Integrand size = 25, antiderivative size = 416 \[ \int \frac {(d+i c d x)^3 (a+b \arctan (c x))^2}{x^3} \, dx=-\frac {b c d^3 (a+b \arctan (c x))}{x}+\frac {7}{2} c^2 d^3 (a+b \arctan (c x))^2-\frac {d^3 (a+b \arctan (c x))^2}{2 x^2}-\frac {3 i c d^3 (a+b \arctan (c x))^2}{x}-i c^3 d^3 x (a+b \arctan (c x))^2-6 c^2 d^3 (a+b \arctan (c x))^2 \text {arctanh}\left (1-\frac {2}{1+i c x}\right )+b^2 c^2 d^3 \log (x)-2 i b c^2 d^3 (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )-\frac {1}{2} b^2 c^2 d^3 \log \left (1+c^2 x^2\right )+6 i b c^2 d^3 (a+b \arctan (c x)) \log \left (2-\frac {2}{1-i c x}\right )+3 b^2 c^2 d^3 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i c x}\right )+b^2 c^2 d^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )+3 i b c^2 d^3 (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )-3 i b c^2 d^3 (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i c x}\right )+\frac {3}{2} b^2 c^2 d^3 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i c x}\right )-\frac {3}{2} b^2 c^2 d^3 \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i c x}\right ) \]
-b*c*d^3*(a+b*arctan(c*x))/x+7/2*c^2*d^3*(a+b*arctan(c*x))^2-1/2*d^3*(a+b* arctan(c*x))^2/x^2-2*I*b*c^2*d^3*(a+b*arctan(c*x))*ln(2/(1+I*c*x))+3*I*b*c ^2*d^3*(a+b*arctan(c*x))*polylog(2,1-2/(1+I*c*x))+6*c^2*d^3*(a+b*arctan(c* x))^2*arctanh(-1+2/(1+I*c*x))+b^2*c^2*d^3*ln(x)+6*I*b*c^2*d^3*(a+b*arctan( c*x))*ln(2-2/(1-I*c*x))-1/2*b^2*c^2*d^3*ln(c^2*x^2+1)-I*c^3*d^3*x*(a+b*arc tan(c*x))^2+3*b^2*c^2*d^3*polylog(2,-1+2/(1-I*c*x))+b^2*c^2*d^3*polylog(2, 1-2/(1+I*c*x))-3*I*c*d^3*(a+b*arctan(c*x))^2/x-3*I*b*c^2*d^3*(a+b*arctan(c *x))*polylog(2,-1+2/(1+I*c*x))+3/2*b^2*c^2*d^3*polylog(3,1-2/(1+I*c*x))-3/ 2*b^2*c^2*d^3*polylog(3,-1+2/(1+I*c*x))
Time = 0.95 (sec) , antiderivative size = 500, normalized size of antiderivative = 1.20 \[ \int \frac {(d+i c d x)^3 (a+b \arctan (c x))^2}{x^3} \, dx=\frac {1}{2} d^3 \left (-\frac {a^2}{x^2}-\frac {6 i a^2 c}{x}-2 i a^2 c^3 x-\frac {2 a b (\arctan (c x)+c x (1+c x \arctan (c x)))}{x^2}-6 a^2 c^2 \log (x)-\frac {b^2 \left (2 c x \arctan (c x)+\left (1+c^2 x^2\right ) \arctan (c x)^2-2 c^2 x^2 \log \left (\frac {c x}{\sqrt {1+c^2 x^2}}\right )\right )}{x^2}-2 i a b c^2 \left (2 c x \arctan (c x)-\log \left (1+c^2 x^2\right )\right )-\frac {6 i a b c \left (2 \arctan (c x)+c x \left (-2 \log (c x)+\log \left (1+c^2 x^2\right )\right )\right )}{x}-2 i b^2 c^2 \left (\arctan (c x) \left ((-i+c x) \arctan (c x)+2 \log \left (1+e^{2 i \arctan (c x)}\right )\right )-i \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )\right )+\frac {6 b^2 c \left (\arctan (c x) \left ((-i+c x) \arctan (c x)+2 i c x \log \left (1-e^{2 i \arctan (c x)}\right )\right )+c x \operatorname {PolyLog}\left (2,e^{2 i \arctan (c x)}\right )\right )}{x}-6 i a b c^2 (\operatorname {PolyLog}(2,-i c x)-\operatorname {PolyLog}(2,i c x))+6 b^2 c^2 \left (\frac {i \pi ^3}{24}-\frac {2}{3} i \arctan (c x)^3-\arctan (c x)^2 \log \left (1-e^{-2 i \arctan (c x)}\right )+\arctan (c x)^2 \log \left (1+e^{2 i \arctan (c x)}\right )-i \arctan (c x) \operatorname {PolyLog}\left (2,e^{-2 i \arctan (c x)}\right )-i \arctan (c x) \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )-\frac {1}{2} \operatorname {PolyLog}\left (3,e^{-2 i \arctan (c x)}\right )+\frac {1}{2} \operatorname {PolyLog}\left (3,-e^{2 i \arctan (c x)}\right )\right )\right ) \]
(d^3*(-(a^2/x^2) - ((6*I)*a^2*c)/x - (2*I)*a^2*c^3*x - (2*a*b*(ArcTan[c*x] + c*x*(1 + c*x*ArcTan[c*x])))/x^2 - 6*a^2*c^2*Log[x] - (b^2*(2*c*x*ArcTan [c*x] + (1 + c^2*x^2)*ArcTan[c*x]^2 - 2*c^2*x^2*Log[(c*x)/Sqrt[1 + c^2*x^2 ]]))/x^2 - (2*I)*a*b*c^2*(2*c*x*ArcTan[c*x] - Log[1 + c^2*x^2]) - ((6*I)*a *b*c*(2*ArcTan[c*x] + c*x*(-2*Log[c*x] + Log[1 + c^2*x^2])))/x - (2*I)*b^2 *c^2*(ArcTan[c*x]*((-I + c*x)*ArcTan[c*x] + 2*Log[1 + E^((2*I)*ArcTan[c*x] )]) - I*PolyLog[2, -E^((2*I)*ArcTan[c*x])]) + (6*b^2*c*(ArcTan[c*x]*((-I + c*x)*ArcTan[c*x] + (2*I)*c*x*Log[1 - E^((2*I)*ArcTan[c*x])]) + c*x*PolyLo g[2, E^((2*I)*ArcTan[c*x])]))/x - (6*I)*a*b*c^2*(PolyLog[2, (-I)*c*x] - Po lyLog[2, I*c*x]) + 6*b^2*c^2*((I/24)*Pi^3 - ((2*I)/3)*ArcTan[c*x]^3 - ArcT an[c*x]^2*Log[1 - E^((-2*I)*ArcTan[c*x])] + ArcTan[c*x]^2*Log[1 + E^((2*I) *ArcTan[c*x])] - I*ArcTan[c*x]*PolyLog[2, E^((-2*I)*ArcTan[c*x])] - I*ArcT an[c*x]*PolyLog[2, -E^((2*I)*ArcTan[c*x])] - PolyLog[3, E^((-2*I)*ArcTan[c *x])]/2 + PolyLog[3, -E^((2*I)*ArcTan[c*x])]/2)))/2
Time = 0.96 (sec) , antiderivative size = 416, 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.
| \(\displaystyle \int \frac {(d+i c d x)^3 (a+b \arctan (c x))^2}{x^3} \, dx\) |
| \(\Big \downarrow \) 5411 |
| \(\displaystyle \int \left (-i c^3 d^3 (a+b \arctan (c x))^2-\frac {3 c^2 d^3 (a+b \arctan (c x))^2}{x}+\frac {d^3 (a+b \arctan (c x))^2}{x^3}+\frac {3 i c d^3 (a+b \arctan (c x))^2}{x^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -6 c^2 d^3 \text {arctanh}\left (1-\frac {2}{1+i c x}\right ) (a+b \arctan (c x))^2-i c^3 d^3 x (a+b \arctan (c x))^2+3 i b c^2 d^3 \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right ) (a+b \arctan (c x))-3 i b c^2 d^3 \operatorname {PolyLog}\left (2,\frac {2}{i c x+1}-1\right ) (a+b \arctan (c x))+\frac {7}{2} c^2 d^3 (a+b \arctan (c x))^2-2 i b c^2 d^3 \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))+6 i b c^2 d^3 \log \left (2-\frac {2}{1-i c x}\right ) (a+b \arctan (c x))-\frac {d^3 (a+b \arctan (c x))^2}{2 x^2}-\frac {3 i c d^3 (a+b \arctan (c x))^2}{x}-\frac {b c d^3 (a+b \arctan (c x))}{x}+3 b^2 c^2 d^3 \operatorname {PolyLog}\left (2,\frac {2}{1-i c x}-1\right )+b^2 c^2 d^3 \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right )+\frac {3}{2} b^2 c^2 d^3 \operatorname {PolyLog}\left (3,1-\frac {2}{i c x+1}\right )-\frac {3}{2} b^2 c^2 d^3 \operatorname {PolyLog}\left (3,\frac {2}{i c x+1}-1\right )-\frac {1}{2} b^2 c^2 d^3 \log \left (c^2 x^2+1\right )+b^2 c^2 d^3 \log (x)\) |
-((b*c*d^3*(a + b*ArcTan[c*x]))/x) + (7*c^2*d^3*(a + b*ArcTan[c*x])^2)/2 - (d^3*(a + b*ArcTan[c*x])^2)/(2*x^2) - ((3*I)*c*d^3*(a + b*ArcTan[c*x])^2) /x - I*c^3*d^3*x*(a + b*ArcTan[c*x])^2 - 6*c^2*d^3*(a + b*ArcTan[c*x])^2*A rcTanh[1 - 2/(1 + I*c*x)] + b^2*c^2*d^3*Log[x] - (2*I)*b*c^2*d^3*(a + b*Ar cTan[c*x])*Log[2/(1 + I*c*x)] - (b^2*c^2*d^3*Log[1 + c^2*x^2])/2 + (6*I)*b *c^2*d^3*(a + b*ArcTan[c*x])*Log[2 - 2/(1 - I*c*x)] + 3*b^2*c^2*d^3*PolyLo g[2, -1 + 2/(1 - I*c*x)] + b^2*c^2*d^3*PolyLog[2, 1 - 2/(1 + I*c*x)] + (3* I)*b*c^2*d^3*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 + I*c*x)] - (3*I)*b*c ^2*d^3*(a + b*ArcTan[c*x])*PolyLog[2, -1 + 2/(1 + I*c*x)] + (3*b^2*c^2*d^3 *PolyLog[3, 1 - 2/(1 + I*c*x)])/2 - (3*b^2*c^2*d^3*PolyLog[3, -1 + 2/(1 + I*c*x)])/2
3.1.90.3.1 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 = 29.44 (sec) , antiderivative size = 1523, normalized size of antiderivative = 3.66
| method | result | size |
| derivativedivides | \(\text {Expression too large to display}\) | \(1523\) |
| default | \(\text {Expression too large to display}\) | \(1523\) |
| parts | \(\text {Expression too large to display}\) | \(1523\) |
c^2*(d^3*a^2*(-I*c*x-3*ln(c*x)-3*I/c/x-1/2/c^2/x^2)+b^2*d^3*(-1/2/c^2/x^2* arctan(c*x)^2+3/2*polylog(3,-(1+I*c*x)^2/(c^2*x^2+1))-6*polylog(3,(1+I*c*x )/(c^2*x^2+1)^(1/2))+ln(1+(1+I*c*x)/(c^2*x^2+1)^(1/2))-6*polylog(3,-(1+I*c *x)/(c^2*x^2+1)^(1/2))+ln((1+I*c*x)/(c^2*x^2+1)^(1/2)-1)+3/2*arctan(c*x)^2 -3*I*arctan(c*x)*polylog(2,-(1+I*c*x)^2/(c^2*x^2+1))-2*dilog(1+I*(1+I*c*x) /(c^2*x^2+1)^(1/2))-2*dilog(1-I*(1+I*c*x)/(c^2*x^2+1)^(1/2))+3*arctan(c*x) ^2*ln((1+I*c*x)^2/(c^2*x^2+1)-1)-3*arctan(c*x)^2*ln(c*x)-3*arctan(c*x)^2*l n(1-(1+I*c*x)/(c^2*x^2+1)^(1/2))-3*arctan(c*x)^2*ln(1+(1+I*c*x)/(c^2*x^2+1 )^(1/2))+6*dilog(1+(1+I*c*x)/(c^2*x^2+1)^(1/2))-6*dilog((1+I*c*x)/(c^2*x^2 +1)^(1/2))-1/2*arctan(c*x)*(I*c*x-(c^2*x^2+1)^(1/2)+1)/c/x-1/2*arctan(c*x) *(I*c*x+(c^2*x^2+1)^(1/2)+1)/c/x-3/2*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+6*I*arctan(c*x)*polylog(2,(1+I *c*x)/(c^2*x^2+1)^(1/2))+6*I*arctan(c*x)*polylog(2,-(1+I*c*x)/(c^2*x^2+1)^ (1/2))-3/2*I*Pi*arctan(c*x)^2-2*I*arctan(c*x)*ln(1+I*(1+I*c*x)/(c^2*x^2+1) ^(1/2))-2*I*arctan(c*x)*ln(1-I*(1+I*c*x)/(c^2*x^2+1)^(1/2))+6*I*arctan(c*x )*ln(1+(1+I*c*x)/(c^2*x^2+1)^(1/2))-3/2*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)))^3*arctan(c*x)^2-3/2*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+3/2*I*Pi*c sgn(((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...
\[ \int \frac {(d+i c d x)^3 (a+b \arctan (c x))^2}{x^3} \, dx=\int { \frac {{\left (i \, c d x + d\right )}^{3} {\left (b \arctan \left (c x\right ) + a\right )}^{2}}{x^{3}} \,d x } \]
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^3, x)
Timed out. \[ \int \frac {(d+i c d x)^3 (a+b \arctan (c x))^2}{x^3} \, dx=\text {Timed out} \]
\[ \int \frac {(d+i c d x)^3 (a+b \arctan (c x))^2}{x^3} \, dx=\int { \frac {{\left (i \, c d x + d\right )}^{3} {\left (b \arctan \left (c x\right ) + a\right )}^{2}}{x^{3}} \,d x } \]
-I*a^2*c^3*d^3*x - I*(2*c*x*arctan(c*x) - log(c^2*x^2 + 1))*a*b*c^2*d^3 - 3*a^2*c^2*d^3*log(x) - 3*I*(c*(log(c^2*x^2 + 1) - log(x^2)) + 2*arctan(c*x )/x)*a*b*c*d^3 - ((c*arctan(c*x) + 1/x)*c + arctan(c*x)/x^2)*a*b*d^3 - 3*I *a^2*c*d^3/x - 1/2*a^2*d^3/x^2 - 1/32*(16*I*(24*b^2*c^5*d^3*integrate(1/16 *x^5*arctan(c*x)^2/(c^2*x^5 + x^3), x) + 2*b^2*c^5*d^3*integrate(1/16*x^5* log(c^2*x^2 + 1)^2/(c^2*x^5 + x^3), x) + 8*b^2*c^5*d^3*integrate(1/16*x^5* log(c^2*x^2 + 1)/(c^2*x^5 + x^3), x) - b^2*c^2*d^3*arctan(c*x)^3 - 24*b^2* c^4*d^3*integrate(1/16*x^4*arctan(c*x)*log(c^2*x^2 + 1)/(c^2*x^5 + x^3), x ) - 16*b^2*c^4*d^3*integrate(1/16*x^4*arctan(c*x)/(c^2*x^5 + x^3), x) - 4* b^2*c^3*d^3*integrate(1/16*x^3*log(c^2*x^2 + 1)^2/(c^2*x^5 + x^3), x) + 24 *b^2*c^3*d^3*integrate(1/16*x^3*log(c^2*x^2 + 1)/(c^2*x^5 + x^3), x) - 16* b^2*c^2*d^3*integrate(1/16*x^2*arctan(c*x)*log(c^2*x^2 + 1)/(c^2*x^5 + x^3 ), x) - 56*b^2*c^2*d^3*integrate(1/16*x^2*arctan(c*x)/(c^2*x^5 + x^3), x) - 72*b^2*c*d^3*integrate(1/16*x*arctan(c*x)^2/(c^2*x^5 + x^3), x) - 6*b^2* c*d^3*integrate(1/16*x*log(c^2*x^2 + 1)^2/(c^2*x^5 + x^3), x) - 4*b^2*c*d^ 3*integrate(1/16*x*log(c^2*x^2 + 1)/(c^2*x^5 + x^3), x) + 8*b^2*d^3*integr ate(1/16*arctan(c*x)*log(c^2*x^2 + 1)/(c^2*x^5 + x^3), x))*x^2 + (128*b^2* c^5*d^3*integrate(1/16*x^5*arctan(c*x)*log(c^2*x^2 + 1)/(c^2*x^5 + x^3), x ) + 256*b^2*c^5*d^3*integrate(1/16*x^5*arctan(c*x)/(c^2*x^5 + x^3), x) + 1 152*b^2*c^4*d^3*integrate(1/16*x^4*arctan(c*x)^2/(c^2*x^5 + x^3), x) + ...
Timed out. \[ \int \frac {(d+i c d x)^3 (a+b \arctan (c x))^2}{x^3} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {(d+i c d x)^3 (a+b \arctan (c x))^2}{x^3} \, dx=\int \frac {{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^2\,{\left (d+c\,d\,x\,1{}\mathrm {i}\right )}^3}{x^3} \,d x \]