Integrand size = 25, antiderivative size = 513 \[ \int \frac {(d+i c d x)^3 (a+b \arctan (c x))^2}{x^7} \, dx=-\frac {b^2 c^2 d^3}{60 x^4}-\frac {i b^2 c^3 d^3}{10 x^3}+\frac {61 b^2 c^4 d^3}{180 x^2}+\frac {37 i b^2 c^5 d^3}{30 x}+\frac {37}{30} i b^2 c^6 d^3 \arctan (c x)-\frac {b c d^3 (a+b \arctan (c x))}{15 x^5}-\frac {3 i b c^2 d^3 (a+b \arctan (c x))}{10 x^4}+\frac {11 b c^3 d^3 (a+b \arctan (c x))}{18 x^3}+\frac {14 i b c^4 d^3 (a+b \arctan (c x))}{15 x^2}-\frac {11 b c^5 d^3 (a+b \arctan (c x))}{6 x}-\frac {d^3 (a+b \arctan (c x))^2}{6 x^6}-\frac {3 i c d^3 (a+b \arctan (c x))^2}{5 x^5}+\frac {3 c^2 d^3 (a+b \arctan (c x))^2}{4 x^4}+\frac {i c^3 d^3 (a+b \arctan (c x))^2}{3 x^3}+\frac {28}{15} i a b c^6 d^3 \log (x)+\frac {113}{45} b^2 c^6 d^3 \log (x)+\frac {37}{20} i b c^6 d^3 (a+b \arctan (c x)) \log \left (\frac {2}{1-i c x}\right )+\frac {1}{60} i b c^6 d^3 (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )-\frac {113}{90} b^2 c^6 d^3 \log \left (1+c^2 x^2\right )-\frac {14}{15} b^2 c^6 d^3 \operatorname {PolyLog}(2,-i c x)+\frac {14}{15} b^2 c^6 d^3 \operatorname {PolyLog}(2,i c x)+\frac {37}{40} b^2 c^6 d^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )-\frac {1}{120} b^2 c^6 d^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right ) \] Output:
-1/60*b^2*c^2*d^3/x^4-1/10*I*b^2*c^3*d^3/x^3+61/180*b^2*c^4*d^3/x^2+28/15* I*a*b*c^6*d^3*ln(x)-3/5*I*c*d^3*(a+b*arctan(c*x))^2/x^5-1/15*b*c*d^3*(a+b* arctan(c*x))/x^5+1/3*I*c^3*d^3*(a+b*arctan(c*x))^2/x^3+11/18*b*c^3*d^3*(a+ b*arctan(c*x))/x^3+37/30*I*b^2*c^6*d^3*arctan(c*x)-11/6*b*c^5*d^3*(a+b*arc tan(c*x))/x-1/6*d^3*(a+b*arctan(c*x))^2/x^6+14/15*I*b*c^4*d^3*(a+b*arctan( c*x))/x^2+3/4*c^2*d^3*(a+b*arctan(c*x))^2/x^4+37/30*I*b^2*c^5*d^3/x+1/60*I *b*c^6*d^3*(a+b*arctan(c*x))*ln(2/(1+I*c*x))+113/45*b^2*c^6*d^3*ln(x)+37/2 0*I*b*c^6*d^3*(a+b*arctan(c*x))*ln(2/(1-I*c*x))-3/10*I*b*c^2*d^3*(a+b*arct an(c*x))/x^4-113/90*b^2*c^6*d^3*ln(c^2*x^2+1)-14/15*b^2*c^6*d^3*polylog(2, -I*c*x)+14/15*b^2*c^6*d^3*polylog(2,I*c*x)+37/40*b^2*c^6*d^3*polylog(2,1-2 /(1-I*c*x))-1/120*b^2*c^6*d^3*polylog(2,1-2/(1+I*c*x))
Time = 1.06 (sec) , antiderivative size = 401, normalized size of antiderivative = 0.78 \[ \int \frac {(d+i c d x)^3 (a+b \arctan (c x))^2}{x^7} \, dx=\frac {d^3 \left (-30 a^2-108 i a^2 c x-12 a b c x+135 a^2 c^2 x^2-54 i a b c^2 x^2-3 b^2 c^2 x^2+60 i a^2 c^3 x^3+110 a b c^3 x^3-18 i b^2 c^3 x^3+168 i a b c^4 x^4+61 b^2 c^4 x^4-330 a b c^5 x^5+222 i b^2 c^5 x^5+64 b^2 c^6 x^6+3 b^2 (-i+c x)^4 \left (-10+4 i c x+c^2 x^2\right ) \arctan (c x)^2+2 b \arctan (c x) \left (b c x \left (-6-27 i c x+55 c^2 x^2+84 i c^3 x^3-165 c^4 x^4+111 i c^5 x^5\right )-3 a \left (10+36 i c x-45 c^2 x^2-20 i c^3 x^3+55 c^6 x^6\right )+168 i b c^6 x^6 \log \left (1-e^{2 i \arctan (c x)}\right )\right )+336 i a b c^6 x^6 \log (c x)+452 b^2 c^6 x^6 \log \left (\frac {c x}{\sqrt {1+c^2 x^2}}\right )-168 i a b c^6 x^6 \log \left (1+c^2 x^2\right )+168 b^2 c^6 x^6 \operatorname {PolyLog}\left (2,e^{2 i \arctan (c x)}\right )\right )}{180 x^6} \] Input:
Integrate[((d + I*c*d*x)^3*(a + b*ArcTan[c*x])^2)/x^7,x]
Output:
(d^3*(-30*a^2 - (108*I)*a^2*c*x - 12*a*b*c*x + 135*a^2*c^2*x^2 - (54*I)*a* b*c^2*x^2 - 3*b^2*c^2*x^2 + (60*I)*a^2*c^3*x^3 + 110*a*b*c^3*x^3 - (18*I)* b^2*c^3*x^3 + (168*I)*a*b*c^4*x^4 + 61*b^2*c^4*x^4 - 330*a*b*c^5*x^5 + (22 2*I)*b^2*c^5*x^5 + 64*b^2*c^6*x^6 + 3*b^2*(-I + c*x)^4*(-10 + (4*I)*c*x + c^2*x^2)*ArcTan[c*x]^2 + 2*b*ArcTan[c*x]*(b*c*x*(-6 - (27*I)*c*x + 55*c^2* x^2 + (84*I)*c^3*x^3 - 165*c^4*x^4 + (111*I)*c^5*x^5) - 3*a*(10 + (36*I)*c *x - 45*c^2*x^2 - (20*I)*c^3*x^3 + 55*c^6*x^6) + (168*I)*b*c^6*x^6*Log[1 - E^((2*I)*ArcTan[c*x])]) + (336*I)*a*b*c^6*x^6*Log[c*x] + 452*b^2*c^6*x^6* Log[(c*x)/Sqrt[1 + c^2*x^2]] - (168*I)*a*b*c^6*x^6*Log[1 + c^2*x^2] + 168* b^2*c^6*x^6*PolyLog[2, E^((2*I)*ArcTan[c*x])]))/(180*x^6)
Time = 0.88 (sec) , antiderivative size = 483, 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.
| \(\displaystyle \int \frac {(d+i c d x)^3 (a+b \arctan (c x))^2}{x^7} \, dx\) |
| \(\Big \downarrow \) 5409 |
| \(\displaystyle -2 b c \int \left (-\frac {i d^3 (a+b \arctan (c x)) c^6}{120 (i-c x)}+\frac {37 i d^3 (a+b \arctan (c x)) c^6}{40 (c x+i)}-\frac {14 i d^3 (a+b \arctan (c x)) c^5}{15 x}-\frac {11 d^3 (a+b \arctan (c x)) c^4}{12 x^2}+\frac {14 i d^3 (a+b \arctan (c x)) c^3}{15 x^3}+\frac {11 d^3 (a+b \arctan (c x)) c^2}{12 x^4}-\frac {3 i d^3 (a+b \arctan (c x)) c}{5 x^5}-\frac {d^3 (a+b \arctan (c x))}{6 x^6}\right )dx+\frac {i c^3 d^3 (a+b \arctan (c x))^2}{3 x^3}+\frac {3 c^2 d^3 (a+b \arctan (c x))^2}{4 x^4}-\frac {d^3 (a+b \arctan (c x))^2}{6 x^6}-\frac {3 i c d^3 (a+b \arctan (c x))^2}{5 x^5}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {i c^3 d^3 (a+b \arctan (c x))^2}{3 x^3}+\frac {3 c^2 d^3 (a+b \arctan (c x))^2}{4 x^4}-2 b c \left (-\frac {37}{40} i c^5 d^3 \log \left (\frac {2}{1-i c x}\right ) (a+b \arctan (c x))-\frac {1}{120} i c^5 d^3 \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))+\frac {11 c^4 d^3 (a+b \arctan (c x))}{12 x}-\frac {7 i c^3 d^3 (a+b \arctan (c x))}{15 x^2}-\frac {11 c^2 d^3 (a+b \arctan (c x))}{36 x^3}+\frac {d^3 (a+b \arctan (c x))}{30 x^5}+\frac {3 i c d^3 (a+b \arctan (c x))}{20 x^4}-\frac {14}{15} i a c^5 d^3 \log (x)-\frac {37}{60} i b c^5 d^3 \arctan (c x)+\frac {7}{15} b c^5 d^3 \operatorname {PolyLog}(2,-i c x)-\frac {7}{15} b c^5 d^3 \operatorname {PolyLog}(2,i c x)-\frac {37}{80} b c^5 d^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )+\frac {1}{240} b c^5 d^3 \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right )-\frac {113}{90} b c^5 d^3 \log (x)-\frac {37 i b c^4 d^3}{60 x}-\frac {61 b c^3 d^3}{360 x^2}+\frac {i b c^2 d^3}{20 x^3}+\frac {113}{180} b c^5 d^3 \log \left (c^2 x^2+1\right )+\frac {b c d^3}{120 x^4}\right )-\frac {d^3 (a+b \arctan (c x))^2}{6 x^6}-\frac {3 i c d^3 (a+b \arctan (c x))^2}{5 x^5}\) |
Input:
Int[((d + I*c*d*x)^3*(a + b*ArcTan[c*x])^2)/x^7,x]
Output:
-1/6*(d^3*(a + b*ArcTan[c*x])^2)/x^6 - (((3*I)/5)*c*d^3*(a + b*ArcTan[c*x] )^2)/x^5 + (3*c^2*d^3*(a + b*ArcTan[c*x])^2)/(4*x^4) + ((I/3)*c^3*d^3*(a + b*ArcTan[c*x])^2)/x^3 - 2*b*c*((b*c*d^3)/(120*x^4) + ((I/20)*b*c^2*d^3)/x ^3 - (61*b*c^3*d^3)/(360*x^2) - (((37*I)/60)*b*c^4*d^3)/x - ((37*I)/60)*b* c^5*d^3*ArcTan[c*x] + (d^3*(a + b*ArcTan[c*x]))/(30*x^5) + (((3*I)/20)*c*d ^3*(a + b*ArcTan[c*x]))/x^4 - (11*c^2*d^3*(a + b*ArcTan[c*x]))/(36*x^3) - (((7*I)/15)*c^3*d^3*(a + b*ArcTan[c*x]))/x^2 + (11*c^4*d^3*(a + b*ArcTan[c *x]))/(12*x) - ((14*I)/15)*a*c^5*d^3*Log[x] - (113*b*c^5*d^3*Log[x])/90 - ((37*I)/40)*c^5*d^3*(a + b*ArcTan[c*x])*Log[2/(1 - I*c*x)] - (I/120)*c^5*d ^3*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)] + (113*b*c^5*d^3*Log[1 + c^2*x^2 ])/180 + (7*b*c^5*d^3*PolyLog[2, (-I)*c*x])/15 - (7*b*c^5*d^3*PolyLog[2, I *c*x])/15 - (37*b*c^5*d^3*PolyLog[2, 1 - 2/(1 - I*c*x)])/80 + (b*c^5*d^3*P olyLog[2, 1 - 2/(1 + I*c*x)])/240)
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 = 2.11 (sec) , antiderivative size = 564, normalized size of antiderivative = 1.10
| method | result | size |
| parts | \(d^{3} a^{2} \left (\frac {i c^{3}}{3 x^{3}}-\frac {1}{6 x^{6}}-\frac {3 i c}{5 x^{5}}+\frac {3 c^{2}}{4 x^{4}}\right )+d^{3} b^{2} c^{6} \left (-\frac {11 \arctan \left (c x \right )}{6 c x}+\frac {11 \arctan \left (c x \right )}{18 c^{3} x^{3}}-\frac {\arctan \left (c x \right )}{15 c^{5} x^{5}}+\frac {37 i \arctan \left (c x \right )}{30}+\frac {113 \ln \left (c x \right )}{45}-\frac {1}{60 c^{4} x^{4}}+\frac {61}{180 c^{2} x^{2}}-\frac {11 \arctan \left (c x \right )^{2}}{12}-\frac {113 \ln \left (c^{2} x^{2}+1\right )}{90}-\frac {3 i \arctan \left (c x \right )}{10 c^{4} x^{4}}+\frac {3 \arctan \left (c x \right )^{2}}{4 c^{4} x^{4}}-\frac {14 \ln \left (c x \right ) \ln \left (i c x +1\right )}{15}+\frac {14 \ln \left (c x \right ) \ln \left (-i c x +1\right )}{15}+\frac {7 \ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )}{15}-\frac {7 \ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )}{15}-\frac {7 \ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )}{15}+\frac {7 \ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )}{15}-\frac {\arctan \left (c x \right )^{2}}{6 c^{6} x^{6}}-\frac {7 \ln \left (c x -i\right )^{2}}{30}-\frac {7 \operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )}{15}+\frac {7 \ln \left (c x +i\right )^{2}}{30}+\frac {7 \operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )}{15}+\frac {14 \operatorname {dilog}\left (-i c x +1\right )}{15}-\frac {14 \operatorname {dilog}\left (i c x +1\right )}{15}+\frac {28 i \arctan \left (c x \right ) \ln \left (c x \right )}{15}+\frac {37 i}{30 c x}-\frac {14 i \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right )}{15}-\frac {i}{10 c^{3} x^{3}}+\frac {14 i \arctan \left (c x \right )}{15 c^{2} x^{2}}-\frac {3 i \arctan \left (c x \right )^{2}}{5 c^{5} x^{5}}+\frac {i \arctan \left (c x \right )^{2}}{3 c^{3} x^{3}}\right )+2 d^{3} a b \,c^{6} \left (\frac {3 \arctan \left (c x \right )}{4 c^{4} x^{4}}-\frac {3 i \arctan \left (c x \right )}{5 c^{5} x^{5}}+\frac {i \arctan \left (c x \right )}{3 c^{3} x^{3}}-\frac {\arctan \left (c x \right )}{6 c^{6} x^{6}}-\frac {7 i \ln \left (c^{2} x^{2}+1\right )}{15}-\frac {11 \arctan \left (c x \right )}{12}+\frac {14 i \ln \left (c x \right )}{15}-\frac {3 i}{20 c^{4} x^{4}}+\frac {7 i}{15 c^{2} x^{2}}-\frac {1}{30 c^{5} x^{5}}+\frac {11}{36 c^{3} x^{3}}-\frac {11}{12 c x}\right )\) | \(564\) |
| derivativedivides | \(c^{6} \left (d^{3} a^{2} \left (\frac {3}{4 c^{4} x^{4}}-\frac {3 i}{5 c^{5} x^{5}}+\frac {i}{3 c^{3} x^{3}}-\frac {1}{6 c^{6} x^{6}}\right )+d^{3} b^{2} \left (-\frac {11 \arctan \left (c x \right )}{6 c x}+\frac {11 \arctan \left (c x \right )}{18 c^{3} x^{3}}-\frac {\arctan \left (c x \right )}{15 c^{5} x^{5}}+\frac {37 i \arctan \left (c x \right )}{30}+\frac {113 \ln \left (c x \right )}{45}-\frac {1}{60 c^{4} x^{4}}+\frac {61}{180 c^{2} x^{2}}-\frac {11 \arctan \left (c x \right )^{2}}{12}-\frac {113 \ln \left (c^{2} x^{2}+1\right )}{90}-\frac {3 i \arctan \left (c x \right )}{10 c^{4} x^{4}}+\frac {3 \arctan \left (c x \right )^{2}}{4 c^{4} x^{4}}-\frac {14 \ln \left (c x \right ) \ln \left (i c x +1\right )}{15}+\frac {14 \ln \left (c x \right ) \ln \left (-i c x +1\right )}{15}+\frac {7 \ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )}{15}-\frac {7 \ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )}{15}-\frac {7 \ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )}{15}+\frac {7 \ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )}{15}-\frac {\arctan \left (c x \right )^{2}}{6 c^{6} x^{6}}-\frac {7 \ln \left (c x -i\right )^{2}}{30}-\frac {7 \operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )}{15}+\frac {7 \ln \left (c x +i\right )^{2}}{30}+\frac {7 \operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )}{15}+\frac {14 \operatorname {dilog}\left (-i c x +1\right )}{15}-\frac {14 \operatorname {dilog}\left (i c x +1\right )}{15}+\frac {28 i \arctan \left (c x \right ) \ln \left (c x \right )}{15}+\frac {37 i}{30 c x}-\frac {14 i \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right )}{15}-\frac {i}{10 c^{3} x^{3}}+\frac {14 i \arctan \left (c x \right )}{15 c^{2} x^{2}}-\frac {3 i \arctan \left (c x \right )^{2}}{5 c^{5} x^{5}}+\frac {i \arctan \left (c x \right )^{2}}{3 c^{3} x^{3}}\right )+2 d^{3} a b \left (\frac {3 \arctan \left (c x \right )}{4 c^{4} x^{4}}-\frac {3 i \arctan \left (c x \right )}{5 c^{5} x^{5}}+\frac {i \arctan \left (c x \right )}{3 c^{3} x^{3}}-\frac {\arctan \left (c x \right )}{6 c^{6} x^{6}}-\frac {7 i \ln \left (c^{2} x^{2}+1\right )}{15}-\frac {11 \arctan \left (c x \right )}{12}+\frac {14 i \ln \left (c x \right )}{15}-\frac {3 i}{20 c^{4} x^{4}}+\frac {7 i}{15 c^{2} x^{2}}-\frac {1}{30 c^{5} x^{5}}+\frac {11}{36 c^{3} x^{3}}-\frac {11}{12 c x}\right )\right )\) | \(567\) |
| default | \(c^{6} \left (d^{3} a^{2} \left (\frac {3}{4 c^{4} x^{4}}-\frac {3 i}{5 c^{5} x^{5}}+\frac {i}{3 c^{3} x^{3}}-\frac {1}{6 c^{6} x^{6}}\right )+d^{3} b^{2} \left (-\frac {11 \arctan \left (c x \right )}{6 c x}+\frac {11 \arctan \left (c x \right )}{18 c^{3} x^{3}}-\frac {\arctan \left (c x \right )}{15 c^{5} x^{5}}+\frac {37 i \arctan \left (c x \right )}{30}+\frac {113 \ln \left (c x \right )}{45}-\frac {1}{60 c^{4} x^{4}}+\frac {61}{180 c^{2} x^{2}}-\frac {11 \arctan \left (c x \right )^{2}}{12}-\frac {113 \ln \left (c^{2} x^{2}+1\right )}{90}-\frac {3 i \arctan \left (c x \right )}{10 c^{4} x^{4}}+\frac {3 \arctan \left (c x \right )^{2}}{4 c^{4} x^{4}}-\frac {14 \ln \left (c x \right ) \ln \left (i c x +1\right )}{15}+\frac {14 \ln \left (c x \right ) \ln \left (-i c x +1\right )}{15}+\frac {7 \ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )}{15}-\frac {7 \ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )}{15}-\frac {7 \ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )}{15}+\frac {7 \ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )}{15}-\frac {\arctan \left (c x \right )^{2}}{6 c^{6} x^{6}}-\frac {7 \ln \left (c x -i\right )^{2}}{30}-\frac {7 \operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )}{15}+\frac {7 \ln \left (c x +i\right )^{2}}{30}+\frac {7 \operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )}{15}+\frac {14 \operatorname {dilog}\left (-i c x +1\right )}{15}-\frac {14 \operatorname {dilog}\left (i c x +1\right )}{15}+\frac {28 i \arctan \left (c x \right ) \ln \left (c x \right )}{15}+\frac {37 i}{30 c x}-\frac {14 i \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right )}{15}-\frac {i}{10 c^{3} x^{3}}+\frac {14 i \arctan \left (c x \right )}{15 c^{2} x^{2}}-\frac {3 i \arctan \left (c x \right )^{2}}{5 c^{5} x^{5}}+\frac {i \arctan \left (c x \right )^{2}}{3 c^{3} x^{3}}\right )+2 d^{3} a b \left (\frac {3 \arctan \left (c x \right )}{4 c^{4} x^{4}}-\frac {3 i \arctan \left (c x \right )}{5 c^{5} x^{5}}+\frac {i \arctan \left (c x \right )}{3 c^{3} x^{3}}-\frac {\arctan \left (c x \right )}{6 c^{6} x^{6}}-\frac {7 i \ln \left (c^{2} x^{2}+1\right )}{15}-\frac {11 \arctan \left (c x \right )}{12}+\frac {14 i \ln \left (c x \right )}{15}-\frac {3 i}{20 c^{4} x^{4}}+\frac {7 i}{15 c^{2} x^{2}}-\frac {1}{30 c^{5} x^{5}}+\frac {11}{36 c^{3} x^{3}}-\frac {11}{12 c x}\right )\right )\) | \(567\) |
Input:
int((d+I*c*d*x)^3*(a+b*arctan(c*x))^2/x^7,x,method=_RETURNVERBOSE)
Output:
d^3*a^2*(1/3*I*c^3/x^3-1/6/x^6-3/5*I*c/x^5+3/4*c^2/x^4)+d^3*b^2*c^6*(-11/6 /c/x*arctan(c*x)+11/18/c^3/x^3*arctan(c*x)-1/15/c^5/x^5*arctan(c*x)-14/15* dilog(1+I*c*x)+14/15*dilog(1-I*c*x)+113/45*ln(c*x)-1/60/c^4/x^4+61/180/c^2 /x^2-11/12*arctan(c*x)^2-113/90*ln(c^2*x^2+1)+3/4/c^4/x^4*arctan(c*x)^2-14 /15*I*arctan(c*x)*ln(c^2*x^2+1)+28/15*I*arctan(c*x)*ln(c*x)-1/6*arctan(c*x )^2/c^6/x^6+37/30*I/c/x-1/10*I/c^3/x^3-7/30*ln(c*x-I)^2-7/15*dilog(-1/2*I* (c*x+I))+7/30*ln(c*x+I)^2+7/15*dilog(1/2*I*(c*x-I))+7/15*ln(c*x-I)*ln(c^2* x^2+1)-7/15*ln(c*x-I)*ln(-1/2*I*(c*x+I))-7/15*ln(c*x+I)*ln(c^2*x^2+1)+7/15 *ln(c*x+I)*ln(1/2*I*(c*x-I))+37/30*I*arctan(c*x)+1/3*I*arctan(c*x)^2/c^3/x ^3-3/5*I*arctan(c*x)^2/c^5/x^5-3/10*I*arctan(c*x)/c^4/x^4+14/15*I*arctan(c *x)/c^2/x^2-14/15*ln(c*x)*ln(1+I*c*x)+14/15*ln(c*x)*ln(1-I*c*x))+2*d^3*a*b *c^6*(3/4/c^4/x^4*arctan(c*x)-3/5*I*arctan(c*x)/c^5/x^5+1/3*I*arctan(c*x)/ c^3/x^3-1/6*arctan(c*x)/c^6/x^6-7/15*I*ln(c^2*x^2+1)-11/12*arctan(c*x)+14/ 15*I*ln(c*x)-3/20*I/c^4/x^4+7/15*I/c^2/x^2-1/30/c^5/x^5+11/36/c^3/x^3-11/1 2/c/x)
\[ \int \frac {(d+i c d x)^3 (a+b \arctan (c x))^2}{x^7} \, dx=\int { \frac {{\left (i \, c d x + d\right )}^{3} {\left (b \arctan \left (c x\right ) + a\right )}^{2}}{x^{7}} \,d x } \] Input:
integrate((d+I*c*d*x)^3*(a+b*arctan(c*x))^2/x^7,x, algorithm="fricas")
Output:
1/240*(240*x^6*integral(1/60*(-60*I*a^2*c^5*d^3*x^5 - 180*a^2*c^4*d^3*x^4 + 120*I*a^2*c^3*d^3*x^3 - 120*a^2*c^2*d^3*x^2 + 180*I*a^2*c*d^3*x + 60*a^2 *d^3 + (60*a*b*c^5*d^3*x^5 - 20*(9*I*a*b - b^2)*c^4*d^3*x^4 - 15*(8*a*b + 3*I*b^2)*c^3*d^3*x^3 - 12*(10*I*a*b + 3*b^2)*c^2*d^3*x^2 - 10*(18*a*b - I* b^2)*c*d^3*x + 60*I*a*b*d^3)*log(-(c*x + I)/(c*x - I)))/(c^2*x^9 + x^7), x ) + (-20*I*b^2*c^3*d^3*x^3 - 45*b^2*c^2*d^3*x^2 + 36*I*b^2*c*d^3*x + 10*b^ 2*d^3)*log(-(c*x + I)/(c*x - I))^2)/x^6
Timed out. \[ \int \frac {(d+i c d x)^3 (a+b \arctan (c x))^2}{x^7} \, dx=\text {Timed out} \] Input:
integrate((d+I*c*d*x)**3*(a+b*atan(c*x))**2/x**7,x)
Output:
Timed out
\[ \int \frac {(d+i c d x)^3 (a+b \arctan (c x))^2}{x^7} \, dx=\int { \frac {{\left (i \, c d x + d\right )}^{3} {\left (b \arctan \left (c x\right ) + a\right )}^{2}}{x^{7}} \,d x } \] Input:
integrate((d+I*c*d*x)^3*(a+b*arctan(c*x))^2/x^7,x, algorithm="maxima")
Output:
-1/3*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^3*d^3 - 1/2*((3*c^3*arctan(c*x) + (3*c^2*x^2 - 1)/x^3)*c - 3*arct an(c*x)/x^4)*a*b*c^2*d^3 - 3/10*I*((2*c^4*log(c^2*x^2 + 1) - 2*c^4*log(x^2 ) - (2*c^2*x^2 - 1)/x^4)*c + 4*arctan(c*x)/x^5)*a*b*c*d^3 - 1/45*((15*c^5* arctan(c*x) + (15*c^4*x^4 - 5*c^2*x^2 + 3)/x^5)*c + 15*arctan(c*x)/x^6)*a* b*d^3 - 1/180*(4*(15*c^5*arctan(c*x) + (15*c^4*x^4 - 5*c^2*x^2 + 3)/x^5)*c *arctan(c*x) - (30*c^4*x^4*arctan(c*x)^2 - 46*c^4*x^4*log(c^2*x^2 + 1) + 9 2*c^4*x^4*log(x) + 16*c^2*x^2 - 3)*c^2/x^4)*b^2*d^3 + 1/3*I*a^2*c^3*d^3/x^ 3 + 3/4*a^2*c^2*d^3/x^4 - 3/5*I*a^2*c*d^3/x^5 - 1/6*b^2*d^3*arctan(c*x)^2/ x^6 - 1/6*a^2*d^3/x^6 - 1/960*(960*I*x^5*integrate(1/240*(180*(b^2*c^5*d^3 *x^4 - 2*b^2*c^3*d^3*x^2 - 3*b^2*c*d^3)*arctan(c*x)^2 + 15*(b^2*c^5*d^3*x^ 4 - 2*b^2*c^3*d^3*x^2 - 3*b^2*c*d^3)*log(c^2*x^2 + 1)^2 + 2*(65*b^2*c^4*d^ 3*x^3 - 36*b^2*c^2*d^3*x)*arctan(c*x) - (20*b^2*c^5*d^3*x^4 - 81*b^2*c^3*d ^3*x^2 + 180*(b^2*c^4*d^3*x^3 + b^2*c^2*d^3*x)*arctan(c*x))*log(c^2*x^2 + 1))/(c^2*x^8 + x^6), x) + 960*x^5*integrate(1/240*(540*(b^2*c^4*d^3*x^3 + b^2*c^2*d^3*x)*arctan(c*x)^2 + 45*(b^2*c^4*d^3*x^3 + b^2*c^2*d^3*x)*log(c^ 2*x^2 + 1)^2 - 2*(20*b^2*c^5*d^3*x^4 - 81*b^2*c^3*d^3*x^2)*arctan(c*x) - ( 65*b^2*c^4*d^3*x^3 - 36*b^2*c^2*d^3*x - 60*(b^2*c^5*d^3*x^4 - 2*b^2*c^3*d^ 3*x^2 - 3*b^2*c*d^3)*arctan(c*x))*log(c^2*x^2 + 1))/(c^2*x^8 + x^6), x) - 4*(20*I*b^2*c^3*d^3*x^2 + 45*b^2*c^2*d^3*x - 36*I*b^2*c*d^3)*arctan(c*x...
\[ \int \frac {(d+i c d x)^3 (a+b \arctan (c x))^2}{x^7} \, dx=\int { \frac {{\left (i \, c d x + d\right )}^{3} {\left (b \arctan \left (c x\right ) + a\right )}^{2}}{x^{7}} \,d x } \] Input:
integrate((d+I*c*d*x)^3*(a+b*arctan(c*x))^2/x^7,x, algorithm="giac")
Output:
integrate((I*c*d*x + d)^3*(b*arctan(c*x) + a)^2/x^7, x)
Timed out. \[ \int \frac {(d+i c d x)^3 (a+b \arctan (c x))^2}{x^7} \, dx=\int \frac {{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^2\,{\left (d+c\,d\,x\,1{}\mathrm {i}\right )}^3}{x^7} \,d x \] Input:
int(((a + b*atan(c*x))^2*(d + c*d*x*1i)^3)/x^7,x)
Output:
int(((a + b*atan(c*x))^2*(d + c*d*x*1i)^3)/x^7, x)
\[ \int \frac {(d+i c d x)^3 (a+b \arctan (c x))^2}{x^7} \, dx=\frac {d^{3} \left (-216 \mathit {atan} \left (c x \right ) a b c i x -168 \,\mathrm {log}\left (c^{2} x^{2}+1\right ) a b \,c^{6} i \,x^{6}+336 \,\mathrm {log}\left (x \right ) a b \,c^{6} i \,x^{6}+60 \mathit {atan} \left (c x \right )^{2} b^{2} c^{3} i \,x^{3}-108 \mathit {atan} \left (c x \right )^{2} b^{2} c i x +168 \mathit {atan} \left (c x \right ) b^{2} c^{4} i \,x^{4}-54 \mathit {atan} \left (c x \right ) b^{2} c^{2} i \,x^{2}+168 a b \,c^{4} i \,x^{4}-54 a b \,c^{2} i \,x^{2}-12 \mathit {atan} \left (c x \right ) b^{2} c x +110 a b \,c^{3} x^{3}-12 a b c x +336 \left (\int \frac {\mathit {atan} \left (c x \right )}{c^{2} x^{3}+x}d x \right ) b^{2} c^{6} i \,x^{6}+222 b^{2} c^{5} i \,x^{5}-30 a^{2}-165 \mathit {atan} \left (c x \right )^{2} b^{2} c^{6} x^{6}-330 \mathit {atan} \left (c x \right ) b^{2} c^{5} x^{5}-330 a b \,c^{5} x^{5}+135 \mathit {atan} \left (c x \right )^{2} b^{2} c^{2} x^{2}-226 \,\mathrm {log}\left (c^{2} x^{2}+1\right ) b^{2} c^{6} x^{6}+120 \mathit {atan} \left (c x \right ) a b \,c^{3} i \,x^{3}+60 a^{2} c^{3} i \,x^{3}+135 a^{2} c^{2} x^{2}+270 \mathit {atan} \left (c x \right ) a b \,c^{2} x^{2}+61 b^{2} c^{4} x^{4}-108 a^{2} c i x +222 \mathit {atan} \left (c x \right ) b^{2} c^{6} i \,x^{6}+110 \mathit {atan} \left (c x \right ) b^{2} c^{3} x^{3}-18 b^{2} c^{3} i \,x^{3}-330 \mathit {atan} \left (c x \right ) a b \,c^{6} x^{6}-30 \mathit {atan} \left (c x \right )^{2} b^{2}+452 \,\mathrm {log}\left (x \right ) b^{2} c^{6} x^{6}-60 \mathit {atan} \left (c x \right ) a b -3 b^{2} c^{2} x^{2}\right )}{180 x^{6}} \] Input:
int((d+I*c*d*x)^3*(a+b*atan(c*x))^2/x^7,x)
Output:
(d**3*( - 165*atan(c*x)**2*b**2*c**6*x**6 + 60*atan(c*x)**2*b**2*c**3*i*x* *3 + 135*atan(c*x)**2*b**2*c**2*x**2 - 108*atan(c*x)**2*b**2*c*i*x - 30*at an(c*x)**2*b**2 - 330*atan(c*x)*a*b*c**6*x**6 + 120*atan(c*x)*a*b*c**3*i*x **3 + 270*atan(c*x)*a*b*c**2*x**2 - 216*atan(c*x)*a*b*c*i*x - 60*atan(c*x) *a*b + 222*atan(c*x)*b**2*c**6*i*x**6 - 330*atan(c*x)*b**2*c**5*x**5 + 168 *atan(c*x)*b**2*c**4*i*x**4 + 110*atan(c*x)*b**2*c**3*x**3 - 54*atan(c*x)* b**2*c**2*i*x**2 - 12*atan(c*x)*b**2*c*x + 336*int(atan(c*x)/(c**2*x**3 + x),x)*b**2*c**6*i*x**6 - 168*log(c**2*x**2 + 1)*a*b*c**6*i*x**6 - 226*log( c**2*x**2 + 1)*b**2*c**6*x**6 + 336*log(x)*a*b*c**6*i*x**6 + 452*log(x)*b* *2*c**6*x**6 + 60*a**2*c**3*i*x**3 + 135*a**2*c**2*x**2 - 108*a**2*c*i*x - 30*a**2 - 330*a*b*c**5*x**5 + 168*a*b*c**4*i*x**4 + 110*a*b*c**3*x**3 - 5 4*a*b*c**2*i*x**2 - 12*a*b*c*x + 222*b**2*c**5*i*x**5 + 61*b**2*c**4*x**4 - 18*b**2*c**3*i*x**3 - 3*b**2*c**2*x**2))/(180*x**6)