\(\int \frac {\text {arctanh}(1+b x)^2}{x} \, dx\) [8]

Optimal result

Integrand size = 12, antiderivative size = 56 \[ \int \frac {\text {arctanh}(1+b x)^2}{x} \, dx=-\text {arctanh}(1+b x)^2 \log \left (-\frac {2}{b x}\right )-\text {arctanh}(1+b x) \operatorname {PolyLog}\left (2,1+\frac {2}{b x}\right )+\frac {1}{2} \operatorname {PolyLog}\left (3,1+\frac {2}{b x}\right ) \] Output:

-arctanh(b*x+1)^2*ln(-2/b/x)-arctanh(b*x+1)*polylog(2,1+2/b/x)+1/2*polylog 
(3,1+2/b/x)
 

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.34 \[ \int \frac {\text {arctanh}(1+b x)^2}{x} \, dx=-\frac {2}{3} \text {arctanh}(1+b x)^3-\text {arctanh}(1+b x)^2 \log \left (1+e^{-2 \text {arctanh}(1+b x)}\right )+\text {arctanh}(1+b x) \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(1+b x)}\right )+\frac {1}{2} \operatorname {PolyLog}\left (3,-e^{-2 \text {arctanh}(1+b x)}\right ) \] Input:

Integrate[ArcTanh[1 + b*x]^2/x,x]
 

Output:

(-2*ArcTanh[1 + b*x]^3)/3 - ArcTanh[1 + b*x]^2*Log[1 + E^(-2*ArcTanh[1 + b 
*x])] + ArcTanh[1 + b*x]*PolyLog[2, -E^(-2*ArcTanh[1 + b*x])] + PolyLog[3, 
 -E^(-2*ArcTanh[1 + b*x])]/2
 

Rubi [A] (verified)

Time = 0.52 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.09, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6661, 25, 27, 6470, 6620, 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.

\(\displaystyle \int \frac {\text {arctanh}(b x+1)^2}{x} \, dx\)

\(\Big \downarrow \) 6661

\(\displaystyle \frac {\int \frac {\text {arctanh}(b x+1)^2}{x}d(b x+1)}{b}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\int -\frac {\text {arctanh}(b x+1)^2}{x}d(b x+1)}{b}\)

\(\Big \downarrow \) 27

\(\displaystyle -\int -\frac {\text {arctanh}(b x+1)^2}{b x}d(b x+1)\)

\(\Big \downarrow \) 6470

\(\displaystyle 2 \int \frac {\text {arctanh}(b x+1) \log \left (-\frac {2}{b x}\right )}{1-(b x+1)^2}d(b x+1)-\text {arctanh}(b x+1)^2 \log \left (-\frac {2}{b x}\right )\)

\(\Big \downarrow \) 6620

\(\displaystyle 2 \left (\frac {1}{2} \int \frac {\operatorname {PolyLog}\left (2,1+\frac {2}{b x}\right )}{1-(b x+1)^2}d(b x+1)-\frac {1}{2} \text {arctanh}(b x+1) \operatorname {PolyLog}\left (2,1+\frac {2}{b x}\right )\right )-\text {arctanh}(b x+1)^2 \log \left (-\frac {2}{b x}\right )\)

\(\Big \downarrow \) 7164

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

Input:

Int[ArcTanh[1 + b*x]^2/x,x]
 

Output:

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

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 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 6661

Int[((a_.) + ArcTanh[(c_) + (d_.)*(x_)]*(b_.))^(p_.)*((e_.) + (f_.)*(x_))^( 
m_.), x_Symbol] :> Simp[1/d   Subst[Int[((d*e - c*f)/d + f*(x/d))^m*(a + b* 
ArcTanh[x])^p, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IG 
tQ[p, 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 = 9.16 (sec) , antiderivative size = 158, normalized size of antiderivative = 2.82

method	result	size
derivativedivides	\(\ln \left (b x \right ) \operatorname {arctanh}\left (b x +1\right )^{2}-\operatorname {arctanh}\left (b x +1\right ) \operatorname {polylog}\left (2, -\frac {\left (b x +2\right )^{2}}{-\left (b x +1\right )^{2}+1}\right )+\frac {\operatorname {polylog}\left (3, -\frac {\left (b x +2\right )^{2}}{-\left (b x +1\right )^{2}+1}\right )}{2}-\left (-i \pi {\operatorname {csgn}\left (\frac {i}{1-\frac {\left (b x +2\right )^{2}}{\left (b x +1\right )^{2}-1}}\right )}^{2}+i \pi {\operatorname {csgn}\left (\frac {i}{1-\frac {\left (b x +2\right )^{2}}{\left (b x +1\right )^{2}-1}}\right )}^{3}+i \pi +\ln \left (2\right )\right ) \operatorname {arctanh}\left (b x +1\right )^{2}\)	\(158\)
default	\(\ln \left (b x \right ) \operatorname {arctanh}\left (b x +1\right )^{2}-\operatorname {arctanh}\left (b x +1\right ) \operatorname {polylog}\left (2, -\frac {\left (b x +2\right )^{2}}{-\left (b x +1\right )^{2}+1}\right )+\frac {\operatorname {polylog}\left (3, -\frac {\left (b x +2\right )^{2}}{-\left (b x +1\right )^{2}+1}\right )}{2}-\left (-i \pi {\operatorname {csgn}\left (\frac {i}{1-\frac {\left (b x +2\right )^{2}}{\left (b x +1\right )^{2}-1}}\right )}^{2}+i \pi {\operatorname {csgn}\left (\frac {i}{1-\frac {\left (b x +2\right )^{2}}{\left (b x +1\right )^{2}-1}}\right )}^{3}+i \pi +\ln \left (2\right )\right ) \operatorname {arctanh}\left (b x +1\right )^{2}\)	\(158\)
parts	\(\ln \left (x \right ) \operatorname {arctanh}\left (b x +1\right )^{2}-2 b \left (\frac {\operatorname {arctanh}\left (b x +1\right ) \operatorname {polylog}\left (2, -\frac {\left (b x +2\right )^{2}}{-\left (b x +1\right )^{2}+1}\right )}{2 b}-\frac {\operatorname {polylog}\left (3, -\frac {\left (b x +2\right )^{2}}{-\left (b x +1\right )^{2}+1}\right )}{4 b}-\frac {\left (-i \pi \,\operatorname {csgn}\left (\frac {i}{b}\right ) {\operatorname {csgn}\left (\frac {i}{\left (1-\frac {\left (b x +2\right )^{2}}{\left (b x +1\right )^{2}-1}\right ) b}\right )}^{2}+i \pi \,\operatorname {csgn}\left (\frac {i}{b}\right ) \operatorname {csgn}\left (\frac {i}{1-\frac {\left (b x +2\right )^{2}}{\left (b x +1\right )^{2}-1}}\right ) \operatorname {csgn}\left (\frac {i}{\left (1-\frac {\left (b x +2\right )^{2}}{\left (b x +1\right )^{2}-1}\right ) b}\right )-i \pi {\operatorname {csgn}\left (\frac {i}{\left (1-\frac {\left (b x +2\right )^{2}}{\left (b x +1\right )^{2}-1}\right ) b}\right )}^{3}-i \pi \,\operatorname {csgn}\left (\frac {i}{1-\frac {\left (b x +2\right )^{2}}{\left (b x +1\right )^{2}-1}}\right ) {\operatorname {csgn}\left (\frac {i}{\left (1-\frac {\left (b x +2\right )^{2}}{\left (b x +1\right )^{2}-1}\right ) b}\right )}^{2}+2 i \pi {\operatorname {csgn}\left (\frac {i}{\left (1-\frac {\left (b x +2\right )^{2}}{\left (b x +1\right )^{2}-1}\right ) b}\right )}^{2}-2 i \pi -2 \ln \left (2\right )+2 \ln \left (b \right )\right ) \operatorname {arctanh}\left (b x +1\right )^{2}}{4 b}\right )\)	\(360\)
risch	\(\text {Expression too large to display}\)	\(2291\)

Input:

int(arctanh(b*x+1)^2/x,x,method=_RETURNVERBOSE)
 

Output:

ln(b*x)*arctanh(b*x+1)^2-arctanh(b*x+1)*polylog(2,-(b*x+2)^2/(-(b*x+1)^2+1 
))+1/2*polylog(3,-(b*x+2)^2/(-(b*x+1)^2+1))-(-I*Pi*csgn(I/(1-(b*x+2)^2/((b 
*x+1)^2-1)))^2+I*Pi*csgn(I/(1-(b*x+2)^2/((b*x+1)^2-1)))^3+I*Pi+ln(2))*arct 
anh(b*x+1)^2
 

Fricas [F]

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

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

Output:

integral(arctanh(b*x + 1)^2/x, x)
 

Sympy [F]

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

integrate(atanh(b*x+1)**2/x,x)
 

Output:

Integral(atanh(b*x + 1)**2/x, x)
 

Maxima [F]

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

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

Output:

1/12*log(-b*x)^3 + 1/4*log(b*x + 2)^2*log(-x) - 1/4*integrate(2*(b*x*log(b 
) + 2*(b*x + 1)*log(-x) + 2*log(b))*log(b*x + 2)/(b*x^2 + 2*x), x)
 

Giac [F]

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

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

Output:

integrate(arctanh(b*x + 1)^2/x, x)
 

Mupad [F(-1)]

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

int(atanh(b*x + 1)^2/x,x)
 

Output:

int(atanh(b*x + 1)^2/x, x)
                                                                                    
                                                                                    
 

Reduce [F]

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

int(atanh(b*x+1)^2/x,x)
 

Output:

int(atanh(b*x + 1)**2/x,x)