Integrand size = 23, antiderivative size = 150 \[ \int \frac {a+b \arctan (c x)}{x (d+i c d x)^2} \, dx=\frac {b}{2 d^2 (i-c x)}-\frac {b \arctan (c x)}{2 d^2}+\frac {i (a+b \arctan (c x))}{d^2 (i-c x)}+\frac {a \log (x)}{d^2}+\frac {(a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{d^2}+\frac {i b \operatorname {PolyLog}(2,-i c x)}{2 d^2}-\frac {i b \operatorname {PolyLog}(2,i c x)}{2 d^2}+\frac {i b \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{2 d^2} \]
1/2*b/d^2/(I-c*x)-1/2*b*arctan(c*x)/d^2+I*(a+b*arctan(c*x))/d^2/(I-c*x)+a* ln(x)/d^2+(a+b*arctan(c*x))*ln(2/(1+I*c*x))/d^2+1/2*I*b*polylog(2,-I*c*x)/ d^2-1/2*I*b*polylog(2,I*c*x)/d^2+1/2*I*b*polylog(2,1-2/(1+I*c*x))/d^2
Time = 0.12 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.85 \[ \int \frac {a+b \arctan (c x)}{x (d+i c d x)^2} \, dx=\frac {b \left (\frac {1}{i-c x}-\arctan (c x)\right )-\frac {2 i (a+b \arctan (c x))}{-i+c x}+2 a \log (x)+2 (a+b \arctan (c x)) \log \left (\frac {2 i}{i-c x}\right )+i b \operatorname {PolyLog}(2,-i c x)-i b \operatorname {PolyLog}(2,i c x)+i b \operatorname {PolyLog}\left (2,\frac {i+c x}{-i+c x}\right )}{2 d^2} \]
(b*((I - c*x)^(-1) - ArcTan[c*x]) - ((2*I)*(a + b*ArcTan[c*x]))/(-I + c*x) + 2*a*Log[x] + 2*(a + b*ArcTan[c*x])*Log[(2*I)/(I - c*x)] + I*b*PolyLog[2 , (-I)*c*x] - I*b*PolyLog[2, I*c*x] + I*b*PolyLog[2, (I + c*x)/(-I + c*x)] )/(2*d^2)
Time = 0.37 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, 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.
| \(\displaystyle \int \frac {a+b \arctan (c x)}{x (d+i c d x)^2} \, dx\) |
| \(\Big \downarrow \) 5411 |
| \(\displaystyle \int \left (\frac {a+b \arctan (c x)}{d^2 x}-\frac {c (a+b \arctan (c x))}{d^2 (c x-i)}+\frac {i c (a+b \arctan (c x))}{d^2 (c x-i)^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {i (a+b \arctan (c x))}{d^2 (-c x+i)}+\frac {\log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{d^2}+\frac {a \log (x)}{d^2}-\frac {b \arctan (c x)}{2 d^2}+\frac {i b \operatorname {PolyLog}(2,-i c x)}{2 d^2}-\frac {i b \operatorname {PolyLog}(2,i c x)}{2 d^2}+\frac {i b \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right )}{2 d^2}+\frac {b}{2 d^2 (-c x+i)}\) |
b/(2*d^2*(I - c*x)) - (b*ArcTan[c*x])/(2*d^2) + (I*(a + b*ArcTan[c*x]))/(d ^2*(I - c*x)) + (a*Log[x])/d^2 + ((a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/ d^2 + ((I/2)*b*PolyLog[2, (-I)*c*x])/d^2 - ((I/2)*b*PolyLog[2, I*c*x])/d^2 + ((I/2)*b*PolyLog[2, 1 - 2/(1 + I*c*x)])/d^2
3.1.55.3.1 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])
Time = 0.81 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.38
| method | result | size |
| parts | \(\frac {a \ln \left (x \right )}{d^{2}}+\frac {i a}{d^{2} \left (-c x +i\right )}-\frac {a \ln \left (c^{2} x^{2}+1\right )}{2 d^{2}}-\frac {i a \arctan \left (c x \right )}{d^{2}}+\frac {b \left (\arctan \left (c x \right ) \ln \left (c x \right )-\frac {i \arctan \left (c x \right )}{c x -i}-\arctan \left (c x \right ) \ln \left (c x -i\right )+\frac {i \ln \left (c x \right ) \ln \left (i c x +1\right )}{2}-\frac {i \ln \left (c x \right ) \ln \left (-i c x +1\right )}{2}+\frac {i \operatorname {dilog}\left (i c x +1\right )}{2}-\frac {i \operatorname {dilog}\left (-i c x +1\right )}{2}-\frac {\arctan \left (c x \right )}{2}-\frac {1}{2 \left (c x -i\right )}+\frac {i \left (\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )+\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right )}{2}-\frac {i \ln \left (c x -i\right )^{2}}{4}\right )}{d^{2}}\) | \(207\) |
| derivativedivides | \(\frac {a \ln \left (c x \right )}{d^{2}}-\frac {i a}{d^{2} \left (c x -i\right )}-\frac {a \ln \left (c^{2} x^{2}+1\right )}{2 d^{2}}-\frac {i a \arctan \left (c x \right )}{d^{2}}+\frac {b \left (\arctan \left (c x \right ) \ln \left (c x \right )-\frac {i \arctan \left (c x \right )}{c x -i}-\arctan \left (c x \right ) \ln \left (c x -i\right )+\frac {i \ln \left (c x \right ) \ln \left (i c x +1\right )}{2}-\frac {i \ln \left (c x \right ) \ln \left (-i c x +1\right )}{2}+\frac {i \operatorname {dilog}\left (i c x +1\right )}{2}-\frac {i \operatorname {dilog}\left (-i c x +1\right )}{2}-\frac {\arctan \left (c x \right )}{2}-\frac {1}{2 \left (c x -i\right )}+\frac {i \left (\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )+\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right )}{2}-\frac {i \ln \left (c x -i\right )^{2}}{4}\right )}{d^{2}}\) | \(208\) |
| default | \(\frac {a \ln \left (c x \right )}{d^{2}}-\frac {i a}{d^{2} \left (c x -i\right )}-\frac {a \ln \left (c^{2} x^{2}+1\right )}{2 d^{2}}-\frac {i a \arctan \left (c x \right )}{d^{2}}+\frac {b \left (\arctan \left (c x \right ) \ln \left (c x \right )-\frac {i \arctan \left (c x \right )}{c x -i}-\arctan \left (c x \right ) \ln \left (c x -i\right )+\frac {i \ln \left (c x \right ) \ln \left (i c x +1\right )}{2}-\frac {i \ln \left (c x \right ) \ln \left (-i c x +1\right )}{2}+\frac {i \operatorname {dilog}\left (i c x +1\right )}{2}-\frac {i \operatorname {dilog}\left (-i c x +1\right )}{2}-\frac {\arctan \left (c x \right )}{2}-\frac {1}{2 \left (c x -i\right )}+\frac {i \left (\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )+\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right )}{2}-\frac {i \ln \left (c x -i\right )^{2}}{4}\right )}{d^{2}}\) | \(208\) |
| risch | \(\frac {i b \operatorname {dilog}\left (\frac {1}{2}-\frac {i c x}{2}\right )}{2 d^{2}}+\frac {i b \ln \left (\frac {1}{2}+\frac {i c x}{2}\right ) \ln \left (\frac {1}{2}-\frac {i c x}{2}\right )}{2 d^{2}}-\frac {i b \operatorname {dilog}\left (-i c x +1\right )}{2 d^{2}}-\frac {i b \ln \left (\frac {1}{2}+\frac {i c x}{2}\right ) \ln \left (-i c x +1\right )}{2 d^{2}}-\frac {i b \ln \left (-i c x +1\right )}{4 d^{2} \left (-i c x -1\right )}-\frac {b \arctan \left (c x \right )}{4 d^{2}}-\frac {b \ln \left (-i c x +1\right ) c x}{4 d^{2} \left (-i c x -1\right )}-\frac {i a \arctan \left (c x \right )}{d^{2}}+\frac {a \ln \left (-i c x \right )}{d^{2}}-\frac {a}{d^{2} \left (-i c x -1\right )}-\frac {a \ln \left (c^{2} x^{2}+1\right )}{2 d^{2}}+\frac {i b \ln \left (c^{2} x^{2}+1\right )}{8 d^{2}}-\frac {i b \ln \left (i c x +1\right )}{2 d^{2} \left (i c x +1\right )}-\frac {i b}{2 d^{2} \left (i c x +1\right )}+\frac {i b \operatorname {dilog}\left (i c x +1\right )}{2 d^{2}}+\frac {i b \ln \left (i c x +1\right )^{2}}{4 d^{2}}\) | \(279\) |
a/d^2*ln(x)+I*a/d^2/(-c*x+I)-1/2*a/d^2*ln(c^2*x^2+1)-I*a/d^2*arctan(c*x)+b /d^2*(arctan(c*x)*ln(c*x)-I*arctan(c*x)/(c*x-I)-arctan(c*x)*ln(c*x-I)+1/2* I*ln(c*x)*ln(1+I*c*x)-1/2*I*ln(c*x)*ln(1-I*c*x)+1/2*I*dilog(1+I*c*x)-1/2*I *dilog(1-I*c*x)-1/2*arctan(c*x)-1/2/(c*x-I)+1/2*I*(dilog(-1/2*I*(c*x+I))+l n(c*x-I)*ln(-1/2*I*(c*x+I)))-1/4*I*ln(c*x-I)^2)
Time = 0.26 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.85 \[ \int \frac {a+b \arctan (c x)}{x (d+i c d x)^2} \, dx=-\frac {2 \, {\left (i \, b c x + b\right )} {\rm Li}_2\left (\frac {c x + i}{c x - i} + 1\right ) - 4 \, {\left (a c x - i \, a\right )} \log \left (x\right ) - 2 \, b \log \left (-\frac {c x + i}{c x - i}\right ) - {\left (-i \, b c x - b\right )} \log \left (\frac {c x + i}{c}\right ) + {\left ({\left (4 \, a - i \, b\right )} c x - 4 i \, a - b\right )} \log \left (\frac {c x - i}{c}\right ) + 4 i \, a + 2 \, b}{4 \, {\left (c d^{2} x - i \, d^{2}\right )}} \]
-1/4*(2*(I*b*c*x + b)*dilog((c*x + I)/(c*x - I) + 1) - 4*(a*c*x - I*a)*log (x) - 2*b*log(-(c*x + I)/(c*x - I)) - (-I*b*c*x - b)*log((c*x + I)/c) + (( 4*a - I*b)*c*x - 4*I*a - b)*log((c*x - I)/c) + 4*I*a + 2*b)/(c*d^2*x - I*d ^2)
Timed out. \[ \int \frac {a+b \arctan (c x)}{x (d+i c d x)^2} \, dx=\text {Timed out} \]
\[ \int \frac {a+b \arctan (c x)}{x (d+i c d x)^2} \, dx=\int { \frac {b \arctan \left (c x\right ) + a}{{\left (i \, c d x + d\right )}^{2} x} \,d x } \]
(-2*I*c*integrate(arctan(c*x)/(c^4*d^2*x^4 + 2*c^2*d^2*x^2 + d^2), x) - in tegrate((c^2*x^2 - 1)*arctan(c*x)/(c^4*d^2*x^5 + 2*c^2*d^2*x^3 + d^2*x), x ))*b + a*(-I/(c*d^2*x - I*d^2) - log(c*x - I)/d^2 + log(x)/d^2)
\[ \int \frac {a+b \arctan (c x)}{x (d+i c d x)^2} \, dx=\int { \frac {b \arctan \left (c x\right ) + a}{{\left (i \, c d x + d\right )}^{2} x} \,d x } \]
Timed out. \[ \int \frac {a+b \arctan (c x)}{x (d+i c d x)^2} \, dx=\int \frac {a+b\,\mathrm {atan}\left (c\,x\right )}{x\,{\left (d+c\,d\,x\,1{}\mathrm {i}\right )}^2} \,d x \]