\(\int \frac {\text {arctanh}(a x)^4}{c-a c x} \, dx\) [136]

Optimal result

Integrand size = 16, antiderivative size = 131 \[ \int \frac {\text {arctanh}(a x)^4}{c-a c x} \, dx=\frac {\text {arctanh}(a x)^4 \log \left (\frac {2}{1-a x}\right )}{a c}+\frac {2 \text {arctanh}(a x)^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{a c}-\frac {3 \text {arctanh}(a x)^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1-a x}\right )}{a c}+\frac {3 \text {arctanh}(a x) \operatorname {PolyLog}\left (4,1-\frac {2}{1-a x}\right )}{a c}-\frac {3 \operatorname {PolyLog}\left (5,1-\frac {2}{1-a x}\right )}{2 a c} \] Output:

arctanh(a*x)^4*ln(2/(-a*x+1))/a/c+2*arctanh(a*x)^3*polylog(2,1-2/(-a*x+1)) 
/a/c-3*arctanh(a*x)^2*polylog(3,1-2/(-a*x+1))/a/c+3*arctanh(a*x)*polylog(4 
,1-2/(-a*x+1))/a/c-3/2*polylog(5,1-2/(-a*x+1))/a/c
 

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.85 \[ \int \frac {\text {arctanh}(a x)^4}{c-a c x} \, dx=-\frac {-\frac {2}{5} \text {arctanh}(a x)^5-\text {arctanh}(a x)^4 \log \left (1+e^{-2 \text {arctanh}(a x)}\right )+2 \text {arctanh}(a x)^3 \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(a x)}\right )+3 \text {arctanh}(a x)^2 \operatorname {PolyLog}\left (3,-e^{-2 \text {arctanh}(a x)}\right )+3 \text {arctanh}(a x) \operatorname {PolyLog}\left (4,-e^{-2 \text {arctanh}(a x)}\right )+\frac {3}{2} \operatorname {PolyLog}\left (5,-e^{-2 \text {arctanh}(a x)}\right )}{a c} \] Input:

Integrate[ArcTanh[a*x]^4/(c - a*c*x),x]
 

Output:

-(((-2*ArcTanh[a*x]^5)/5 - ArcTanh[a*x]^4*Log[1 + E^(-2*ArcTanh[a*x])] + 2 
*ArcTanh[a*x]^3*PolyLog[2, -E^(-2*ArcTanh[a*x])] + 3*ArcTanh[a*x]^2*PolyLo 
g[3, -E^(-2*ArcTanh[a*x])] + 3*ArcTanh[a*x]*PolyLog[4, -E^(-2*ArcTanh[a*x] 
)] + (3*PolyLog[5, -E^(-2*ArcTanh[a*x])])/2)/(a*c))
 

Rubi [A] (verified)

Time = 0.81 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.04, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {6470, 6620, 6624, 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.

\(\displaystyle \int \frac {\text {arctanh}(a x)^4}{c-a c x} \, dx\)

\(\Big \downarrow \) 6470

\(\displaystyle \frac {\text {arctanh}(a x)^4 \log \left (\frac {2}{1-a x}\right )}{a c}-\frac {4 \int \frac {\text {arctanh}(a x)^3 \log \left (\frac {2}{1-a x}\right )}{1-a^2 x^2}dx}{c}\)

\(\Big \downarrow \) 6620

\(\displaystyle \frac {\text {arctanh}(a x)^4 \log \left (\frac {2}{1-a x}\right )}{a c}-\frac {4 \left (\frac {3}{2} \int \frac {\text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{1-a^2 x^2}dx-\frac {\text {arctanh}(a x)^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{2 a}\right )}{c}\)

\(\Big \downarrow \) 6624

\(\displaystyle \frac {\text {arctanh}(a x)^4 \log \left (\frac {2}{1-a x}\right )}{a c}-\frac {4 \left (\frac {3}{2} \left (\frac {\text {arctanh}(a x)^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1-a x}\right )}{2 a}-\int \frac {\text {arctanh}(a x) \operatorname {PolyLog}\left (3,1-\frac {2}{1-a x}\right )}{1-a^2 x^2}dx\right )-\frac {\text {arctanh}(a x)^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{2 a}\right )}{c}\)

\(\Big \downarrow \) 6624

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

\(\Big \downarrow \) 7164

\(\displaystyle \frac {\text {arctanh}(a x)^4 \log \left (\frac {2}{1-a x}\right )}{a c}-\frac {4 \left (\frac {3}{2} \left (\frac {\text {arctanh}(a x)^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1-a x}\right )}{2 a}-\frac {\text {arctanh}(a x) \operatorname {PolyLog}\left (4,1-\frac {2}{1-a x}\right )}{2 a}+\frac {\operatorname {PolyLog}\left (5,1-\frac {2}{1-a x}\right )}{4 a}\right )-\frac {\text {arctanh}(a x)^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{2 a}\right )}{c}\)

Input:

Int[ArcTanh[a*x]^4/(c - a*c*x),x]
 

Output:

(ArcTanh[a*x]^4*Log[2/(1 - a*x)])/(a*c) - (4*(-1/2*(ArcTanh[a*x]^3*PolyLog 
[2, 1 - 2/(1 - a*x)])/a + (3*((ArcTanh[a*x]^2*PolyLog[3, 1 - 2/(1 - a*x)]) 
/(2*a) - (ArcTanh[a*x]*PolyLog[4, 1 - 2/(1 - a*x)])/(2*a) + PolyLog[5, 1 - 
 2/(1 - a*x)]/(4*a)))/2))/c
 

Defintions of rubi rules used

rule 6470

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

Maple [C] (warning: unable to verify)

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

Time = 0.75 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.74

method	result	size
derivativedivides	\(\frac {-\frac {\operatorname {arctanh}\left (a x \right )^{4} \ln \left (a x -1\right )}{c}+\frac {\left (-i \pi {\operatorname {csgn}\left (\frac {i}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right )}^{2}+i \pi {\operatorname {csgn}\left (\frac {i}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right )}^{3}+i \pi +\ln \left (2\right )\right ) \operatorname {arctanh}\left (a x \right )^{4}+2 \operatorname {arctanh}\left (a x \right )^{3} \operatorname {polylog}\left (2, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )-3 \operatorname {arctanh}\left (a x \right )^{2} \operatorname {polylog}\left (3, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )+3 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (4, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )-\frac {3 \operatorname {polylog}\left (5, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{2}}{c}}{a}\)	\(228\)
default	\(\frac {-\frac {\operatorname {arctanh}\left (a x \right )^{4} \ln \left (a x -1\right )}{c}+\frac {\left (-i \pi {\operatorname {csgn}\left (\frac {i}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right )}^{2}+i \pi {\operatorname {csgn}\left (\frac {i}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right )}^{3}+i \pi +\ln \left (2\right )\right ) \operatorname {arctanh}\left (a x \right )^{4}+2 \operatorname {arctanh}\left (a x \right )^{3} \operatorname {polylog}\left (2, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )-3 \operatorname {arctanh}\left (a x \right )^{2} \operatorname {polylog}\left (3, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )+3 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (4, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )-\frac {3 \operatorname {polylog}\left (5, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{2}}{c}}{a}\)	\(228\)
parts	\(-\frac {\ln \left (a x -1\right ) \operatorname {arctanh}\left (a x \right )^{4}}{a c}+\frac {\frac {\left (-i \pi {\operatorname {csgn}\left (\frac {i}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right )}^{2}+i \pi {\operatorname {csgn}\left (\frac {i}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right )}^{3}+i \pi +\ln \left (2\right )\right ) \operatorname {arctanh}\left (a x \right )^{4}}{a}+\frac {2 \operatorname {arctanh}\left (a x \right )^{3} \operatorname {polylog}\left (2, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{a}-\frac {3 \operatorname {arctanh}\left (a x \right )^{2} \operatorname {polylog}\left (3, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{a}+\frac {3 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (4, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{a}-\frac {3 \operatorname {polylog}\left (5, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{2 a}}{c}\)	\(242\)

Input:

int(arctanh(a*x)^4/(-a*c*x+c),x,method=_RETURNVERBOSE)
 

Output:

1/a*(-1/c*arctanh(a*x)^4*ln(a*x-1)+4/c*(1/4*(-I*Pi*csgn(I/(-(a*x+1)^2/(a^2 
*x^2-1)+1))^2+I*Pi*csgn(I/(-(a*x+1)^2/(a^2*x^2-1)+1))^3+I*Pi+ln(2))*arctan 
h(a*x)^4+1/2*arctanh(a*x)^3*polylog(2,-(a*x+1)^2/(-a^2*x^2+1))-3/4*arctanh 
(a*x)^2*polylog(3,-(a*x+1)^2/(-a^2*x^2+1))+3/4*arctanh(a*x)*polylog(4,-(a* 
x+1)^2/(-a^2*x^2+1))-3/8*polylog(5,-(a*x+1)^2/(-a^2*x^2+1))))
 

Fricas [F]

\[ \int \frac {\text {arctanh}(a x)^4}{c-a c x} \, dx=\int { -\frac {\operatorname {artanh}\left (a x\right )^{4}}{a c x - c} \,d x } \] Input:

integrate(arctanh(a*x)^4/(-a*c*x+c),x, algorithm="fricas")
 

Output:

integral(-arctanh(a*x)^4/(a*c*x - c), x)
 

Sympy [F]

\[ \int \frac {\text {arctanh}(a x)^4}{c-a c x} \, dx=- \frac {\int \frac {\operatorname {atanh}^{4}{\left (a x \right )}}{a x - 1}\, dx}{c} \] Input:

integrate(atanh(a*x)**4/(-a*c*x+c),x)
 

Output:

-Integral(atanh(a*x)**4/(a*x - 1), x)/c
 

Maxima [F]

\[ \int \frac {\text {arctanh}(a x)^4}{c-a c x} \, dx=\int { -\frac {\operatorname {artanh}\left (a x\right )^{4}}{a c x - c} \,d x } \] Input:

integrate(arctanh(a*x)^4/(-a*c*x+c),x, algorithm="maxima")
 

Output:

-1/80*log(-a*x + 1)^5/(a*c) + 1/16*integrate(-(log(a*x + 1)^4 - 4*log(a*x 
+ 1)^3*log(-a*x + 1) + 6*log(a*x + 1)^2*log(-a*x + 1)^2 - 4*log(a*x + 1)*l 
og(-a*x + 1)^3)/(a*c*x - c), x)
 

Giac [F]

\[ \int \frac {\text {arctanh}(a x)^4}{c-a c x} \, dx=\int { -\frac {\operatorname {artanh}\left (a x\right )^{4}}{a c x - c} \,d x } \] Input:

integrate(arctanh(a*x)^4/(-a*c*x+c),x, algorithm="giac")
 

Output:

integrate(-arctanh(a*x)^4/(a*c*x - c), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\text {arctanh}(a x)^4}{c-a c x} \, dx=\int \frac {{\mathrm {atanh}\left (a\,x\right )}^4}{c-a\,c\,x} \,d x \] Input:

int(atanh(a*x)^4/(c - a*c*x),x)
 

Output:

int(atanh(a*x)^4/(c - a*c*x), x)
 

Reduce [F]

\[ \int \frac {\text {arctanh}(a x)^4}{c-a c x} \, dx=-\frac {\int \frac {\mathit {atanh} \left (a x \right )^{4}}{a x -1}d x}{c} \] Input:

int(atanh(a*x)^4/(-a*c*x+c),x)
                                                                                    
                                                                                    
 

Output:

( - int(atanh(a*x)**4/(a*x - 1),x))/c