Integrand size = 23, antiderivative size = 195 \[ \int \frac {a+b \arctan (c x)}{x (d+i c d x)^3} \, dx=\frac {i b}{8 d^3 (i-c x)^2}+\frac {5 b}{8 d^3 (i-c x)}-\frac {5 b \arctan (c x)}{8 d^3}-\frac {a+b \arctan (c x)}{2 d^3 (i-c x)^2}+\frac {i (a+b \arctan (c x))}{d^3 (i-c x)}+\frac {a \log (x)}{d^3}+\frac {(a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{d^3}+\frac {i b \operatorname {PolyLog}(2,-i c x)}{2 d^3}-\frac {i b \operatorname {PolyLog}(2,i c x)}{2 d^3}+\frac {i b \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{2 d^3} \]
1/8*I*b/d^3/(I-c*x)^2+5/8*b/d^3/(I-c*x)-5/8*b*arctan(c*x)/d^3+1/2*(-a-b*ar ctan(c*x))/d^3/(I-c*x)^2+I*(a+b*arctan(c*x))/d^3/(I-c*x)+a*ln(x)/d^3+(a+b* arctan(c*x))*ln(2/(1+I*c*x))/d^3+1/2*I*b*polylog(2,-I*c*x)/d^3-1/2*I*b*pol ylog(2,I*c*x)/d^3+1/2*I*b*polylog(2,1-2/(1+I*c*x))/d^3
Time = 0.19 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.83 \[ \int \frac {a+b \arctan (c x)}{x (d+i c d x)^3} \, dx=\frac {\frac {5 b}{i-c x}+\frac {i b}{(-i+c x)^2}-5 b \arctan (c x)-\frac {4 (a+b \arctan (c x))}{(-i+c x)^2}-\frac {8 i (a+b \arctan (c x))}{-i+c x}+8 a \log (x)+8 (a+b \arctan (c x)) \log \left (\frac {2 i}{i-c x}\right )+4 i b \operatorname {PolyLog}(2,-i c x)-4 i b \operatorname {PolyLog}(2,i c x)+4 i b \operatorname {PolyLog}\left (2,\frac {i+c x}{-i+c x}\right )}{8 d^3} \]
((5*b)/(I - c*x) + (I*b)/(-I + c*x)^2 - 5*b*ArcTan[c*x] - (4*(a + b*ArcTan [c*x]))/(-I + c*x)^2 - ((8*I)*(a + b*ArcTan[c*x]))/(-I + c*x) + 8*a*Log[x] + 8*(a + b*ArcTan[c*x])*Log[(2*I)/(I - c*x)] + (4*I)*b*PolyLog[2, (-I)*c* x] - (4*I)*b*PolyLog[2, I*c*x] + (4*I)*b*PolyLog[2, (I + c*x)/(-I + c*x)]) /(8*d^3)
Time = 0.43 (sec) , antiderivative size = 195, 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.
| \(\displaystyle \int \frac {a+b \arctan (c x)}{x (d+i c d x)^3} \, dx\) |
| \(\Big \downarrow \) 5411 |
| \(\displaystyle \int \left (\frac {a+b \arctan (c x)}{d^3 x}-\frac {c (a+b \arctan (c x))}{d^3 (c x-i)}+\frac {i c (a+b \arctan (c x))}{d^3 (c x-i)^2}+\frac {c (a+b \arctan (c x))}{d^3 (c x-i)^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {i (a+b \arctan (c x))}{d^3 (-c x+i)}-\frac {a+b \arctan (c x)}{2 d^3 (-c x+i)^2}+\frac {\log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{d^3}+\frac {a \log (x)}{d^3}-\frac {5 b \arctan (c x)}{8 d^3}+\frac {i b \operatorname {PolyLog}(2,-i c x)}{2 d^3}-\frac {i b \operatorname {PolyLog}(2,i c x)}{2 d^3}+\frac {i b \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right )}{2 d^3}+\frac {5 b}{8 d^3 (-c x+i)}+\frac {i b}{8 d^3 (-c x+i)^2}\) |
((I/8)*b)/(d^3*(I - c*x)^2) + (5*b)/(8*d^3*(I - c*x)) - (5*b*ArcTan[c*x])/ (8*d^3) - (a + b*ArcTan[c*x])/(2*d^3*(I - c*x)^2) + (I*(a + b*ArcTan[c*x]) )/(d^3*(I - c*x)) + (a*Log[x])/d^3 + ((a + b*ArcTan[c*x])*Log[2/(1 + I*c*x )])/d^3 + ((I/2)*b*PolyLog[2, (-I)*c*x])/d^3 - ((I/2)*b*PolyLog[2, I*c*x]) /d^3 + ((I/2)*b*PolyLog[2, 1 - 2/(1 + I*c*x)])/d^3
3.1.63.3.1 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 = 1.41 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.34
| method | result | size |
| derivativedivides | \(\frac {a \ln \left (c x \right )}{d^{3}}-\frac {a}{2 d^{3} \left (c x -i\right )^{2}}-\frac {i a}{d^{3} \left (c x -i\right )}-\frac {a \ln \left (c^{2} x^{2}+1\right )}{2 d^{3}}-\frac {i a \arctan \left (c x \right )}{d^{3}}+\frac {b \left (\arctan \left (c x \right ) \ln \left (c x \right )-\frac {\arctan \left (c x \right )}{2 \left (c x -i\right )^{2}}-\frac {i \arctan \left (c x \right )}{c x -i}-\arctan \left (c x \right ) \ln \left (c x -i\right )-\frac {5 \arctan \left (c x \right )}{8}+\frac {i}{8 \left (c x -i\right )^{2}}-\frac {5}{8 \left (c x -i\right )}-\frac {i \left (\operatorname {dilog}\left (-i \left (c x +i\right )\right )+\ln \left (c x \right ) \ln \left (-i \left (c x +i\right )\right )\right )}{2}+\frac {i \left (\left (\ln \left (c x \right )-\ln \left (-i c x \right )\right ) \ln \left (-i \left (-c x +i\right )\right )-\operatorname {dilog}\left (-i c x \right )\right )}{2}+\frac {i \left (\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )+\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right )}{2}-\frac {i \ln \left (c x -i\right )^{2}}{4}\right )}{d^{3}}\) | \(261\) |
| default | \(\frac {a \ln \left (c x \right )}{d^{3}}-\frac {a}{2 d^{3} \left (c x -i\right )^{2}}-\frac {i a}{d^{3} \left (c x -i\right )}-\frac {a \ln \left (c^{2} x^{2}+1\right )}{2 d^{3}}-\frac {i a \arctan \left (c x \right )}{d^{3}}+\frac {b \left (\arctan \left (c x \right ) \ln \left (c x \right )-\frac {\arctan \left (c x \right )}{2 \left (c x -i\right )^{2}}-\frac {i \arctan \left (c x \right )}{c x -i}-\arctan \left (c x \right ) \ln \left (c x -i\right )-\frac {5 \arctan \left (c x \right )}{8}+\frac {i}{8 \left (c x -i\right )^{2}}-\frac {5}{8 \left (c x -i\right )}-\frac {i \left (\operatorname {dilog}\left (-i \left (c x +i\right )\right )+\ln \left (c x \right ) \ln \left (-i \left (c x +i\right )\right )\right )}{2}+\frac {i \left (\left (\ln \left (c x \right )-\ln \left (-i c x \right )\right ) \ln \left (-i \left (-c x +i\right )\right )-\operatorname {dilog}\left (-i c x \right )\right )}{2}+\frac {i \left (\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )+\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right )}{2}-\frac {i \ln \left (c x -i\right )^{2}}{4}\right )}{d^{3}}\) | \(261\) |
| parts | \(\frac {a \ln \left (x \right )}{d^{3}}-\frac {a}{2 d^{3} \left (-c x +i\right )^{2}}+\frac {i a}{d^{3} \left (-c x +i\right )}-\frac {a \ln \left (c^{2} x^{2}+1\right )}{2 d^{3}}-\frac {i a \arctan \left (c x \right )}{d^{3}}+\frac {b \left (\arctan \left (c x \right ) \ln \left (c x \right )-\frac {\arctan \left (c x \right )}{2 \left (c x -i\right )^{2}}-\frac {i \arctan \left (c x \right )}{c x -i}-\arctan \left (c x \right ) \ln \left (c x -i\right )-\frac {5 \arctan \left (c x \right )}{8}+\frac {i}{8 \left (c x -i\right )^{2}}-\frac {5}{8 \left (c x -i\right )}-\frac {i \left (\operatorname {dilog}\left (-i \left (c x +i\right )\right )+\ln \left (c x \right ) \ln \left (-i \left (c x +i\right )\right )\right )}{2}+\frac {i \left (\left (\ln \left (c x \right )-\ln \left (-i c x \right )\right ) \ln \left (-i \left (-c x +i\right )\right )-\operatorname {dilog}\left (-i c x \right )\right )}{2}+\frac {i \left (\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )+\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right )}{2}-\frac {i \ln \left (c x -i\right )^{2}}{4}\right )}{d^{3}}\) | \(261\) |
| risch | \(\frac {i b \operatorname {dilog}\left (\frac {1}{2}-\frac {i c x}{2}\right )}{2 d^{3}}-\frac {i b \ln \left (i c x +1\right )}{2 d^{3} \left (i c x +1\right )}+\frac {i b \ln \left (-i c x +1\right ) c^{2} x^{2}}{16 d^{3} \left (-i c x -1\right )^{2}}-\frac {i a \arctan \left (c x \right )}{d^{3}}+\frac {i \ln \left (\frac {1}{2}-\frac {i c x}{2}\right ) \ln \left (\frac {1}{2}+\frac {i c x}{2}\right ) b}{2 d^{3}}-\frac {i b \ln \left (i c x +1\right )}{4 d^{3} \left (i c x +1\right )^{2}}+\frac {a \ln \left (-i c x \right )}{d^{3}}+\frac {a}{2 d^{3} \left (-i c x -1\right )^{2}}-\frac {a}{d^{3} \left (-i c x -1\right )}+\frac {5 i b \ln \left (c^{2} x^{2}+1\right )}{32 d^{3}}-\frac {b \ln \left (-i c x +1\right ) c x}{4 d^{3} \left (-i c x -1\right )}+\frac {i b \ln \left (i c x +1\right )^{2}}{4 d^{3}}-\frac {i \operatorname {dilog}\left (-i c x +1\right ) b}{2 d^{3}}+\frac {3 i b \ln \left (-i c x +1\right )}{16 d^{3} \left (-i c x -1\right )^{2}}-\frac {i b}{8 d^{3} \left (i c x +1\right )^{2}}-\frac {i b}{2 d^{3} \left (i c x +1\right )}+\frac {b \ln \left (-i c x +1\right ) c x}{8 d^{3} \left (-i c x -1\right )^{2}}-\frac {a \ln \left (c^{2} x^{2}+1\right )}{2 d^{3}}-\frac {i b \ln \left (\frac {1}{2}+\frac {i c x}{2}\right ) \ln \left (-i c x +1\right )}{2 d^{3}}+\frac {i b \operatorname {dilog}\left (i c x +1\right )}{2 d^{3}}-\frac {i b \ln \left (-i c x +1\right )}{4 d^{3} \left (-i c x -1\right )}-\frac {5 b \arctan \left (c x \right )}{16 d^{3}}+\frac {i b}{8 d^{3} \left (-i c x -1\right )}\) | \(429\) |
a/d^3*ln(c*x)-1/2*a/d^3/(c*x-I)^2-I*a/d^3/(c*x-I)-1/2*a/d^3*ln(c^2*x^2+1)- I*a/d^3*arctan(c*x)+b/d^3*(arctan(c*x)*ln(c*x)-1/2*arctan(c*x)/(c*x-I)^2-I *arctan(c*x)/(c*x-I)-arctan(c*x)*ln(c*x-I)-5/8*arctan(c*x)+1/8*I/(c*x-I)^2 -5/8/(c*x-I)-1/2*I*(dilog(-I*(c*x+I))+ln(c*x)*ln(-I*(c*x+I)))+1/2*I*((ln(c *x)-ln(-I*c*x))*ln(-I*(-c*x+I))-dilog(-I*c*x))+1/2*I*(dilog(-1/2*I*(c*x+I) )+ln(c*x-I)*ln(-1/2*I*(c*x+I)))-1/4*I*ln(c*x-I)^2)
Time = 0.27 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.03 \[ \int \frac {a+b \arctan (c x)}{x (d+i c d x)^3} \, dx=-\frac {2 \, {\left (8 i \, a + 5 \, b\right )} c x + 8 \, {\left (i \, b c^{2} x^{2} + 2 \, b c x - i \, b\right )} {\rm Li}_2\left (\frac {c x + i}{c x - i} + 1\right ) - 16 \, {\left (a c^{2} x^{2} - 2 i \, a c x - a\right )} \log \left (x\right ) - 4 \, {\left (2 \, b c x - 3 i \, b\right )} \log \left (-\frac {c x + i}{c x - i}\right ) + 5 \, {\left (i \, b c^{2} x^{2} + 2 \, b c x - i \, b\right )} \log \left (\frac {c x + i}{c}\right ) + {\left ({\left (16 \, a - 5 i \, b\right )} c^{2} x^{2} + 2 \, {\left (-16 i \, a - 5 \, b\right )} c x - 16 \, a + 5 i \, b\right )} \log \left (\frac {c x - i}{c}\right ) + 24 \, a - 12 i \, b}{16 \, {\left (c^{2} d^{3} x^{2} - 2 i \, c d^{3} x - d^{3}\right )}} \]
-1/16*(2*(8*I*a + 5*b)*c*x + 8*(I*b*c^2*x^2 + 2*b*c*x - I*b)*dilog((c*x + I)/(c*x - I) + 1) - 16*(a*c^2*x^2 - 2*I*a*c*x - a)*log(x) - 4*(2*b*c*x - 3 *I*b)*log(-(c*x + I)/(c*x - I)) + 5*(I*b*c^2*x^2 + 2*b*c*x - I*b)*log((c*x + I)/c) + ((16*a - 5*I*b)*c^2*x^2 + 2*(-16*I*a - 5*b)*c*x - 16*a + 5*I*b) *log((c*x - I)/c) + 24*a - 12*I*b)/(c^2*d^3*x^2 - 2*I*c*d^3*x - d^3)
Exception generated. \[ \int \frac {a+b \arctan (c x)}{x (d+i c d x)^3} \, dx=\text {Exception raised: RecursionError} \]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 406 vs. \(2 (151) = 302\).
Time = 0.29 (sec) , antiderivative size = 406, normalized size of antiderivative = 2.08 \[ \int \frac {a+b \arctan (c x)}{x (d+i c d x)^3} \, dx=\frac {2 \, {\left (-8 i \, a - 5 \, b\right )} c x + 4 \, {\left (-i \, b c^{2} x^{2} - 2 \, b c x + i \, b\right )} \arctan \left (c x\right )^{2} - {\left (i \, b c^{2} x^{2} + 2 \, b c x - i \, b\right )} \log \left (c^{2} x^{2} + 1\right )^{2} - 4 \, {\left (b c^{2} x^{2} - 2 i \, b c x - b\right )} \arctan \left (c x\right ) \log \left (\frac {1}{4} \, c^{2} x^{2} + \frac {1}{4}\right ) + 16 \, {\left (b c^{2} x^{2} - 2 i \, b c x - b\right )} \arctan \left (c x\right ) \log \left (c x\right ) - {\left ({\left (16 i \, a + 5 \, b\right )} c^{2} x^{2} + 2 \, {\left (16 \, a + 3 i \, b\right )} c x - 16 i \, a + 19 \, b\right )} \arctan \left (c x\right ) + 5 \, {\left (b c^{2} x^{2} - 2 i \, b c x - b\right )} \arctan \left (c x, -1\right ) + 8 \, {\left (-i \, b c^{2} x^{2} - 2 \, b c x + i \, b\right )} {\rm Li}_2\left (i \, c x + 1\right ) + 8 \, {\left (i \, b c^{2} x^{2} + 2 \, b c x - i \, b\right )} {\rm Li}_2\left (\frac {1}{2} i \, c x + \frac {1}{2}\right ) + 8 \, {\left (i \, b c^{2} x^{2} + 2 \, b c x - i \, b\right )} {\rm Li}_2\left (-i \, c x + 1\right ) - 2 \, {\left (2 \, {\left (\pi b + 2 \, a\right )} c^{2} x^{2} - 4 \, {\left (i \, \pi b + 2 i \, a\right )} c x - 2 \, \pi b - {\left (i \, b c^{2} x^{2} + 2 \, b c x - i \, b\right )} \log \left (\frac {1}{4} \, c^{2} x^{2} + \frac {1}{4}\right ) - 4 \, a\right )} \log \left (c^{2} x^{2} + 1\right ) + 16 \, {\left (a c^{2} x^{2} - 2 i \, a c x - a\right )} \log \left (x\right ) - 24 \, a + 12 i \, b}{16 \, {\left (c^{2} d^{3} x^{2} - 2 i \, c d^{3} x - d^{3}\right )}} \]
1/16*(2*(-8*I*a - 5*b)*c*x + 4*(-I*b*c^2*x^2 - 2*b*c*x + I*b)*arctan(c*x)^ 2 - (I*b*c^2*x^2 + 2*b*c*x - I*b)*log(c^2*x^2 + 1)^2 - 4*(b*c^2*x^2 - 2*I* b*c*x - b)*arctan(c*x)*log(1/4*c^2*x^2 + 1/4) + 16*(b*c^2*x^2 - 2*I*b*c*x - b)*arctan(c*x)*log(c*x) - ((16*I*a + 5*b)*c^2*x^2 + 2*(16*a + 3*I*b)*c*x - 16*I*a + 19*b)*arctan(c*x) + 5*(b*c^2*x^2 - 2*I*b*c*x - b)*arctan2(c*x, -1) + 8*(-I*b*c^2*x^2 - 2*b*c*x + I*b)*dilog(I*c*x + 1) + 8*(I*b*c^2*x^2 + 2*b*c*x - I*b)*dilog(1/2*I*c*x + 1/2) + 8*(I*b*c^2*x^2 + 2*b*c*x - I*b)* dilog(-I*c*x + 1) - 2*(2*(pi*b + 2*a)*c^2*x^2 - 4*(I*pi*b + 2*I*a)*c*x - 2 *pi*b - (I*b*c^2*x^2 + 2*b*c*x - I*b)*log(1/4*c^2*x^2 + 1/4) - 4*a)*log(c^ 2*x^2 + 1) + 16*(a*c^2*x^2 - 2*I*a*c*x - a)*log(x) - 24*a + 12*I*b)/(c^2*d ^3*x^2 - 2*I*c*d^3*x - d^3)
\[ \int \frac {a+b \arctan (c x)}{x (d+i c d x)^3} \, dx=\int { \frac {b \arctan \left (c x\right ) + a}{{\left (i \, c d x + d\right )}^{3} x} \,d x } \]
Timed out. \[ \int \frac {a+b \arctan (c x)}{x (d+i c d x)^3} \, dx=\int \frac {a+b\,\mathrm {atan}\left (c\,x\right )}{x\,{\left (d+c\,d\,x\,1{}\mathrm {i}\right )}^3} \,d x \]