Integrand size = 25, antiderivative size = 385 \[ \int \frac {(d+i c d x)^3 (a+b \arctan (c x))^2}{x} \, dx=3 a b c d^3 x-\frac {1}{3} i b^2 c d^3 x+\frac {1}{3} i b^2 d^3 \arctan (c x)+3 b^2 c d^3 x \arctan (c x)+\frac {1}{3} i b c^2 d^3 x^2 (a+b \arctan (c x))-\frac {29}{6} d^3 (a+b \arctan (c x))^2+3 i c d^3 x (a+b \arctan (c x))^2-\frac {3}{2} c^2 d^3 x^2 (a+b \arctan (c x))^2-\frac {1}{3} i c^3 d^3 x^3 (a+b \arctan (c x))^2+2 d^3 (a+b \arctan (c x))^2 \text {arctanh}\left (1-\frac {2}{1+i c x}\right )+\frac {20}{3} i b d^3 (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )-\frac {3}{2} b^2 d^3 \log \left (1+c^2 x^2\right )-\frac {10}{3} b^2 d^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )-i b d^3 (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )+i b d^3 (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i c x}\right )-\frac {1}{2} b^2 d^3 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i c x}\right )+\frac {1}{2} b^2 d^3 \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i c x}\right ) \]
3*a*b*c*d^3*x+1/3*I*b^2*d^3*arctan(c*x)-1/3*I*c^3*d^3*x^3*(a+b*arctan(c*x) )^2+3*b^2*c*d^3*x*arctan(c*x)+1/3*I*b*c^2*d^3*x^2*(a+b*arctan(c*x))-29/6*d ^3*(a+b*arctan(c*x))^2-1/3*I*b^2*c*d^3*x-3/2*c^2*d^3*x^2*(a+b*arctan(c*x)) ^2+I*b*d^3*(a+b*arctan(c*x))*polylog(2,-1+2/(1+I*c*x))-2*d^3*(a+b*arctan(c *x))^2*arctanh(-1+2/(1+I*c*x))+3*I*c*d^3*x*(a+b*arctan(c*x))^2-3/2*b^2*d^3 *ln(c^2*x^2+1)-10/3*b^2*d^3*polylog(2,1-2/(1+I*c*x))+20/3*I*b*d^3*(a+b*arc tan(c*x))*ln(2/(1+I*c*x))-I*b*d^3*(a+b*arctan(c*x))*polylog(2,1-2/(1+I*c*x ))-1/2*b^2*d^3*polylog(3,1-2/(1+I*c*x))+1/2*b^2*d^3*polylog(3,-1+2/(1+I*c* x))
Time = 0.82 (sec) , antiderivative size = 465, normalized size of antiderivative = 1.21 \[ \int \frac {(d+i c d x)^3 (a+b \arctan (c x))^2}{x} \, dx=-\frac {1}{24} i d^3 \left (b^2 \pi ^3-72 a^2 c x+72 i a b c x+8 b^2 c x-36 i a^2 c^2 x^2-8 a b c^2 x^2+8 a^2 c^3 x^3-72 i a b \arctan (c x)-8 b^2 \arctan (c x)-144 a b c x \arctan (c x)+72 i b^2 c x \arctan (c x)-72 i a b c^2 x^2 \arctan (c x)-8 b^2 c^2 x^2 \arctan (c x)+16 a b c^3 x^3 \arctan (c x)+44 i b^2 \arctan (c x)^2-72 b^2 c x \arctan (c x)^2-36 i b^2 c^2 x^2 \arctan (c x)^2+8 b^2 c^3 x^3 \arctan (c x)^2-16 b^2 \arctan (c x)^3+24 i b^2 \arctan (c x)^2 \log \left (1-e^{-2 i \arctan (c x)}\right )-160 b^2 \arctan (c x) \log \left (1+e^{2 i \arctan (c x)}\right )-24 i b^2 \arctan (c x)^2 \log \left (1+e^{2 i \arctan (c x)}\right )+24 i a^2 \log (c x)+80 a b \log \left (1+c^2 x^2\right )-36 i b^2 \log \left (1+c^2 x^2\right )-24 b^2 \arctan (c x) \operatorname {PolyLog}\left (2,e^{-2 i \arctan (c x)}\right )-8 b^2 (-10 i+3 \arctan (c x)) \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )-24 a b \operatorname {PolyLog}(2,-i c x)+24 a b \operatorname {PolyLog}(2,i c x)+12 i b^2 \operatorname {PolyLog}\left (3,e^{-2 i \arctan (c x)}\right )-12 i b^2 \operatorname {PolyLog}\left (3,-e^{2 i \arctan (c x)}\right )\right ) \]
(-1/24*I)*d^3*(b^2*Pi^3 - 72*a^2*c*x + (72*I)*a*b*c*x + 8*b^2*c*x - (36*I) *a^2*c^2*x^2 - 8*a*b*c^2*x^2 + 8*a^2*c^3*x^3 - (72*I)*a*b*ArcTan[c*x] - 8* b^2*ArcTan[c*x] - 144*a*b*c*x*ArcTan[c*x] + (72*I)*b^2*c*x*ArcTan[c*x] - ( 72*I)*a*b*c^2*x^2*ArcTan[c*x] - 8*b^2*c^2*x^2*ArcTan[c*x] + 16*a*b*c^3*x^3 *ArcTan[c*x] + (44*I)*b^2*ArcTan[c*x]^2 - 72*b^2*c*x*ArcTan[c*x]^2 - (36*I )*b^2*c^2*x^2*ArcTan[c*x]^2 + 8*b^2*c^3*x^3*ArcTan[c*x]^2 - 16*b^2*ArcTan[ c*x]^3 + (24*I)*b^2*ArcTan[c*x]^2*Log[1 - E^((-2*I)*ArcTan[c*x])] - 160*b^ 2*ArcTan[c*x]*Log[1 + E^((2*I)*ArcTan[c*x])] - (24*I)*b^2*ArcTan[c*x]^2*Lo g[1 + E^((2*I)*ArcTan[c*x])] + (24*I)*a^2*Log[c*x] + 80*a*b*Log[1 + c^2*x^ 2] - (36*I)*b^2*Log[1 + c^2*x^2] - 24*b^2*ArcTan[c*x]*PolyLog[2, E^((-2*I) *ArcTan[c*x])] - 8*b^2*(-10*I + 3*ArcTan[c*x])*PolyLog[2, -E^((2*I)*ArcTan [c*x])] - 24*a*b*PolyLog[2, (-I)*c*x] + 24*a*b*PolyLog[2, I*c*x] + (12*I)* b^2*PolyLog[3, E^((-2*I)*ArcTan[c*x])] - (12*I)*b^2*PolyLog[3, -E^((2*I)*A rcTan[c*x])])
Time = 0.95 (sec) , antiderivative size = 385, 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.
| \(\displaystyle \int \frac {(d+i c d x)^3 (a+b \arctan (c x))^2}{x} \, dx\) |
| \(\Big \downarrow \) 5411 |
| \(\displaystyle \int \left (-i c^3 d^3 x^2 (a+b \arctan (c x))^2-3 c^2 d^3 x (a+b \arctan (c x))^2+3 i c d^3 (a+b \arctan (c x))^2+\frac {d^3 (a+b \arctan (c x))^2}{x}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 d^3 \text {arctanh}\left (1-\frac {2}{1+i c x}\right ) (a+b \arctan (c x))^2-\frac {1}{3} i c^3 d^3 x^3 (a+b \arctan (c x))^2-\frac {3}{2} c^2 d^3 x^2 (a+b \arctan (c x))^2+\frac {1}{3} i b c^2 d^3 x^2 (a+b \arctan (c x))-i b d^3 \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right ) (a+b \arctan (c x))+i b d^3 \operatorname {PolyLog}\left (2,\frac {2}{i c x+1}-1\right ) (a+b \arctan (c x))+3 i c d^3 x (a+b \arctan (c x))^2-\frac {29}{6} d^3 (a+b \arctan (c x))^2+\frac {20}{3} i b d^3 \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))+3 a b c d^3 x+\frac {1}{3} i b^2 d^3 \arctan (c x)+3 b^2 c d^3 x \arctan (c x)-\frac {3}{2} b^2 d^3 \log \left (c^2 x^2+1\right )-\frac {10}{3} b^2 d^3 \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right )-\frac {1}{2} b^2 d^3 \operatorname {PolyLog}\left (3,1-\frac {2}{i c x+1}\right )+\frac {1}{2} b^2 d^3 \operatorname {PolyLog}\left (3,\frac {2}{i c x+1}-1\right )-\frac {1}{3} i b^2 c d^3 x\) |
3*a*b*c*d^3*x - (I/3)*b^2*c*d^3*x + (I/3)*b^2*d^3*ArcTan[c*x] + 3*b^2*c*d^ 3*x*ArcTan[c*x] + (I/3)*b*c^2*d^3*x^2*(a + b*ArcTan[c*x]) - (29*d^3*(a + b *ArcTan[c*x])^2)/6 + (3*I)*c*d^3*x*(a + b*ArcTan[c*x])^2 - (3*c^2*d^3*x^2* (a + b*ArcTan[c*x])^2)/2 - (I/3)*c^3*d^3*x^3*(a + b*ArcTan[c*x])^2 + 2*d^3 *(a + b*ArcTan[c*x])^2*ArcTanh[1 - 2/(1 + I*c*x)] + ((20*I)/3)*b*d^3*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)] - (3*b^2*d^3*Log[1 + c^2*x^2])/2 - (10*b ^2*d^3*PolyLog[2, 1 - 2/(1 + I*c*x)])/3 - I*b*d^3*(a + b*ArcTan[c*x])*Poly Log[2, 1 - 2/(1 + I*c*x)] + I*b*d^3*(a + b*ArcTan[c*x])*PolyLog[2, -1 + 2/ (1 + I*c*x)] - (b^2*d^3*PolyLog[3, 1 - 2/(1 + I*c*x)])/2 + (b^2*d^3*PolyLo g[3, -1 + 2/(1 + I*c*x)])/2
3.1.88.3.1 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 = 23.60 (sec) , antiderivative size = 1409, normalized size of antiderivative = 3.66
| method | result | size |
| parts | \(\text {Expression too large to display}\) | \(1409\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(1411\) |
| default | \(\text {Expression too large to display}\) | \(1411\) |
d^3*a^2*(-1/3*I*c^3*x^3-3/2*c^2*x^2+3*I*c*x+ln(x))+b^2*d^3*(-1/2*polylog(3 ,-(1+I*c*x)^2/(c^2*x^2+1))+2*polylog(3,(1+I*c*x)/(c^2*x^2+1)^(1/2))+2*poly log(3,-(1+I*c*x)/(c^2*x^2+1)^(1/2))-3/2*c^2*x^2*arctan(c*x)^2+3*ln(1+(1+I* c*x)^2/(c^2*x^2+1))+11/6*arctan(c*x)^2+I*arctan(c*x)*polylog(2,-(1+I*c*x)^ 2/(c^2*x^2+1))+20/3*dilog(1+I*(1+I*c*x)/(c^2*x^2+1)^(1/2))+20/3*dilog(1-I* (1+I*c*x)/(c^2*x^2+1)^(1/2))+11/3*arctan(c*x)*(c*x-I)-arctan(c*x)^2*ln((1+ I*c*x)^2/(c^2*x^2+1)-1)+arctan(c*x)^2*ln(c*x)+arctan(c*x)^2*ln(1-(1+I*c*x) /(c^2*x^2+1)^(1/2))+arctan(c*x)^2*ln(1+(1+I*c*x)/(c^2*x^2+1)^(1/2))-2*I*ar ctan(c*x)*polylog(2,(1+I*c*x)/(c^2*x^2+1)^(1/2))-2*I*arctan(c*x)*polylog(2 ,-(1+I*c*x)/(c^2*x^2+1)^(1/2))+1/2*I*Pi*arctan(c*x)^2+1/2*I*Pi*csgn(I/(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+I* c*x)^2/(c^2*x^2+1)-1)/(1+(1+I*c*x)^2/(c^2*x^2+1)))*arctan(c*x)^2+1/2*I*Pi* 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)))*arctan(c*x)^2-1/2*I*Pi *csgn(I/(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 +(1+I*c*x)^2/(c^2*x^2+1)))^2*arctan(c*x)^2-1/2*I*Pi*csgn(I*((1+I*c*x)^2/(c ^2*x^2+1)-1))*csgn(I*((1+I*c*x)^2/(c^2*x^2+1)-1)/(1+(1+I*c*x)^2/(c^2*x^2+1 )))^2*arctan(c*x)^2+1/2*I*Pi*csgn(((1+I*c*x)^2/(c^2*x^2+1)-1)/(1+(1+I*c*x) ^2/(c^2*x^2+1)))^3*arctan(c*x)^2-1/2*I*Pi*csgn(((1+I*c*x)^2/(c^2*x^2+1)-1) /(1+(1+I*c*x)^2/(c^2*x^2+1)))^2*arctan(c*x)^2+1/2*I*Pi*csgn(I*((1+I*c*x...
\[ \int \frac {(d+i c d x)^3 (a+b \arctan (c x))^2}{x} \, dx=\int { \frac {{\left (i \, c d x + d\right )}^{3} {\left (b \arctan \left (c x\right ) + a\right )}^{2}}{x} \,d x } \]
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, x)
Timed out. \[ \int \frac {(d+i c d x)^3 (a+b \arctan (c x))^2}{x} \, dx=\text {Timed out} \]
\[ \int \frac {(d+i c d x)^3 (a+b \arctan (c x))^2}{x} \, dx=\int { \frac {{\left (i \, c d x + d\right )}^{3} {\left (b \arctan \left (c x\right ) + a\right )}^{2}}{x} \,d x } \]
-1/3*I*a^2*c^3*d^3*x^3 - 36*I*b^2*c^5*d^3*integrate(1/48*x^5*arctan(c*x)^2 /(c^2*x^3 + x), x) - 12*b^2*c^5*d^3*integrate(1/48*x^5*arctan(c*x)*log(c^2 *x^2 + 1)/(c^2*x^3 + x), x) - 3*I*b^2*c^5*d^3*integrate(1/48*x^5*log(c^2*x ^2 + 1)^2/(c^2*x^3 + x), x) - 96*I*a*b*c^5*d^3*integrate(1/48*x^5*arctan(c *x)/(c^2*x^3 + x), x) - 8*b^2*c^5*d^3*integrate(1/48*x^5*arctan(c*x)/(c^2* x^3 + x), x) - 4*I*b^2*c^5*d^3*integrate(1/48*x^5*log(c^2*x^2 + 1)/(c^2*x^ 3 + x), x) - 108*b^2*c^4*d^3*integrate(1/48*x^4*arctan(c*x)^2/(c^2*x^3 + x ), x) + 36*I*b^2*c^4*d^3*integrate(1/48*x^4*arctan(c*x)*log(c^2*x^2 + 1)/( c^2*x^3 + x), x) - 9*b^2*c^4*d^3*integrate(1/48*x^4*log(c^2*x^2 + 1)^2/(c^ 2*x^3 + x), x) - 288*a*b*c^4*d^3*integrate(1/48*x^4*arctan(c*x)/(c^2*x^3 + x), x) + 44*I*b^2*c^4*d^3*integrate(1/48*x^4*arctan(c*x)/(c^2*x^3 + x), x ) - 22*b^2*c^4*d^3*integrate(1/48*x^4*log(c^2*x^2 + 1)/(c^2*x^3 + x), x) - 3/2*a^2*c^2*d^3*x^2 + 72*I*b^2*c^3*d^3*integrate(1/48*x^3*arctan(c*x)^2/( c^2*x^3 + x), x) + 24*b^2*c^3*d^3*integrate(1/48*x^3*arctan(c*x)*log(c^2*x ^2 + 1)/(c^2*x^3 + x), x) + 6*I*b^2*c^3*d^3*integrate(1/48*x^3*log(c^2*x^2 + 1)^2/(c^2*x^3 + x), x) - 96*I*a*b*c^3*d^3*integrate(1/48*x^3*arctan(c*x )/(c^2*x^3 + x), x) + 108*b^2*c^3*d^3*integrate(1/48*x^3*arctan(c*x)/(c^2* x^3 + x), x) + 54*I*b^2*c^3*d^3*integrate(1/48*x^3*log(c^2*x^2 + 1)/(c^2*x ^3 + x), x) + 3/4*I*b^2*d^3*arctan(c*x)^3 - 72*b^2*c^2*d^3*integrate(1/48* x^2*arctan(c*x)^2/(c^2*x^3 + x), x) + 24*I*b^2*c^2*d^3*integrate(1/48*x...
Timed out. \[ \int \frac {(d+i c d x)^3 (a+b \arctan (c x))^2}{x} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {(d+i c d x)^3 (a+b \arctan (c x))^2}{x} \, dx=\int \frac {{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^2\,{\left (d+c\,d\,x\,1{}\mathrm {i}\right )}^3}{x} \,d x \]