\(\int \frac {\text {arctanh}(a x)^3}{x^2 (c+a c x)} \, dx\) [132]

Optimal result

Integrand size = 18, antiderivative size = 191 \[ \int \frac {\text {arctanh}(a x)^3}{x^2 (c+a c x)} \, dx=\frac {a \text {arctanh}(a x)^3}{c}-\frac {\text {arctanh}(a x)^3}{c x}+\frac {3 a \text {arctanh}(a x)^2 \log \left (2-\frac {2}{1+a x}\right )}{c}-\frac {a \text {arctanh}(a x)^3 \log \left (2-\frac {2}{1+a x}\right )}{c}-\frac {3 a \text {arctanh}(a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+a x}\right )}{c}+\frac {3 a \text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1+a x}\right )}{2 c}-\frac {3 a \operatorname {PolyLog}\left (3,-1+\frac {2}{1+a x}\right )}{2 c}+\frac {3 a \text {arctanh}(a x) \operatorname {PolyLog}\left (3,-1+\frac {2}{1+a x}\right )}{2 c}+\frac {3 a \operatorname {PolyLog}\left (4,-1+\frac {2}{1+a x}\right )}{4 c} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.27 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.81 \[ \int \frac {\text {arctanh}(a x)^3}{x^2 (c+a c x)} \, dx=\frac {a \left (\frac {i \pi ^3}{8}-\frac {\pi ^4}{64}-\text {arctanh}(a x)^3-\frac {\text {arctanh}(a x)^3}{a x}+\frac {1}{2} \text {arctanh}(a x)^4+3 \text {arctanh}(a x)^2 \log \left (1-e^{2 \text {arctanh}(a x)}\right )-\text {arctanh}(a x)^3 \log \left (1-e^{2 \text {arctanh}(a x)}\right )-\frac {3}{2} (-2+\text {arctanh}(a x)) \text {arctanh}(a x) \operatorname {PolyLog}\left (2,e^{2 \text {arctanh}(a x)}\right )+\frac {3}{2} (-1+\text {arctanh}(a x)) \operatorname {PolyLog}\left (3,e^{2 \text {arctanh}(a x)}\right )-\frac {3}{4} \operatorname {PolyLog}\left (4,e^{2 \text {arctanh}(a x)}\right )\right )}{c} \] Input:

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

Output:

(a*((I/8)*Pi^3 - Pi^4/64 - ArcTanh[a*x]^3 - ArcTanh[a*x]^3/(a*x) + ArcTanh 
[a*x]^4/2 + 3*ArcTanh[a*x]^2*Log[1 - E^(2*ArcTanh[a*x])] - ArcTanh[a*x]^3* 
Log[1 - E^(2*ArcTanh[a*x])] - (3*(-2 + ArcTanh[a*x])*ArcTanh[a*x]*PolyLog[ 
2, E^(2*ArcTanh[a*x])])/2 + (3*(-1 + ArcTanh[a*x])*PolyLog[3, E^(2*ArcTanh 
[a*x])])/2 - (3*PolyLog[4, E^(2*ArcTanh[a*x])])/4))/c
 

Rubi [A] (verified)

Time = 1.60 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.04, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6496, 27, 6452, 6494, 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.

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

\(\Big \downarrow \) 6496

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 6452

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

\(\Big \downarrow \) 6494

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

\(\Big \downarrow \) 6550

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

\(\Big \downarrow \) 6494

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

\(\Big \downarrow \) 6618

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

\(\Big \downarrow \) 6622

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

\(\Big \downarrow \) 7164

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

Input:

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

Output:

(-(ArcTanh[a*x]^3/x) + 3*a*(ArcTanh[a*x]^3/3 + ArcTanh[a*x]^2*Log[2 - 2/(1 
 + a*x)] - 2*a*((ArcTanh[a*x]*PolyLog[2, -1 + 2/(1 + a*x)])/(2*a) + PolyLo 
g[3, -1 + 2/(1 + a*x)]/(4*a))))/c - (a*(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 
))))/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 6452

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

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 6496

Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + ( 
e_.)*(x_)), x_Symbol] :> Simp[1/d   Int[(f*x)^m*(a + b*ArcTanh[c*x])^p, x], 
 x] - Simp[e/(d*f)   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] && LtQ[m, -1]
 

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 = 3.66 (sec) , antiderivative size = 1339, normalized size of antiderivative = 7.01

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1339\)
default	\(\text {Expression too large to display}\)	\(1339\)
parts	\(\text {Expression too large to display}\)	\(1717\)

Input:

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

Output:

a*(-1/c*arctanh(a*x)^3/a/x-1/c*arctanh(a*x)^3*ln(a*x)+1/c*arctanh(a*x)^3*l 
n(a*x+1)-3/c*(-1/6*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))*ar 
ctanh(a*x)^3+1/6*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))*arctanh(a*x)^3-1/6*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*arctanh(a*x)^3+1/6*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*arctanh(a*x)^3+2*polylog(4,(a*x+1)/(-a^2*x^2+1)^(1/2))+2*po 
lylog(4,-(a*x+1)/(-a^2*x^2+1)^(1/2))+2/3*arctanh(a*x)^3*ln((a*x+1)/(-a^2*x 
^2+1)^(1/2))+2*polylog(3,-(a*x+1)/(-a^2*x^2+1)^(1/2))+2*polylog(3,(a*x+1)/ 
(-a^2*x^2+1)^(1/2))+1/3*ln(2)*arctanh(a*x)^3-1/6*arctanh(a*x)^4+1/3*arctan 
h(a*x)^3-2*arctanh(a*x)*polylog(2,-(a*x+1)/(-a^2*x^2+1)^(1/2))-2*arctanh(a 
*x)*polylog(2,(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))+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*arctanh(a*x)*poly 
log(3,(a*x+1)/(-a^2*x^2+1)^(1/2))-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))+1/3*arctanh(a* 
x)^3*ln(1-(a*x+1)/(-a^2*x^2+1)^(1/2))-arctanh(a*x)^2*ln(1-(a*x+1)/(-a^2*x^ 
2+1)^(1/2))-arctanh(a*x)^2*ln(1+(a*x+1)/(-a^2*x^2+1)^(1/2))+1/6*I*Pi*cs...
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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