\(\int \frac {\coth ^{-1}(a+b x)^2}{x} \, dx\) [12]

Optimal result

Integrand size = 12, antiderivative size = 148 \[ \int \frac {\coth ^{-1}(a+b x)^2}{x} \, dx=-\coth ^{-1}(a+b x)^2 \log \left (\frac {2}{1+a+b x}\right )+\coth ^{-1}(a+b x)^2 \log \left (\frac {2 b x}{(1-a) (1+a+b x)}\right )+\coth ^{-1}(a+b x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+a+b x}\right )-\coth ^{-1}(a+b x) \operatorname {PolyLog}\left (2,1-\frac {2 b x}{(1-a) (1+a+b x)}\right )+\frac {1}{2} \operatorname {PolyLog}\left (3,1-\frac {2}{1+a+b x}\right )-\frac {1}{2} \operatorname {PolyLog}\left (3,1-\frac {2 b x}{(1-a) (1+a+b x)}\right ) \] Output:

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

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 2.25 (sec) , antiderivative size = 777, normalized size of antiderivative = 5.25 \[ \int \frac {\coth ^{-1}(a+b x)^2}{x} \, dx =\text {Too large to display} \] Input:

Integrate[ArcCoth[a + b*x]^2/x,x]
 

Output:

(-1/24*I)*Pi^3 - (2*ArcCoth[a + b*x]^3)/3 - (2*a*ArcCoth[a + b*x]^3)/3 + ( 
2*Sqrt[1 - a^(-2)]*a*E^ArcTanh[a^(-1)]*ArcCoth[a + b*x]^3)/3 - I*Pi*ArcCot 
h[a + b*x]*Log[(E^(-ArcCoth[a + b*x]) + E^ArcCoth[a + b*x])/2] - ArcCoth[a 
 + b*x]^2*Log[1 - Sqrt[(-1 + a)/(1 + a)]*E^ArcCoth[a + b*x]] - ArcCoth[a + 
 b*x]^2*Log[1 + Sqrt[(-1 + a)/(1 + a)]*E^ArcCoth[a + b*x]] - ArcCoth[a + b 
*x]^2*Log[1 - E^(2*ArcCoth[a + b*x])] + ArcCoth[a + b*x]^2*Log[1 - E^(2*Ar 
cCoth[a + b*x] - 2*ArcTanh[a^(-1)])] + ArcCoth[a + b*x]^2*Log[1 - E^(ArcCo 
th[a + b*x] - ArcTanh[a^(-1)])] + ArcCoth[a + b*x]^2*Log[1 + E^(ArcCoth[a 
+ b*x] - ArcTanh[a^(-1)])] - 2*ArcCoth[a + b*x]*ArcTanh[a^(-1)]*Log[(I/2)* 
(E^(ArcCoth[a + b*x] - ArcTanh[a^(-1)]) - E^(-ArcCoth[a + b*x] + ArcTanh[a 
^(-1)]))] + ArcCoth[a + b*x]^2*Log[(-1 - E^(2*ArcCoth[a + b*x]) + a*(-1 + 
E^(2*ArcCoth[a + b*x])))/(2*E^ArcCoth[a + b*x])] + I*Pi*ArcCoth[a + b*x]*L 
og[1/Sqrt[1 - (a + b*x)^(-2)]] - ArcCoth[a + b*x]^2*Log[-((b*x)/((a + b*x) 
*Sqrt[1 - (a + b*x)^(-2)]))] + 2*ArcCoth[a + b*x]*ArcTanh[a^(-1)]*Log[I*Si 
nh[ArcCoth[a + b*x] - ArcTanh[a^(-1)]]] - 2*ArcCoth[a + b*x]*PolyLog[2, -( 
Sqrt[(-1 + a)/(1 + a)]*E^ArcCoth[a + b*x])] - 2*ArcCoth[a + b*x]*PolyLog[2 
, Sqrt[(-1 + a)/(1 + a)]*E^ArcCoth[a + b*x]] - ArcCoth[a + b*x]*PolyLog[2, 
 E^(2*ArcCoth[a + b*x])] + ArcCoth[a + b*x]*PolyLog[2, E^(2*ArcCoth[a + b* 
x] - 2*ArcTanh[a^(-1)])] + 2*ArcCoth[a + b*x]*PolyLog[2, -E^(ArcCoth[a + b 
*x] - ArcTanh[a^(-1)])] + 2*ArcCoth[a + b*x]*PolyLog[2, E^(ArcCoth[a + ...
 

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6662, 25, 27, 6475}

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.

\(\displaystyle \int \frac {\coth ^{-1}(a+b x)^2}{x} \, dx\)

\(\Big \downarrow \) 6662

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 6475

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

Input:

Int[ArcCoth[a + b*x]^2/x,x]
 

Output:

-(ArcCoth[a + b*x]^2*Log[2/(1 + a + b*x)]) + ArcCoth[a + b*x]^2*Log[(2*b*x 
)/((1 - a)*(1 + a + b*x))] + ArcCoth[a + b*x]*PolyLog[2, 1 - 2/(1 + a + b* 
x)] - ArcCoth[a + b*x]*PolyLog[2, 1 - (2*b*x)/((1 - a)*(1 + a + b*x))] + P 
olyLog[3, 1 - 2/(1 + a + b*x)]/2 - PolyLog[3, 1 - (2*b*x)/((1 - a)*(1 + a 
+ b*x))]/2
 

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 6475

Int[((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^2/((d_) + (e_.)*(x_)), x_Symbol] :> 
 Simp[(-(a + b*ArcCoth[c*x])^2)*(Log[2/(1 + c*x)]/e), x] + (Simp[(a + b*Arc 
Coth[c*x])^2*(Log[2*c*((d + e*x)/((c*d + e)*(1 + c*x)))]/e), x] + Simp[b*(a 
 + b*ArcCoth[c*x])*(PolyLog[2, 1 - 2/(1 + c*x)]/e), x] - Simp[b*(a + b*ArcC 
oth[c*x])*(PolyLog[2, 1 - 2*c*((d + e*x)/((c*d + e)*(1 + c*x)))]/e), x] + S 
imp[b^2*(PolyLog[3, 1 - 2/(1 + c*x)]/(2*e)), x] - Simp[b^2*(PolyLog[3, 1 - 
2*c*((d + e*x)/((c*d + e)*(1 + c*x)))]/(2*e)), x]) /; FreeQ[{a, b, c, d, e} 
, x] && NeQ[c^2*d^2 - e^2, 0]
 

rule 6662

Int[((a_.) + ArcCoth[(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* 
ArcCoth[x])^p, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IG 
tQ[p, 0]
 

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 6.32 (sec) , antiderivative size = 866, normalized size of antiderivative = 5.85

method	result	size
derivativedivides	\(\ln \left (-b x \right ) \operatorname {arccoth}\left (b x +a \right )^{2}-\operatorname {arccoth}\left (b x +a \right )^{2} \ln \left (-\frac {b x +a +1}{b x +a -1}-1+a \left (\frac {b x +a +1}{b x +a -1}-1\right )\right )+\frac {i \pi \,\operatorname {csgn}\left (\frac {i \left (-\frac {b x +a +1}{b x +a -1}-1+a \left (\frac {b x +a +1}{b x +a -1}-1\right )\right )}{\frac {b x +a +1}{b x +a -1}-1}\right ) \left (\operatorname {csgn}\left (i \left (-\frac {b x +a +1}{b x +a -1}-1+a \left (\frac {b x +a +1}{b x +a -1}-1\right )\right )\right ) \operatorname {csgn}\left (\frac {i}{\frac {b x +a +1}{b x +a -1}-1}\right )-\operatorname {csgn}\left (\frac {i \left (-\frac {b x +a +1}{b x +a -1}-1+a \left (\frac {b x +a +1}{b x +a -1}-1\right )\right )}{\frac {b x +a +1}{b x +a -1}-1}\right ) \operatorname {csgn}\left (\frac {i}{\frac {b x +a +1}{b x +a -1}-1}\right )-\operatorname {csgn}\left (i \left (-\frac {b x +a +1}{b x +a -1}-1+a \left (\frac {b x +a +1}{b x +a -1}-1\right )\right )\right ) \operatorname {csgn}\left (\frac {i \left (-\frac {b x +a +1}{b x +a -1}-1+a \left (\frac {b x +a +1}{b x +a -1}-1\right )\right )}{\frac {b x +a +1}{b x +a -1}-1}\right )+\operatorname {csgn}\left (\frac {i \left (-\frac {b x +a +1}{b x +a -1}-1+a \left (\frac {b x +a +1}{b x +a -1}-1\right )\right )}{\frac {b x +a +1}{b x +a -1}-1}\right )^{2}\right ) \operatorname {arccoth}\left (b x +a \right )^{2}}{2}+\operatorname {arccoth}\left (b x +a \right )^{2} \ln \left (\frac {b x +a +1}{b x +a -1}-1\right )-\operatorname {arccoth}\left (b x +a \right )^{2} \ln \left (1+\frac {1}{\sqrt {\frac {b x +a -1}{b x +a +1}}}\right )-2 \,\operatorname {arccoth}\left (b x +a \right ) \operatorname {polylog}\left (2, -\frac {1}{\sqrt {\frac {b x +a -1}{b x +a +1}}}\right )+2 \operatorname {polylog}\left (3, -\frac {1}{\sqrt {\frac {b x +a -1}{b x +a +1}}}\right )-\operatorname {arccoth}\left (b x +a \right )^{2} \ln \left (1-\frac {1}{\sqrt {\frac {b x +a -1}{b x +a +1}}}\right )-2 \,\operatorname {arccoth}\left (b x +a \right ) \operatorname {polylog}\left (2, \frac {1}{\sqrt {\frac {b x +a -1}{b x +a +1}}}\right )+2 \operatorname {polylog}\left (3, \frac {1}{\sqrt {\frac {b x +a -1}{b x +a +1}}}\right )+\frac {a \operatorname {arccoth}\left (b x +a \right )^{2} \ln \left (1-\frac {\left (a -1\right ) \left (b x +a +1\right )}{\left (b x +a -1\right ) \left (1+a \right )}\right )}{a -1}+\frac {a \,\operatorname {arccoth}\left (b x +a \right ) \operatorname {polylog}\left (2, \frac {\left (a -1\right ) \left (b x +a +1\right )}{\left (b x +a -1\right ) \left (1+a \right )}\right )}{a -1}-\frac {a \operatorname {polylog}\left (3, \frac {\left (a -1\right ) \left (b x +a +1\right )}{\left (b x +a -1\right ) \left (1+a \right )}\right )}{2 \left (a -1\right )}-\frac {\operatorname {arccoth}\left (b x +a \right )^{2} \ln \left (1-\frac {\left (a -1\right ) \left (b x +a +1\right )}{\left (b x +a -1\right ) \left (1+a \right )}\right )}{a -1}-\frac {\operatorname {arccoth}\left (b x +a \right ) \operatorname {polylog}\left (2, \frac {\left (a -1\right ) \left (b x +a +1\right )}{\left (b x +a -1\right ) \left (1+a \right )}\right )}{a -1}+\frac {\operatorname {polylog}\left (3, \frac {\left (a -1\right ) \left (b x +a +1\right )}{\left (b x +a -1\right ) \left (1+a \right )}\right )}{2 a -2}\)	\(866\)
default	\(\ln \left (-b x \right ) \operatorname {arccoth}\left (b x +a \right )^{2}-\operatorname {arccoth}\left (b x +a \right )^{2} \ln \left (-\frac {b x +a +1}{b x +a -1}-1+a \left (\frac {b x +a +1}{b x +a -1}-1\right )\right )+\frac {i \pi \,\operatorname {csgn}\left (\frac {i \left (-\frac {b x +a +1}{b x +a -1}-1+a \left (\frac {b x +a +1}{b x +a -1}-1\right )\right )}{\frac {b x +a +1}{b x +a -1}-1}\right ) \left (\operatorname {csgn}\left (i \left (-\frac {b x +a +1}{b x +a -1}-1+a \left (\frac {b x +a +1}{b x +a -1}-1\right )\right )\right ) \operatorname {csgn}\left (\frac {i}{\frac {b x +a +1}{b x +a -1}-1}\right )-\operatorname {csgn}\left (\frac {i \left (-\frac {b x +a +1}{b x +a -1}-1+a \left (\frac {b x +a +1}{b x +a -1}-1\right )\right )}{\frac {b x +a +1}{b x +a -1}-1}\right ) \operatorname {csgn}\left (\frac {i}{\frac {b x +a +1}{b x +a -1}-1}\right )-\operatorname {csgn}\left (i \left (-\frac {b x +a +1}{b x +a -1}-1+a \left (\frac {b x +a +1}{b x +a -1}-1\right )\right )\right ) \operatorname {csgn}\left (\frac {i \left (-\frac {b x +a +1}{b x +a -1}-1+a \left (\frac {b x +a +1}{b x +a -1}-1\right )\right )}{\frac {b x +a +1}{b x +a -1}-1}\right )+\operatorname {csgn}\left (\frac {i \left (-\frac {b x +a +1}{b x +a -1}-1+a \left (\frac {b x +a +1}{b x +a -1}-1\right )\right )}{\frac {b x +a +1}{b x +a -1}-1}\right )^{2}\right ) \operatorname {arccoth}\left (b x +a \right )^{2}}{2}+\operatorname {arccoth}\left (b x +a \right )^{2} \ln \left (\frac {b x +a +1}{b x +a -1}-1\right )-\operatorname {arccoth}\left (b x +a \right )^{2} \ln \left (1+\frac {1}{\sqrt {\frac {b x +a -1}{b x +a +1}}}\right )-2 \,\operatorname {arccoth}\left (b x +a \right ) \operatorname {polylog}\left (2, -\frac {1}{\sqrt {\frac {b x +a -1}{b x +a +1}}}\right )+2 \operatorname {polylog}\left (3, -\frac {1}{\sqrt {\frac {b x +a -1}{b x +a +1}}}\right )-\operatorname {arccoth}\left (b x +a \right )^{2} \ln \left (1-\frac {1}{\sqrt {\frac {b x +a -1}{b x +a +1}}}\right )-2 \,\operatorname {arccoth}\left (b x +a \right ) \operatorname {polylog}\left (2, \frac {1}{\sqrt {\frac {b x +a -1}{b x +a +1}}}\right )+2 \operatorname {polylog}\left (3, \frac {1}{\sqrt {\frac {b x +a -1}{b x +a +1}}}\right )+\frac {a \operatorname {arccoth}\left (b x +a \right )^{2} \ln \left (1-\frac {\left (a -1\right ) \left (b x +a +1\right )}{\left (b x +a -1\right ) \left (1+a \right )}\right )}{a -1}+\frac {a \,\operatorname {arccoth}\left (b x +a \right ) \operatorname {polylog}\left (2, \frac {\left (a -1\right ) \left (b x +a +1\right )}{\left (b x +a -1\right ) \left (1+a \right )}\right )}{a -1}-\frac {a \operatorname {polylog}\left (3, \frac {\left (a -1\right ) \left (b x +a +1\right )}{\left (b x +a -1\right ) \left (1+a \right )}\right )}{2 \left (a -1\right )}-\frac {\operatorname {arccoth}\left (b x +a \right )^{2} \ln \left (1-\frac {\left (a -1\right ) \left (b x +a +1\right )}{\left (b x +a -1\right ) \left (1+a \right )}\right )}{a -1}-\frac {\operatorname {arccoth}\left (b x +a \right ) \operatorname {polylog}\left (2, \frac {\left (a -1\right ) \left (b x +a +1\right )}{\left (b x +a -1\right ) \left (1+a \right )}\right )}{a -1}+\frac {\operatorname {polylog}\left (3, \frac {\left (a -1\right ) \left (b x +a +1\right )}{\left (b x +a -1\right ) \left (1+a \right )}\right )}{2 a -2}\)	\(866\)
parts	\(\text {Expression too large to display}\)	\(1411\)

Input:

int(arccoth(b*x+a)^2/x,x,method=_RETURNVERBOSE)
 

Output:

ln(-b*x)*arccoth(b*x+a)^2-arccoth(b*x+a)^2*ln(-(b*x+a+1)/(b*x+a-1)-1+a*((b 
*x+a+1)/(b*x+a-1)-1))+1/2*I*Pi*csgn(I*(-(b*x+a+1)/(b*x+a-1)-1+a*((b*x+a+1) 
/(b*x+a-1)-1))/((b*x+a+1)/(b*x+a-1)-1))*(csgn(I*(-(b*x+a+1)/(b*x+a-1)-1+a* 
((b*x+a+1)/(b*x+a-1)-1)))*csgn(I/((b*x+a+1)/(b*x+a-1)-1))-csgn(I*(-(b*x+a+ 
1)/(b*x+a-1)-1+a*((b*x+a+1)/(b*x+a-1)-1))/((b*x+a+1)/(b*x+a-1)-1))*csgn(I/ 
((b*x+a+1)/(b*x+a-1)-1))-csgn(I*(-(b*x+a+1)/(b*x+a-1)-1+a*((b*x+a+1)/(b*x+ 
a-1)-1)))*csgn(I*(-(b*x+a+1)/(b*x+a-1)-1+a*((b*x+a+1)/(b*x+a-1)-1))/((b*x+ 
a+1)/(b*x+a-1)-1))+csgn(I*(-(b*x+a+1)/(b*x+a-1)-1+a*((b*x+a+1)/(b*x+a-1)-1 
))/((b*x+a+1)/(b*x+a-1)-1))^2)*arccoth(b*x+a)^2+arccoth(b*x+a)^2*ln((b*x+a 
+1)/(b*x+a-1)-1)-arccoth(b*x+a)^2*ln(1+1/((b*x+a-1)/(b*x+a+1))^(1/2))-2*ar 
ccoth(b*x+a)*polylog(2,-1/((b*x+a-1)/(b*x+a+1))^(1/2))+2*polylog(3,-1/((b* 
x+a-1)/(b*x+a+1))^(1/2))-arccoth(b*x+a)^2*ln(1-1/((b*x+a-1)/(b*x+a+1))^(1/ 
2))-2*arccoth(b*x+a)*polylog(2,1/((b*x+a-1)/(b*x+a+1))^(1/2))+2*polylog(3, 
1/((b*x+a-1)/(b*x+a+1))^(1/2))+a/(a-1)*arccoth(b*x+a)^2*ln(1-(a-1)/(b*x+a- 
1)*(b*x+a+1)/(1+a))+a/(a-1)*arccoth(b*x+a)*polylog(2,(a-1)/(b*x+a-1)*(b*x+ 
a+1)/(1+a))-1/2*a/(a-1)*polylog(3,(a-1)/(b*x+a-1)*(b*x+a+1)/(1+a))-1/(a-1) 
*arccoth(b*x+a)^2*ln(1-(a-1)/(b*x+a-1)*(b*x+a+1)/(1+a))-1/(a-1)*arccoth(b* 
x+a)*polylog(2,(a-1)/(b*x+a-1)*(b*x+a+1)/(1+a))+1/2/(a-1)*polylog(3,(a-1)/ 
(b*x+a-1)*(b*x+a+1)/(1+a))
 

Fricas [F]

\[ \int \frac {\coth ^{-1}(a+b x)^2}{x} \, dx=\int { \frac {\operatorname {arcoth}\left (b x + a\right )^{2}}{x} \,d x } \] Input:

integrate(arccoth(b*x+a)^2/x,x, algorithm="fricas")
 

Output:

integral(arccoth(b*x + a)^2/x, x)
 

Sympy [F]

\[ \int \frac {\coth ^{-1}(a+b x)^2}{x} \, dx=\int \frac {\operatorname {acoth}^{2}{\left (a + b x \right )}}{x}\, dx \] Input:

integrate(acoth(b*x+a)**2/x,x)
 

Output:

Integral(acoth(a + b*x)**2/x, x)
 

Maxima [F]

\[ \int \frac {\coth ^{-1}(a+b x)^2}{x} \, dx=\int { \frac {\operatorname {arcoth}\left (b x + a\right )^{2}}{x} \,d x } \] Input:

integrate(arccoth(b*x+a)^2/x,x, algorithm="maxima")
 

Output:

integrate(arccoth(b*x + a)^2/x, x)
 

Giac [F]

\[ \int \frac {\coth ^{-1}(a+b x)^2}{x} \, dx=\int { \frac {\operatorname {arcoth}\left (b x + a\right )^{2}}{x} \,d x } \] Input:

integrate(arccoth(b*x+a)^2/x,x, algorithm="giac")
 

Output:

integrate(arccoth(b*x + a)^2/x, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\coth ^{-1}(a+b x)^2}{x} \, dx=\int \frac {{\mathrm {acoth}\left (a+b\,x\right )}^2}{x} \,d x \] Input:

int(acoth(a + b*x)^2/x,x)
 

Output:

int(acoth(a + b*x)^2/x, x)
 

Reduce [F]

\[ \int \frac {\coth ^{-1}(a+b x)^2}{x} \, dx=\int \frac {\mathit {acoth} \left (b x +a \right )^{2}}{x}d x \] Input:

int(acoth(b*x+a)^2/x,x)
                                                                                    
                                                                                    
 

Output:

int(acoth(a + b*x)**2/x,x)