Integrand size = 22, antiderivative size = 98 \[ \int \frac {(a+b \arctan (c x))^2}{d+i c d x} \, dx=\frac {i (a+b \arctan (c x))^2 \log \left (\frac {2}{1+i c x}\right )}{c d}-\frac {b (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{c d}+\frac {i b^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i c x}\right )}{2 c d} \]
I*(a+b*arctan(c*x))^2*ln(2/(1+I*c*x))/c/d-b*(a+b*arctan(c*x))*polylog(2,1- 2/(1+I*c*x))/c/d+1/2*I*b^2*polylog(3,1-2/(1+I*c*x))/c/d
Time = 0.03 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.97 \[ \int \frac {(a+b \arctan (c x))^2}{d+i c d x} \, dx=\frac {i \left (2 (a+b \arctan (c x))^2 \log \left (\frac {2 d}{d+i c d x}\right )+2 i b (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,\frac {i+c x}{-i+c x}\right )+b^2 \operatorname {PolyLog}\left (3,\frac {i+c x}{-i+c x}\right )\right )}{2 c d} \]
((I/2)*(2*(a + b*ArcTan[c*x])^2*Log[(2*d)/(d + I*c*d*x)] + (2*I)*b*(a + b* ArcTan[c*x])*PolyLog[2, (I + c*x)/(-I + c*x)] + b^2*PolyLog[3, (I + c*x)/( -I + c*x)]))/(c*d)
Time = 0.45 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.02, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {5379, 5529, 7164}
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.
| \(\displaystyle \int \frac {(a+b \arctan (c x))^2}{d+i c d x} \, dx\) |
| \(\Big \downarrow \) 5379 |
| \(\displaystyle \frac {i \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))^2}{c d}-\frac {2 i b \int \frac {(a+b \arctan (c x)) \log \left (\frac {2}{i c x+1}\right )}{c^2 x^2+1}dx}{d}\) |
| \(\Big \downarrow \) 5529 |
| \(\displaystyle \frac {i \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))^2}{c d}-\frac {2 i b \left (\frac {1}{2} i b \int \frac {\operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right )}{c^2 x^2+1}dx-\frac {i \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right ) (a+b \arctan (c x))}{2 c}\right )}{d}\) |
| \(\Big \downarrow \) 7164 |
| \(\displaystyle \frac {i \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))^2}{c d}-\frac {2 i b \left (-\frac {i \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right ) (a+b \arctan (c x))}{2 c}-\frac {b \operatorname {PolyLog}\left (3,1-\frac {2}{i c x+1}\right )}{4 c}\right )}{d}\) |
(I*(a + b*ArcTan[c*x])^2*Log[2/(1 + I*c*x)])/(c*d) - ((2*I)*b*(((-1/2*I)*( a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 + I*c*x)])/c - (b*PolyLog[3, 1 - 2/ (1 + I*c*x)])/(4*c)))/d
3.1.98.3.1 Defintions of rubi rules used
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-(a + b*ArcTan[c*x])^p)*(Log[2/(1 + e*(x/d))]/e), x] + Simp[b*c*( p/e) Int[(a + b*ArcTan[c*x])^(p - 1)*(Log[2/(1 + e*(x/d))]/(1 + c^2*x^2)) , x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 + e^2, 0 ]
Int[(Log[u_]*((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2 ), x_Symbol] :> Simp[(-I)*(a + b*ArcTan[c*x])^p*(PolyLog[2, 1 - u]/(2*c*d)) , x] + Simp[b*p*(I/2) Int[(a + b*ArcTan[c*x])^(p - 1)*(PolyLog[2, 1 - u]/ (d + e*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[e, c ^2*d] && EqQ[(1 - u)^2 - (1 - 2*(I/(I - c*x)))^2, 0]
Int[(u_)*PolyLog[n_, v_], x_Symbol] :> With[{w = DerivativeDivides[v, u*v, x]}, Simp[w*PolyLog[n + 1, v], x] /; !FalseQ[w]] /; FreeQ[n, x]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 24.67 (sec) , antiderivative size = 851, normalized size of antiderivative = 8.68
| method | result | size |
| derivativedivides | \(\frac {-\frac {i a^{2} \ln \left (c^{2} x^{2}+1\right )}{2 d}+\frac {a^{2} \arctan \left (c x \right )}{d}+\frac {b^{2} \left (-i \ln \left (i c x +1\right ) \arctan \left (c x \right )^{2}+2 i \left (\frac {\arctan \left (c x \right )^{2} \ln \left (\frac {2 i \left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}{2}-\frac {i \arctan \left (c x \right )^{3}}{3}+\frac {i \pi \left (\operatorname {csgn}\left (\frac {i}{1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}}\right ) \operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}\right )^{2}-\operatorname {csgn}\left (\frac {i}{1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}}\right ) \operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right ) \operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}\right )+\operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}\right )^{3}-\operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right ) \operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}\right )^{2}-\operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}\right ) \operatorname {csgn}\left (\frac {i \left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}\right )^{2}-\operatorname {csgn}\left (\frac {i \left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}\right )^{3}+\operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}\right ) \operatorname {csgn}\left (\frac {i \left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}\right )+\operatorname {csgn}\left (\frac {i \left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}\right )^{2}-1\right ) \arctan \left (c x \right )^{2}}{4}-\frac {i \arctan \left (c x \right ) \operatorname {polylog}\left (2, -\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}{2}+\frac {\operatorname {polylog}\left (3, -\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}{4}\right )\right )}{d}+\frac {2 a b \left (-i \ln \left (i c x +1\right ) \arctan \left (c x \right )-\frac {\left (\ln \left (i c x +1\right )-\ln \left (\frac {1}{2}+\frac {i c x}{2}\right )\right ) \ln \left (\frac {1}{2}-\frac {i c x}{2}\right )}{2}+\frac {\operatorname {dilog}\left (\frac {1}{2}+\frac {i c x}{2}\right )}{2}+\frac {\ln \left (i c x +1\right )^{2}}{4}\right )}{d}}{c}\) | \(851\) |
| default | \(\frac {-\frac {i a^{2} \ln \left (c^{2} x^{2}+1\right )}{2 d}+\frac {a^{2} \arctan \left (c x \right )}{d}+\frac {b^{2} \left (-i \ln \left (i c x +1\right ) \arctan \left (c x \right )^{2}+2 i \left (\frac {\arctan \left (c x \right )^{2} \ln \left (\frac {2 i \left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}{2}-\frac {i \arctan \left (c x \right )^{3}}{3}+\frac {i \pi \left (\operatorname {csgn}\left (\frac {i}{1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}}\right ) \operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}\right )^{2}-\operatorname {csgn}\left (\frac {i}{1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}}\right ) \operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right ) \operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}\right )+\operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}\right )^{3}-\operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right ) \operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}\right )^{2}-\operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}\right ) \operatorname {csgn}\left (\frac {i \left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}\right )^{2}-\operatorname {csgn}\left (\frac {i \left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}\right )^{3}+\operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}\right ) \operatorname {csgn}\left (\frac {i \left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}\right )+\operatorname {csgn}\left (\frac {i \left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}\right )^{2}-1\right ) \arctan \left (c x \right )^{2}}{4}-\frac {i \arctan \left (c x \right ) \operatorname {polylog}\left (2, -\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}{2}+\frac {\operatorname {polylog}\left (3, -\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}{4}\right )\right )}{d}+\frac {2 a b \left (-i \ln \left (i c x +1\right ) \arctan \left (c x \right )-\frac {\left (\ln \left (i c x +1\right )-\ln \left (\frac {1}{2}+\frac {i c x}{2}\right )\right ) \ln \left (\frac {1}{2}-\frac {i c x}{2}\right )}{2}+\frac {\operatorname {dilog}\left (\frac {1}{2}+\frac {i c x}{2}\right )}{2}+\frac {\ln \left (i c x +1\right )^{2}}{4}\right )}{d}}{c}\) | \(851\) |
| parts | \(-\frac {i a^{2} \ln \left (c^{2} x^{2}+1\right )}{2 d c}+\frac {a^{2} \arctan \left (c x \right )}{d c}+\frac {b^{2} \left (-i \ln \left (i c x +1\right ) \arctan \left (c x \right )^{2}+2 i \left (\frac {\arctan \left (c x \right )^{2} \ln \left (\frac {2 i \left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}{2}-\frac {i \pi \left (-\operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}\right ) \operatorname {csgn}\left (\frac {i \left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}\right )-\operatorname {csgn}\left (\frac {i \left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}\right )^{2}+\operatorname {csgn}\left (\frac {i}{1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}}\right ) \operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right ) \operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}\right )-\operatorname {csgn}\left (\frac {i}{1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}}\right ) \operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}\right )^{2}+\operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right ) \operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}\right )^{2}-\operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}\right )^{3}+\operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}\right ) \operatorname {csgn}\left (\frac {i \left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}\right )^{2}+\operatorname {csgn}\left (\frac {i \left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}\right )^{3}+1\right ) \arctan \left (c x \right )^{2}}{4}-\frac {i \arctan \left (c x \right ) \operatorname {polylog}\left (2, -\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}{2}+\frac {\operatorname {polylog}\left (3, -\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}{4}-\frac {i \arctan \left (c x \right )^{3}}{3}\right )\right )}{d c}+\frac {2 a b \left (-i \ln \left (i c x +1\right ) \arctan \left (c x \right )-\frac {\left (\ln \left (i c x +1\right )-\ln \left (\frac {1}{2}+\frac {i c x}{2}\right )\right ) \ln \left (\frac {1}{2}-\frac {i c x}{2}\right )}{2}+\frac {\operatorname {dilog}\left (\frac {1}{2}+\frac {i c x}{2}\right )}{2}+\frac {\ln \left (i c x +1\right )^{2}}{4}\right )}{d c}\) | \(860\) |
1/c*(-1/2*I*a^2/d*ln(c^2*x^2+1)+a^2/d*arctan(c*x)+b^2/d*(-I*ln(1+I*c*x)*ar ctan(c*x)^2+2*I*(1/2*arctan(c*x)^2*ln(2*I*(1+I*c*x)^2/(c^2*x^2+1))-1/3*I*a rctan(c*x)^3+1/4*I*Pi*(csgn(I/(1+(1+I*c*x)^2/(c^2*x^2+1)))*csgn((1+I*c*x)^ 2/(c^2*x^2+1)/(1+(1+I*c*x)^2/(c^2*x^2+1)))^2-csgn(I/(1+(1+I*c*x)^2/(c^2*x^ 2+1)))*csgn((1+I*c*x)^2/(c^2*x^2+1))*csgn((1+I*c*x)^2/(c^2*x^2+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+I*c*x)^2/(c^2*x^2+ 1)))^3-csgn((1+I*c*x)^2/(c^2*x^2+1))*csgn((1+I*c*x)^2/(c^2*x^2+1)/(1+(1+I* c*x)^2/(c^2*x^2+1)))^2-csgn((1+I*c*x)^2/(c^2*x^2+1)/(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+I*c*x)^2/(c^2*x^2+1)))^2-csgn( I*(1+I*c*x)^2/(c^2*x^2+1)/(1+(1+I*c*x)^2/(c^2*x^2+1)))^3+csgn((1+I*c*x)^2/ (c^2*x^2+1)/(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+I*c*x)^2/(c^2*x^2+1)))+csgn(I*(1+I*c*x)^2/(c^2*x^2+1)/(1+(1+I*c*x)^2/( c^2*x^2+1)))^2-1)*arctan(c*x)^2-1/2*I*arctan(c*x)*polylog(2,-(1+I*c*x)^2/( c^2*x^2+1))+1/4*polylog(3,-(1+I*c*x)^2/(c^2*x^2+1))))+2/d*a*b*(-I*ln(1+I*c *x)*arctan(c*x)-1/2*(ln(1+I*c*x)-ln(1/2+1/2*I*c*x))*ln(1/2-1/2*I*c*x)+1/2* dilog(1/2+1/2*I*c*x)+1/4*ln(1+I*c*x)^2))
\[ \int \frac {(a+b \arctan (c x))^2}{d+i c d x} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )}^{2}}{i \, c d x + d} \,d x } \]
integral(1/4*(I*b^2*log(-(c*x + I)/(c*x - I))^2 + 4*a*b*log(-(c*x + I)/(c* x - I)) - 4*I*a^2)/(c*d*x - I*d), x)
Timed out. \[ \int \frac {(a+b \arctan (c x))^2}{d+i c d x} \, dx=\text {Timed out} \]
\[ \int \frac {(a+b \arctan (c x))^2}{d+i c d x} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )}^{2}}{i \, c d x + d} \,d x } \]
-I*a^2*log(I*c*d*x + d)/(c*d) + 1/96*(24*b^2*arctan(c*x)^3 + 12*I*b^2*arct an(c*x)^2*log(c^2*x^2 + 1) + 6*b^2*arctan(c*x)*log(c^2*x^2 + 1)^2 + 3*I*b^ 2*log(c^2*x^2 + 1)^3 - 8*(48*b^2*c*integrate(1/16*x*arctan(c*x)*log(c^2*x^ 2 + 1)/(c^2*d*x^2 + d), x) - b^2*arctan(c*x)^3/(c*d) + 12*b^2*integrate(1/ 16*log(c^2*x^2 + 1)^2/(c^2*d*x^2 + d), x) - 12*a*b*arctan(c*x)^2/(c*d))*c* d - 96*I*c*d*integrate(1/16*(20*b^2*c*x*arctan(c*x)^2 + 3*b^2*c*x*log(c^2* x^2 + 1)^2 + 32*a*b*c*x*arctan(c*x) + 4*b^2*arctan(c*x)*log(c^2*x^2 + 1))/ (c^2*d*x^2 + d), x))/(c*d)
\[ \int \frac {(a+b \arctan (c x))^2}{d+i c d x} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )}^{2}}{i \, c d x + d} \,d x } \]
Timed out. \[ \int \frac {(a+b \arctan (c x))^2}{d+i c d x} \, dx=\int \frac {{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^2}{d+c\,d\,x\,1{}\mathrm {i}} \,d x \]