Integrand size = 25, antiderivative size = 429 \[ \int \frac {(d+i c d x)^3 (a+b \arctan (c x))^2}{x^4} \, dx=-\frac {b^2 c^2 d^3}{3 x}-\frac {1}{3} b^2 c^3 d^3 \arctan (c x)-\frac {b c d^3 (a+b \arctan (c x))}{3 x^2}-\frac {3 i b c^2 d^3 (a+b \arctan (c x))}{x}+\frac {11}{6} i c^3 d^3 (a+b \arctan (c x))^2-\frac {d^3 (a+b \arctan (c x))^2}{3 x^3}-\frac {3 i c d^3 (a+b \arctan (c x))^2}{2 x^2}+\frac {3 c^2 d^3 (a+b \arctan (c x))^2}{x}-2 i c^3 d^3 (a+b \arctan (c x))^2 \text {arctanh}\left (1-\frac {2}{1+i c x}\right )+3 i b^2 c^3 d^3 \log (x)-\frac {3}{2} i b^2 c^3 d^3 \log \left (1+c^2 x^2\right )-\frac {20}{3} b c^3 d^3 (a+b \arctan (c x)) \log \left (2-\frac {2}{1-i c x}\right )+\frac {10}{3} i b^2 c^3 d^3 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i c x}\right )-b c^3 d^3 (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )+b c^3 d^3 (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i c x}\right )+\frac {1}{2} i b^2 c^3 d^3 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i c x}\right )-\frac {1}{2} i b^2 c^3 d^3 \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i c x}\right ) \]
-1/3*b^2*c^2*d^3/x-1/3*b^2*c^3*d^3*arctan(c*x)-1/3*b*c*d^3*(a+b*arctan(c*x ))/x^2-3*I*b*c^2*d^3*(a+b*arctan(c*x))/x+10/3*I*b^2*c^3*d^3*polylog(2,-1+2 /(1-I*c*x))-1/3*d^3*(a+b*arctan(c*x))^2/x^3+11/6*I*c^3*d^3*(a+b*arctan(c*x ))^2+3*c^2*d^3*(a+b*arctan(c*x))^2/x+2*I*c^3*d^3*(a+b*arctan(c*x))^2*arcta nh(-1+2/(1+I*c*x))-1/2*I*b^2*c^3*d^3*polylog(3,-1+2/(1+I*c*x))-3/2*I*b^2*c ^3*d^3*ln(c^2*x^2+1)-20/3*b*c^3*d^3*(a+b*arctan(c*x))*ln(2-2/(1-I*c*x))-3/ 2*I*c*d^3*(a+b*arctan(c*x))^2/x^2-b*c^3*d^3*(a+b*arctan(c*x))*polylog(2,1- 2/(1+I*c*x))+b*c^3*d^3*(a+b*arctan(c*x))*polylog(2,-1+2/(1+I*c*x))+3*I*b^2 *c^3*d^3*ln(x)+1/2*I*b^2*c^3*d^3*polylog(3,1-2/(1+I*c*x))
Time = 0.64 (sec) , antiderivative size = 595, normalized size of antiderivative = 1.39 \[ \int \frac {(d+i c d x)^3 (a+b \arctan (c x))^2}{x^4} \, dx=\frac {d^3 \left (-8 a^2-36 i a^2 c x-8 a b c x+72 a^2 c^2 x^2-72 i a b c^2 x^2-8 b^2 c^2 x^2-b^2 c^3 \pi ^3 x^3-16 a b \arctan (c x)-72 i a b c x \arctan (c x)-8 b^2 c x \arctan (c x)+144 a b c^2 x^2 \arctan (c x)-72 i b^2 c^2 x^2 \arctan (c x)-72 i a b c^3 x^3 \arctan (c x)-8 b^2 c^3 x^3 \arctan (c x)-8 b^2 \arctan (c x)^2-36 i b^2 c x \arctan (c x)^2+72 b^2 c^2 x^2 \arctan (c x)^2+44 i b^2 c^3 x^3 \arctan (c x)^2+16 b^2 c^3 x^3 \arctan (c x)^3-24 i b^2 c^3 x^3 \arctan (c x)^2 \log \left (1-e^{-2 i \arctan (c x)}\right )-160 b^2 c^3 x^3 \arctan (c x) \log \left (1-e^{2 i \arctan (c x)}\right )+24 i b^2 c^3 x^3 \arctan (c x)^2 \log \left (1+e^{2 i \arctan (c x)}\right )-24 i a^2 c^3 x^3 \log (x)-160 a b c^3 x^3 \log (c x)+72 i b^2 c^3 x^3 \log \left (\frac {c x}{\sqrt {1+c^2 x^2}}\right )+80 a b c^3 x^3 \log \left (1+c^2 x^2\right )+24 b^2 c^3 x^3 \arctan (c x) \operatorname {PolyLog}\left (2,e^{-2 i \arctan (c x)}\right )+24 b^2 c^3 x^3 \arctan (c x) \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )+80 i b^2 c^3 x^3 \operatorname {PolyLog}\left (2,e^{2 i \arctan (c x)}\right )+24 a b c^3 x^3 \operatorname {PolyLog}(2,-i c x)-24 a b c^3 x^3 \operatorname {PolyLog}(2,i c x)-12 i b^2 c^3 x^3 \operatorname {PolyLog}\left (3,e^{-2 i \arctan (c x)}\right )+12 i b^2 c^3 x^3 \operatorname {PolyLog}\left (3,-e^{2 i \arctan (c x)}\right )\right )}{24 x^3} \]
(d^3*(-8*a^2 - (36*I)*a^2*c*x - 8*a*b*c*x + 72*a^2*c^2*x^2 - (72*I)*a*b*c^ 2*x^2 - 8*b^2*c^2*x^2 - b^2*c^3*Pi^3*x^3 - 16*a*b*ArcTan[c*x] - (72*I)*a*b *c*x*ArcTan[c*x] - 8*b^2*c*x*ArcTan[c*x] + 144*a*b*c^2*x^2*ArcTan[c*x] - ( 72*I)*b^2*c^2*x^2*ArcTan[c*x] - (72*I)*a*b*c^3*x^3*ArcTan[c*x] - 8*b^2*c^3 *x^3*ArcTan[c*x] - 8*b^2*ArcTan[c*x]^2 - (36*I)*b^2*c*x*ArcTan[c*x]^2 + 72 *b^2*c^2*x^2*ArcTan[c*x]^2 + (44*I)*b^2*c^3*x^3*ArcTan[c*x]^2 + 16*b^2*c^3 *x^3*ArcTan[c*x]^3 - (24*I)*b^2*c^3*x^3*ArcTan[c*x]^2*Log[1 - E^((-2*I)*Ar cTan[c*x])] - 160*b^2*c^3*x^3*ArcTan[c*x]*Log[1 - E^((2*I)*ArcTan[c*x])] + (24*I)*b^2*c^3*x^3*ArcTan[c*x]^2*Log[1 + E^((2*I)*ArcTan[c*x])] - (24*I)* a^2*c^3*x^3*Log[x] - 160*a*b*c^3*x^3*Log[c*x] + (72*I)*b^2*c^3*x^3*Log[(c* x)/Sqrt[1 + c^2*x^2]] + 80*a*b*c^3*x^3*Log[1 + c^2*x^2] + 24*b^2*c^3*x^3*A rcTan[c*x]*PolyLog[2, E^((-2*I)*ArcTan[c*x])] + 24*b^2*c^3*x^3*ArcTan[c*x] *PolyLog[2, -E^((2*I)*ArcTan[c*x])] + (80*I)*b^2*c^3*x^3*PolyLog[2, E^((2* I)*ArcTan[c*x])] + 24*a*b*c^3*x^3*PolyLog[2, (-I)*c*x] - 24*a*b*c^3*x^3*Po lyLog[2, I*c*x] - (12*I)*b^2*c^3*x^3*PolyLog[3, E^((-2*I)*ArcTan[c*x])] + (12*I)*b^2*c^3*x^3*PolyLog[3, -E^((2*I)*ArcTan[c*x])]))/(24*x^3)
Time = 1.10 (sec) , antiderivative size = 429, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, 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.
| \(\displaystyle \int \frac {(d+i c d x)^3 (a+b \arctan (c x))^2}{x^4} \, dx\) |
| \(\Big \downarrow \) 5411 |
| \(\displaystyle \int \left (-\frac {i c^3 d^3 (a+b \arctan (c x))^2}{x}-\frac {3 c^2 d^3 (a+b \arctan (c x))^2}{x^2}+\frac {d^3 (a+b \arctan (c x))^2}{x^4}+\frac {3 i c d^3 (a+b \arctan (c x))^2}{x^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -2 i c^3 d^3 \text {arctanh}\left (1-\frac {2}{1+i c x}\right ) (a+b \arctan (c x))^2-b c^3 d^3 \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right ) (a+b \arctan (c x))+b c^3 d^3 \operatorname {PolyLog}\left (2,\frac {2}{i c x+1}-1\right ) (a+b \arctan (c x))+\frac {11}{6} i c^3 d^3 (a+b \arctan (c x))^2-\frac {20}{3} b c^3 d^3 \log \left (2-\frac {2}{1-i c x}\right ) (a+b \arctan (c x))+\frac {3 c^2 d^3 (a+b \arctan (c x))^2}{x}-\frac {3 i b c^2 d^3 (a+b \arctan (c x))}{x}-\frac {d^3 (a+b \arctan (c x))^2}{3 x^3}-\frac {3 i c d^3 (a+b \arctan (c x))^2}{2 x^2}-\frac {b c d^3 (a+b \arctan (c x))}{3 x^2}-\frac {1}{3} b^2 c^3 d^3 \arctan (c x)+\frac {10}{3} i b^2 c^3 d^3 \operatorname {PolyLog}\left (2,\frac {2}{1-i c x}-1\right )+\frac {1}{2} i b^2 c^3 d^3 \operatorname {PolyLog}\left (3,1-\frac {2}{i c x+1}\right )-\frac {1}{2} i b^2 c^3 d^3 \operatorname {PolyLog}\left (3,\frac {2}{i c x+1}-1\right )+3 i b^2 c^3 d^3 \log (x)-\frac {b^2 c^2 d^3}{3 x}-\frac {3}{2} i b^2 c^3 d^3 \log \left (c^2 x^2+1\right )\) |
-1/3*(b^2*c^2*d^3)/x - (b^2*c^3*d^3*ArcTan[c*x])/3 - (b*c*d^3*(a + b*ArcTa n[c*x]))/(3*x^2) - ((3*I)*b*c^2*d^3*(a + b*ArcTan[c*x]))/x + ((11*I)/6)*c^ 3*d^3*(a + b*ArcTan[c*x])^2 - (d^3*(a + b*ArcTan[c*x])^2)/(3*x^3) - (((3*I )/2)*c*d^3*(a + b*ArcTan[c*x])^2)/x^2 + (3*c^2*d^3*(a + b*ArcTan[c*x])^2)/ x - (2*I)*c^3*d^3*(a + b*ArcTan[c*x])^2*ArcTanh[1 - 2/(1 + I*c*x)] + (3*I) *b^2*c^3*d^3*Log[x] - ((3*I)/2)*b^2*c^3*d^3*Log[1 + c^2*x^2] - (20*b*c^3*d ^3*(a + b*ArcTan[c*x])*Log[2 - 2/(1 - I*c*x)])/3 + ((10*I)/3)*b^2*c^3*d^3* PolyLog[2, -1 + 2/(1 - I*c*x)] - b*c^3*d^3*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 + I*c*x)] + b*c^3*d^3*(a + b*ArcTan[c*x])*PolyLog[2, -1 + 2/(1 + I*c*x)] + (I/2)*b^2*c^3*d^3*PolyLog[3, 1 - 2/(1 + I*c*x)] - (I/2)*b^2*c^3* d^3*PolyLog[3, -1 + 2/(1 + I*c*x)]
3.1.91.3.1 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])
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 33.32 (sec) , antiderivative size = 1655, normalized size of antiderivative = 3.86
| method | result | size |
| derivativedivides | \(\text {Expression too large to display}\) | \(1655\) |
| default | \(\text {Expression too large to display}\) | \(1655\) |
| parts | \(\text {Expression too large to display}\) | \(1727\) |
c^3*(8/3*b^2*d^3*arctan(c*x)+1/2*b^2*d^3*Pi*arctan(c*x)^2+b^2*d^3*arctan(c *x)*polylog(2,-(1+I*c*x)^2/(c^2*x^2+1))+3*b^2*d^3*arctan(c*x)^2/c/x+1/2*b^ 2*d^3*Pi*arctan(c*x)^2*csgn(I*((1+I*c*x)^2/(c^2*x^2+1)-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+(1+I*c*x)^2/(c^2*x^2+1))) -2*b^2*d^3*arctan(c*x)*polylog(2,(1+I*c*x)/(c^2*x^2+1)^(1/2))-2*b^2*d^3*ar ctan(c*x)*polylog(2,-(1+I*c*x)/(c^2*x^2+1)^(1/2))-20/3*b^2*d^3*arctan(c*x) *ln(1+(1+I*c*x)/(c^2*x^2+1)^(1/2))+2*a*d^3*b*(-1/3*arctan(c*x)/c^3/x^3-I*a rctan(c*x)*ln(c*x)+3/c/x*arctan(c*x)-3/2*I*arctan(c*x)/c^2/x^2+1/2*ln(c*x) *ln(1+I*c*x)-1/2*ln(c*x)*ln(1-I*c*x)+1/2*dilog(1+I*c*x)-1/2*dilog(1-I*c*x) -3/2*I/c/x-1/6/c^2/x^2-10/3*ln(c*x)+5/3*ln(c^2*x^2+1)-3/2*I*arctan(c*x))+1 1/6*I*b^2*d^3*arctan(c*x)^2+1/2*I*b^2*d^3*polylog(3,-(1+I*c*x)^2/(c^2*x^2+ 1))-2*I*b^2*d^3*polylog(3,-(1+I*c*x)/(c^2*x^2+1)^(1/2))-2*I*b^2*d^3*polylo g(3,(1+I*c*x)/(c^2*x^2+1)^(1/2))+20/3*I*b^2*d^3*dilog(1+(1+I*c*x)/(c^2*x^2 +1)^(1/2))-20/3*I*b^2*d^3*dilog((1+I*c*x)/(c^2*x^2+1)^(1/2))+3*I*b^2*d^3*l n(1+(1+I*c*x)/(c^2*x^2+1)^(1/2))+3*I*b^2*d^3*ln((1+I*c*x)/(c^2*x^2+1)^(1/2 )-1)+d^3*a^2*(-1/3/c^3/x^3-I*ln(c*x)+3/c/x-3/2*I/c^2/x^2)+1/2*b^2*d^3*Pi*a rctan(c*x)^2*csgn(I*((1+I*c*x)^2/(c^2*x^2+1)-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))*csgn(I/(1+(1+I*c*x)^2/(c^2*x^2+1))) +1/2*b^2*d^3*Pi*arctan(c*x)^2*csgn(((1+I*c*x)^2/(c^2*x^2+1)-1)/(1+(1+I*c*x )^2/(c^2*x^2+1)))^3-1/2*b^2*d^3*Pi*arctan(c*x)^2*csgn(((1+I*c*x)^2/(c^2...
\[ \int \frac {(d+i c d x)^3 (a+b \arctan (c x))^2}{x^4} \, dx=\int { \frac {{\left (i \, c d x + d\right )}^{3} {\left (b \arctan \left (c x\right ) + a\right )}^{2}}{x^{4}} \,d x } \]
integral(1/4*(-4*I*a^2*c^3*d^3*x^3 - 12*a^2*c^2*d^3*x^2 + 12*I*a^2*c*d^3*x + 4*a^2*d^3 + (I*b^2*c^3*d^3*x^3 + 3*b^2*c^2*d^3*x^2 - 3*I*b^2*c*d^3*x - b^2*d^3)*log(-(c*x + I)/(c*x - I))^2 + 4*(a*b*c^3*d^3*x^3 - 3*I*a*b*c^2*d^ 3*x^2 - 3*a*b*c*d^3*x + I*a*b*d^3)*log(-(c*x + I)/(c*x - I)))/x^4, x)
Timed out. \[ \int \frac {(d+i c d x)^3 (a+b \arctan (c x))^2}{x^4} \, dx=\text {Timed out} \]
\[ \int \frac {(d+i c d x)^3 (a+b \arctan (c x))^2}{x^4} \, dx=\int { \frac {{\left (i \, c d x + d\right )}^{3} {\left (b \arctan \left (c x\right ) + a\right )}^{2}}{x^{4}} \,d x } \]
-I*a^2*c^3*d^3*log(x) + 3*(c*(log(c^2*x^2 + 1) - log(x^2)) + 2*arctan(c*x) /x)*a*b*c^2*d^3 - 3*I*((c*arctan(c*x) + 1/x)*c + arctan(c*x)/x^2)*a*b*c*d^ 3 + 1/3*((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*d^3 + 3*a^2*c^2*d^3/x - 3/2*I*a^2*c*d^3/x^2 - 1/3*a^2*d^3/x^3 - 1/ 96*(24*(3*b^2*c^3*d^3*arctan(c*x)^3 + 48*b^2*c^5*d^3*integrate(1/48*x^5*ar ctan(c*x)*log(c^2*x^2 + 1)/(c^2*x^6 + x^4), x) + 36*b^2*c^4*d^3*integrate( 1/48*x^4*log(c^2*x^2 + 1)^2/(c^2*x^6 + x^4), x) - 144*b^2*c^4*d^3*integrat e(1/48*x^4*log(c^2*x^2 + 1)/(c^2*x^6 + x^4), x) - 96*b^2*c^3*d^3*integrate (1/48*x^3*arctan(c*x)*log(c^2*x^2 + 1)/(c^2*x^6 + x^4), x) + 432*b^2*c^3*d ^3*integrate(1/48*x^3*arctan(c*x)/(c^2*x^6 + x^4), x) + 288*b^2*c^2*d^3*in tegrate(1/48*x^2*arctan(c*x)^2/(c^2*x^6 + x^4), x) + 24*b^2*c^2*d^3*integr ate(1/48*x^2*log(c^2*x^2 + 1)^2/(c^2*x^6 + x^4), x) + 88*b^2*c^2*d^3*integ rate(1/48*x^2*log(c^2*x^2 + 1)/(c^2*x^6 + x^4), x) - 144*b^2*c*d^3*integra te(1/48*x*arctan(c*x)*log(c^2*x^2 + 1)/(c^2*x^6 + x^4), x) - 32*b^2*c*d^3* integrate(1/48*x*arctan(c*x)/(c^2*x^6 + x^4), x) - 144*b^2*d^3*integrate(1 /48*arctan(c*x)^2/(c^2*x^6 + x^4), x) - 12*b^2*d^3*integrate(1/48*log(c^2* x^2 + 1)^2/(c^2*x^6 + x^4), x))*x^3 + I*(3456*b^2*c^5*d^3*integrate(1/48*x ^5*arctan(c*x)^2/(c^2*x^6 + x^4), x) + 9216*a*b*c^5*d^3*integrate(1/48*x^5 *arctan(c*x)/(c^2*x^6 + x^4), x) + b^2*c^3*d^3*log(c^2*x^2 + 1)^3 + 72*b^2 *c^3*d^3*arctan(c*x)^2 - 3456*b^2*c^4*d^3*integrate(1/48*x^4*arctan(c*x...
Timed out. \[ \int \frac {(d+i c d x)^3 (a+b \arctan (c x))^2}{x^4} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {(d+i c d x)^3 (a+b \arctan (c x))^2}{x^4} \, dx=\int \frac {{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^2\,{\left (d+c\,d\,x\,1{}\mathrm {i}\right )}^3}{x^4} \,d x \]