Integrand size = 25, antiderivative size = 300 \[ \int \frac {(d+i c d x)^2 (a+b \arctan (c x))^2}{x} \, dx=a b c d^2 x+b^2 c d^2 x \arctan (c x)-\frac {5}{2} d^2 (a+b \arctan (c x))^2+2 i c d^2 x (a+b \arctan (c x))^2-\frac {1}{2} c^2 d^2 x^2 (a+b \arctan (c x))^2+2 d^2 (a+b \arctan (c x))^2 \text {arctanh}\left (1-\frac {2}{1+i c x}\right )+4 i b d^2 (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )-\frac {1}{2} b^2 d^2 \log \left (1+c^2 x^2\right )-2 b^2 d^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )-i b d^2 (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )+i b d^2 (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i c x}\right )-\frac {1}{2} b^2 d^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i c x}\right )+\frac {1}{2} b^2 d^2 \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i c x}\right ) \]
a*b*c*d^2*x+b^2*c*d^2*x*arctan(c*x)-5/2*d^2*(a+b*arctan(c*x))^2+2*I*c*d^2* x*(a+b*arctan(c*x))^2-1/2*c^2*d^2*x^2*(a+b*arctan(c*x))^2-2*d^2*(a+b*arcta n(c*x))^2*arctanh(-1+2/(1+I*c*x))+4*I*b*d^2*(a+b*arctan(c*x))*ln(2/(1+I*c* x))-1/2*b^2*d^2*ln(c^2*x^2+1)-2*b^2*d^2*polylog(2,1-2/(1+I*c*x))-I*b*d^2*( a+b*arctan(c*x))*polylog(2,1-2/(1+I*c*x))+I*b*d^2*(a+b*arctan(c*x))*polylo g(2,-1+2/(1+I*c*x))-1/2*b^2*d^2*polylog(3,1-2/(1+I*c*x))+1/2*b^2*d^2*polyl og(3,-1+2/(1+I*c*x))
Time = 0.54 (sec) , antiderivative size = 360, normalized size of antiderivative = 1.20 \[ \int \frac {(d+i c d x)^2 (a+b \arctan (c x))^2}{x} \, dx=\frac {1}{2} d^2 \left (4 i a^2 c x-a^2 c^2 x^2+2 b^2 c x \arctan (c x)-b^2 \left (1+c^2 x^2\right ) \arctan (c x)^2-2 a b \left (-c x+\left (1+c^2 x^2\right ) \arctan (c x)\right )+2 a^2 \log (c x)+4 i a b \left (2 c x \arctan (c x)-\log \left (1+c^2 x^2\right )\right )-b^2 \log \left (1+c^2 x^2\right )+4 b^2 \left (\arctan (c x) \left ((1+i c x) \arctan (c x)+2 i \log \left (1+e^{2 i \arctan (c x)}\right )\right )+\operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )\right )+2 i a b (\operatorname {PolyLog}(2,-i c x)-\operatorname {PolyLog}(2,i c x))+2 b^2 \left (-\frac {i \pi ^3}{24}+\frac {2}{3} i \arctan (c x)^3+\arctan (c x)^2 \log \left (1-e^{-2 i \arctan (c x)}\right )-\arctan (c x)^2 \log \left (1+e^{2 i \arctan (c x)}\right )+i \arctan (c x) \operatorname {PolyLog}\left (2,e^{-2 i \arctan (c x)}\right )+i \arctan (c x) \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )+\frac {1}{2} \operatorname {PolyLog}\left (3,e^{-2 i \arctan (c x)}\right )-\frac {1}{2} \operatorname {PolyLog}\left (3,-e^{2 i \arctan (c x)}\right )\right )\right ) \]
(d^2*((4*I)*a^2*c*x - a^2*c^2*x^2 + 2*b^2*c*x*ArcTan[c*x] - b^2*(1 + c^2*x ^2)*ArcTan[c*x]^2 - 2*a*b*(-(c*x) + (1 + c^2*x^2)*ArcTan[c*x]) + 2*a^2*Log [c*x] + (4*I)*a*b*(2*c*x*ArcTan[c*x] - Log[1 + c^2*x^2]) - b^2*Log[1 + c^2 *x^2] + 4*b^2*(ArcTan[c*x]*((1 + I*c*x)*ArcTan[c*x] + (2*I)*Log[1 + E^((2* I)*ArcTan[c*x])]) + PolyLog[2, -E^((2*I)*ArcTan[c*x])]) + (2*I)*a*b*(PolyL og[2, (-I)*c*x] - PolyLog[2, I*c*x]) + 2*b^2*((-1/24*I)*Pi^3 + ((2*I)/3)*A rcTan[c*x]^3 + ArcTan[c*x]^2*Log[1 - E^((-2*I)*ArcTan[c*x])] - ArcTan[c*x] ^2*Log[1 + E^((2*I)*ArcTan[c*x])] + I*ArcTan[c*x]*PolyLog[2, E^((-2*I)*Arc Tan[c*x])] + I*ArcTan[c*x]*PolyLog[2, -E^((2*I)*ArcTan[c*x])] + PolyLog[3, E^((-2*I)*ArcTan[c*x])]/2 - PolyLog[3, -E^((2*I)*ArcTan[c*x])]/2)))/2
Time = 0.76 (sec) , antiderivative size = 300, 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.
| \(\displaystyle \int \frac {(d+i c d x)^2 (a+b \arctan (c x))^2}{x} \, dx\) |
| \(\Big \downarrow \) 5411 |
| \(\displaystyle \int \left (-c^2 d^2 x (a+b \arctan (c x))^2+2 i c d^2 (a+b \arctan (c x))^2+\frac {d^2 (a+b \arctan (c x))^2}{x}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 d^2 \text {arctanh}\left (1-\frac {2}{1+i c x}\right ) (a+b \arctan (c x))^2-\frac {1}{2} c^2 d^2 x^2 (a+b \arctan (c x))^2-i b d^2 \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right ) (a+b \arctan (c x))+i b d^2 \operatorname {PolyLog}\left (2,\frac {2}{i c x+1}-1\right ) (a+b \arctan (c x))+2 i c d^2 x (a+b \arctan (c x))^2-\frac {5}{2} d^2 (a+b \arctan (c x))^2+4 i b d^2 \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))+a b c d^2 x+b^2 c d^2 x \arctan (c x)-\frac {1}{2} b^2 d^2 \log \left (c^2 x^2+1\right )-2 b^2 d^2 \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right )-\frac {1}{2} b^2 d^2 \operatorname {PolyLog}\left (3,1-\frac {2}{i c x+1}\right )+\frac {1}{2} b^2 d^2 \operatorname {PolyLog}\left (3,\frac {2}{i c x+1}-1\right )\) |
a*b*c*d^2*x + b^2*c*d^2*x*ArcTan[c*x] - (5*d^2*(a + b*ArcTan[c*x])^2)/2 + (2*I)*c*d^2*x*(a + b*ArcTan[c*x])^2 - (c^2*d^2*x^2*(a + b*ArcTan[c*x])^2)/ 2 + 2*d^2*(a + b*ArcTan[c*x])^2*ArcTanh[1 - 2/(1 + I*c*x)] + (4*I)*b*d^2*( a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)] - (b^2*d^2*Log[1 + c^2*x^2])/2 - 2*b ^2*d^2*PolyLog[2, 1 - 2/(1 + I*c*x)] - I*b*d^2*(a + b*ArcTan[c*x])*PolyLog [2, 1 - 2/(1 + I*c*x)] + I*b*d^2*(a + b*ArcTan[c*x])*PolyLog[2, -1 + 2/(1 + I*c*x)] - (b^2*d^2*PolyLog[3, 1 - 2/(1 + I*c*x)])/2 + (b^2*d^2*PolyLog[3 , -1 + 2/(1 + I*c*x)])/2
3.1.80.3.1 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 = 29.47 (sec) , antiderivative size = 1317, normalized size of antiderivative = 4.39
| method | result | size |
| parts | \(\text {Expression too large to display}\) | \(1317\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(1319\) |
| default | \(\text {Expression too large to display}\) | \(1319\) |
a^2*d^2*(-1/2*c^2*x^2+2*I*c*x+ln(x))+b^2*d^2*(-1/2*polylog(3,-(1+I*c*x)^2/ (c^2*x^2+1))+2*polylog(3,(1+I*c*x)/(c^2*x^2+1)^(1/2))+2*polylog(3,-(1+I*c* x)/(c^2*x^2+1)^(1/2))-1/2*c^2*x^2*arctan(c*x)^2+ln(1+(1+I*c*x)^2/(c^2*x^2+ 1))+3/2*arctan(c*x)^2+I*arctan(c*x)*polylog(2,-(1+I*c*x)^2/(c^2*x^2+1))+4* dilog(1+I*(1+I*c*x)/(c^2*x^2+1)^(1/2))+4*dilog(1-I*(1+I*c*x)/(c^2*x^2+1)^( 1/2))+arctan(c*x)*(c*x-I)-arctan(c*x)^2*ln((1+I*c*x)^2/(c^2*x^2+1)-1)+arct an(c*x)^2*ln(c*x)+arctan(c*x)^2*ln(1-(1+I*c*x)/(c^2*x^2+1)^(1/2))+arctan(c *x)^2*ln(1+(1+I*c*x)/(c^2*x^2+1)^(1/2))-2*I*arctan(c*x)*polylog(2,(1+I*c*x )/(c^2*x^2+1)^(1/2))-2*I*arctan(c*x)*polylog(2,-(1+I*c*x)/(c^2*x^2+1)^(1/2 ))+1/2*I*Pi*arctan(c*x)^2+1/2*I*Pi*csgn(I/(1+(1+I*c*x)^2/(c^2*x^2+1)))*csg n(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+1/2*I*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)^2-1/2*I*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)^2-1/2*I*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+2*I*arct an(c*x)^2*c*x+1/2*I*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)^2-1/2*I*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+1/2*I*Pi*csgn(I*((1+I*c*x)^2/(c...
\[ \int \frac {(d+i c d x)^2 (a+b \arctan (c x))^2}{x} \, dx=\int { \frac {{\left (i \, c d x + d\right )}^{2} {\left (b \arctan \left (c x\right ) + a\right )}^{2}}{x} \,d x } \]
integral(-1/4*(4*a^2*c^2*d^2*x^2 - 8*I*a^2*c*d^2*x - 4*a^2*d^2 - (b^2*c^2* d^2*x^2 - 2*I*b^2*c*d^2*x - b^2*d^2)*log(-(c*x + I)/(c*x - I))^2 + 4*(I*a* b*c^2*d^2*x^2 + 2*a*b*c*d^2*x - I*a*b*d^2)*log(-(c*x + I)/(c*x - I)))/x, x )
\[ \int \frac {(d+i c d x)^2 (a+b \arctan (c x))^2}{x} \, dx=- d^{2} \left (\int \left (- \frac {a^{2}}{x}\right )\, dx + \int \left (- 2 i a^{2} c\right )\, dx + \int a^{2} c^{2} x\, dx + \int \left (- \frac {b^{2} \operatorname {atan}^{2}{\left (c x \right )}}{x}\right )\, dx + \int \left (- 2 i b^{2} c \operatorname {atan}^{2}{\left (c x \right )}\right )\, dx + \int \left (- \frac {2 a b \operatorname {atan}{\left (c x \right )}}{x}\right )\, dx + \int b^{2} c^{2} x \operatorname {atan}^{2}{\left (c x \right )}\, dx + \int \left (- 4 i a b c \operatorname {atan}{\left (c x \right )}\right )\, dx + \int 2 a b c^{2} x \operatorname {atan}{\left (c x \right )}\, dx\right ) \]
-d**2*(Integral(-a**2/x, x) + Integral(-2*I*a**2*c, x) + Integral(a**2*c** 2*x, x) + Integral(-b**2*atan(c*x)**2/x, x) + Integral(-2*I*b**2*c*atan(c* x)**2, x) + Integral(-2*a*b*atan(c*x)/x, x) + Integral(b**2*c**2*x*atan(c* x)**2, x) + Integral(-4*I*a*b*c*atan(c*x), x) + Integral(2*a*b*c**2*x*atan (c*x), x))
\[ \int \frac {(d+i c d x)^2 (a+b \arctan (c x))^2}{x} \, dx=\int { \frac {{\left (i \, c d x + d\right )}^{2} {\left (b \arctan \left (c x\right ) + a\right )}^{2}}{x} \,d x } \]
-12*b^2*c^4*d^2*integrate(1/16*x^4*arctan(c*x)^2/(c^2*x^3 + x), x) + 2*I*b ^2*c^4*d^2*integrate(1/8*x^4*arctan(c*x)*log(c^2*x^2 + 1)/(c^2*x^3 + x), x ) - b^2*c^4*d^2*integrate(1/16*x^4*log(c^2*x^2 + 1)^2/(c^2*x^3 + x), x) + 2*I*b^2*c^4*d^2*integrate(1/8*x^4*arctan(c*x)/(c^2*x^3 + x), x) - 32*a*b*c ^4*d^2*integrate(1/16*x^4*arctan(c*x)/(c^2*x^3 + x), x) - 2*b^2*c^4*d^2*in tegrate(1/16*x^4*log(c^2*x^2 + 1)/(c^2*x^3 + x), x) - 1/2*a^2*c^2*d^2*x^2 + 12*I*b^2*c^3*d^2*integrate(1/8*x^3*arctan(c*x)^2/(c^2*x^3 + x), x) + 8*b ^2*c^3*d^2*integrate(1/16*x^3*arctan(c*x)*log(c^2*x^2 + 1)/(c^2*x^3 + x), x) + I*b^2*c^3*d^2*integrate(1/8*x^3*log(c^2*x^2 + 1)^2/(c^2*x^3 + x), x) + 20*b^2*c^3*d^2*integrate(1/16*x^3*arctan(c*x)/(c^2*x^3 + x), x) + 5*I*b^ 2*c^3*d^2*integrate(1/8*x^3*log(c^2*x^2 + 1)/(c^2*x^3 + x), x) + 1/2*I*b^2 *d^2*arctan(c*x)^3 - 8*I*b^2*c^2*d^2*integrate(1/8*x^2*arctan(c*x)/(c^2*x^ 3 + x), x) + 2*I*a^2*c*d^2*x + 8*b^2*c*d^2*integrate(1/16*x*arctan(c*x)*lo g(c^2*x^2 + 1)/(c^2*x^3 + x), x) + I*b^2*c*d^2*integrate(1/8*x*log(c^2*x^2 + 1)^2/(c^2*x^3 + x), x) + 1/8*b^2*d^2*log(c^2*x^2 + 1)^2 + 2*I*(2*c*x*ar ctan(c*x) - log(c^2*x^2 + 1))*a*b*d^2 + 12*b^2*d^2*integrate(1/16*arctan(c *x)^2/(c^2*x^3 + x), x) - 2*I*b^2*d^2*integrate(1/8*arctan(c*x)*log(c^2*x^ 2 + 1)/(c^2*x^3 + x), x) + b^2*d^2*integrate(1/16*log(c^2*x^2 + 1)^2/(c^2* x^3 + x), x) + 32*a*b*d^2*integrate(1/16*arctan(c*x)/(c^2*x^3 + x), x) + a ^2*d^2*log(x) - 1/8*(b^2*c^2*d^2*x^2 - 4*I*b^2*c*d^2*x)*arctan(c*x)^2 +...
Timed out. \[ \int \frac {(d+i c d x)^2 (a+b \arctan (c x))^2}{x} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {(d+i c d x)^2 (a+b \arctan (c x))^2}{x} \, dx=\int \frac {{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^2\,{\left (d+c\,d\,x\,1{}\mathrm {i}\right )}^2}{x} \,d x \]