\(\int \frac {\text {arctanh}(a x)^3}{c x+a c x^2} \, dx\) [131]

Optimal result

Integrand size = 19, antiderivative size = 93 \[ \int \frac {\text {arctanh}(a x)^3}{c x+a c x^2} \, dx=\frac {\text {arctanh}(a x)^3 \log \left (2-\frac {2}{1+a x}\right )}{c}-\frac {3 \text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1+a x}\right )}{2 c}-\frac {3 \text {arctanh}(a x) \operatorname {PolyLog}\left (3,-1+\frac {2}{1+a x}\right )}{2 c}-\frac {3 \operatorname {PolyLog}\left (4,-1+\frac {2}{1+a x}\right )}{4 c} \] Output:

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

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.92 \[ \int \frac {\text {arctanh}(a x)^3}{c x+a c x^2} \, dx=\frac {\pi ^4-32 \text {arctanh}(a x)^4+64 \text {arctanh}(a x)^3 \log \left (1-e^{2 \text {arctanh}(a x)}\right )+96 \text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,e^{2 \text {arctanh}(a x)}\right )-96 \text {arctanh}(a x) \operatorname {PolyLog}\left (3,e^{2 \text {arctanh}(a x)}\right )+48 \operatorname {PolyLog}\left (4,e^{2 \text {arctanh}(a x)}\right )}{64 c} \] Input:

Integrate[ArcTanh[a*x]^3/(c*x + a*c*x^2),x]
 

Output:

(Pi^4 - 32*ArcTanh[a*x]^4 + 64*ArcTanh[a*x]^3*Log[1 - E^(2*ArcTanh[a*x])] 
+ 96*ArcTanh[a*x]^2*PolyLog[2, E^(2*ArcTanh[a*x])] - 96*ArcTanh[a*x]*PolyL 
og[3, E^(2*ArcTanh[a*x])] + 48*PolyLog[4, E^(2*ArcTanh[a*x])])/(64*c)
 

Rubi [A] (verified)

Time = 0.64 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.08, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {2026, 6494, 6618, 6622, 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.

\(\displaystyle \int \frac {\text {arctanh}(a x)^3}{a c x^2+c x} \, dx\)

\(\Big \downarrow \) 2026

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

\(\Big \downarrow \) 6494

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

\(\Big \downarrow \) 6618

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

\(\Big \downarrow \) 6622

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

\(\Big \downarrow \) 7164

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

Input:

Int[ArcTanh[a*x]^3/(c*x + a*c*x^2),x]
 

Output:

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

Defintions of rubi rules used

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 6494

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_))), x 
_Symbol] :> Simp[(a + b*ArcTanh[c*x])^p*(Log[2 - 2/(1 + e*(x/d))]/d), x] - 
Simp[b*c*(p/d)   Int[(a + b*ArcTanh[c*x])^(p - 1)*(Log[2 - 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 6618

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 6622

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] & 
& EqQ[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 = 0.32 (sec) , antiderivative size = 1101, normalized size of antiderivative = 11.84

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1101\)
default	\(\text {Expression too large to display}\)	\(1101\)
parts	\(\text {Expression too large to display}\)	\(1481\)

Input:

int(arctanh(a*x)^3/(a*c*x^2+c*x),x,method=_RETURNVERBOSE)
 

Output:

1/a*(a/c*arctanh(a*x)^3*ln(a*x)-a/c*arctanh(a*x)^3*ln(a*x+1)-3*a/c*(-2/3*a 
rctanh(a*x)^3*ln((a*x+1)/(-a^2*x^2+1)^(1/2))+1/6*arctanh(a*x)^4-1/6*(I*Pi* 
csgn(I*(a*x+1)^2/(a^2*x^2-1))^3+2*I*Pi*csgn(I*(a*x+1)/(-a^2*x^2+1)^(1/2))* 
csgn(I*(a*x+1)^2/(a^2*x^2-1))^2+I*Pi*csgn(I*(a*x+1)/(-a^2*x^2+1)^(1/2))^2* 
csgn(I*(a*x+1)^2/(a^2*x^2-1))-I*Pi*csgn(I*(a*x+1)^2/(a^2*x^2-1))*csgn(I*(a 
*x+1)^2/(a^2*x^2-1)/(-(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))*csgn(I*(a*x+1)^2/(a^2*x^2-1))*csgn(I*(a*x+1)^2/(a^2*x^2-1) 
/(-(a*x+1)^2/(a^2*x^2-1)+1))-I*Pi*csgn(I*(-(a*x+1)^2/(a^2*x^2-1)-1))*csgn( 
I*(-(a*x+1)^2/(a^2*x^2-1)-1)/(-(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))*csgn(I/(-(a*x+1)^2/(a^2*x^2-1)+1))*csgn(I*(-(a*x+ 
1)^2/(a^2*x^2-1)-1)/(-(a*x+1)^2/(a^2*x^2-1)+1))+I*Pi*csgn(I*(-(a*x+1)^2/(a 
^2*x^2-1)-1)/(-(a*x+1)^2/(a^2*x^2-1)+1))^3-I*Pi*csgn(I/(-(a*x+1)^2/(a^2*x^ 
2-1)+1))*csgn(I*(-(a*x+1)^2/(a^2*x^2-1)-1)/(-(a*x+1)^2/(a^2*x^2-1)+1))^2+I 
*Pi*csgn(I*(a*x+1)^2/(a^2*x^2-1)/(-(a*x+1)^2/(a^2*x^2-1)+1))^3+I*Pi*csgn(I 
/(-(a*x+1)^2/(a^2*x^2-1)+1))*csgn(I*(a*x+1)^2/(a^2*x^2-1)/(-(a*x+1)^2/(a^2 
*x^2-1)+1))^2+2*ln(2))*arctanh(a*x)^3+1/3*arctanh(a*x)^3*ln((a*x+1)^2/(-a^ 
2*x^2+1)-1)-1/3*arctanh(a*x)^3*ln(1+(a*x+1)/(-a^2*x^2+1)^(1/2))-arctanh(a* 
x)^2*polylog(2,-(a*x+1)/(-a^2*x^2+1)^(1/2))+2*arctanh(a*x)*polylog(3,-(a*x 
+1)/(-a^2*x^2+1)^(1/2))-2*polylog(4,-(a*x+1)/(-a^2*x^2+1)^(1/2))-1/3*arcta 
nh(a*x)^3*ln(1-(a*x+1)/(-a^2*x^2+1)^(1/2))-arctanh(a*x)^2*polylog(2,(a*...
 

Fricas [F]

\[ \int \frac {\text {arctanh}(a x)^3}{c x+a c x^2} \, dx=\int { \frac {\operatorname {artanh}\left (a x\right )^{3}}{a c x^{2} + c x} \,d x } \] Input:

integrate(arctanh(a*x)^3/(a*c*x^2+c*x),x, algorithm="fricas")
 

Output:

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

Sympy [F]

\[ \int \frac {\text {arctanh}(a x)^3}{c x+a c x^2} \, dx=\frac {\int \frac {\operatorname {atanh}^{3}{\left (a x \right )}}{a x^{2} + x}\, dx}{c} \] Input:

integrate(atanh(a*x)**3/(a*c*x**2+c*x),x)
 

Output:

Integral(atanh(a*x)**3/(a*x**2 + x), x)/c
 

Maxima [F]

\[ \int \frac {\text {arctanh}(a x)^3}{c x+a c x^2} \, dx=\int { \frac {\operatorname {artanh}\left (a x\right )^{3}}{a c x^{2} + c x} \,d x } \] Input:

integrate(arctanh(a*x)^3/(a*c*x^2+c*x),x, algorithm="maxima")
 

Output:

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

Giac [F]

\[ \int \frac {\text {arctanh}(a x)^3}{c x+a c x^2} \, dx=\int { \frac {\operatorname {artanh}\left (a x\right )^{3}}{a c x^{2} + c x} \,d x } \] Input:

integrate(arctanh(a*x)^3/(a*c*x^2+c*x),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\text {arctanh}(a x)^3}{c x+a c x^2} \, dx=\int \frac {{\mathrm {atanh}\left (a\,x\right )}^3}{a\,c\,x^2+c\,x} \,d x \] Input:

int(atanh(a*x)^3/(c*x + a*c*x^2),x)
 

Output:

int(atanh(a*x)^3/(c*x + a*c*x^2), x)
 

Reduce [F]

\[ \int \frac {\text {arctanh}(a x)^3}{c x+a c x^2} \, dx=\frac {-\mathit {atanh} \left (a x \right )^{4}-4 \left (\int \frac {\mathit {atanh} \left (a x \right )^{3}}{a^{2} x^{3}-x}d x \right )}{4 c} \] Input:

int(atanh(a*x)^3/(a*c*x^2+c*x),x)
 

Output:

( - atanh(a*x)**4 - 4*int(atanh(a*x)**3/(a**2*x**3 - x),x))/(4*c)