + E^((2*I)*ArcTan[c*x])] - (24*I)*b^2*c*x*ArcTan[c*x]^2*Log[1 + E^((2* I)*ArcTan[c*x])] + (24*I)*a^2*c*x*Log[x] + 16*a*b*c*x*Log[c*x] + 16*a*b*c* x*Log[1 + c^2*x^2] - (4*I)*b^2*c*x*Log[1 + c^2*x^2] - 24*b^2*c*x*ArcTan[c* x]*PolyLog[2, E^((-2*I)*ArcTan[c*x])] - 24*b^2*c*x*(-I + ArcTan[c*x])*Poly Log[2, -E^((2*I)*ArcTan[c*x])] - (8*I)*b^2*c*x*PolyLog[2, E^((2*I)*ArcTan[ c*x])] - 24*a*b*c*x*PolyLog[2, (-I)*c*x] + 24*a*b*c*x*PolyLog[2, I*c*x] + (12*I)*b^2*c*x*PolyLog[3, E^((-2*I)*ArcTan[c*x])] - (12*I)*b^2*c*x*PolyLog [3, -E^((2*I)*ArcTan[c*x])]))/(8*x)
Time = 0.92 (sec) , antiderivative size = 402, 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.
| \(\displaystyle \int \frac {(d+i c d x)^3 (a+b \arctan (c x))^2}{x^2} \, dx\) |
| \(\Big \downarrow \) 5411 |
| \(\displaystyle \int \left (-i c^3 d^3 x (a+b \arctan (c x))^2-3 c^2 d^3 (a+b \arctan (c x))^2+\frac {d^3 (a+b \arctan (c x))^2}{x^2}+\frac {3 i c d^3 (a+b \arctan (c x))^2}{x}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 6 i c d^3 \text {arctanh}\left (1-\frac {2}{1+i c x}\right ) (a+b \arctan (c x))^2-\frac {1}{2} i c^3 d^3 x^2 (a+b \arctan (c x))^2-3 c^2 d^3 x (a+b \arctan (c x))^2+3 b c d^3 \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right ) (a+b \arctan (c x))-3 b c d^3 \operatorname {PolyLog}\left (2,\frac {2}{i c x+1}-1\right ) (a+b \arctan (c x))-\frac {9}{2} i c d^3 (a+b \arctan (c x))^2-\frac {d^3 (a+b \arctan (c x))^2}{x}-6 b c d^3 \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))+2 b c d^3 \log \left (2-\frac {2}{1-i c x}\right ) (a+b \arctan (c x))+i a b c^2 d^3 x+i b^2 c^2 d^3 x \arctan (c x)-\frac {1}{2} i b^2 c d^3 \log \left (c^2 x^2+1\right )-i b^2 c d^3 \operatorname {PolyLog}\left (2,\frac {2}{1-i c x}-1\right )-3 i b^2 c d^3 \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right )-\frac {3}{2} i b^2 c d^3 \operatorname {PolyLog}\left (3,1-\frac {2}{i c x+1}\right )+\frac {3}{2} i b^2 c d^3 \operatorname {PolyLog}\left (3,\frac {2}{i c x+1}-1\right )\) |
I*a*b*c^2*d^3*x + I*b^2*c^2*d^3*x*ArcTan[c*x] - ((9*I)/2)*c*d^3*(a + b*Arc Tan[c*x])^2 - (d^3*(a + b*ArcTan[c*x])^2)/x - 3*c^2*d^3*x*(a + b*ArcTan[c* x])^2 - (I/2)*c^3*d^3*x^2*(a + b*ArcTan[c*x])^2 + (6*I)*c*d^3*(a + b*ArcTa n[c*x])^2*ArcTanh[1 - 2/(1 + I*c*x)] - 6*b*c*d^3*(a + b*ArcTan[c*x])*Log[2 /(1 + I*c*x)] - (I/2)*b^2*c*d^3*Log[1 + c^2*x^2] + 2*b*c*d^3*(a + b*ArcTan [c*x])*Log[2 - 2/(1 - I*c*x)] - I*b^2*c*d^3*PolyLog[2, -1 + 2/(1 - I*c*x)] - (3*I)*b^2*c*d^3*PolyLog[2, 1 - 2/(1 + I*c*x)] + 3*b*c*d^3*(a + b*ArcTan [c*x])*PolyLog[2, 1 - 2/(1 + I*c*x)] - 3*b*c*d^3*(a + b*ArcTan[c*x])*PolyL og[2, -1 + 2/(1 + I*c*x)] - ((3*I)/2)*b^2*c*d^3*PolyLog[3, 1 - 2/(1 + I*c* x)] + ((3*I)/2)*b^2*c*d^3*PolyLog[3, -1 + 2/(1 + I*c*x)]
3.1.89.3.1 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 = 32.12 (sec) , antiderivative size = 1437, normalized size of antiderivative = 3.57
| method | result | size |
| parts | \(\text {Expression too large to display}\) | \(1437\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(1439\) |
| default | \(\text {Expression too large to display}\) | \(1439\) |
d^3*a^2*(-1/2*I*c^3*x^2-3*c^2*x+3*I*c*ln(x)-1/x)+b^2*d^3*c*(-3*arctan(c*x) ^2*c*x-3*arctan(c*x)*polylog(2,-(1+I*c*x)^2/(c^2*x^2+1))-6*arctan(c*x)*ln( 1+I*(1+I*c*x)/(c^2*x^2+1)^(1/2))-6*arctan(c*x)*ln(1-I*(1+I*c*x)/(c^2*x^2+1 )^(1/2))+6*arctan(c*x)*polylog(2,(1+I*c*x)/(c^2*x^2+1)^(1/2))+6*arctan(c*x )*polylog(2,-(1+I*c*x)/(c^2*x^2+1)^(1/2))-3/2*Pi*arctan(c*x)^2-3/2*Pi*csgn (I/(1+(1+I*c*x)^2/(c^2*x^2+1)))*csgn(I*((1+I*c*x)^2/(c^2*x^2+1)-1))*csgn(I *((1+I*c*x)^2/(c^2*x^2+1)-1)/(1+(1+I*c*x)^2/(c^2*x^2+1)))*arctan(c*x)^2+3/ 2*Pi*csgn(I*((1+I*c*x)^2/(c^2*x^2+1)-1))*csgn(I*((1+I*c*x)^2/(c^2*x^2+1)-1 )/(1+(1+I*c*x)^2/(c^2*x^2+1)))^2*arctan(c*x)^2-3/2*Pi*csgn(I*((1+I*c*x)^2/ (c^2*x^2+1)-1)/(1+(1+I*c*x)^2/(c^2*x^2+1)))*csgn(((1+I*c*x)^2/(c^2*x^2+1)- 1)/(1+(1+I*c*x)^2/(c^2*x^2+1)))*arctan(c*x)^2+3/2*Pi*csgn(I*((1+I*c*x)^2/( c^2*x^2+1)-1)/(1+(1+I*c*x)^2/(c^2*x^2+1)))*csgn(((1+I*c*x)^2/(c^2*x^2+1)-1 )/(1+(1+I*c*x)^2/(c^2*x^2+1)))^2*arctan(c*x)^2+3/2*Pi*csgn(I/(1+(1+I*c*x)^ 2/(c^2*x^2+1)))*csgn(I*((1+I*c*x)^2/(c^2*x^2+1)-1)/(1+(1+I*c*x)^2/(c^2*x^2 +1)))^2*arctan(c*x)^2-3/2*Pi*csgn(I*((1+I*c*x)^2/(c^2*x^2+1)-1)/(1+(1+I*c* x)^2/(c^2*x^2+1)))^3*arctan(c*x)^2+3/2*Pi*csgn(((1+I*c*x)^2/(c^2*x^2+1)-1) /(1+(1+I*c*x)^2/(c^2*x^2+1)))^2*arctan(c*x)^2-3/2*Pi*csgn(((1+I*c*x)^2/(c^ 2*x^2+1)-1)/(1+(1+I*c*x)^2/(c^2*x^2+1)))^3*arctan(c*x)^2-arctan(c*x)^2/c/x +2*arctan(c*x)*ln(1+(1+I*c*x)/(c^2*x^2+1)^(1/2))-1/2*I*arctan(c*x)^2*c^2*x ^2+I*ln(1+(1+I*c*x)^2/(c^2*x^2+1))-3/2*I*polylog(3,-(1+I*c*x)^2/(c^2*x^...
\[ \int \frac {(d+i c d x)^3 (a+b \arctan (c x))^2}{x^2} \, dx=\int { \frac {{\left (i \, c d x + d\right )}^{3} {\left (b \arctan \left (c x\right ) + a\right )}^{2}}{x^{2}} \,d x } \]
integral(1/4*(-4*I*a^2*c^3*d^3*x^3 - 12*a^2*c^2*d^3*x^2 + 12*I*a^2*c*d^3*x + 4*a^2*d^3 + (I*b^2*c^3*d^3*x^3 + 3*b^2*c^2*d^3*x^2 - 3*I*b^2*c*d^3*x - b^2*d^3)*log(-(c*x + I)/(c*x - I))^2 + 4*(a*b*c^3*d^3*x^3 - 3*I*a*b*c^2*d^ 3*x^2 - 3*a*b*c*d^3*x + I*a*b*d^3)*log(-(c*x + I)/(c*x - I)))/x^2, x)
Timed out. \[ \int \frac {(d+i c d x)^3 (a+b \arctan (c x))^2}{x^2} \, dx=\text {Timed out} \]
\[ \int \frac {(d+i c d x)^3 (a+b \arctan (c x))^2}{x^2} \, dx=\int { \frac {{\left (i \, c d x + d\right )}^{3} {\left (b \arctan \left (c x\right ) + a\right )}^{2}}{x^{2}} \,d x } \]
-1/2*I*a^2*c^3*d^3*x^2 - 3*a^2*c^2*d^3*x - 3*(2*c*x*arctan(c*x) - log(c^2* x^2 + 1))*a*b*c*d^3 + 3*I*a^2*c*d^3*log(x) - (c*(log(c^2*x^2 + 1) - log(x^ 2)) + 2*arctan(c*x)/x)*a*b*d^3 - a^2*d^3/x + 1/96*(12*(-I*b^2*c^3*d^3*x^3 - 6*b^2*c^2*d^3*x^2 - 2*b^2*d^3)*arctan(c*x)^2 + 12*(b^2*c^3*d^3*x^3 - 6*I *b^2*c^2*d^3*x^2 - 2*I*b^2*d^3)*arctan(c*x)*log(c^2*x^2 + 1) - 3*(-I*b^2*c ^3*d^3*x^3 - 6*b^2*c^2*d^3*x^2 - 2*b^2*d^3)*log(c^2*x^2 + 1)^2 - 2*I*(576* b^2*c^5*d^3*integrate(1/16*x^5*arctan(c*x)^2/(c^2*x^4 + x^2), x) + 48*b^2* c^5*d^3*integrate(1/16*x^5*log(c^2*x^2 + 1)^2/(c^2*x^4 + x^2), x) + 1536*a *b*c^5*d^3*integrate(1/16*x^5*arctan(c*x)/(c^2*x^4 + x^2), x) + 96*b^2*c^5 *d^3*integrate(1/16*x^5*log(c^2*x^2 + 1)/(c^2*x^4 + x^2), x) - 576*b^2*c^4 *d^3*integrate(1/16*x^4*arctan(c*x)*log(c^2*x^2 + 1)/(c^2*x^4 + x^2), x) - 1344*b^2*c^4*d^3*integrate(1/16*x^4*arctan(c*x)/(c^2*x^4 + x^2), x) - 115 2*b^2*c^3*d^3*integrate(1/16*x^3*arctan(c*x)^2/(c^2*x^4 + x^2), x) - 3072* a*b*c^3*d^3*integrate(1/16*x^3*arctan(c*x)/(c^2*x^4 + x^2), x) - b^2*c*d^3 *log(c^2*x^2 + 1)^3 - 12*b^2*c*d^3*arctan(c*x)^2 - 384*b^2*c^2*d^3*integra te(1/16*x^2*arctan(c*x)*log(c^2*x^2 + 1)/(c^2*x^4 + x^2), x) - 9*b^2*c*d^3 *log(c^2*x^2 + 1)^2 - 1728*b^2*c*d^3*integrate(1/16*x*arctan(c*x)^2/(c^2*x ^4 + x^2), x) - 144*b^2*c*d^3*integrate(1/16*x*log(c^2*x^2 + 1)^2/(c^2*x^4 + x^2), x) - 4608*a*b*c*d^3*integrate(1/16*x*arctan(c*x)/(c^2*x^4 + x^2), x) - 192*b^2*c*d^3*integrate(1/16*x*log(c^2*x^2 + 1)/(c^2*x^4 + x^2), ...
Timed out. \[ \int \frac {(d+i c d x)^3 (a+b \arctan (c x))^2}{x^2} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {(d+i c d x)^3 (a+b \arctan (c x))^2}{x^2} \, dx=\int \frac {{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^2\,{\left (d+c\,d\,x\,1{}\mathrm {i}\right )}^3}{x^2} \,d x \]