3.1.30 \(\int \frac {(a+b \text {arctanh}(c x))^3}{x} \, dx\) [30]

3.1.30.1 Optimal result

Integrand size = 14, antiderivative size = 184 \[ \int \frac {(a+b \text {arctanh}(c x))^3}{x} \, dx=2 (a+b \text {arctanh}(c x))^3 \text {arctanh}\left (1-\frac {2}{1-c x}\right )-\frac {3}{2} b (a+b \text {arctanh}(c x))^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )+\frac {3}{2} b (a+b \text {arctanh}(c x))^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-c x}\right )+\frac {3}{2} b^2 (a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (3,1-\frac {2}{1-c x}\right )-\frac {3}{2} b^2 (a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (3,-1+\frac {2}{1-c x}\right )-\frac {3}{4} b^3 \operatorname {PolyLog}\left (4,1-\frac {2}{1-c x}\right )+\frac {3}{4} b^3 \operatorname {PolyLog}\left (4,-1+\frac {2}{1-c x}\right ) \]

output

-2*(a+b*arctanh(c*x))^3*arctanh(-1+2/(-c*x+1))-3/2*b*(a+b*arctanh(c*x))^2* 
polylog(2,1-2/(-c*x+1))+3/2*b*(a+b*arctanh(c*x))^2*polylog(2,-1+2/(-c*x+1) 
)+3/2*b^2*(a+b*arctanh(c*x))*polylog(3,1-2/(-c*x+1))-3/2*b^2*(a+b*arctanh( 
c*x))*polylog(3,-1+2/(-c*x+1))-3/4*b^3*polylog(4,1-2/(-c*x+1))+3/4*b^3*pol 
ylog(4,-1+2/(-c*x+1))
 

3.1.30.2 Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.51 (sec) , antiderivative size = 315, normalized size of antiderivative = 1.71 \[ \int \frac {(a+b \text {arctanh}(c x))^3}{x} \, dx=a^3 \log (c x)+\frac {3}{2} a^2 b (-\operatorname {PolyLog}(2,-c x)+\operatorname {PolyLog}(2,c x))+3 a b^2 \left (\frac {i \pi ^3}{24}-\frac {2}{3} \text {arctanh}(c x)^3-\text {arctanh}(c x)^2 \log \left (1+e^{-2 \text {arctanh}(c x)}\right )+\text {arctanh}(c x)^2 \log \left (1-e^{2 \text {arctanh}(c x)}\right )+\text {arctanh}(c x) \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(c x)}\right )+\text {arctanh}(c x) \operatorname {PolyLog}\left (2,e^{2 \text {arctanh}(c x)}\right )+\frac {1}{2} \operatorname {PolyLog}\left (3,-e^{-2 \text {arctanh}(c x)}\right )-\frac {1}{2} \operatorname {PolyLog}\left (3,e^{2 \text {arctanh}(c x)}\right )\right )+\frac {1}{64} b^3 \left (\pi ^4-32 \text {arctanh}(c x)^4-64 \text {arctanh}(c x)^3 \log \left (1+e^{-2 \text {arctanh}(c x)}\right )+64 \text {arctanh}(c x)^3 \log \left (1-e^{2 \text {arctanh}(c x)}\right )+96 \text {arctanh}(c x)^2 \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(c x)}\right )+96 \text {arctanh}(c x)^2 \operatorname {PolyLog}\left (2,e^{2 \text {arctanh}(c x)}\right )+96 \text {arctanh}(c x) \operatorname {PolyLog}\left (3,-e^{-2 \text {arctanh}(c x)}\right )-96 \text {arctanh}(c x) \operatorname {PolyLog}\left (3,e^{2 \text {arctanh}(c x)}\right )+48 \operatorname {PolyLog}\left (4,-e^{-2 \text {arctanh}(c x)}\right )+48 \operatorname {PolyLog}\left (4,e^{2 \text {arctanh}(c x)}\right )\right ) \]

input

Integrate[(a + b*ArcTanh[c*x])^3/x,x]
 

output

a^3*Log[c*x] + (3*a^2*b*(-PolyLog[2, -(c*x)] + PolyLog[2, c*x]))/2 + 3*a*b 
^2*((I/24)*Pi^3 - (2*ArcTanh[c*x]^3)/3 - ArcTanh[c*x]^2*Log[1 + E^(-2*ArcT 
anh[c*x])] + ArcTanh[c*x]^2*Log[1 - E^(2*ArcTanh[c*x])] + ArcTanh[c*x]*Pol 
yLog[2, -E^(-2*ArcTanh[c*x])] + ArcTanh[c*x]*PolyLog[2, E^(2*ArcTanh[c*x]) 
] + PolyLog[3, -E^(-2*ArcTanh[c*x])]/2 - PolyLog[3, E^(2*ArcTanh[c*x])]/2) 
 + (b^3*(Pi^4 - 32*ArcTanh[c*x]^4 - 64*ArcTanh[c*x]^3*Log[1 + E^(-2*ArcTan 
h[c*x])] + 64*ArcTanh[c*x]^3*Log[1 - E^(2*ArcTanh[c*x])] + 96*ArcTanh[c*x] 
^2*PolyLog[2, -E^(-2*ArcTanh[c*x])] + 96*ArcTanh[c*x]^2*PolyLog[2, E^(2*Ar 
cTanh[c*x])] + 96*ArcTanh[c*x]*PolyLog[3, -E^(-2*ArcTanh[c*x])] - 96*ArcTa 
nh[c*x]*PolyLog[3, E^(2*ArcTanh[c*x])] + 48*PolyLog[4, -E^(-2*ArcTanh[c*x] 
)] + 48*PolyLog[4, E^(2*ArcTanh[c*x])]))/64
 

3.1.30.3 Rubi [A] (verified)

Time = 1.14 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.15, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {6448, 6614, 6620, 6624, 7164}

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.

\(\displaystyle \int \frac {(a+b \text {arctanh}(c x))^3}{x} \, dx\)

\(\Big \downarrow \) 6448

\(\displaystyle 2 \text {arctanh}\left (1-\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))^3-6 b c \int \frac {(a+b \text {arctanh}(c x))^2 \text {arctanh}\left (1-\frac {2}{1-c x}\right )}{1-c^2 x^2}dx\)

\(\Big \downarrow \) 6614

\(\displaystyle 2 \text {arctanh}\left (1-\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))^3-6 b c \left (\frac {1}{2} \int \frac {(a+b \text {arctanh}(c x))^2 \log \left (2-\frac {2}{1-c x}\right )}{1-c^2 x^2}dx-\frac {1}{2} \int \frac {(a+b \text {arctanh}(c x))^2 \log \left (\frac {2}{1-c x}\right )}{1-c^2 x^2}dx\right )\)

\(\Big \downarrow \) 6620

\(\displaystyle 2 \text {arctanh}\left (1-\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))^3-6 b c \left (\frac {1}{2} \left (\frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))^2}{2 c}-b \int \frac {(a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{1-c^2 x^2}dx\right )+\frac {1}{2} \left (b \int \frac {(a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,\frac {2}{1-c x}-1\right )}{1-c^2 x^2}dx-\frac {\operatorname {PolyLog}\left (2,\frac {2}{1-c x}-1\right ) (a+b \text {arctanh}(c x))^2}{2 c}\right )\right )\)

\(\Big \downarrow \) 6624

\(\displaystyle 2 \text {arctanh}\left (1-\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))^3-6 b c \left (\frac {1}{2} \left (\frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))^2}{2 c}-b \left (\frac {\operatorname {PolyLog}\left (3,1-\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))}{2 c}-\frac {1}{2} b \int \frac {\operatorname {PolyLog}\left (3,1-\frac {2}{1-c x}\right )}{1-c^2 x^2}dx\right )\right )+\frac {1}{2} \left (b \left (\frac {\operatorname {PolyLog}\left (3,\frac {2}{1-c x}-1\right ) (a+b \text {arctanh}(c x))}{2 c}-\frac {1}{2} b \int \frac {\operatorname {PolyLog}\left (3,\frac {2}{1-c x}-1\right )}{1-c^2 x^2}dx\right )-\frac {\operatorname {PolyLog}\left (2,\frac {2}{1-c x}-1\right ) (a+b \text {arctanh}(c x))^2}{2 c}\right )\right )\)

\(\Big \downarrow \) 7164

\(\displaystyle 2 \text {arctanh}\left (1-\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))^3-6 b c \left (\frac {1}{2} \left (\frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))^2}{2 c}-b \left (\frac {\operatorname {PolyLog}\left (3,1-\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))}{2 c}-\frac {b \operatorname {PolyLog}\left (4,1-\frac {2}{1-c x}\right )}{4 c}\right )\right )+\frac {1}{2} \left (b \left (\frac {\operatorname {PolyLog}\left (3,\frac {2}{1-c x}-1\right ) (a+b \text {arctanh}(c x))}{2 c}-\frac {b \operatorname {PolyLog}\left (4,\frac {2}{1-c x}-1\right )}{4 c}\right )-\frac {\operatorname {PolyLog}\left (2,\frac {2}{1-c x}-1\right ) (a+b \text {arctanh}(c x))^2}{2 c}\right )\right )\)

input

Int[(a + b*ArcTanh[c*x])^3/x,x]
 

output

2*(a + b*ArcTanh[c*x])^3*ArcTanh[1 - 2/(1 - c*x)] - 6*b*c*((((a + b*ArcTan 
h[c*x])^2*PolyLog[2, 1 - 2/(1 - c*x)])/(2*c) - b*(((a + b*ArcTanh[c*x])*Po 
lyLog[3, 1 - 2/(1 - c*x)])/(2*c) - (b*PolyLog[4, 1 - 2/(1 - c*x)])/(4*c))) 
/2 + (-1/2*((a + b*ArcTanh[c*x])^2*PolyLog[2, -1 + 2/(1 - c*x)])/c + b*((( 
a + b*ArcTanh[c*x])*PolyLog[3, -1 + 2/(1 - c*x)])/(2*c) - (b*PolyLog[4, -1 
 + 2/(1 - c*x)])/(4*c)))/2)
 

3.1.30.3.1 Defintions of rubi rules used

rule 6448

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_)/(x_), x_Symbol] :> Simp[2*(a + 
 b*ArcTanh[c*x])^p*ArcTanh[1 - 2/(1 - c*x)], x] - Simp[2*b*c*p   Int[(a + b 
*ArcTanh[c*x])^(p - 1)*(ArcTanh[1 - 2/(1 - c*x)]/(1 - c^2*x^2)), x], x] /; 
FreeQ[{a, b, c}, x] && IGtQ[p, 1]
 

rule 6614

Int[(ArcTanh[u_]*((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*( 
x_)^2), x_Symbol] :> Simp[1/2   Int[Log[1 + u]*((a + b*ArcTanh[c*x])^p/(d + 
 e*x^2)), x], x] - Simp[1/2   Int[Log[1 - u]*((a + b*ArcTanh[c*x])^p/(d + e 
*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d + e, 
 0] && EqQ[u^2 - (1 - 2/(1 - c*x))^2, 0]
 

rule 6620

Int[(Log[u_]*((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^ 
2), x_Symbol] :> Simp[(-(a + b*ArcTanh[c*x])^p)*(PolyLog[2, 1 - u]/(2*c*d)) 
, x] + Simp[b*(p/2)   Int[(a + b*ArcTanh[c*x])^(p - 1)*(PolyLog[2, 1 - u]/( 
d + e*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d 
 + e, 0] && EqQ[(1 - u)^2 - (1 - 2/(1 - c*x))^2, 0]
 

rule 6624

Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*PolyLog[k_, u_])/((d_) + (e_ 
.)*(x_)^2), x_Symbol] :> Simp[(a + b*ArcTanh[c*x])^p*(PolyLog[k + 1, u]/(2* 
c*d)), x] - Simp[b*(p/2)   Int[(a + b*ArcTanh[c*x])^(p - 1)*(PolyLog[k + 1, 
 u]/(d + e*x^2)), x], x] /; FreeQ[{a, b, c, d, e, k}, x] && IGtQ[p, 0] && E 
qQ[c^2*d + e, 0] && EqQ[u^2 - (1 - 2/(1 - c*x))^2, 0]
 

rule 7164

Int[(u_)*PolyLog[n_, v_], x_Symbol] :> With[{w = DerivativeDivides[v, u*v, 
x]}, Simp[w*PolyLog[n + 1, v], x] /;  !FalseQ[w]] /; FreeQ[n, x]
 

3.1.30.4 Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 16.52 (sec) , antiderivative size = 1304, normalized size of antiderivative = 7.09

method	result	size
parts	\(\text {Expression too large to display}\)	\(1304\)
derivativedivides	\(\text {Expression too large to display}\)	\(1306\)
default	\(\text {Expression too large to display}\)	\(1306\)

input

int((a+b*arctanh(c*x))^3/x,x,method=_RETURNVERBOSE)
 

output

a^3*ln(x)+b^3*(ln(c*x)*arctanh(c*x)^3-arctanh(c*x)^3*ln((c*x+1)^2/(-c^2*x^ 
2+1)-1)+arctanh(c*x)^3*ln(1+(c*x+1)/(-c^2*x^2+1)^(1/2))+3*arctanh(c*x)^2*p 
olylog(2,-(c*x+1)/(-c^2*x^2+1)^(1/2))-6*arctanh(c*x)*polylog(3,-(c*x+1)/(- 
c^2*x^2+1)^(1/2))+6*polylog(4,-(c*x+1)/(-c^2*x^2+1)^(1/2))+arctanh(c*x)^3* 
ln(1-(c*x+1)/(-c^2*x^2+1)^(1/2))+3*arctanh(c*x)^2*polylog(2,(c*x+1)/(-c^2* 
x^2+1)^(1/2))-6*arctanh(c*x)*polylog(3,(c*x+1)/(-c^2*x^2+1)^(1/2))+6*polyl 
og(4,(c*x+1)/(-c^2*x^2+1)^(1/2))+1/2*I*Pi*csgn(I*(-(c*x+1)^2/(c^2*x^2-1)-1 
)/(1-(c*x+1)^2/(c^2*x^2-1)))*(csgn(I*(-(c*x+1)^2/(c^2*x^2-1)-1))*csgn(I/(1 
-(c*x+1)^2/(c^2*x^2-1)))-csgn(I*(-(c*x+1)^2/(c^2*x^2-1)-1))*csgn(I*(-(c*x+ 
1)^2/(c^2*x^2-1)-1)/(1-(c*x+1)^2/(c^2*x^2-1)))-csgn(I*(-(c*x+1)^2/(c^2*x^2 
-1)-1)/(1-(c*x+1)^2/(c^2*x^2-1)))*csgn(I/(1-(c*x+1)^2/(c^2*x^2-1)))+csgn(I 
*(-(c*x+1)^2/(c^2*x^2-1)-1)/(1-(c*x+1)^2/(c^2*x^2-1)))^2)*arctanh(c*x)^3-3 
/2*arctanh(c*x)^2*polylog(2,-(c*x+1)^2/(-c^2*x^2+1))+3/2*arctanh(c*x)*poly 
log(3,-(c*x+1)^2/(-c^2*x^2+1))-3/4*polylog(4,-(c*x+1)^2/(-c^2*x^2+1)))+3*a 
*b^2*(ln(c*x)*arctanh(c*x)^2-arctanh(c*x)*polylog(2,-(c*x+1)^2/(-c^2*x^2+1 
))+1/2*polylog(3,-(c*x+1)^2/(-c^2*x^2+1))-arctanh(c*x)^2*ln((c*x+1)^2/(-c^ 
2*x^2+1)-1)+arctanh(c*x)^2*ln(1+(c*x+1)/(-c^2*x^2+1)^(1/2))+2*arctanh(c*x) 
*polylog(2,-(c*x+1)/(-c^2*x^2+1)^(1/2))-2*polylog(3,-(c*x+1)/(-c^2*x^2+1)^ 
(1/2))+arctanh(c*x)^2*ln(1-(c*x+1)/(-c^2*x^2+1)^(1/2))+2*arctanh(c*x)*poly 
log(2,(c*x+1)/(-c^2*x^2+1)^(1/2))-2*polylog(3,(c*x+1)/(-c^2*x^2+1)^(1/2...
 

3.1.30.5 Fricas [F]

\[ \int \frac {(a+b \text {arctanh}(c x))^3}{x} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{3}}{x} \,d x } \]

input

integrate((a+b*arctanh(c*x))^3/x,x, algorithm="fricas")
 

output

integral((b^3*arctanh(c*x)^3 + 3*a*b^2*arctanh(c*x)^2 + 3*a^2*b*arctanh(c* 
x) + a^3)/x, x)
 

3.1.30.6 Sympy [F]

\[ \int \frac {(a+b \text {arctanh}(c x))^3}{x} \, dx=\int \frac {\left (a + b \operatorname {atanh}{\left (c x \right )}\right )^{3}}{x}\, dx \]

input

integrate((a+b*atanh(c*x))**3/x,x)
 

output

Integral((a + b*atanh(c*x))**3/x, x)
 

3.1.30.7 Maxima [F]

\[ \int \frac {(a+b \text {arctanh}(c x))^3}{x} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{3}}{x} \,d x } \]

input

integrate((a+b*arctanh(c*x))^3/x,x, algorithm="maxima")
 

output

a^3*log(x) + integrate(1/8*b^3*(log(c*x + 1) - log(-c*x + 1))^3/x + 3/4*a* 
b^2*(log(c*x + 1) - log(-c*x + 1))^2/x + 3/2*a^2*b*(log(c*x + 1) - log(-c* 
x + 1))/x, x)
 

3.1.30.8 Giac [F]

\[ \int \frac {(a+b \text {arctanh}(c x))^3}{x} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{3}}{x} \,d x } \]

input

integrate((a+b*arctanh(c*x))^3/x,x, algorithm="giac")
 

output

integrate((b*arctanh(c*x) + a)^3/x, x)
 

3.1.30.9 Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b \text {arctanh}(c x))^3}{x} \, dx=\int \frac {{\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}^3}{x} \,d x \]

input

int((a + b*atanh(c*x))^3/x,x)
 

output

int((a + b*atanh(c*x))^3/x, x)