\(\int \frac {x \text {arctanh}(a x)^4}{c-a c x} \, dx\) [135]

Optimal result

Integrand size = 17, antiderivative size = 261 \[ \int \frac {x \text {arctanh}(a x)^4}{c-a c x} \, dx=-\frac {\text {arctanh}(a x)^4}{a^2 c}-\frac {x \text {arctanh}(a x)^4}{a c}+\frac {4 \text {arctanh}(a x)^3 \log \left (\frac {2}{1-a x}\right )}{a^2 c}+\frac {\text {arctanh}(a x)^4 \log \left (\frac {2}{1-a x}\right )}{a^2 c}+\frac {6 \text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{a^2 c}+\frac {2 \text {arctanh}(a x)^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{a^2 c}-\frac {6 \text {arctanh}(a x) \operatorname {PolyLog}\left (3,1-\frac {2}{1-a x}\right )}{a^2 c}-\frac {3 \text {arctanh}(a x)^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1-a x}\right )}{a^2 c}+\frac {3 \operatorname {PolyLog}\left (4,1-\frac {2}{1-a x}\right )}{a^2 c}+\frac {3 \text {arctanh}(a x) \operatorname {PolyLog}\left (4,1-\frac {2}{1-a x}\right )}{a^2 c}-\frac {3 \operatorname {PolyLog}\left (5,1-\frac {2}{1-a x}\right )}{2 a^2 c} \] Output:

-arctanh(a*x)^4/a^2/c-x*arctanh(a*x)^4/a/c+4*arctanh(a*x)^3*ln(2/(-a*x+1)) 
/a^2/c+arctanh(a*x)^4*ln(2/(-a*x+1))/a^2/c+6*arctanh(a*x)^2*polylog(2,1-2/ 
(-a*x+1))/a^2/c+2*arctanh(a*x)^3*polylog(2,1-2/(-a*x+1))/a^2/c-6*arctanh(a 
*x)*polylog(3,1-2/(-a*x+1))/a^2/c-3*arctanh(a*x)^2*polylog(3,1-2/(-a*x+1)) 
/a^2/c+3*polylog(4,1-2/(-a*x+1))/a^2/c+3*arctanh(a*x)*polylog(4,1-2/(-a*x+ 
1))/a^2/c-3/2*polylog(5,1-2/(-a*x+1))/a^2/c
 

Mathematica [A] (verified)

Time = 0.27 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.66 \[ \int \frac {x \text {arctanh}(a x)^4}{c-a c x} \, dx=-\frac {-\text {arctanh}(a x)^4+a x \text {arctanh}(a x)^4-\frac {2}{5} \text {arctanh}(a x)^5-4 \text {arctanh}(a x)^3 \log \left (1+e^{-2 \text {arctanh}(a x)}\right )-\text {arctanh}(a x)^4 \log \left (1+e^{-2 \text {arctanh}(a x)}\right )+2 \text {arctanh}(a x)^2 (3+\text {arctanh}(a x)) \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(a x)}\right )+3 \text {arctanh}(a x) (2+\text {arctanh}(a x)) \operatorname {PolyLog}\left (3,-e^{-2 \text {arctanh}(a x)}\right )+3 \operatorname {PolyLog}\left (4,-e^{-2 \text {arctanh}(a x)}\right )+3 \text {arctanh}(a x) \operatorname {PolyLog}\left (4,-e^{-2 \text {arctanh}(a x)}\right )+\frac {3}{2} \operatorname {PolyLog}\left (5,-e^{-2 \text {arctanh}(a x)}\right )}{a^2 c} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 1.98 (sec) , antiderivative size = 274, normalized size of antiderivative = 1.05, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.588, Rules used = {6492, 27, 6436, 6470, 6546, 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.

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

\(\Big \downarrow \) 6492

\(\displaystyle \frac {\int \frac {\text {arctanh}(a x)^4}{c (1-a x)}dx}{a}-\frac {\int \text {arctanh}(a x)^4dx}{a c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {\text {arctanh}(a x)^4}{1-a x}dx}{a c}-\frac {\int \text {arctanh}(a x)^4dx}{a c}\)

\(\Big \downarrow \) 6436

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

\(\Big \downarrow \) 6470

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

\(\Big \downarrow \) 6546

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

\(\Big \downarrow \) 6470

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

\(\Big \downarrow \) 6620

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

\(\Big \downarrow \) 6624

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

\(\Big \downarrow \) 6624

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

\(\Big \downarrow \) 7164

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 6436

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

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 6492

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

rule 6546

Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*(x_))/((d_) + (e_.)*(x_)^2), 
 x_Symbol] :> Simp[(a + b*ArcTanh[c*x])^(p + 1)/(b*e*(p + 1)), x] + Simp[1/ 
(c*d)   Int[(a + b*ArcTanh[c*x])^p/(1 - c*x), x], x] /; FreeQ[{a, b, c, d, 
e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 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 = 1.24 (sec) , antiderivative size = 381, normalized size of antiderivative = 1.46

method	result	size
derivativedivides	\(\frac {-\frac {\operatorname {arctanh}\left (a x \right )^{4} a x}{c}-\frac {\operatorname {arctanh}\left (a x \right )^{4} \ln \left (a x -1\right )}{c}+\frac {2 \operatorname {arctanh}\left (a x \right )^{3} \operatorname {polylog}\left (2, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )-3 \operatorname {arctanh}\left (a x \right )^{2} \operatorname {polylog}\left (3, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )+3 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (4, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )-\frac {3 \operatorname {polylog}\left (5, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{2}+i \pi \operatorname {arctanh}\left (a x \right )^{4}-i \pi {\operatorname {csgn}\left (\frac {i}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right )}^{2} \operatorname {arctanh}\left (a x \right )^{4}+i \pi {\operatorname {csgn}\left (\frac {i}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right )}^{3} \operatorname {arctanh}\left (a x \right )^{4}+\ln \left (2\right ) \operatorname {arctanh}\left (a x \right )^{4}-\operatorname {arctanh}\left (a x \right )^{4}+3 \operatorname {polylog}\left (4, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )+4 \operatorname {arctanh}\left (a x \right )^{3} \ln \left (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}+1\right )+6 \operatorname {arctanh}\left (a x \right )^{2} \operatorname {polylog}\left (2, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )-6 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (3, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{c}}{a^{2}}\)	\(381\)
default	\(\frac {-\frac {\operatorname {arctanh}\left (a x \right )^{4} a x}{c}-\frac {\operatorname {arctanh}\left (a x \right )^{4} \ln \left (a x -1\right )}{c}+\frac {2 \operatorname {arctanh}\left (a x \right )^{3} \operatorname {polylog}\left (2, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )-3 \operatorname {arctanh}\left (a x \right )^{2} \operatorname {polylog}\left (3, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )+3 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (4, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )-\frac {3 \operatorname {polylog}\left (5, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{2}+i \pi \operatorname {arctanh}\left (a x \right )^{4}-i \pi {\operatorname {csgn}\left (\frac {i}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right )}^{2} \operatorname {arctanh}\left (a x \right )^{4}+i \pi {\operatorname {csgn}\left (\frac {i}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right )}^{3} \operatorname {arctanh}\left (a x \right )^{4}+\ln \left (2\right ) \operatorname {arctanh}\left (a x \right )^{4}-\operatorname {arctanh}\left (a x \right )^{4}+3 \operatorname {polylog}\left (4, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )+4 \operatorname {arctanh}\left (a x \right )^{3} \ln \left (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}+1\right )+6 \operatorname {arctanh}\left (a x \right )^{2} \operatorname {polylog}\left (2, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )-6 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (3, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{c}}{a^{2}}\)	\(381\)
parts	\(-\frac {x \operatorname {arctanh}\left (a x \right )^{4}}{a c}-\frac {\operatorname {arctanh}\left (a x \right )^{4} \ln \left (a x -1\right )}{c \,a^{2}}+\frac {4 a \left (\frac {\operatorname {arctanh}\left (a x \right )^{3} \operatorname {polylog}\left (2, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{2 a^{3}}-\frac {3 \operatorname {arctanh}\left (a x \right )^{2} \operatorname {polylog}\left (3, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{4 a^{3}}+\frac {3 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (4, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{4 a^{3}}-\frac {3 \operatorname {polylog}\left (5, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{8 a^{3}}+\frac {\operatorname {arctanh}\left (a x \right )^{3} \ln \left (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}+1\right )}{a^{3}}+\frac {3 \operatorname {arctanh}\left (a x \right )^{2} \operatorname {polylog}\left (2, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{2 a^{3}}-\frac {3 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (3, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{2 a^{3}}+\frac {i \pi \operatorname {arctanh}\left (a x \right )^{4}}{4 a^{3}}-\frac {i \pi {\operatorname {csgn}\left (\frac {i}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right )}^{2} \operatorname {arctanh}\left (a x \right )^{4}}{4 a^{3}}+\frac {\ln \left (2\right ) \operatorname {arctanh}\left (a x \right )^{4}}{4 a^{3}}+\frac {i \pi {\operatorname {csgn}\left (\frac {i}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right )}^{3} \operatorname {arctanh}\left (a x \right )^{4}}{4 a^{3}}-\frac {\operatorname {arctanh}\left (a x \right )^{4}}{4 a^{3}}+\frac {3 \operatorname {polylog}\left (4, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{4 a^{3}}\right )}{c}\)	\(422\)

Input:

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

Output:

1/a^2*(-1/c*arctanh(a*x)^4*a*x-1/c*arctanh(a*x)^4*ln(a*x-1)+4/c*(1/2*arcta 
nh(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))+1/4*I*Pi*arctanh(a*x)^4-1/4*I*Pi 
*csgn(I/(-(a*x+1)^2/(a^2*x^2-1)+1))^2*arctanh(a*x)^4+1/4*I*Pi*csgn(I/(-(a* 
x+1)^2/(a^2*x^2-1)+1))^3*arctanh(a*x)^4+1/4*ln(2)*arctanh(a*x)^4-1/4*arcta 
nh(a*x)^4+3/4*polylog(4,-(a*x+1)^2/(-a^2*x^2+1))+arctanh(a*x)^3*ln((a*x+1) 
^2/(-a^2*x^2+1)+1)+3/2*arctanh(a*x)^2*polylog(2,-(a*x+1)^2/(-a^2*x^2+1))-3 
/2*arctanh(a*x)*polylog(3,-(a*x+1)^2/(-a^2*x^2+1))))
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

-1/80*(log(-a*x + 1)^5 + 5*(log(-a*x + 1)^4 - 4*log(-a*x + 1)^3 + 12*log(- 
a*x + 1)^2 - 24*log(-a*x + 1) + 24)*(a*x - 1))/(a^2*c) + 1/16*integrate(-( 
x*log(a*x + 1)^4 - 4*x*log(a*x + 1)^3*log(-a*x + 1) + 6*x*log(a*x + 1)^2*l 
og(-a*x + 1)^2 - 4*x*log(a*x + 1)*log(-a*x + 1)^3)/(a*c*x - c), x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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