\(\int \frac {-192+56 x-64 (i \pi +\log (3))+(-96+30 x-32 (i \pi +\log (3))) \log (3+i \pi -x+\log (3))+(-12+4 x-4 (i \pi +\log (3))) \log ^2(3+i \pi -x+\log (3))}{3 x^5-x^6+x^5 (i \pi +\log (3))} \, dx\) [245]

Optimal result

Integrand size = 101, antiderivative size = 23 \[ \int \frac {-192+56 x-64 (i \pi +\log (3))+(-96+30 x-32 (i \pi +\log (3))) \log (3+i \pi -x+\log (3))+(-12+4 x-4 (i \pi +\log (3))) \log ^2(3+i \pi -x+\log (3))}{3 x^5-x^6+x^5 (i \pi +\log (3))} \, dx=\frac {(-4-\log (3+i \pi -x+\log (3)))^2}{x^4} \] Output:

(-4-ln(ln(3)+I*Pi+3-x))^2/x^4
                                                                                    
                                                                                    
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.

Time = 1.06 (sec) , antiderivative size = 917, normalized size of antiderivative = 39.87 \[ \int \frac {-192+56 x-64 (i \pi +\log (3))+(-96+30 x-32 (i \pi +\log (3))) \log (3+i \pi -x+\log (3))+(-12+4 x-4 (i \pi +\log (3))) \log ^2(3+i \pi -x+\log (3))}{3 x^5-x^6+x^5 (i \pi +\log (3))} \, dx =\text {Too large to display} \] Input:

Integrate[(-192 + 56*x - 64*(I*Pi + Log[3]) + (-96 + 30*x - 32*(I*Pi + Log 
[3]))*Log[3 + I*Pi - x + Log[3]] + (-12 + 4*x - 4*(I*Pi + Log[3]))*Log[3 + 
 I*Pi - x + Log[3]]^2)/(3*x^5 - x^6 + x^5*(I*Pi + Log[3])),x]
 

Output:

(96/x^4 + (12*Log[x])/(Pi - I*(3 + Log[3]))^4 - (12*Log[-Pi - I*(-3 + x - 
Log[3])])/(Pi - I*(3 + Log[3]))^4 + (2*(Pi - I*(3 + Log[3]))*(Pi - I*(3 + 
2*x + Log[3])) + 4*x^2*Log[x] - 4*x^2*Log[-Pi - I*(-3 + x - Log[3])])/(x^2 
*(Pi - I*(3 + Log[3]))^4) + (2*(12 + (4*I)*Pi + Log[81])*((Pi - I*(3 + Log 
[3]))*(2*Pi^2 - 6*x^2 - 3*x*(3 + Log[3]) - 2*(3 + Log[3])^2 - I*Pi*(12 + 3 
*x + Log[81])) - (6*I)*x^3*Log[x] + (6*I)*x^3*Log[-Pi - I*(-3 + x - Log[3] 
)]))/(x^3*(Pi - I*(3 + Log[3]))^5) + (12*(4*Pi - I*(12 + Log[81]))*Log[3 + 
 I*Pi - x + Log[3]])/(x^4*(Pi - I*(3 + Log[3]))) + (3*(2*Pi - I*(6 + Log[9 
]))*Log[3 + I*Pi - x + Log[3]]^2)/(x^4*(Pi - I*(3 + Log[3]))) + (12*(4 + L 
og[3 + I*Pi - x + Log[3]]))/(x*(3 + I*Pi + Log[3])^3) + (4*(4 + Log[3 + I* 
Pi - x + Log[3]]))/(x^3*(3 + I*Pi + Log[3])) - (6*(4 + Log[3 + I*Pi - x + 
Log[3]]))/(x^2*(Pi - I*(3 + Log[3]))^2) - (12*Log[x/(3 + I*Pi + Log[3])]*( 
4 + Log[3 + I*Pi - x + Log[3]]))/(Pi - I*(3 + Log[3]))^4 + (6*(4 + Log[3 + 
 I*Pi - x + Log[3]])^2)/(Pi - I*(3 + Log[3]))^4 + (6*(-3 - I*Pi - Log[3] + 
 x*Log[x] - x*Log[-Pi - I*x + I*(3 + Log[3])]))/(x*(Pi - I*(3 + Log[3]))^4 
) - (12*PolyLog[2, (3 + I*Pi - x + Log[3])/(3 + I*Pi + Log[3])])/(Pi - I*( 
3 + Log[3]))^4 + ((2*Pi - I*(6 + Log[9]))*(-(1/(x^2*(3 + I*Pi + Log[3])^2) 
) - (5*I)/(x*(Pi - I*(3 + Log[3]))^3) + (11*Log[x])/(Pi - I*(3 + Log[3]))^ 
4 - (2*Log[-Pi - I*(-3 + x - Log[3])])/(3 + I*Pi + Log[3])^4 - (9*Log[-Pi 
- I*(-3 + x - Log[3])])/(Pi - I*(3 + Log[3]))^4 + (6*Log[3 + I*Pi - x +...
 

Rubi [C] (verified)

Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.

Time = 3.11 (sec) , antiderivative size = 505, normalized size of antiderivative = 21.96, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {6, 2026, 7292, 7293, 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.

\(\displaystyle \int \frac {56 x+(4 x-12-4 (\log (3)+i \pi )) \log ^2(-x+i \pi +3+\log (3))+(30 x-96-32 (\log (3)+i \pi )) \log (-x+i \pi +3+\log (3))-192-64 (\log (3)+i \pi )}{-x^6+3 x^5+x^5 (\log (3)+i \pi )} \, dx\)

\(\Big \downarrow \) 6

\(\displaystyle \int \frac {56 x+(4 x-12-4 (\log (3)+i \pi )) \log ^2(-x+i \pi +3+\log (3))+(30 x-96-32 (\log (3)+i \pi )) \log (-x+i \pi +3+\log (3))-192-64 (\log (3)+i \pi )}{-x^6+x^5 (3+i \pi +\log (3))}dx\)

\(\Big \downarrow \) 2026

\(\displaystyle \int \frac {56 x+(4 x-12-4 (\log (3)+i \pi )) \log ^2(-x+i \pi +3+\log (3))+(30 x-96-32 (\log (3)+i \pi )) \log (-x+i \pi +3+\log (3))-192-64 (\log (3)+i \pi )}{x^5 (-x+i \pi +3+\log (3))}dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {56 x+(4 x-12-4 (\log (3)+i \pi )) \log ^2(-x+i \pi +3+\log (3))+(30 x-96-32 (\log (3)+i \pi )) \log (-x+i \pi +3+\log (3))-192 \left (1+\frac {1}{3} (\log (3)+i \pi )\right )}{x^5 (-x+i \pi +3+\log (3))}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (-\frac {4 \log ^2(-x+i \pi +3+\log (3))}{x^5}-\frac {2 (15 x-16 i \pi -48-16 \log (3)) \log (-x+i \pi +3+\log (3))}{x^5 (x-i \pi -3-\log (3))}-\frac {8 (7 x-8 i \pi -24-8 \log (3))}{x^5 (x-i \pi -3-\log (3))}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {2 \operatorname {PolyLog}\left (2,\frac {3+i \pi +\log (3)}{-x+\log (3)+i \pi +3}\right )}{(3+i \pi +\log (3))^4}-\frac {2 \operatorname {PolyLog}\left (2,1-\frac {x}{3+i \pi +\log (3)}\right )}{(\pi -i (3+\log (3)))^4}+\frac {16}{x^4}+\frac {\log ^2(-x+i \pi +3+\log (3))}{x^4}+\frac {8 \log (-x+i \pi +3+\log (3))}{x^4}-\frac {\log (-x+i \pi +3+\log (3))}{x^2 (\pi -i (3+\log (3)))^2}-\frac {\log (-x+i \pi +3+\log (3))}{x^2 (3+i \pi +\log (3))^2}+\frac {1}{3 x^2 (\pi -i (3+\log (3)))^2}+\frac {1}{3 x^2 (3+i \pi +\log (3))^2}+\frac {\log ^2(-x+i \pi +3+\log (3))}{(\pi -i (3+\log (3)))^4}-\frac {2 (-x+i \pi +3+\log (3)) \log (-x+i \pi +3+\log (3))}{x (3+i \pi +\log (3))^4}-\frac {2 \log \left (\frac {x}{3+i \pi +\log (3)}\right ) \log (-x+i \pi +3+\log (3))}{(\pi -i (3+\log (3)))^4}+\frac {2 \log \left (1-\frac {3+i \pi +\log (3)}{-x+i \pi +3+\log (3)}\right ) \log (-x+i \pi +3+\log (3))}{(3+i \pi +\log (3))^4}+\frac {2 \log (-x+i \pi +3+\log (3))}{x (3+i \pi +\log (3))^3}+\frac {11 \log (x)}{3 (\pi -i (3+\log (3)))^4}-\frac {11 \log (x)}{3 (3+i \pi +\log (3))^4}-\frac {11 \log (-i x-\pi +i (3+\log (3)))}{3 (\pi -i (3+\log (3)))^4}+\frac {5 \log (-i x-\pi +i (3+\log (3)))}{3 (3+i \pi +\log (3))^4}\)

Input:

Int[(-192 + 56*x - 64*(I*Pi + Log[3]) + (-96 + 30*x - 32*(I*Pi + Log[3]))* 
Log[3 + I*Pi - x + Log[3]] + (-12 + 4*x - 4*(I*Pi + Log[3]))*Log[3 + I*Pi 
- x + Log[3]]^2)/(3*x^5 - x^6 + x^5*(I*Pi + Log[3])),x]
 

Output:

16/x^4 + 1/(3*x^2*(3 + I*Pi + Log[3])^2) + 1/(3*x^2*(Pi - I*(3 + Log[3]))^ 
2) - (11*Log[x])/(3*(3 + I*Pi + Log[3])^4) + (11*Log[x])/(3*(Pi - I*(3 + L 
og[3]))^4) + (8*Log[3 + I*Pi - x + Log[3]])/x^4 + (2*Log[3 + I*Pi - x + Lo 
g[3]])/(x*(3 + I*Pi + Log[3])^3) - Log[3 + I*Pi - x + Log[3]]/(x^2*(3 + I* 
Pi + Log[3])^2) - (2*(3 + I*Pi - x + Log[3])*Log[3 + I*Pi - x + Log[3]])/( 
x*(3 + I*Pi + Log[3])^4) - Log[3 + I*Pi - x + Log[3]]/(x^2*(Pi - I*(3 + Lo 
g[3]))^2) - (2*Log[x/(3 + I*Pi + Log[3])]*Log[3 + I*Pi - x + Log[3]])/(Pi 
- I*(3 + Log[3]))^4 + Log[3 + I*Pi - x + Log[3]]^2/x^4 + Log[3 + I*Pi - x 
+ Log[3]]^2/(Pi - I*(3 + Log[3]))^4 + (5*Log[-Pi - I*x + I*(3 + Log[3])])/ 
(3*(3 + I*Pi + Log[3])^4) - (11*Log[-Pi - I*x + I*(3 + Log[3])])/(3*(Pi - 
I*(3 + Log[3]))^4) + (2*Log[3 + I*Pi - x + Log[3]]*Log[1 - (3 + I*Pi + Log 
[3])/(3 + I*Pi - x + Log[3])])/(3 + I*Pi + Log[3])^4 - (2*PolyLog[2, (3 + 
I*Pi + Log[3])/(3 + I*Pi - x + Log[3])])/(3 + I*Pi + Log[3])^4 - (2*PolyLo 
g[2, 1 - x/(3 + I*Pi + Log[3])])/(Pi - I*(3 + Log[3]))^4
 

Defintions of rubi rules used

rule 6

Int[(u_.)*((v_.) + (a_.)*(Fx_) + (b_.)*(Fx_))^(p_.), x_Symbol] :> Int[u*(v 
+ (a + b)*Fx)^p, x] /; FreeQ[{a, b}, x] &&  !FreeQ[Fx, x]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2026

Int[(Fx_.)*(Px_)^(p_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Int[x^(p 
*r)*ExpandToSum[Px/x^r, x]^p*Fx, x] /; IGtQ[r, 0]] /; PolyQ[Px, x] && Integ 
erQ[p] &&  !MonomialQ[Px, x] && (ILtQ[p, 0] ||  !PolyQ[u, x])
 

rule 7292

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =! 
= u]
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

Maple [A] (verified)

Time = 4.95 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.52

method	result	size
norman	\(\frac {16+\ln \left (\ln \left (3\right )+i \pi +3-x \right )^{2}+8 \ln \left (\ln \left (3\right )+i \pi +3-x \right )}{x^{4}}\)	\(35\)
risch	\(\frac {\ln \left (\ln \left (3\right )+i \pi +3-x \right )^{2}}{x^{4}}+\frac {8 \ln \left (\ln \left (3\right )+i \pi +3-x \right )}{x^{4}}+\frac {16}{x^{4}}\)	\(42\)
parallelrisch	\(\text {Expression too large to display}\)	\(1473\)

Input:

int(((-4*ln(3)-4*I*Pi+4*x-12)*ln(ln(3)+I*Pi+3-x)^2+(-32*ln(3)-32*I*Pi+30*x 
-96)*ln(ln(3)+I*Pi+3-x)-64*ln(3)-64*I*Pi+56*x-192)/(x^5*(ln(3)+I*Pi)-x^6+3 
*x^5),x,method=_RETURNVERBOSE)
 

Output:

(16+ln(ln(3)+I*Pi+3-x)^2+8*ln(ln(3)+I*Pi+3-x))/x^4
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.39 \[ \int \frac {-192+56 x-64 (i \pi +\log (3))+(-96+30 x-32 (i \pi +\log (3))) \log (3+i \pi -x+\log (3))+(-12+4 x-4 (i \pi +\log (3))) \log ^2(3+i \pi -x+\log (3))}{3 x^5-x^6+x^5 (i \pi +\log (3))} \, dx=\frac {\log \left (i \, \pi - x + \log \left (3\right ) + 3\right )^{2} + 8 \, \log \left (i \, \pi - x + \log \left (3\right ) + 3\right ) + 16}{x^{4}} \] Input:

integrate(((-4*log(3)-4*I*pi+4*x-12)*log(log(3)+I*pi+3-x)^2+(-32*log(3)-32 
*I*pi+30*x-96)*log(log(3)+I*pi+3-x)-64*log(3)-64*I*pi+56*x-192)/(x^5*(log( 
3)+I*pi)-x^6+3*x^5),x, algorithm="fricas")
 

Output:

(log(I*pi - x + log(3) + 3)^2 + 8*log(I*pi - x + log(3) + 3) + 16)/x^4
                                                                                    
                                                                                    
 

Sympy [A] (verification not implemented)

Time = 0.56 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.61 \[ \int \frac {-192+56 x-64 (i \pi +\log (3))+(-96+30 x-32 (i \pi +\log (3))) \log (3+i \pi -x+\log (3))+(-12+4 x-4 (i \pi +\log (3))) \log ^2(3+i \pi -x+\log (3))}{3 x^5-x^6+x^5 (i \pi +\log (3))} \, dx=\frac {\log {\left (- x + \log {\left (3 \right )} + 3 + i \pi \right )}^{2}}{x^{4}} + \frac {8 \log {\left (- x + \log {\left (3 \right )} + 3 + i \pi \right )}}{x^{4}} + \frac {16}{x^{4}} \] Input:

integrate(((-4*ln(3)-4*I*pi+4*x-12)*ln(ln(3)+I*pi+3-x)**2+(-32*ln(3)-32*I* 
pi+30*x-96)*ln(ln(3)+I*pi+3-x)-64*ln(3)-64*I*pi+56*x-192)/(x**5*(ln(3)+I*p 
i)-x**6+3*x**5),x)
 

Output:

log(-x + log(3) + 3 + I*pi)**2/x**4 + 8*log(-x + log(3) + 3 + I*pi)/x**4 + 
 16/x**4
 

Maxima [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1585 vs. \(2 (19) = 38\).

Time = 0.27 (sec) , antiderivative size = 1585, normalized size of antiderivative = 68.91 \[ \int \frac {-192+56 x-64 (i \pi +\log (3))+(-96+30 x-32 (i \pi +\log (3))) \log (3+i \pi -x+\log (3))+(-12+4 x-4 (i \pi +\log (3))) \log ^2(3+i \pi -x+\log (3))}{3 x^5-x^6+x^5 (i \pi +\log (3))} \, dx=\text {Too large to display} \] Input:

integrate(((-4*log(3)-4*I*pi+4*x-12)*log(log(3)+I*pi+3-x)^2+(-32*log(3)-32 
*I*pi+30*x-96)*log(log(3)+I*pi+3-x)-64*log(3)-64*I*pi+56*x-192)/(x^5*(log( 
3)+I*pi)-x^6+3*x^5),x, algorithm="maxima")
 

Output:

64*I*pi*(log(-I*pi + x - log(3) - 3)/(405*I*pi + I*pi^5 - 5*(-I*pi - 3)*lo 
g(3)^4 + log(3)^5 + 15*pi^4 - 10*(-6*I*pi + pi^2 - 9)*log(3)^3 - 90*I*pi^3 
 - 10*(-27*I*pi + I*pi^3 + 9*pi^2 - 27)*log(3)^2 - 270*pi^2 - 5*(-108*I*pi 
 - pi^4 + 12*I*pi^3 + 54*pi^2 - 81)*log(3) + 243) - log(x)/(405*I*pi + I*p 
i^5 - 5*(-I*pi - 3)*log(3)^4 + log(3)^5 + 15*pi^4 - 10*(-6*I*pi + pi^2 - 9 
)*log(3)^3 - 90*I*pi^3 - 10*(-27*I*pi + I*pi^3 + 9*pi^2 - 27)*log(3)^2 - 2 
70*pi^2 - 5*(-108*I*pi - pi^4 + 12*I*pi^3 + 54*pi^2 - 81)*log(3) + 243) + 
(81*I*pi - 3*I*pi^3 - 6*(-I*pi - log(3) - 3)*x^2 + 12*x^3 - 9*(-I*pi - 3)* 
log(3)^2 + 3*log(3)^3 - 27*pi^2 - 4*(-6*I*pi + pi^2 + 2*(-I*pi - 3)*log(3) 
 - log(3)^2 - 9)*x - 9*(-6*I*pi + pi^2 - 9)*log(3) + 81)/((1296*I*pi + 12* 
pi^4 - 48*(-I*pi - 3)*log(3)^3 + 12*log(3)^4 - 144*I*pi^3 - 72*(-6*I*pi + 
pi^2 - 9)*log(3)^2 - 648*pi^2 - 48*(-27*I*pi + I*pi^3 + 9*pi^2 - 27)*log(3 
) + 972)*x^4)) + 64*(log(-I*pi + x - log(3) - 3)/(405*I*pi + I*pi^5 - 5*(- 
I*pi - 3)*log(3)^4 + log(3)^5 + 15*pi^4 - 10*(-6*I*pi + pi^2 - 9)*log(3)^3 
 - 90*I*pi^3 - 10*(-27*I*pi + I*pi^3 + 9*pi^2 - 27)*log(3)^2 - 270*pi^2 - 
5*(-108*I*pi - pi^4 + 12*I*pi^3 + 54*pi^2 - 81)*log(3) + 243) - log(x)/(40 
5*I*pi + I*pi^5 - 5*(-I*pi - 3)*log(3)^4 + log(3)^5 + 15*pi^4 - 10*(-6*I*p 
i + pi^2 - 9)*log(3)^3 - 90*I*pi^3 - 10*(-27*I*pi + I*pi^3 + 9*pi^2 - 27)* 
log(3)^2 - 270*pi^2 - 5*(-108*I*pi - pi^4 + 12*I*pi^3 + 54*pi^2 - 81)*log( 
3) + 243) + (81*I*pi - 3*I*pi^3 - 6*(-I*pi - log(3) - 3)*x^2 + 12*x^3 -...
                                                                                    
                                                                                    
 

Giac [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1159 vs. \(2 (19) = 38\).

Time = 0.15 (sec) , antiderivative size = 1159, normalized size of antiderivative = 50.39 \[ \int \frac {-192+56 x-64 (i \pi +\log (3))+(-96+30 x-32 (i \pi +\log (3))) \log (3+i \pi -x+\log (3))+(-12+4 x-4 (i \pi +\log (3))) \log ^2(3+i \pi -x+\log (3))}{3 x^5-x^6+x^5 (i \pi +\log (3))} \, dx=\text {Too large to display} \] Input:

integrate(((-4*log(3)-4*I*pi+4*x-12)*log(log(3)+I*pi+3-x)^2+(-32*log(3)-32 
*I*pi+30*x-96)*log(log(3)+I*pi+3-x)-64*log(3)-64*I*pi+56*x-192)/(x^5*(log( 
3)+I*pi)-x^6+3*x^5),x, algorithm="giac")
 

Output:

-6*I*log(I*pi - x + log(3) + 3)^2/(-6*I*pi^4 + 36*I*pi^2*(I*pi - x + log(3 
) + 3)^2 - 24*pi*(I*pi - x + log(3) + 3)^3 - 6*I*(I*pi - x + log(3) + 3)^4 
 - 24*pi^3*(-I*pi + x - log(3) - 3) - 24*pi^3*log(3) + 72*pi^2*(pi + I*x - 
 I*log(3) - 3*I)*log(3) + 72*pi*(I*pi - x + log(3) + 3)^2*log(3) + 24*I*(I 
*pi - x + log(3) + 3)^3*log(3) + 36*I*pi^2*log(3)^2 - 36*I*(I*pi - x + log 
(3) + 3)^2*log(3)^2 + 72*pi*(-I*pi + x - log(3) - 3)*log(3)^2 + 24*pi*log( 
3)^3 - 24*(pi + I*x - I*log(3) - 3*I)*log(3)^3 - 6*I*log(3)^4 - 72*pi^3 + 
216*pi^2*(pi + I*x - I*log(3) - 3*I) + 216*pi*(I*pi - x + log(3) + 3)^2 + 
72*I*(I*pi - x + log(3) + 3)^3 + 216*I*pi^2*log(3) - 216*I*(I*pi - x + log 
(3) + 3)^2*log(3) + 432*pi*(-I*pi + x - log(3) - 3)*log(3) + 216*pi*log(3) 
^2 - 216*(pi + I*x - I*log(3) - 3*I)*log(3)^2 - 72*I*log(3)^3 + 324*I*pi^2 
 - 324*I*(I*pi - x + log(3) + 3)^2 + 648*pi*(-I*pi + x - log(3) - 3) + 648 
*pi*log(3) - 648*(pi + I*x - I*log(3) - 3*I)*log(3) - 324*I*log(3)^2 - 648 
*I*x + 1458*I) - 48*I*log(I*pi - x + log(3) + 3)/(-6*I*pi^4 + 36*I*pi^2*(I 
*pi - x + log(3) + 3)^2 - 24*pi*(I*pi - x + log(3) + 3)^3 - 6*I*(I*pi - x 
+ log(3) + 3)^4 - 24*pi^3*(-I*pi + x - log(3) - 3) - 24*pi^3*log(3) + 72*p 
i^2*(pi + I*x - I*log(3) - 3*I)*log(3) + 72*pi*(I*pi - x + log(3) + 3)^2*l 
og(3) + 24*I*(I*pi - x + log(3) + 3)^3*log(3) + 36*I*pi^2*log(3)^2 - 36*I* 
(I*pi - x + log(3) + 3)^2*log(3)^2 + 72*pi*(-I*pi + x - log(3) - 3)*log(3) 
^2 + 24*pi*log(3)^3 - 24*(pi + I*x - I*log(3) - 3*I)*log(3)^3 - 6*I*log...
 

Mupad [B] (verification not implemented)

Time = 1.22 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87 \[ \int \frac {-192+56 x-64 (i \pi +\log (3))+(-96+30 x-32 (i \pi +\log (3))) \log (3+i \pi -x+\log (3))+(-12+4 x-4 (i \pi +\log (3))) \log ^2(3+i \pi -x+\log (3))}{3 x^5-x^6+x^5 (i \pi +\log (3))} \, dx=\frac {{\left (\ln \left (\ln \left (3\right )-x+3+\Pi \,1{}\mathrm {i}\right )+4\right )}^2}{x^4} \] Input:

int(-(Pi*64i - 56*x + 64*log(3) + log(Pi*1i - x + log(3) + 3)^2*(Pi*4i - 4 
*x + 4*log(3) + 12) + log(Pi*1i - x + log(3) + 3)*(Pi*32i - 30*x + 32*log( 
3) + 96) + 192)/(x^5*(Pi*1i + log(3)) + 3*x^5 - x^6),x)
 

Output:

(log(Pi*1i - x + log(3) + 3) + 4)^2/x^4
                                                                                    
                                                                                    
 

Reduce [F]

\[ \int \frac {-192+56 x-64 (i \pi +\log (3))+(-96+30 x-32 (i \pi +\log (3))) \log (3+i \pi -x+\log (3))+(-12+4 x-4 (i \pi +\log (3))) \log ^2(3+i \pi -x+\log (3))}{3 x^5-x^6+x^5 (i \pi +\log (3))} \, dx=\int \frac {\left (-4 \,\mathrm {log}\left (3\right )-4 i \pi +4 x -12\right ) \mathrm {log}\left (\mathrm {log}\left (3\right )+i \pi +3-x \right )^{2}+\left (-32 \,\mathrm {log}\left (3\right )-32 i \pi +30 x -96\right ) \mathrm {log}\left (\mathrm {log}\left (3\right )+i \pi +3-x \right )-64 \,\mathrm {log}\left (3\right )-64 i \pi +56 x -192}{x^{5} \left (\mathrm {log}\left (3\right )+i \pi \right )-x^{6}+3 x^{5}}d x \] Input:

int(((-4*log(3)-4*I*Pi+4*x-12)*log(log(3)+I*Pi+3-x)^2+(-32*log(3)-32*I*Pi+ 
30*x-96)*log(log(3)+I*Pi+3-x)-64*log(3)-64*I*Pi+56*x-192)/(x^5*(log(3)+I*P 
i)-x^6+3*x^5),x)
 

Output:

int(((-4*log(3)-4*I*Pi+4*x-12)*log(log(3)+I*Pi+3-x)^2+(-32*log(3)-32*I*Pi+ 
30*x-96)*log(log(3)+I*Pi+3-x)-64*log(3)-64*I*Pi+56*x-192)/(x^5*(log(3)+I*P 
i)-x^6+3*x^5),x)