Integrand size = 25, antiderivative size = 293 \[ \int \frac {(d+i c d x)^3 (a+b \arctan (c x))^2}{x^5} \, dx=-\frac {b^2 c^2 d^3}{12 x^2}-\frac {i b^2 c^3 d^3}{x}-i b^2 c^4 d^3 \arctan (c x)-\frac {b c d^3 (a+b \arctan (c x))}{6 x^3}-\frac {i b c^2 d^3 (a+b \arctan (c x))}{x^2}+\frac {7 b c^3 d^3 (a+b \arctan (c x))}{2 x}-\frac {d^3 (1+i c x)^4 (a+b \arctan (c x))^2}{4 x^4}-4 i a b c^4 d^3 \log (x)-\frac {11}{3} b^2 c^4 d^3 \log (x)-4 i b c^4 d^3 (a+b \arctan (c x)) \log \left (\frac {2}{1-i c x}\right )+\frac {11}{6} b^2 c^4 d^3 \log \left (1+c^2 x^2\right )+2 b^2 c^4 d^3 \operatorname {PolyLog}(2,-i c x)-2 b^2 c^4 d^3 \operatorname {PolyLog}(2,i c x)-2 b^2 c^4 d^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right ) \]
-1/12*b^2*c^2*d^3/x^2-I*b^2*c^3*d^3/x-I*b^2*c^4*d^3*arctan(c*x)-1/6*b*c*d^ 3*(a+b*arctan(c*x))/x^3-I*b*c^2*d^3*(a+b*arctan(c*x))/x^2+7/2*b*c^3*d^3*(a +b*arctan(c*x))/x-1/4*d^3*(1+I*c*x)^4*(a+b*arctan(c*x))^2/x^4-4*I*a*b*c^4* d^3*ln(x)-11/3*b^2*c^4*d^3*ln(x)-4*I*b*c^4*d^3*(a+b*arctan(c*x))*ln(2/(1-I *c*x))+11/6*b^2*c^4*d^3*ln(c^2*x^2+1)+2*b^2*c^4*d^3*polylog(2,-I*c*x)-2*b^ 2*c^4*d^3*polylog(2,I*c*x)-2*b^2*c^4*d^3*polylog(2,1-2/(1-I*c*x))
Time = 0.69 (sec) , antiderivative size = 322, normalized size of antiderivative = 1.10 \[ \int \frac {(d+i c d x)^3 (a+b \arctan (c x))^2}{x^5} \, dx=\frac {d^3 \left (-3 a^2-12 i a^2 c x-2 a b c x+18 a^2 c^2 x^2-12 i a b c^2 x^2-b^2 c^2 x^2+12 i a^2 c^3 x^3+42 a b c^3 x^3-12 i b^2 c^3 x^3-b^2 c^4 x^4-3 b^2 (-i+c x)^4 \arctan (c x)^2+2 b \arctan (c x) \left (b c x \left (-1-6 i c x+21 c^2 x^2-6 i c^3 x^3\right )+3 a \left (-1-4 i c x+6 c^2 x^2+4 i c^3 x^3+7 c^4 x^4\right )-24 i b c^4 x^4 \log \left (1-e^{2 i \arctan (c x)}\right )\right )-48 i a b c^4 x^4 \log (c x)-44 b^2 c^4 x^4 \log \left (\frac {c x}{\sqrt {1+c^2 x^2}}\right )+24 i a b c^4 x^4 \log \left (1+c^2 x^2\right )-24 b^2 c^4 x^4 \operatorname {PolyLog}\left (2,e^{2 i \arctan (c x)}\right )\right )}{12 x^4} \]
(d^3*(-3*a^2 - (12*I)*a^2*c*x - 2*a*b*c*x + 18*a^2*c^2*x^2 - (12*I)*a*b*c^ 2*x^2 - b^2*c^2*x^2 + (12*I)*a^2*c^3*x^3 + 42*a*b*c^3*x^3 - (12*I)*b^2*c^3 *x^3 - b^2*c^4*x^4 - 3*b^2*(-I + c*x)^4*ArcTan[c*x]^2 + 2*b*ArcTan[c*x]*(b *c*x*(-1 - (6*I)*c*x + 21*c^2*x^2 - (6*I)*c^3*x^3) + 3*a*(-1 - (4*I)*c*x + 6*c^2*x^2 + (4*I)*c^3*x^3 + 7*c^4*x^4) - (24*I)*b*c^4*x^4*Log[1 - E^((2*I )*ArcTan[c*x])]) - (48*I)*a*b*c^4*x^4*Log[c*x] - 44*b^2*c^4*x^4*Log[(c*x)/ Sqrt[1 + c^2*x^2]] + (24*I)*a*b*c^4*x^4*Log[1 + c^2*x^2] - 24*b^2*c^4*x^4* PolyLog[2, E^((2*I)*ArcTan[c*x])]))/(12*x^4)
Time = 0.53 (sec) , antiderivative size = 276, normalized size of antiderivative = 0.94, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {5409, 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.
| \(\displaystyle \int \frac {(d+i c d x)^3 (a+b \arctan (c x))^2}{x^5} \, dx\) |
| \(\Big \downarrow \) 5409 |
| \(\displaystyle -2 b c \int \left (-\frac {2 i d^3 (a+b \arctan (c x)) c^4}{c x+i}+\frac {2 i d^3 (a+b \arctan (c x)) c^3}{x}+\frac {7 d^3 (a+b \arctan (c x)) c^2}{4 x^2}-\frac {i d^3 (a+b \arctan (c x)) c}{x^3}-\frac {d^3 (a+b \arctan (c x))}{4 x^4}\right )dx-\frac {d^3 (1+i c x)^4 (a+b \arctan (c x))^2}{4 x^4}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -2 b c \left (2 i c^3 d^3 \log \left (\frac {2}{1-i c x}\right ) (a+b \arctan (c x))-\frac {7 c^2 d^3 (a+b \arctan (c x))}{4 x}+\frac {d^3 (a+b \arctan (c x))}{12 x^3}+\frac {i c d^3 (a+b \arctan (c x))}{2 x^2}+2 i a c^3 d^3 \log (x)+\frac {1}{2} i b c^3 d^3 \arctan (c x)-b c^3 d^3 \operatorname {PolyLog}(2,-i c x)+b c^3 d^3 \operatorname {PolyLog}(2,i c x)+b c^3 d^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )+\frac {11}{6} b c^3 d^3 \log (x)+\frac {i b c^2 d^3}{2 x}-\frac {11}{12} b c^3 d^3 \log \left (c^2 x^2+1\right )+\frac {b c d^3}{24 x^2}\right )-\frac {d^3 (1+i c x)^4 (a+b \arctan (c x))^2}{4 x^4}\) |
-1/4*(d^3*(1 + I*c*x)^4*(a + b*ArcTan[c*x])^2)/x^4 - 2*b*c*((b*c*d^3)/(24* x^2) + ((I/2)*b*c^2*d^3)/x + (I/2)*b*c^3*d^3*ArcTan[c*x] + (d^3*(a + b*Arc Tan[c*x]))/(12*x^3) + ((I/2)*c*d^3*(a + b*ArcTan[c*x]))/x^2 - (7*c^2*d^3*( a + b*ArcTan[c*x]))/(4*x) + (2*I)*a*c^3*d^3*Log[x] + (11*b*c^3*d^3*Log[x]) /6 + (2*I)*c^3*d^3*(a + b*ArcTan[c*x])*Log[2/(1 - I*c*x)] - (11*b*c^3*d^3* Log[1 + c^2*x^2])/12 - b*c^3*d^3*PolyLog[2, (-I)*c*x] + b*c^3*d^3*PolyLog[ 2, I*c*x] + b*c^3*d^3*PolyLog[2, 1 - 2/(1 - I*c*x)])
3.1.92.3.1 Defintions of rubi rules used
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_)*((f_.)*(x_))^(m_.)*((d_.) + (e_ .)*(x_))^(q_), x_Symbol] :> With[{u = IntHide[(f*x)^m*(d + e*x)^q, x]}, Sim p[(a + b*ArcTan[c*x])^p u, x] - Simp[b*c*p Int[ExpandIntegrand[(a + b*A rcTan[c*x])^(p - 1), u/(1 + c^2*x^2), x], x], x]] /; FreeQ[{a, b, c, d, e, f, q}, x] && IGtQ[p, 1] && EqQ[c^2*d^2 + e^2, 0] && IntegersQ[m, q] && NeQ[ m, -1] && NeQ[q, -1] && ILtQ[m + q + 1, 0] && LtQ[m*q, 0]
Time = 4.46 (sec) , antiderivative size = 501, normalized size of antiderivative = 1.71
| method | result | size |
| parts | \(d^{3} a^{2} \left (\frac {3 c^{2}}{2 x^{2}}-\frac {1}{4 x^{4}}+\frac {i c^{3}}{x}-\frac {i c}{x^{3}}\right )+b^{2} d^{3} c^{4} \left (\frac {3 \arctan \left (c x \right )^{2}}{2 c^{2} x^{2}}+\frac {11 \ln \left (c^{2} x^{2}+1\right )}{6}+\frac {7 \arctan \left (c x \right )^{2}}{4}-\frac {11 \ln \left (c x \right )}{3}-\frac {1}{12 c^{2} x^{2}}-\frac {\arctan \left (c x \right )^{2}}{4 c^{4} x^{4}}+\frac {\ln \left (c x -i\right )^{2}}{2}+\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )-\frac {\ln \left (c x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )+\frac {7 \arctan \left (c x \right )}{2 c x}-2 \operatorname {dilog}\left (-i c x +1\right )+2 \operatorname {dilog}\left (i c x +1\right )-\ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )+\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )+\ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )-\ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )-i \arctan \left (c x \right )+2 \ln \left (c x \right ) \ln \left (i c x +1\right )-2 \ln \left (c x \right ) \ln \left (-i c x +1\right )-4 i \arctan \left (c x \right ) \ln \left (c x \right )-\frac {\arctan \left (c x \right )}{6 c^{3} x^{3}}-\frac {i}{c x}-\frac {i \arctan \left (c x \right )}{c^{2} x^{2}}+2 i \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right )+\frac {i \arctan \left (c x \right )^{2}}{c x}-\frac {i \arctan \left (c x \right )^{2}}{c^{3} x^{3}}\right )+2 a \,d^{3} b \,c^{4} \left (-\frac {\arctan \left (c x \right )}{4 c^{4} x^{4}}-\frac {i \arctan \left (c x \right )}{c^{3} x^{3}}+\frac {i \arctan \left (c x \right )}{c x}+\frac {3 \arctan \left (c x \right )}{2 c^{2} x^{2}}-\frac {i}{2 c^{2} x^{2}}-2 i \ln \left (c x \right )-\frac {1}{12 c^{3} x^{3}}+\frac {7}{4 c x}+i \ln \left (c^{2} x^{2}+1\right )+\frac {7 \arctan \left (c x \right )}{4}\right )\) | \(501\) |
| derivativedivides | \(c^{4} \left (d^{3} a^{2} \left (-\frac {1}{4 c^{4} x^{4}}-\frac {i}{c^{3} x^{3}}+\frac {i}{c x}+\frac {3}{2 c^{2} x^{2}}\right )+b^{2} d^{3} \left (\frac {3 \arctan \left (c x \right )^{2}}{2 c^{2} x^{2}}+\frac {11 \ln \left (c^{2} x^{2}+1\right )}{6}+\frac {7 \arctan \left (c x \right )^{2}}{4}-\frac {11 \ln \left (c x \right )}{3}-\frac {1}{12 c^{2} x^{2}}-\frac {\arctan \left (c x \right )^{2}}{4 c^{4} x^{4}}+\frac {\ln \left (c x -i\right )^{2}}{2}+\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )-\frac {\ln \left (c x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )+\frac {7 \arctan \left (c x \right )}{2 c x}-2 \operatorname {dilog}\left (-i c x +1\right )+2 \operatorname {dilog}\left (i c x +1\right )-\ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )+\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )+\ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )-\ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )-i \arctan \left (c x \right )+2 \ln \left (c x \right ) \ln \left (i c x +1\right )-2 \ln \left (c x \right ) \ln \left (-i c x +1\right )-4 i \arctan \left (c x \right ) \ln \left (c x \right )-\frac {\arctan \left (c x \right )}{6 c^{3} x^{3}}-\frac {i}{c x}-\frac {i \arctan \left (c x \right )}{c^{2} x^{2}}+2 i \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right )+\frac {i \arctan \left (c x \right )^{2}}{c x}-\frac {i \arctan \left (c x \right )^{2}}{c^{3} x^{3}}\right )+2 a \,d^{3} b \left (-\frac {\arctan \left (c x \right )}{4 c^{4} x^{4}}-\frac {i \arctan \left (c x \right )}{c^{3} x^{3}}+\frac {i \arctan \left (c x \right )}{c x}+\frac {3 \arctan \left (c x \right )}{2 c^{2} x^{2}}-\frac {i}{2 c^{2} x^{2}}-2 i \ln \left (c x \right )-\frac {1}{12 c^{3} x^{3}}+\frac {7}{4 c x}+i \ln \left (c^{2} x^{2}+1\right )+\frac {7 \arctan \left (c x \right )}{4}\right )\right )\) | \(504\) |
| default | \(c^{4} \left (d^{3} a^{2} \left (-\frac {1}{4 c^{4} x^{4}}-\frac {i}{c^{3} x^{3}}+\frac {i}{c x}+\frac {3}{2 c^{2} x^{2}}\right )+b^{2} d^{3} \left (\frac {3 \arctan \left (c x \right )^{2}}{2 c^{2} x^{2}}+\frac {11 \ln \left (c^{2} x^{2}+1\right )}{6}+\frac {7 \arctan \left (c x \right )^{2}}{4}-\frac {11 \ln \left (c x \right )}{3}-\frac {1}{12 c^{2} x^{2}}-\frac {\arctan \left (c x \right )^{2}}{4 c^{4} x^{4}}+\frac {\ln \left (c x -i\right )^{2}}{2}+\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )-\frac {\ln \left (c x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )+\frac {7 \arctan \left (c x \right )}{2 c x}-2 \operatorname {dilog}\left (-i c x +1\right )+2 \operatorname {dilog}\left (i c x +1\right )-\ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )+\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )+\ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )-\ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )-i \arctan \left (c x \right )+2 \ln \left (c x \right ) \ln \left (i c x +1\right )-2 \ln \left (c x \right ) \ln \left (-i c x +1\right )-4 i \arctan \left (c x \right ) \ln \left (c x \right )-\frac {\arctan \left (c x \right )}{6 c^{3} x^{3}}-\frac {i}{c x}-\frac {i \arctan \left (c x \right )}{c^{2} x^{2}}+2 i \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right )+\frac {i \arctan \left (c x \right )^{2}}{c x}-\frac {i \arctan \left (c x \right )^{2}}{c^{3} x^{3}}\right )+2 a \,d^{3} b \left (-\frac {\arctan \left (c x \right )}{4 c^{4} x^{4}}-\frac {i \arctan \left (c x \right )}{c^{3} x^{3}}+\frac {i \arctan \left (c x \right )}{c x}+\frac {3 \arctan \left (c x \right )}{2 c^{2} x^{2}}-\frac {i}{2 c^{2} x^{2}}-2 i \ln \left (c x \right )-\frac {1}{12 c^{3} x^{3}}+\frac {7}{4 c x}+i \ln \left (c^{2} x^{2}+1\right )+\frac {7 \arctan \left (c x \right )}{4}\right )\right )\) | \(504\) |
d^3*a^2*(3/2*c^2/x^2-1/4/x^4+I*c^3/x-I*c/x^3)+b^2*d^3*c^4*(3/2/c^2/x^2*arc tan(c*x)^2+11/6*ln(c^2*x^2+1)+7/4*arctan(c*x)^2-11/3*ln(c*x)+dilog(-1/2*I* (c*x+I))-1/12/c^2/x^2-1/4/c^4/x^4*arctan(c*x)^2-dilog(1/2*I*(c*x-I))+7/2/c /x*arctan(c*x)+2*dilog(1+I*c*x)-2*dilog(1-I*c*x)-I*arctan(c*x)+1/2*ln(c*x- I)^2-1/2*ln(c*x+I)^2-ln(c*x-I)*ln(c^2*x^2+1)+ln(c*x-I)*ln(-1/2*I*(c*x+I))+ ln(c*x+I)*ln(c^2*x^2+1)-ln(c*x+I)*ln(1/2*I*(c*x-I))+2*ln(c*x)*ln(1+I*c*x)- 2*ln(c*x)*ln(1-I*c*x)-I/c/x-1/6*arctan(c*x)/c^3/x^3-I*arctan(c*x)/c^2/x^2- 4*I*arctan(c*x)*ln(c*x)+2*I*arctan(c*x)*ln(c^2*x^2+1)+I*arctan(c*x)^2/c/x- I*arctan(c*x)^2/c^3/x^3)+2*a*d^3*b*c^4*(-1/4*arctan(c*x)/c^4/x^4-I*arctan( c*x)/c^3/x^3+I*arctan(c*x)/c/x+3/2/c^2/x^2*arctan(c*x)-1/2*I/c^2/x^2-2*I*l n(c*x)-1/12/c^3/x^3+7/4/c/x+I*ln(c^2*x^2+1)+7/4*arctan(c*x))
\[ \int \frac {(d+i c d x)^3 (a+b \arctan (c x))^2}{x^5} \, dx=\int { \frac {{\left (i \, c d x + d\right )}^{3} {\left (b \arctan \left (c x\right ) + a\right )}^{2}}{x^{5}} \,d x } \]
1/16*(16*x^4*integral(1/4*(-4*I*a^2*c^5*d^3*x^5 - 12*a^2*c^4*d^3*x^4 + 8*I *a^2*c^3*d^3*x^3 - 8*a^2*c^2*d^3*x^2 + 12*I*a^2*c*d^3*x + 4*a^2*d^3 + (4*a *b*c^5*d^3*x^5 - 4*(3*I*a*b - b^2)*c^4*d^3*x^4 - 2*(4*a*b + 3*I*b^2)*c^3*d ^3*x^3 - 4*(2*I*a*b + b^2)*c^2*d^3*x^2 - (12*a*b - I*b^2)*c*d^3*x + 4*I*a* b*d^3)*log(-(c*x + I)/(c*x - I)))/(c^2*x^7 + x^5), x) + (-4*I*b^2*c^3*d^3* x^3 - 6*b^2*c^2*d^3*x^2 + 4*I*b^2*c*d^3*x + b^2*d^3)*log(-(c*x + I)/(c*x - I))^2)/x^4
Timed out. \[ \int \frac {(d+i c d x)^3 (a+b \arctan (c x))^2}{x^5} \, dx=\text {Timed out} \]
\[ \int \frac {(d+i c d x)^3 (a+b \arctan (c x))^2}{x^5} \, dx=\int { \frac {{\left (i \, c d x + d\right )}^{3} {\left (b \arctan \left (c x\right ) + a\right )}^{2}}{x^{5}} \,d x } \]
I*(c*(log(c^2*x^2 + 1) - log(x^2)) + 2*arctan(c*x)/x)*a*b*c^3*d^3 + 3*((c* arctan(c*x) + 1/x)*c + arctan(c*x)/x^2)*a*b*c^2*d^3 + I*((c^2*log(c^2*x^2 + 1) - c^2*log(x^2) - 1/x^2)*c - 2*arctan(c*x)/x^3)*a*b*c*d^3 + I*a^2*c^3* d^3/x + 1/6*((3*c^3*arctan(c*x) + (3*c^2*x^2 - 1)/x^3)*c - 3*arctan(c*x)/x ^4)*a*b*d^3 + 1/12*(2*(3*c^3*arctan(c*x) + (3*c^2*x^2 - 1)/x^3)*c*arctan(c *x) - (3*c^2*x^2*arctan(c*x)^2 - 4*c^2*x^2*log(c^2*x^2 + 1) + 8*c^2*x^2*lo g(x) + 1)*c^2/x^2)*b^2*d^3 + 3/2*a^2*c^2*d^3/x^2 - I*a^2*c*d^3/x^3 - 1/4*b ^2*d^3*arctan(c*x)^2/x^4 - 1/4*a^2*d^3/x^4 - 1/32*(8*I*(b^2*c^4*d^3*arctan (c*x)^3 + 4*b^2*c^5*d^3*integrate(1/16*x^4*log(c^2*x^2 + 1)^2/(c^2*x^6 + x ^4), x) - 16*b^2*c^5*d^3*integrate(1/16*x^4*log(c^2*x^2 + 1)/(c^2*x^6 + x^ 4), x) - 48*b^2*c^4*d^3*integrate(1/16*x^3*arctan(c*x)*log(c^2*x^2 + 1)/(c ^2*x^6 + x^4), x) + 80*b^2*c^4*d^3*integrate(1/16*x^3*arctan(c*x)/(c^2*x^6 + x^4), x) - 96*b^2*c^3*d^3*integrate(1/16*x^2*arctan(c*x)^2/(c^2*x^6 + x ^4), x) - 8*b^2*c^3*d^3*integrate(1/16*x^2*log(c^2*x^2 + 1)^2/(c^2*x^6 + x ^4), x) + 40*b^2*c^3*d^3*integrate(1/16*x^2*log(c^2*x^2 + 1)/(c^2*x^6 + x^ 4), x) - 48*b^2*c^2*d^3*integrate(1/16*x*arctan(c*x)*log(c^2*x^2 + 1)/(c^2 *x^6 + x^4), x) - 32*b^2*c^2*d^3*integrate(1/16*x*arctan(c*x)/(c^2*x^6 + x ^4), x) - 144*b^2*c*d^3*integrate(1/16*arctan(c*x)^2/(c^2*x^6 + x^4), x) - 12*b^2*c*d^3*integrate(1/16*log(c^2*x^2 + 1)^2/(c^2*x^6 + x^4), x))*x^3 - 8*(b^2*c^4*d^3*arctan(c*x)^2 - 16*b^2*c^5*d^3*integrate(1/16*x^4*arcta...
Timed out. \[ \int \frac {(d+i c d x)^3 (a+b \arctan (c x))^2}{x^5} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {(d+i c d x)^3 (a+b \arctan (c x))^2}{x^5} \, dx=\int \frac {{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^2\,{\left (d+c\,d\,x\,1{}\mathrm {i}\right )}^3}{x^5} \,d x \]