\(\int \frac {\text {arctanh}(a x)^3}{x (1-a^2 x^2)} \, dx\) [245]

Optimal result

Integrand size = 22, antiderivative size = 91 \[ \int \frac {\text {arctanh}(a x)^3}{x \left (1-a^2 x^2\right )} \, dx=\frac {1}{4} \text {arctanh}(a x)^4+\text {arctanh}(a x)^3 \log \left (2-\frac {2}{1+a x}\right )-\frac {3}{2} \text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1+a x}\right )-\frac {3}{2} \text {arctanh}(a x) \operatorname {PolyLog}\left (3,-1+\frac {2}{1+a x}\right )-\frac {3}{4} \operatorname {PolyLog}\left (4,-1+\frac {2}{1+a x}\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.38 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.91 \[ \int \frac {\text {arctanh}(a x)^3}{x \left (1-a^2 x^2\right )} \, dx=-\frac {1}{4} \text {arctanh}(a x)^4+\text {arctanh}(a x)^3 \log \left (1-e^{2 \text {arctanh}(a x)}\right )+\frac {3}{2} \text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,e^{2 \text {arctanh}(a x)}\right )-\frac {3}{2} \text {arctanh}(a x) \operatorname {PolyLog}\left (3,e^{2 \text {arctanh}(a x)}\right )+\frac {3}{4} \operatorname {PolyLog}\left (4,e^{2 \text {arctanh}(a x)}\right ) \] Input:

Integrate[ArcTanh[a*x]^3/(x*(1 - a^2*x^2)),x]
 

Output:

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

Rubi [A] (verified)

Time = 0.75 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.14, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {6550, 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.

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

\(\Big \downarrow \) 6550

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

\(\Big \downarrow \) 6494

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

\(\Big \downarrow \) 6618

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

\(\Big \downarrow \) 6622

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

\(\Big \downarrow \) 7164

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

Input:

Int[ArcTanh[a*x]^3/(x*(1 - a^2*x^2)),x]
 

Output:

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

Defintions of rubi rules used

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 6550

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_)^2)), 
 x_Symbol] :> Simp[(a + b*ArcTanh[c*x])^(p + 1)/(b*d*(p + 1)), x] + Simp[1/ 
d   Int[(a + b*ArcTanh[c*x])^p/(x*(1 + c*x)), x], x] /; FreeQ[{a, b, c, d, 
e}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 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 = 21.55 (sec) , antiderivative size = 1165, normalized size of antiderivative = 12.80

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1165\)
default	\(\text {Expression too large to display}\)	\(1165\)
parts	\(\text {Expression too large to display}\)	\(1558\)

Input:

int(arctanh(a*x)^3/x/(-a^2*x^2+1),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [F]

\[ \int \frac {\text {arctanh}(a x)^3}{x \left (1-a^2 x^2\right )} \, dx=\int { -\frac {\operatorname {artanh}\left (a x\right )^{3}}{{\left (a^{2} x^{2} - 1\right )} x} \,d x } \] Input:

integrate(arctanh(a*x)^3/x/(-a^2*x^2+1),x, algorithm="fricas")
 

Output:

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

Sympy [F]

\[ \int \frac {\text {arctanh}(a x)^3}{x \left (1-a^2 x^2\right )} \, dx=- \int \frac {\operatorname {atanh}^{3}{\left (a x \right )}}{a^{2} x^{3} - x}\, dx \] Input:

integrate(atanh(a*x)**3/x/(-a**2*x**2+1),x)
 

Output:

-Integral(atanh(a*x)**3/(a**2*x**3 - x), x)
 

Maxima [F]

\[ \int \frac {\text {arctanh}(a x)^3}{x \left (1-a^2 x^2\right )} \, dx=\int { -\frac {\operatorname {artanh}\left (a x\right )^{3}}{{\left (a^{2} x^{2} - 1\right )} x} \,d x } \] Input:

integrate(arctanh(a*x)^3/x/(-a^2*x^2+1),x, algorithm="maxima")
 

Output:

1/16*log(a*x + 1)*log(-a*x + 1)^3 + 1/64*log(-a*x + 1)^4 - 1/8*integrate(1 
/2*(3*(a^2*x^2 + a*x + 2)*log(a*x + 1)*log(-a*x + 1)^2 + 2*log(a*x + 1)^3 
- 6*log(a*x + 1)^2*log(-a*x + 1))/(a^2*x^3 - x), x)
 

Giac [F]

\[ \int \frac {\text {arctanh}(a x)^3}{x \left (1-a^2 x^2\right )} \, dx=\int { -\frac {\operatorname {artanh}\left (a x\right )^{3}}{{\left (a^{2} x^{2} - 1\right )} x} \,d x } \] Input:

integrate(arctanh(a*x)^3/x/(-a^2*x^2+1),x, algorithm="giac")
 

Output:

integrate(-arctanh(a*x)^3/((a^2*x^2 - 1)*x), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\text {arctanh}(a x)^3}{x \left (1-a^2 x^2\right )} \, dx=-\int \frac {{\mathrm {atanh}\left (a\,x\right )}^3}{x\,\left (a^2\,x^2-1\right )} \,d x \] Input:

int(-atanh(a*x)^3/(x*(a^2*x^2 - 1)),x)
 

Output:

-int(atanh(a*x)^3/(x*(a^2*x^2 - 1)), x)
 

Reduce [F]

\[ \int \frac {\text {arctanh}(a x)^3}{x \left (1-a^2 x^2\right )} \, dx=-\left (\int \frac {\mathit {atanh} \left (a x \right )^{3}}{a^{2} x^{3}-x}d x \right ) \] Input:

int(atanh(a*x)^3/x/(-a^2*x^2+1),x)
 

Output:

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