Integrand size = 25, antiderivative size = 391 \[ \int \frac {(a+b \arctan (c x))^2}{x^2 (d+i c d x)^3} \, dx=-\frac {i b^2 c}{16 d^3 (i-c x)^2}-\frac {19 b^2 c}{16 d^3 (i-c x)}+\frac {19 b^2 c \arctan (c x)}{16 d^3}+\frac {b c (a+b \arctan (c x))}{4 d^3 (i-c x)^2}-\frac {9 i b c (a+b \arctan (c x))}{4 d^3 (i-c x)}+\frac {i c (a+b \arctan (c x))^2}{8 d^3}-\frac {(a+b \arctan (c x))^2}{d^3 x}+\frac {i c (a+b \arctan (c x))^2}{2 d^3 (i-c x)^2}+\frac {2 c (a+b \arctan (c x))^2}{d^3 (i-c x)}-\frac {6 i c (a+b \arctan (c x))^2 \text {arctanh}\left (1-\frac {2}{1+i c x}\right )}{d^3}-\frac {3 i c (a+b \arctan (c x))^2 \log \left (\frac {2}{1+i c x}\right )}{d^3}+\frac {2 b c (a+b \arctan (c x)) \log \left (2-\frac {2}{1-i c x}\right )}{d^3}-\frac {i b^2 c \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i c x}\right )}{d^3}+\frac {3 b c (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i c x}\right )}{d^3}-\frac {3 i b^2 c \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i c x}\right )}{2 d^3} \]
-1/16*I*b^2*c/d^3/(I-c*x)^2-19/16*b^2*c/d^3/(I-c*x)+19/16*b^2*c*arctan(c*x )/d^3+1/4*b*c*(a+b*arctan(c*x))/d^3/(I-c*x)^2-9/4*I*b*c*(a+b*arctan(c*x))/ d^3/(I-c*x)+1/8*I*c*(a+b*arctan(c*x))^2/d^3-(a+b*arctan(c*x))^2/d^3/x+1/2* I*c*(a+b*arctan(c*x))^2/d^3/(I-c*x)^2+2*c*(a+b*arctan(c*x))^2/d^3/(I-c*x)+ 6*I*c*(a+b*arctan(c*x))^2*arctanh(-1+2/(1+I*c*x))/d^3-3*I*c*(a+b*arctan(c* x))^2*ln(2/(1+I*c*x))/d^3+2*b*c*(a+b*arctan(c*x))*ln(2-2/(1-I*c*x))/d^3-I* b^2*c*polylog(2,-1+2/(1-I*c*x))/d^3+3*b*c*(a+b*arctan(c*x))*polylog(2,-1+2 /(1+I*c*x))/d^3-3/2*I*b^2*c*polylog(3,-1+2/(1+I*c*x))/d^3
Time = 2.96 (sec) , antiderivative size = 548, normalized size of antiderivative = 1.40 \[ \int \frac {(a+b \arctan (c x))^2}{x^2 (d+i c d x)^3} \, dx=-\frac {\frac {64 a^2}{x}-\frac {32 i a^2 c}{(-i+c x)^2}+\frac {128 a^2 c}{-i+c x}+192 a^2 c \arctan (c x)+192 i a^2 c \log (x)-96 i a^2 c \log \left (1+c^2 x^2\right )-i b^2 c \left (8 i \pi ^3-64 \arctan (c x)^2+\frac {64 i \arctan (c x)^2}{c x}+40 \cos (2 \arctan (c x))+80 i \arctan (c x) \cos (2 \arctan (c x))-80 \arctan (c x)^2 \cos (2 \arctan (c x))+\cos (4 \arctan (c x))+4 i \arctan (c x) \cos (4 \arctan (c x))-8 \arctan (c x)^2 \cos (4 \arctan (c x))-192 \arctan (c x)^2 \log \left (1-e^{-2 i \arctan (c x)}\right )-128 i \arctan (c x) \log \left (1-e^{2 i \arctan (c x)}\right )-192 i \arctan (c x) \operatorname {PolyLog}\left (2,e^{-2 i \arctan (c x)}\right )-64 \operatorname {PolyLog}\left (2,e^{2 i \arctan (c x)}\right )-96 \operatorname {PolyLog}\left (3,e^{-2 i \arctan (c x)}\right )-40 i \sin (2 \arctan (c x))+80 \arctan (c x) \sin (2 \arctan (c x))+80 i \arctan (c x)^2 \sin (2 \arctan (c x))-i \sin (4 \arctan (c x))+4 \arctan (c x) \sin (4 \arctan (c x))+8 i \arctan (c x)^2 \sin (4 \arctan (c x))\right )+\frac {4 a b \left (96 c x \arctan (c x)^2+48 c x \operatorname {PolyLog}\left (2,e^{2 i \arctan (c x)}\right )+c x \left (20 \cos (2 \arctan (c x))+\cos (4 \arctan (c x))-32 \log (c x)+16 \log \left (1+c^2 x^2\right )-20 i \sin (2 \arctan (c x))-i \sin (4 \arctan (c x))\right )+4 \arctan (c x) \left (8+10 i c x \cos (2 \arctan (c x))+i c x \cos (4 \arctan (c x))+24 i c x \log \left (1-e^{2 i \arctan (c x)}\right )+10 c x \sin (2 \arctan (c x))+c x \sin (4 \arctan (c x))\right )\right )}{x}}{64 d^3} \]
-1/64*((64*a^2)/x - ((32*I)*a^2*c)/(-I + c*x)^2 + (128*a^2*c)/(-I + c*x) + 192*a^2*c*ArcTan[c*x] + (192*I)*a^2*c*Log[x] - (96*I)*a^2*c*Log[1 + c^2*x ^2] - I*b^2*c*((8*I)*Pi^3 - 64*ArcTan[c*x]^2 + ((64*I)*ArcTan[c*x]^2)/(c*x ) + 40*Cos[2*ArcTan[c*x]] + (80*I)*ArcTan[c*x]*Cos[2*ArcTan[c*x]] - 80*Arc Tan[c*x]^2*Cos[2*ArcTan[c*x]] + Cos[4*ArcTan[c*x]] + (4*I)*ArcTan[c*x]*Cos [4*ArcTan[c*x]] - 8*ArcTan[c*x]^2*Cos[4*ArcTan[c*x]] - 192*ArcTan[c*x]^2*L og[1 - E^((-2*I)*ArcTan[c*x])] - (128*I)*ArcTan[c*x]*Log[1 - E^((2*I)*ArcT an[c*x])] - (192*I)*ArcTan[c*x]*PolyLog[2, E^((-2*I)*ArcTan[c*x])] - 64*Po lyLog[2, E^((2*I)*ArcTan[c*x])] - 96*PolyLog[3, E^((-2*I)*ArcTan[c*x])] - (40*I)*Sin[2*ArcTan[c*x]] + 80*ArcTan[c*x]*Sin[2*ArcTan[c*x]] + (80*I)*Arc Tan[c*x]^2*Sin[2*ArcTan[c*x]] - I*Sin[4*ArcTan[c*x]] + 4*ArcTan[c*x]*Sin[4 *ArcTan[c*x]] + (8*I)*ArcTan[c*x]^2*Sin[4*ArcTan[c*x]]) + (4*a*b*(96*c*x*A rcTan[c*x]^2 + 48*c*x*PolyLog[2, E^((2*I)*ArcTan[c*x])] + c*x*(20*Cos[2*Ar cTan[c*x]] + Cos[4*ArcTan[c*x]] - 32*Log[c*x] + 16*Log[1 + c^2*x^2] - (20* I)*Sin[2*ArcTan[c*x]] - I*Sin[4*ArcTan[c*x]]) + 4*ArcTan[c*x]*(8 + (10*I)* c*x*Cos[2*ArcTan[c*x]] + I*c*x*Cos[4*ArcTan[c*x]] + (24*I)*c*x*Log[1 - E^( (2*I)*ArcTan[c*x])] + 10*c*x*Sin[2*ArcTan[c*x]] + c*x*Sin[4*ArcTan[c*x]])) )/x)/d^3
Time = 1.11 (sec) , antiderivative size = 391, 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.
| \(\displaystyle \int \frac {(a+b \arctan (c x))^2}{x^2 (d+i c d x)^3} \, dx\) |
| \(\Big \downarrow \) 5411 |
| \(\displaystyle \int \left (\frac {3 i c^2 (a+b \arctan (c x))^2}{d^3 (c x-i)}+\frac {2 c^2 (a+b \arctan (c x))^2}{d^3 (c x-i)^2}-\frac {i c^2 (a+b \arctan (c x))^2}{d^3 (c x-i)^3}+\frac {(a+b \arctan (c x))^2}{d^3 x^2}-\frac {3 i c (a+b \arctan (c x))^2}{d^3 x}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {6 i c \text {arctanh}\left (1-\frac {2}{1+i c x}\right ) (a+b \arctan (c x))^2}{d^3}+\frac {3 b c \operatorname {PolyLog}\left (2,\frac {2}{i c x+1}-1\right ) (a+b \arctan (c x))}{d^3}-\frac {9 i b c (a+b \arctan (c x))}{4 d^3 (-c x+i)}+\frac {b c (a+b \arctan (c x))}{4 d^3 (-c x+i)^2}-\frac {(a+b \arctan (c x))^2}{d^3 x}+\frac {2 c (a+b \arctan (c x))^2}{d^3 (-c x+i)}+\frac {i c (a+b \arctan (c x))^2}{2 d^3 (-c x+i)^2}+\frac {i c (a+b \arctan (c x))^2}{8 d^3}+\frac {2 b c \log \left (2-\frac {2}{1-i c x}\right ) (a+b \arctan (c x))}{d^3}-\frac {3 i c \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))^2}{d^3}+\frac {19 b^2 c \arctan (c x)}{16 d^3}-\frac {i b^2 c \operatorname {PolyLog}\left (2,\frac {2}{1-i c x}-1\right )}{d^3}-\frac {3 i b^2 c \operatorname {PolyLog}\left (3,\frac {2}{i c x+1}-1\right )}{2 d^3}-\frac {19 b^2 c}{16 d^3 (-c x+i)}-\frac {i b^2 c}{16 d^3 (-c x+i)^2}\) |
((-1/16*I)*b^2*c)/(d^3*(I - c*x)^2) - (19*b^2*c)/(16*d^3*(I - c*x)) + (19* b^2*c*ArcTan[c*x])/(16*d^3) + (b*c*(a + b*ArcTan[c*x]))/(4*d^3*(I - c*x)^2 ) - (((9*I)/4)*b*c*(a + b*ArcTan[c*x]))/(d^3*(I - c*x)) + ((I/8)*c*(a + b* ArcTan[c*x])^2)/d^3 - (a + b*ArcTan[c*x])^2/(d^3*x) + ((I/2)*c*(a + b*ArcT an[c*x])^2)/(d^3*(I - c*x)^2) + (2*c*(a + b*ArcTan[c*x])^2)/(d^3*(I - c*x) ) - ((6*I)*c*(a + b*ArcTan[c*x])^2*ArcTanh[1 - 2/(1 + I*c*x)])/d^3 - ((3*I )*c*(a + b*ArcTan[c*x])^2*Log[2/(1 + I*c*x)])/d^3 + (2*b*c*(a + b*ArcTan[c *x])*Log[2 - 2/(1 - I*c*x)])/d^3 - (I*b^2*c*PolyLog[2, -1 + 2/(1 - I*c*x)] )/d^3 + (3*b*c*(a + b*ArcTan[c*x])*PolyLog[2, -1 + 2/(1 + I*c*x)])/d^3 - ( ((3*I)/2)*b^2*c*PolyLog[3, -1 + 2/(1 + I*c*x)])/d^3
3.2.17.3.1 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 = 14.81 (sec) , antiderivative size = 8688, normalized size of antiderivative = 22.22
| method | result | size |
| derivativedivides | \(\text {Expression too large to display}\) | \(8688\) |
| default | \(\text {Expression too large to display}\) | \(8688\) |
| parts | \(\text {Expression too large to display}\) | \(8690\) |
\[ \int \frac {(a+b \arctan (c x))^2}{x^2 (d+i c d x)^3} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )}^{2}}{{\left (i \, c d x + d\right )}^{3} x^{2}} \,d x } \]
-1/8*(6*(-I*b^2*c^3*x^3 - 2*b^2*c^2*x^2 + I*b^2*c*x)*log(2*c*x/(c*x - I))* log(-(c*x + I)/(c*x - I))^2 + 12*(-I*b^2*c^3*x^3 - 2*b^2*c^2*x^2 + I*b^2*c *x)*dilog(-2*c*x/(c*x - I) + 1)*log(-(c*x + I)/(c*x - I)) - (6*b^2*c^2*x^2 - 9*I*b^2*c*x - 2*b^2)*log(-(c*x + I)/(c*x - I))^2 - 8*(c^2*d^3*x^3 - 2*I *c*d^3*x^2 - d^3*x)*integral(1/2*(2*I*a^2*c*x - 2*a^2 + (6*I*b^2*c^3*x^3 + 9*b^2*c^2*x^2 - 2*(a*b + I*b^2)*c*x - 2*I*a*b)*log(-(c*x + I)/(c*x - I))) /(c^4*d^3*x^6 - 2*I*c^3*d^3*x^5 - 2*I*c*d^3*x^3 - d^3*x^2), x) + 12*(I*b^2 *c^3*x^3 + 2*b^2*c^2*x^2 - I*b^2*c*x)*polylog(3, -(c*x + I)/(c*x - I)))/(c ^2*d^3*x^3 - 2*I*c*d^3*x^2 - d^3*x)
Timed out. \[ \int \frac {(a+b \arctan (c x))^2}{x^2 (d+i c d x)^3} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {(a+b \arctan (c x))^2}{x^2 (d+i c d x)^3} \, dx=\text {Exception raised: RuntimeError} \]
\[ \int \frac {(a+b \arctan (c x))^2}{x^2 (d+i c d x)^3} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )}^{2}}{{\left (i \, c d x + d\right )}^{3} x^{2}} \,d x } \]
Timed out. \[ \int \frac {(a+b \arctan (c x))^2}{x^2 (d+i c d x)^3} \, dx=\int \frac {{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^2}{x^2\,{\left (d+c\,d\,x\,1{}\mathrm {i}\right )}^3} \,d x \]