\(\int \log ^3(a+b \coth (c+d x)) \, dx\) [16]

Optimal result

Integrand size = 13, antiderivative size = 274 \[ \int \log ^3(a+b \coth (c+d x)) \, dx=-\frac {\log \left (\frac {b (1-\coth (c+d x))}{a+b}\right ) \log ^3(a+b \coth (c+d x))}{2 d}+\frac {\log \left (-\frac {b (1+\coth (c+d x))}{a-b}\right ) \log ^3(a+b \coth (c+d x))}{2 d}+\frac {3 \log ^2(a+b \coth (c+d x)) \operatorname {PolyLog}\left (2,\frac {a+b \coth (c+d x)}{a-b}\right )}{2 d}-\frac {3 \log ^2(a+b \coth (c+d x)) \operatorname {PolyLog}\left (2,\frac {a+b \coth (c+d x)}{a+b}\right )}{2 d}-\frac {3 \log (a+b \coth (c+d x)) \operatorname {PolyLog}\left (3,\frac {a+b \coth (c+d x)}{a-b}\right )}{d}+\frac {3 \log (a+b \coth (c+d x)) \operatorname {PolyLog}\left (3,\frac {a+b \coth (c+d x)}{a+b}\right )}{d}+\frac {3 \operatorname {PolyLog}\left (4,\frac {a+b \coth (c+d x)}{a-b}\right )}{d}-\frac {3 \operatorname {PolyLog}\left (4,\frac {a+b \coth (c+d x)}{a+b}\right )}{d} \] Output:

-1/2*ln(b*(1-coth(d*x+c))/(a+b))*ln(a+b*coth(d*x+c))^3/d+1/2*ln(-b*(1+coth 
(d*x+c))/(a-b))*ln(a+b*coth(d*x+c))^3/d+3/2*ln(a+b*coth(d*x+c))^2*polylog( 
2,(a+b*coth(d*x+c))/(a-b))/d-3/2*ln(a+b*coth(d*x+c))^2*polylog(2,(a+b*coth 
(d*x+c))/(a+b))/d-3*ln(a+b*coth(d*x+c))*polylog(3,(a+b*coth(d*x+c))/(a-b)) 
/d+3*ln(a+b*coth(d*x+c))*polylog(3,(a+b*coth(d*x+c))/(a+b))/d+3*polylog(4, 
(a+b*coth(d*x+c))/(a-b))/d-3*polylog(4,(a+b*coth(d*x+c))/(a+b))/d
 

Mathematica [A] (verified)

Time = 1.64 (sec) , antiderivative size = 248, normalized size of antiderivative = 0.91 \[ \int \log ^3(a+b \coth (c+d x)) \, dx=\frac {\log \left (-\frac {b (1+\coth (c+d x))}{a-b}\right ) \log ^3(a+b \coth (c+d x))-\log \left (\frac {b-b \coth (c+d x)}{a+b}\right ) \log ^3(a+b \coth (c+d x))+3 \log ^2(a+b \coth (c+d x)) \operatorname {PolyLog}\left (2,\frac {a+b \coth (c+d x)}{a-b}\right )-3 \log ^2(a+b \coth (c+d x)) \operatorname {PolyLog}\left (2,\frac {a+b \coth (c+d x)}{a+b}\right )-6 \log (a+b \coth (c+d x)) \operatorname {PolyLog}\left (3,\frac {a+b \coth (c+d x)}{a-b}\right )+6 \log (a+b \coth (c+d x)) \operatorname {PolyLog}\left (3,\frac {a+b \coth (c+d x)}{a+b}\right )+6 \operatorname {PolyLog}\left (4,\frac {a+b \coth (c+d x)}{a-b}\right )-6 \operatorname {PolyLog}\left (4,\frac {a+b \coth (c+d x)}{a+b}\right )}{2 d} \] Input:

Integrate[Log[a + b*Coth[c + d*x]]^3,x]
 

Output:

(Log[-((b*(1 + Coth[c + d*x]))/(a - b))]*Log[a + b*Coth[c + d*x]]^3 - Log[ 
(b - b*Coth[c + d*x])/(a + b)]*Log[a + b*Coth[c + d*x]]^3 + 3*Log[a + b*Co 
th[c + d*x]]^2*PolyLog[2, (a + b*Coth[c + d*x])/(a - b)] - 3*Log[a + b*Cot 
h[c + d*x]]^2*PolyLog[2, (a + b*Coth[c + d*x])/(a + b)] - 6*Log[a + b*Coth 
[c + d*x]]*PolyLog[3, (a + b*Coth[c + d*x])/(a - b)] + 6*Log[a + b*Coth[c 
+ d*x]]*PolyLog[3, (a + b*Coth[c + d*x])/(a + b)] + 6*PolyLog[4, (a + b*Co 
th[c + d*x])/(a - b)] - 6*PolyLog[4, (a + b*Coth[c + d*x])/(a + b)])/(2*d)
 

Rubi [A] (verified)

Time = 0.91 (sec) , antiderivative size = 254, normalized size of antiderivative = 0.93, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {4852, 2856, 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.

\(\displaystyle \int \log ^3(a+b \coth (c+d x)) \, dx\)

\(\Big \downarrow \) 4852

\(\displaystyle \frac {\int \frac {\log ^3(a+b \coth (c+d x))}{1-\coth ^2(c+d x)}d\coth (c+d x)}{d}\)

\(\Big \downarrow \) 2856

\(\displaystyle \frac {\int \left (\frac {\log ^3(a+b \coth (c+d x))}{2 (1-\coth (c+d x))}+\frac {\log ^3(a+b \coth (c+d x))}{2 (\coth (c+d x)+1)}\right )d\coth (c+d x)}{d}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {3 \operatorname {PolyLog}\left (4,\frac {a+b \coth (c+d x)}{a-b}\right )-3 \operatorname {PolyLog}\left (4,\frac {a+b \coth (c+d x)}{a+b}\right )+\frac {3}{2} \operatorname {PolyLog}\left (2,\frac {a+b \coth (c+d x)}{a-b}\right ) \log ^2(a+b \coth (c+d x))-\frac {3}{2} \operatorname {PolyLog}\left (2,\frac {a+b \coth (c+d x)}{a+b}\right ) \log ^2(a+b \coth (c+d x))-3 \operatorname {PolyLog}\left (3,\frac {a+b \coth (c+d x)}{a-b}\right ) \log (a+b \coth (c+d x))+3 \operatorname {PolyLog}\left (3,\frac {a+b \coth (c+d x)}{a+b}\right ) \log (a+b \coth (c+d x))-\frac {1}{2} \log \left (\frac {b (1-\coth (c+d x))}{a+b}\right ) \log ^3(a+b \coth (c+d x))+\frac {1}{2} \log \left (-\frac {b (\coth (c+d x)+1)}{a-b}\right ) \log ^3(a+b \coth (c+d x))}{d}\)

Input:

Int[Log[a + b*Coth[c + d*x]]^3,x]
 

Output:

(-1/2*(Log[(b*(1 - Coth[c + d*x]))/(a + b)]*Log[a + b*Coth[c + d*x]]^3) + 
(Log[-((b*(1 + Coth[c + d*x]))/(a - b))]*Log[a + b*Coth[c + d*x]]^3)/2 + ( 
3*Log[a + b*Coth[c + d*x]]^2*PolyLog[2, (a + b*Coth[c + d*x])/(a - b)])/2 
- (3*Log[a + b*Coth[c + d*x]]^2*PolyLog[2, (a + b*Coth[c + d*x])/(a + b)]) 
/2 - 3*Log[a + b*Coth[c + d*x]]*PolyLog[3, (a + b*Coth[c + d*x])/(a - b)] 
+ 3*Log[a + b*Coth[c + d*x]]*PolyLog[3, (a + b*Coth[c + d*x])/(a + b)] + 3 
*PolyLog[4, (a + b*Coth[c + d*x])/(a - b)] - 3*PolyLog[4, (a + b*Coth[c + 
d*x])/(a + b)])/d
 

Defintions of rubi rules used

rule 2009

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

rule 2856

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_. 
)*(x_)^(r_))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x) 
^n])^p, (f + g*x^r)^q, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, r}, x] && I 
GtQ[p, 0] && IntegerQ[q] && (GtQ[q, 0] || (IntegerQ[r] && NeQ[r, 1]))
 

rule 4852

Int[u_, x_Symbol] :> With[{v = FunctionOfTrig[u, x]}, Simp[With[{d = FreeFa 
ctors[Cot[v], x]}, -d/Coefficient[v, x, 1]   Subst[Int[SubstFor[1/(1 + d^2* 
x^2), Cot[v]/d, u, x], x], x, Cot[v]/d]], x] /;  !FalseQ[v] && FunctionOfQ[ 
NonfreeFactors[Cot[v], x], u, x, True] && TryPureTanSubst[ActivateTrig[u], 
x]]
 

Maple [A] (verified)

Time = 10.64 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.04

method	result	size
derivativedivides	\(-\frac {b \left (-\frac {\ln \left (a +b \coth \left (d x +c \right )\right )^{3} \ln \left (1+\frac {a +b \coth \left (d x +c \right )}{-a +b}\right )+3 \ln \left (a +b \coth \left (d x +c \right )\right )^{2} \operatorname {polylog}\left (2, -\frac {a +b \coth \left (d x +c \right )}{-a +b}\right )-6 \ln \left (a +b \coth \left (d x +c \right )\right ) \operatorname {polylog}\left (3, -\frac {a +b \coth \left (d x +c \right )}{-a +b}\right )+6 \operatorname {polylog}\left (4, -\frac {a +b \coth \left (d x +c \right )}{-a +b}\right )}{2 b}+\frac {\ln \left (a +b \coth \left (d x +c \right )\right )^{3} \ln \left (1+\frac {a +b \coth \left (d x +c \right )}{-a -b}\right )+3 \ln \left (a +b \coth \left (d x +c \right )\right )^{2} \operatorname {polylog}\left (2, -\frac {a +b \coth \left (d x +c \right )}{-a -b}\right )-6 \ln \left (a +b \coth \left (d x +c \right )\right ) \operatorname {polylog}\left (3, -\frac {a +b \coth \left (d x +c \right )}{-a -b}\right )+6 \operatorname {polylog}\left (4, -\frac {a +b \coth \left (d x +c \right )}{-a -b}\right )}{2 b}\right )}{d}\)	\(284\)
default	\(-\frac {b \left (-\frac {\ln \left (a +b \coth \left (d x +c \right )\right )^{3} \ln \left (1+\frac {a +b \coth \left (d x +c \right )}{-a +b}\right )+3 \ln \left (a +b \coth \left (d x +c \right )\right )^{2} \operatorname {polylog}\left (2, -\frac {a +b \coth \left (d x +c \right )}{-a +b}\right )-6 \ln \left (a +b \coth \left (d x +c \right )\right ) \operatorname {polylog}\left (3, -\frac {a +b \coth \left (d x +c \right )}{-a +b}\right )+6 \operatorname {polylog}\left (4, -\frac {a +b \coth \left (d x +c \right )}{-a +b}\right )}{2 b}+\frac {\ln \left (a +b \coth \left (d x +c \right )\right )^{3} \ln \left (1+\frac {a +b \coth \left (d x +c \right )}{-a -b}\right )+3 \ln \left (a +b \coth \left (d x +c \right )\right )^{2} \operatorname {polylog}\left (2, -\frac {a +b \coth \left (d x +c \right )}{-a -b}\right )-6 \ln \left (a +b \coth \left (d x +c \right )\right ) \operatorname {polylog}\left (3, -\frac {a +b \coth \left (d x +c \right )}{-a -b}\right )+6 \operatorname {polylog}\left (4, -\frac {a +b \coth \left (d x +c \right )}{-a -b}\right )}{2 b}\right )}{d}\)	\(284\)

Input:

int(ln(a+b*coth(d*x+c))^3,x,method=_RETURNVERBOSE)
 

Output:

-1/d*b*(-1/2/b*(ln(a+b*coth(d*x+c))^3*ln(1+1/(-a+b)*(a+b*coth(d*x+c)))+3*l 
n(a+b*coth(d*x+c))^2*polylog(2,-1/(-a+b)*(a+b*coth(d*x+c)))-6*ln(a+b*coth( 
d*x+c))*polylog(3,-1/(-a+b)*(a+b*coth(d*x+c)))+6*polylog(4,-1/(-a+b)*(a+b* 
coth(d*x+c))))+1/2/b*(ln(a+b*coth(d*x+c))^3*ln(1+1/(-a-b)*(a+b*coth(d*x+c) 
))+3*ln(a+b*coth(d*x+c))^2*polylog(2,-1/(-a-b)*(a+b*coth(d*x+c)))-6*ln(a+b 
*coth(d*x+c))*polylog(3,-1/(-a-b)*(a+b*coth(d*x+c)))+6*polylog(4,-1/(-a-b) 
*(a+b*coth(d*x+c)))))
 

Fricas [F]

\[ \int \log ^3(a+b \coth (c+d x)) \, dx=\int { \log \left (b \coth \left (d x + c\right ) + a\right )^{3} \,d x } \] Input:

integrate(log(a+b*coth(d*x+c))^3,x, algorithm="fricas")
 

Output:

integral(log(b*coth(d*x + c) + a)^3, x)
 

Sympy [F]

\[ \int \log ^3(a+b \coth (c+d x)) \, dx=\int \log {\left (a + b \coth {\left (c + d x \right )} \right )}^{3}\, dx \] Input:

integrate(ln(a+b*coth(d*x+c))**3,x)
 

Output:

Integral(log(a + b*coth(c + d*x))**3, x)
 

Maxima [F]

\[ \int \log ^3(a+b \coth (c+d x)) \, dx=\int { \log \left (b \coth \left (d x + c\right ) + a\right )^{3} \,d x } \] Input:

integrate(log(a+b*coth(d*x+c))^3,x, algorithm="maxima")
 

Output:

x*log((a + b)*e^(2*d*x + 2*c) - a + b)^3 - integrate((((a*e^(2*c) + b*e^(2 
*c))*e^(2*d*x) - a + b)*log(e^(d*x + c) + 1)^3 + 3*((a*e^(2*c) + b*e^(2*c) 
)*e^(2*d*x) - a + b)*log(e^(d*x + c) + 1)^2*log(e^(d*x + c) - 1) + 3*((a*e 
^(2*c) + b*e^(2*c))*e^(2*d*x) - a + b)*log(e^(d*x + c) + 1)*log(e^(d*x + c 
) - 1)^2 + ((a*e^(2*c) + b*e^(2*c))*e^(2*d*x) - a + b)*log(e^(d*x + c) - 1 
)^3 + 3*(2*(a*d*e^(2*c) + b*d*e^(2*c))*x*e^(2*d*x) + ((a*e^(2*c) + b*e^(2* 
c))*e^(2*d*x) - a + b)*log(e^(d*x + c) + 1) + ((a*e^(2*c) + b*e^(2*c))*e^( 
2*d*x) - a + b)*log(e^(d*x + c) - 1))*log((a + b)*e^(2*d*x + 2*c) - a + b) 
^2 - 3*(((a*e^(2*c) + b*e^(2*c))*e^(2*d*x) - a + b)*log(e^(d*x + c) + 1)^2 
 + 2*((a*e^(2*c) + b*e^(2*c))*e^(2*d*x) - a + b)*log(e^(d*x + c) + 1)*log( 
e^(d*x + c) - 1) + ((a*e^(2*c) + b*e^(2*c))*e^(2*d*x) - a + b)*log(e^(d*x 
+ c) - 1)^2)*log((a + b)*e^(2*d*x + 2*c) - a + b))/((a*e^(2*c) + b*e^(2*c) 
)*e^(2*d*x) - a + b), x)
 

Giac [F]

\[ \int \log ^3(a+b \coth (c+d x)) \, dx=\int { \log \left (b \coth \left (d x + c\right ) + a\right )^{3} \,d x } \] Input:

integrate(log(a+b*coth(d*x+c))^3,x, algorithm="giac")
 

Output:

integrate(log(b*coth(d*x + c) + a)^3, x)
 

Mupad [F(-1)]

Timed out. \[ \int \log ^3(a+b \coth (c+d x)) \, dx=\int {\ln \left (a+b\,\mathrm {coth}\left (c+d\,x\right )\right )}^3 \,d x \] Input:

int(log(a + b*coth(c + d*x))^3,x)
 

Output:

int(log(a + b*coth(c + d*x))^3, x)
 

Reduce [F]

\[ \int \log ^3(a+b \coth (c+d x)) \, dx=\int \mathrm {log}\left (\coth \left (d x +c \right ) b +a \right )^{3}d x \] Input:

int(log(a+b*coth(d*x+c))^3,x)
 

Output:

int(log(coth(c + d*x)*b + a)**3,x)