Integrand size = 25, antiderivative size = 221 \[ \int \frac {(a+b \arctan (c x))^2}{x (d+i c d x)^2} \, dx=-\frac {i b^2}{2 d^2 (i-c x)}+\frac {i b^2 \arctan (c x)}{2 d^2}+\frac {b (a+b \arctan (c x))}{d^2 (i-c x)}-\frac {(a+b \arctan (c x))^2}{2 d^2}+\frac {i (a+b \arctan (c x))^2}{d^2 (i-c x)}+\frac {2 (a+b \arctan (c x))^2 \text {arctanh}\left (1-\frac {2}{1+i c x}\right )}{d^2}+\frac {(a+b \arctan (c x))^2 \log \left (\frac {2}{1+i c x}\right )}{d^2}+\frac {i b (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i c x}\right )}{d^2}+\frac {b^2 \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i c x}\right )}{2 d^2} \]
-1/2*I*b^2/d^2/(I-c*x)+1/2*I*b^2*arctan(c*x)/d^2+b*(a+b*arctan(c*x))/d^2/( I-c*x)-1/2*(a+b*arctan(c*x))^2/d^2+I*(a+b*arctan(c*x))^2/d^2/(I-c*x)-2*(a+ b*arctan(c*x))^2*arctanh(-1+2/(1+I*c*x))/d^2+(a+b*arctan(c*x))^2*ln(2/(1+I *c*x))/d^2+I*b*(a+b*arctan(c*x))*polylog(2,-1+2/(1+I*c*x))/d^2+1/2*b^2*pol ylog(3,-1+2/(1+I*c*x))/d^2
Time = 0.81 (sec) , antiderivative size = 299, normalized size of antiderivative = 1.35 \[ \int \frac {(a+b \arctan (c x))^2}{x (d+i c d x)^2} \, dx=\frac {-\frac {24 i a^2}{-i+c x}-24 i a^2 \arctan (c x)+24 a^2 \log (c x)-12 a^2 \log \left (1+c^2 x^2\right )-12 a b \left (4 i \arctan (c x)^2+i \cos (2 \arctan (c x))+2 i \operatorname {PolyLog}\left (2,e^{2 i \arctan (c x)}\right )-2 \arctan (c x) \left (\cos (2 \arctan (c x))+2 \log \left (1-e^{2 i \arctan (c x)}\right )-i \sin (2 \arctan (c x))\right )+\sin (2 \arctan (c x))\right )+b^2 \left (-i \pi ^3-6 \cos (2 \arctan (c x))-12 i \arctan (c x) \cos (2 \arctan (c x))+12 \arctan (c x)^2 \cos (2 \arctan (c x))+24 \arctan (c x)^2 \log \left (1-e^{-2 i \arctan (c x)}\right )+24 i \arctan (c x) \operatorname {PolyLog}\left (2,e^{-2 i \arctan (c x)}\right )+12 \operatorname {PolyLog}\left (3,e^{-2 i \arctan (c x)}\right )+6 i \sin (2 \arctan (c x))-12 \arctan (c x) \sin (2 \arctan (c x))-12 i \arctan (c x)^2 \sin (2 \arctan (c x))\right )}{24 d^2} \]
(((-24*I)*a^2)/(-I + c*x) - (24*I)*a^2*ArcTan[c*x] + 24*a^2*Log[c*x] - 12* a^2*Log[1 + c^2*x^2] - 12*a*b*((4*I)*ArcTan[c*x]^2 + I*Cos[2*ArcTan[c*x]] + (2*I)*PolyLog[2, E^((2*I)*ArcTan[c*x])] - 2*ArcTan[c*x]*(Cos[2*ArcTan[c* x]] + 2*Log[1 - E^((2*I)*ArcTan[c*x])] - I*Sin[2*ArcTan[c*x]]) + Sin[2*Arc Tan[c*x]]) + b^2*((-I)*Pi^3 - 6*Cos[2*ArcTan[c*x]] - (12*I)*ArcTan[c*x]*Co s[2*ArcTan[c*x]] + 12*ArcTan[c*x]^2*Cos[2*ArcTan[c*x]] + 24*ArcTan[c*x]^2* Log[1 - E^((-2*I)*ArcTan[c*x])] + (24*I)*ArcTan[c*x]*PolyLog[2, E^((-2*I)* ArcTan[c*x])] + 12*PolyLog[3, E^((-2*I)*ArcTan[c*x])] + (6*I)*Sin[2*ArcTan [c*x]] - 12*ArcTan[c*x]*Sin[2*ArcTan[c*x]] - (12*I)*ArcTan[c*x]^2*Sin[2*Ar cTan[c*x]]))/(24*d^2)
Time = 0.74 (sec) , antiderivative size = 221, 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.
| \(\displaystyle \int \frac {(a+b \arctan (c x))^2}{x (d+i c d x)^2} \, dx\) |
| \(\Big \downarrow \) 5411 |
| \(\displaystyle \int \left (\frac {(a+b \arctan (c x))^2}{d^2 x}-\frac {c (a+b \arctan (c x))^2}{d^2 (c x-i)}+\frac {i c (a+b \arctan (c x))^2}{d^2 (c x-i)^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 \text {arctanh}\left (1-\frac {2}{1+i c x}\right ) (a+b \arctan (c x))^2}{d^2}+\frac {i b \operatorname {PolyLog}\left (2,\frac {2}{i c x+1}-1\right ) (a+b \arctan (c x))}{d^2}+\frac {b (a+b \arctan (c x))}{d^2 (-c x+i)}+\frac {i (a+b \arctan (c x))^2}{d^2 (-c x+i)}-\frac {(a+b \arctan (c x))^2}{2 d^2}+\frac {\log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))^2}{d^2}+\frac {i b^2 \arctan (c x)}{2 d^2}+\frac {b^2 \operatorname {PolyLog}\left (3,\frac {2}{i c x+1}-1\right )}{2 d^2}-\frac {i b^2}{2 d^2 (-c x+i)}\) |
((-1/2*I)*b^2)/(d^2*(I - c*x)) + ((I/2)*b^2*ArcTan[c*x])/d^2 + (b*(a + b*A rcTan[c*x]))/(d^2*(I - c*x)) - (a + b*ArcTan[c*x])^2/(2*d^2) + (I*(a + b*A rcTan[c*x])^2)/(d^2*(I - c*x)) + (2*(a + b*ArcTan[c*x])^2*ArcTanh[1 - 2/(1 + I*c*x)])/d^2 + ((a + b*ArcTan[c*x])^2*Log[2/(1 + I*c*x)])/d^2 + (I*b*(a + b*ArcTan[c*x])*PolyLog[2, -1 + 2/(1 + I*c*x)])/d^2 + (b^2*PolyLog[3, -1 + 2/(1 + I*c*x)])/(2*d^2)
3.2.8.3.1 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 = 6.57 (sec) , antiderivative size = 1670, normalized size of antiderivative = 7.56
| method | result | size |
| parts | \(\text {Expression too large to display}\) | \(1670\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(1671\) |
| default | \(\text {Expression too large to display}\) | \(1671\) |
a^2/d^2*ln(x)+I*a^2/d^2/(-c*x+I)-1/2*a^2/d^2*ln(c^2*x^2+1)-I*a^2/d^2*arcta n(c*x)+b^2/d^2*(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*arctan(c*x)^2-arctan(c*x)^2*ln((1+I*c*x)^2/(c ^2*x^2+1)-1)+arctan(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)*p olylog(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*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+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*arcta n(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+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*ar ctan(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)))^3*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)))^2*arctan(c*x)^2+arctan(c*x)^2*ln(2*I*(1+I*c*x)^2/(c^2*x^2...
\[ \int \frac {(a+b \arctan (c x))^2}{x (d+i c d x)^2} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )}^{2}}{{\left (i \, c d x + d\right )}^{2} x} \,d x } \]
1/4*(I*b^2*log(-(c*x + I)/(c*x - I))^2 - (b^2*c*x - I*b^2)*log(2*c*x/(c*x - I))*log(-(c*x + I)/(c*x - I))^2 - 2*(b^2*c*x - I*b^2)*dilog(-2*c*x/(c*x - I) + 1)*log(-(c*x + I)/(c*x - I)) + 4*(c*d^2*x - I*d^2)*integral(-(a^2*c *x + I*a^2 - ((-I*a*b - b^2)*c*x + a*b)*log(-(c*x + I)/(c*x - I)))/(c^3*d^ 2*x^4 - I*c^2*d^2*x^3 + c*d^2*x^2 - I*d^2*x), x) + 2*(b^2*c*x - I*b^2)*pol ylog(3, -(c*x + I)/(c*x - I)))/(c*d^2*x - I*d^2)
Timed out. \[ \int \frac {(a+b \arctan (c x))^2}{x (d+i c d x)^2} \, dx=\text {Timed out} \]
\[ \int \frac {(a+b \arctan (c x))^2}{x (d+i c d x)^2} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )}^{2}}{{\left (i \, c d x + d\right )}^{2} x} \,d x } \]
a^2*(-I/(c*d^2*x - I*d^2) - log(c*x - I)/d^2 + log(x)/d^2) - 1/32*(8*I*b^2 *arctan(c*x)^2 - 8*(-I*b^2*c*x - b^2)*arctan(c*x)^3 - (b^2*c*x - I*b^2)*lo g(c^2*x^2 + 1)^3 - 2*(I*b^2 + (-I*b^2*c*x - b^2)*arctan(c*x))*log(c^2*x^2 + 1)^2 - (6*b^2*c^4*(c^2/(c^8*d^2*x^2 + c^6*d^2) + log(c^2*x^2 + 1)/(c^6*d ^2*x^2 + c^4*d^2)) - 256*b^2*c^4*integrate(1/16*x^4*arctan(c*x)^2/(c^4*d^2 *x^5 + 2*c^2*d^2*x^3 + d^2*x), x) - 64*b^2*c^4*integrate(1/16*x^4*log(c^2* x^2 + 1)^2/(c^4*d^2*x^5 + 2*c^2*d^2*x^3 + d^2*x), x) - 256*b^2*c^3*integra te(1/16*x^3*arctan(c*x)/(c^4*d^2*x^5 + 2*c^2*d^2*x^3 + d^2*x), x) - 16*(c* (x/(c^4*d^2*x^2 + c^2*d^2) + arctan(c*x)/(c^3*d^2)) - 2*arctan(c*x)/(c^4*d ^2*x^2 + c^2*d^2))*a*b*c^2 - 640*b^2*c^2*integrate(1/16*x^2*arctan(c*x)^2/ (c^4*d^2*x^5 + 2*c^2*d^2*x^3 + d^2*x), x) + 3*b^2*c^2*log(c^2*x^2 + 1)^2/( c^4*d^2*x^2 + c^2*d^2) - 256*b^2*c*integrate(1/16*x*arctan(c*x)*log(c^2*x^ 2 + 1)/(c^4*d^2*x^5 + 2*c^2*d^2*x^3 + d^2*x), x) - 256*b^2*c*integrate(1/1 6*x*arctan(c*x)/(c^4*d^2*x^5 + 2*c^2*d^2*x^3 + d^2*x), x) + 384*b^2*integr ate(1/16*arctan(c*x)^2/(c^4*d^2*x^5 + 2*c^2*d^2*x^3 + d^2*x), x) + 32*b^2* integrate(1/16*log(c^2*x^2 + 1)^2/(c^4*d^2*x^5 + 2*c^2*d^2*x^3 + d^2*x), x ) + 1024*a*b*integrate(1/16*arctan(c*x)/(c^4*d^2*x^5 + 2*c^2*d^2*x^3 + d^2 *x), x))*(c*d^2*x - I*d^2) - 32*(I*c*d^2*x + d^2)*integrate(1/8*(b^2*c^3*x ^3*log(c^2*x^2 + 1)^2 - 32*a*b*c*x*arctan(c*x) + 4*(b^2*c^3*x^3 - 2*b^2*c* x)*arctan(c*x)^2 - 2*(b^2*c^3*x^3 + b^2*c*x - (b^2*c^2*x^2 - b^2)*arcta...
\[ \int \frac {(a+b \arctan (c x))^2}{x (d+i c d x)^2} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )}^{2}}{{\left (i \, c d x + d\right )}^{2} x} \,d x } \]
Timed out. \[ \int \frac {(a+b \arctan (c x))^2}{x (d+i c d x)^2} \, dx=\int \frac {{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^2}{x\,{\left (d+c\,d\,x\,1{}\mathrm {i}\right )}^2} \,d x \]