Integrand size = 23, antiderivative size = 167 \[ \int \frac {x^2 (a+b \arctan (c x))}{(d+i c d x)^2} \, dx=-\frac {a x}{c^2 d^2}-\frac {i b}{2 c^3 d^2 (i-c x)}+\frac {i b \arctan (c x)}{2 c^3 d^2}-\frac {b x \arctan (c x)}{c^2 d^2}+\frac {a+b \arctan (c x)}{c^3 d^2 (i-c x)}+\frac {2 i (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{c^3 d^2}+\frac {b \log \left (1+c^2 x^2\right )}{2 c^3 d^2}-\frac {b \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{c^3 d^2} \]
-a*x/c^2/d^2-1/2*I*b/c^3/d^2/(I-c*x)+1/2*I*b*arctan(c*x)/c^3/d^2-b*x*arcta n(c*x)/c^2/d^2+(a+b*arctan(c*x))/c^3/d^2/(I-c*x)+2*I*(a+b*arctan(c*x))*ln( 2/(1+I*c*x))/c^3/d^2+1/2*b*ln(c^2*x^2+1)/c^3/d^2-b*polylog(2,1-2/(1+I*c*x) )/c^3/d^2
Time = 0.70 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.92 \[ \int \frac {x^2 (a+b \arctan (c x))}{(d+i c d x)^2} \, dx=-\frac {4 a c x+\frac {4 a}{-i+c x}-8 a \arctan (c x)+4 i a \log \left (1+c^2 x^2\right )+b \left (-8 \arctan (c x)^2+\cos (2 \arctan (c x))-2 \log \left (1+c^2 x^2\right )-4 \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )-i \sin (2 \arctan (c x))+2 \arctan (c x) \left (2 c x+i \cos (2 \arctan (c x))-4 i \log \left (1+e^{2 i \arctan (c x)}\right )+\sin (2 \arctan (c x))\right )\right )}{4 c^3 d^2} \]
-1/4*(4*a*c*x + (4*a)/(-I + c*x) - 8*a*ArcTan[c*x] + (4*I)*a*Log[1 + c^2*x ^2] + b*(-8*ArcTan[c*x]^2 + Cos[2*ArcTan[c*x]] - 2*Log[1 + c^2*x^2] - 4*Po lyLog[2, -E^((2*I)*ArcTan[c*x])] - I*Sin[2*ArcTan[c*x]] + 2*ArcTan[c*x]*(2 *c*x + I*Cos[2*ArcTan[c*x]] - (4*I)*Log[1 + E^((2*I)*ArcTan[c*x])] + Sin[2 *ArcTan[c*x]])))/(c^3*d^2)
Time = 0.38 (sec) , antiderivative size = 167, 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.
| \(\displaystyle \int \frac {x^2 (a+b \arctan (c x))}{(d+i c d x)^2} \, dx\) |
| \(\Big \downarrow \) 5411 |
| \(\displaystyle \int \left (-\frac {2 i (a+b \arctan (c x))}{c^2 d^2 (c x-i)}-\frac {a+b \arctan (c x)}{c^2 d^2}+\frac {a+b \arctan (c x)}{c^2 d^2 (c x-i)^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a+b \arctan (c x)}{c^3 d^2 (-c x+i)}+\frac {2 i \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{c^3 d^2}-\frac {a x}{c^2 d^2}+\frac {i b \arctan (c x)}{2 c^3 d^2}-\frac {b x \arctan (c x)}{c^2 d^2}-\frac {b \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right )}{c^3 d^2}-\frac {i b}{2 c^3 d^2 (-c x+i)}+\frac {b \log \left (c^2 x^2+1\right )}{2 c^3 d^2}\) |
-((a*x)/(c^2*d^2)) - ((I/2)*b)/(c^3*d^2*(I - c*x)) + ((I/2)*b*ArcTan[c*x]) /(c^3*d^2) - (b*x*ArcTan[c*x])/(c^2*d^2) + (a + b*ArcTan[c*x])/(c^3*d^2*(I - c*x)) + ((2*I)*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(c^3*d^2) + (b*L og[1 + c^2*x^2])/(2*c^3*d^2) - (b*PolyLog[2, 1 - 2/(1 + I*c*x)])/(c^3*d^2)
3.1.52.3.1 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 = 0.73 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.62
| method | result | size |
| derivativedivides | \(\frac {-\frac {a c x}{d^{2}}-\frac {a}{d^{2} \left (c x -i\right )}+\frac {2 a \arctan \left (c x \right )}{d^{2}}+\frac {i b \arctan \left (\frac {c x}{2}\right )}{8 d^{2}}-\frac {b \arctan \left (c x \right ) c x}{d^{2}}-\frac {b \arctan \left (c x \right )}{d^{2} \left (c x -i\right )}-\frac {2 i b \arctan \left (c x \right ) \ln \left (c x -i\right )}{d^{2}}-\frac {b \ln \left (-\frac {i \left (c x +i\right )}{2}\right ) \ln \left (c x -i\right )}{d^{2}}-\frac {b \operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )}{d^{2}}+\frac {b \ln \left (c x -i\right )^{2}}{2 d^{2}}+\frac {b \ln \left (c^{4} x^{4}+10 c^{2} x^{2}+9\right )}{16 d^{2}}+\frac {3 i b \arctan \left (c x \right )}{4 d^{2}}-\frac {i a \ln \left (c^{2} x^{2}+1\right )}{d^{2}}+\frac {i b}{2 d^{2} \left (c x -i\right )}-\frac {i b \arctan \left (\frac {c x}{2}-\frac {i}{2}\right )}{4 d^{2}}+\frac {3 b \ln \left (c^{2} x^{2}+1\right )}{8 d^{2}}-\frac {i b \arctan \left (\frac {1}{6} c^{3} x^{3}+\frac {7}{6} c x \right )}{8 d^{2}}}{c^{3}}\) | \(271\) |
| default | \(\frac {-\frac {a c x}{d^{2}}-\frac {a}{d^{2} \left (c x -i\right )}+\frac {2 a \arctan \left (c x \right )}{d^{2}}+\frac {i b \arctan \left (\frac {c x}{2}\right )}{8 d^{2}}-\frac {b \arctan \left (c x \right ) c x}{d^{2}}-\frac {b \arctan \left (c x \right )}{d^{2} \left (c x -i\right )}-\frac {2 i b \arctan \left (c x \right ) \ln \left (c x -i\right )}{d^{2}}-\frac {b \ln \left (-\frac {i \left (c x +i\right )}{2}\right ) \ln \left (c x -i\right )}{d^{2}}-\frac {b \operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )}{d^{2}}+\frac {b \ln \left (c x -i\right )^{2}}{2 d^{2}}+\frac {b \ln \left (c^{4} x^{4}+10 c^{2} x^{2}+9\right )}{16 d^{2}}+\frac {3 i b \arctan \left (c x \right )}{4 d^{2}}-\frac {i a \ln \left (c^{2} x^{2}+1\right )}{d^{2}}+\frac {i b}{2 d^{2} \left (c x -i\right )}-\frac {i b \arctan \left (\frac {c x}{2}-\frac {i}{2}\right )}{4 d^{2}}+\frac {3 b \ln \left (c^{2} x^{2}+1\right )}{8 d^{2}}-\frac {i b \arctan \left (\frac {1}{6} c^{3} x^{3}+\frac {7}{6} c x \right )}{8 d^{2}}}{c^{3}}\) | \(271\) |
| parts | \(-\frac {a x}{c^{2} d^{2}}+\frac {a}{d^{2} c^{3} \left (-c x +i\right )}-\frac {2 i b \arctan \left (c x \right ) \ln \left (c x -i\right )}{c^{3} d^{2}}+\frac {2 a \arctan \left (c x \right )}{d^{2} c^{3}}-\frac {b x \arctan \left (c x \right )}{c^{2} d^{2}}-\frac {b \arctan \left (c x \right )}{c^{3} d^{2} \left (c x -i\right )}+\frac {3 i b \arctan \left (c x \right )}{4 c^{3} d^{2}}-\frac {b \ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )}{c^{3} d^{2}}-\frac {b \operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )}{c^{3} d^{2}}+\frac {b \ln \left (c x -i\right )^{2}}{2 c^{3} d^{2}}+\frac {b \ln \left (c^{4} x^{4}+10 c^{2} x^{2}+9\right )}{16 c^{3} d^{2}}-\frac {i b \arctan \left (\frac {c x}{2}-\frac {i}{2}\right )}{4 c^{3} d^{2}}+\frac {i b \arctan \left (\frac {c x}{2}\right )}{8 c^{3} d^{2}}-\frac {i a \ln \left (c^{2} x^{2}+1\right )}{d^{2} c^{3}}-\frac {i b \arctan \left (\frac {1}{6} c^{3} x^{3}+\frac {7}{6} c x \right )}{8 c^{3} d^{2}}+\frac {3 b \ln \left (c^{2} x^{2}+1\right )}{8 c^{3} d^{2}}+\frac {i b}{2 c^{3} d^{2} \left (c x -i\right )}\) | \(316\) |
| risch | \(-\frac {b \ln \left (i c x +1\right )^{2}}{2 c^{3} d^{2}}+\left (\frac {i b x}{2 c^{2} d^{2}}+\frac {i b}{2 c^{3} d^{2} \left (c x -i\right )}\right ) \ln \left (i c x +1\right )+\frac {3 i b \arctan \left (c x \right )}{4 c^{3} d^{2}}-\frac {b}{2 c^{3} d^{2}}-\frac {b \ln \left (-i c x +1\right )}{4 d^{2} c^{3} \left (-i c x -1\right )}-\frac {i b}{2 c^{3} d^{2} \left (-c x +i\right )}+\frac {3 b \ln \left (c^{2} x^{2}+1\right )}{8 c^{3} d^{2}}+\frac {b \ln \left (\frac {1}{2}+\frac {i c x}{2}\right ) \ln \left (-i c x +1\right )}{d^{2} c^{3}}-\frac {b \ln \left (\frac {1}{2}+\frac {i c x}{2}\right ) \ln \left (\frac {1}{2}-\frac {i c x}{2}\right )}{d^{2} c^{3}}-\frac {b \operatorname {dilog}\left (\frac {1}{2}-\frac {i c x}{2}\right )}{d^{2} c^{3}}-\frac {i a \ln \left (c^{2} x^{2}+1\right )}{d^{2} c^{3}}+\frac {i b \ln \left (-i c x +1\right ) x}{4 d^{2} c^{2} \left (-i c x -1\right )}+\frac {2 a \arctan \left (c x \right )}{d^{2} c^{3}}-\frac {i a}{d^{2} c^{3}}-\frac {a x}{c^{2} d^{2}}-\frac {i \ln \left (-i c x +1\right ) b x}{2 d^{2} c^{2}}+\frac {\ln \left (-i c x +1\right ) b}{2 d^{2} c^{3}}+\frac {i a}{d^{2} c^{3} \left (-i c x -1\right )}\) | \(349\) |
1/c^3*(-a/d^2*c*x-a/d^2/(c*x-I)+2*a/d^2*arctan(c*x)+1/8*I*b/d^2*arctan(1/2 *c*x)-b/d^2*arctan(c*x)*c*x-b/d^2*arctan(c*x)/(c*x-I)-2*I*b/d^2*arctan(c*x )*ln(c*x-I)-b/d^2*ln(-1/2*I*(c*x+I))*ln(c*x-I)-b/d^2*dilog(-1/2*I*(c*x+I)) +1/2*b/d^2*ln(c*x-I)^2+1/16*b/d^2*ln(c^4*x^4+10*c^2*x^2+9)+3/4*I*b/d^2*arc tan(c*x)-I*a/d^2*ln(c^2*x^2+1)+1/2*I*b/d^2/(c*x-I)-1/4*I*b/d^2*arctan(1/2* c*x-1/2*I)+3/8*b/d^2*ln(c^2*x^2+1)-1/8*I*b/d^2*arctan(1/6*c^3*x^3+7/6*c*x) )
\[ \int \frac {x^2 (a+b \arctan (c x))}{(d+i c d x)^2} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )} x^{2}}{{\left (i \, c d x + d\right )}^{2}} \,d x } \]
integral(1/2*(-I*b*x^2*log(-(c*x + I)/(c*x - I)) - 2*a*x^2)/(c^2*d^2*x^2 - 2*I*c*d^2*x - d^2), x)
\[ \int \frac {x^2 (a+b \arctan (c x))}{(d+i c d x)^2} \, dx=\frac {\left (- i b c^{2} x^{2} + 2 b c x \log {\left (i c x + 1 \right )} - b c x - 2 i b \log {\left (i c x + 1 \right )} - i b\right ) \log {\left (- i c x + 1 \right )}}{2 c^{4} d^{2} x - 2 i c^{3} d^{2}} - \frac {\int \left (- \frac {b}{c^{3} x^{3} - i c^{2} x^{2} + c x - i}\right )\, dx + \int \left (- \frac {2 b \log {\left (i c x + 1 \right )}}{c^{3} x^{3} - i c^{2} x^{2} + c x - i}\right )\, dx + \int \frac {2 a c^{3} x^{3}}{c^{3} x^{3} - i c^{2} x^{2} + c x - i}\, dx + \int \left (- \frac {2 b c^{2} x^{2}}{c^{3} x^{3} - i c^{2} x^{2} + c x - i}\right )\, dx + \int \frac {2 i a c^{2} x^{2}}{c^{3} x^{3} - i c^{2} x^{2} + c x - i}\, dx + \int \left (- \frac {i b c^{3} x^{3}}{c^{3} x^{3} - i c^{2} x^{2} + c x - i}\right )\, dx + \int \frac {3 b c^{2} x^{2} \log {\left (i c x + 1 \right )}}{c^{3} x^{3} - i c^{2} x^{2} + c x - i}\, dx + \int \left (- \frac {4 i b c x \log {\left (i c x + 1 \right )}}{c^{3} x^{3} - i c^{2} x^{2} + c x - i}\right )\, dx + \int \left (- \frac {i b c^{3} x^{3} \log {\left (i c x + 1 \right )}}{c^{3} x^{3} - i c^{2} x^{2} + c x - i}\right )\, dx}{2 c^{2} d^{2}} \]
(-I*b*c**2*x**2 + 2*b*c*x*log(I*c*x + 1) - b*c*x - 2*I*b*log(I*c*x + 1) - I*b)*log(-I*c*x + 1)/(2*c**4*d**2*x - 2*I*c**3*d**2) - (Integral(-b/(c**3* x**3 - I*c**2*x**2 + c*x - I), x) + Integral(-2*b*log(I*c*x + 1)/(c**3*x** 3 - I*c**2*x**2 + c*x - I), x) + Integral(2*a*c**3*x**3/(c**3*x**3 - I*c** 2*x**2 + c*x - I), x) + Integral(-2*b*c**2*x**2/(c**3*x**3 - I*c**2*x**2 + c*x - I), x) + Integral(2*I*a*c**2*x**2/(c**3*x**3 - I*c**2*x**2 + c*x - I), x) + Integral(-I*b*c**3*x**3/(c**3*x**3 - I*c**2*x**2 + c*x - I), x) + Integral(3*b*c**2*x**2*log(I*c*x + 1)/(c**3*x**3 - I*c**2*x**2 + c*x - I) , x) + Integral(-4*I*b*c*x*log(I*c*x + 1)/(c**3*x**3 - I*c**2*x**2 + c*x - I), x) + Integral(-I*b*c**3*x**3*log(I*c*x + 1)/(c**3*x**3 - I*c**2*x**2 + c*x - I), x))/(2*c**2*d**2)
\[ \int \frac {x^2 (a+b \arctan (c x))}{(d+i c d x)^2} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )} x^{2}}{{\left (i \, c d x + d\right )}^{2}} \,d x } \]
-a*(1/(c^4*d^2*x - I*c^3*d^2) + x/(c^2*d^2) + 2*I*log(c*x - I)/(c^3*d^2)) + 1/4*(2*I*c^2*x^2 + 4*(c*x - I)*arctan(c*x)^2 + (c*x - I)*log(c^2*x^2 + 1 )^2 - (-I*c^4*d^2*x - c^3*d^2)*((c*(x/(c^6*d^2*x^2 + c^4*d^2) + arctan(c*x )/(c^5*d^2)) - 2*arctan(c*x)/(c^6*d^2*x^2 + c^4*d^2))*c + 8*integrate(1/4* log(c^2*x^2 + 1)/(c^6*d^2*x^4 + 2*c^4*d^2*x^2 + c^2*d^2), x)) - (c^4*d^2*x - I*c^3*d^2)*(c*(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)) + 16*integrate(1/4*arctan(c*x)/(c^6*d^2*x^4 + 2*c^4*d^2*x^ 2 + c^2*d^2), x)) + 2*(-I*c^5*d^2*x - c^4*d^2)*(c*(x/(c^6*d^2*x^2 + c^4*d^ 2) + arctan(c*x)/(c^5*d^2)) - 8*c*integrate(1/4*x^2*log(c^2*x^2 + 1)/(c^6* d^2*x^4 + 2*c^4*d^2*x^2 + c^2*d^2), x) - 2*arctan(c*x)/(c^6*d^2*x^2 + c^4* d^2)) - 2*(c^5*d^2*x - I*c^4*d^2)*(16*c*integrate(1/4*x^2*arctan(c*x)/(c^6 *d^2*x^4 + 2*c^4*d^2*x^2 + c^2*d^2), x) - c^2/(c^8*d^2*x^2 + c^6*d^2) - lo g(c^2*x^2 + 1)/(c^6*d^2*x^2 + c^4*d^2)) + 2*c*x - 2*(c^2*x^2 - I*c*x + 1)* arctan(c*x) - 4*(c^7*d^2*x - I*c^6*d^2)*integrate(1/4*(2*c*x^4*arctan(c*x) + x^3*log(c^2*x^2 + 1))/(c^6*d^2*x^4 + 2*c^4*d^2*x^2 + c^2*d^2), x) - 4*( -I*c^7*d^2*x - c^6*d^2)*integrate(1/4*(c*x^4*log(c^2*x^2 + 1) - 2*x^3*arct an(c*x))/(c^6*d^2*x^4 + 2*c^4*d^2*x^2 + c^2*d^2), x) - 12*(I*c^6*d^2*x + c ^5*d^2)*integrate(1/4*(2*c*x^3*arctan(c*x) + x^2*log(c^2*x^2 + 1))/(c^6*d^ 2*x^4 + 2*c^4*d^2*x^2 + c^2*d^2), x) - 12*(c^6*d^2*x - I*c^5*d^2)*integrat e(1/4*(c*x^3*log(c^2*x^2 + 1) - 2*x^2*arctan(c*x))/(c^6*d^2*x^4 + 2*c^4...
\[ \int \frac {x^2 (a+b \arctan (c x))}{(d+i c d x)^2} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )} x^{2}}{{\left (i \, c d x + d\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {x^2 (a+b \arctan (c x))}{(d+i c d x)^2} \, dx=\int \frac {x^2\,\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}{{\left (d+c\,d\,x\,1{}\mathrm {i}\right )}^2} \,d x \]