\(\int \frac {(a+b \text {arctanh}(c x))^2}{x (d+c d x)} \, dx\) [99]

Optimal result

Integrand size = 22, antiderivative size = 77 \[ \int \frac {(a+b \text {arctanh}(c x))^2}{x (d+c d x)} \, dx=\frac {(a+b \text {arctanh}(c x))^2 \log \left (2-\frac {2}{1+c x}\right )}{d}-\frac {b (a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+c x}\right )}{d}-\frac {b^2 \operatorname {PolyLog}\left (3,-1+\frac {2}{1+c x}\right )}{2 d} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.69 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.71 \[ \int \frac {(a+b \text {arctanh}(c x))^2}{x (d+c d x)} \, dx=\frac {a^2 \log (c x)-a^2 \log (1+c x)+a b \left (2 \text {arctanh}(c x) \log \left (1-e^{-2 \text {arctanh}(c x)}\right )-\operatorname {PolyLog}\left (2,e^{-2 \text {arctanh}(c x)}\right )\right )+b^2 \left (\frac {i \pi ^3}{24}-\frac {2}{3} \text {arctanh}(c x)^3+\text {arctanh}(c x)^2 \log \left (1-e^{2 \text {arctanh}(c x)}\right )+\text {arctanh}(c x) \operatorname {PolyLog}\left (2,e^{2 \text {arctanh}(c x)}\right )-\frac {1}{2} \operatorname {PolyLog}\left (3,e^{2 \text {arctanh}(c x)}\right )\right )}{d} \] Input:

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

Output:

(a^2*Log[c*x] - a^2*Log[1 + c*x] + a*b*(2*ArcTanh[c*x]*Log[1 - E^(-2*ArcTa 
nh[c*x])] - PolyLog[2, E^(-2*ArcTanh[c*x])]) + b^2*((I/24)*Pi^3 - (2*ArcTa 
nh[c*x]^3)/3 + ArcTanh[c*x]^2*Log[1 - E^(2*ArcTanh[c*x])] + ArcTanh[c*x]*P 
olyLog[2, E^(2*ArcTanh[c*x])] - PolyLog[3, E^(2*ArcTanh[c*x])]/2))/d
 

Rubi [A] (verified)

Time = 0.62 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.09, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {6494, 6618, 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.

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

\(\Big \downarrow \) 6494

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

\(\Big \downarrow \) 6618

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

\(\Big \downarrow \) 7164

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

Input:

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

Output:

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

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 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 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 = 1.01 (sec) , antiderivative size = 1148, normalized size of antiderivative = 14.91

method	result	size
parts	\(\text {Expression too large to display}\)	\(1148\)
derivativedivides	\(\text {Expression too large to display}\)	\(1150\)
default	\(\text {Expression too large to display}\)	\(1150\)

Input:

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

Output:

a^2/d*(-ln(c*x+1)+ln(x))+b^2/d*(arctanh(c*x)^2*ln(c*x)-arctanh(c*x)^2*ln(c 
*x+1)+2*arctanh(c*x)^2*ln((c*x+1)/(-c^2*x^2+1)^(1/2))-arctanh(c*x)^2*ln((c 
*x+1)^2/(-c^2*x^2+1)-1)-2/3*arctanh(c*x)^3+1/2*arctanh(c*x)^2*(I*Pi*csgn(I 
*(c*x+1)^2/(c^2*x^2-1))^3+2*I*Pi*csgn(I*(c*x+1)/(-c^2*x^2+1)^(1/2))*csgn(I 
*(c*x+1)^2/(c^2*x^2-1))^2-I*Pi*csgn(I*(c*x+1)^2/(c^2*x^2-1))*csgn(I*(c*x+1 
)^2/(c^2*x^2-1)/(1-(c*x+1)^2/(c^2*x^2-1)))^2-I*Pi*csgn(I*(c*x+1)^2/(c^2*x^ 
2-1))*csgn(I*(c*x+1)^2/(c^2*x^2-1)/(1-(c*x+1)^2/(c^2*x^2-1)))*csgn(I/(1-(c 
*x+1)^2/(c^2*x^2-1)))+I*Pi*csgn(I*(c*x+1)/(-c^2*x^2+1)^(1/2))^2*csgn(I*(c* 
x+1)^2/(c^2*x^2-1))+I*Pi*csgn(I*(c*x+1)^2/(c^2*x^2-1)/(1-(c*x+1)^2/(c^2*x^ 
2-1)))^3+I*Pi*csgn(I*(c*x+1)^2/(c^2*x^2-1)/(1-(c*x+1)^2/(c^2*x^2-1)))^2*cs 
gn(I/(1-(c*x+1)^2/(c^2*x^2-1)))-I*Pi*csgn(I/(1-(c*x+1)^2/(c^2*x^2-1)))*csg 
n(I*(-(c*x+1)^2/(c^2*x^2-1)-1)/(1-(c*x+1)^2/(c^2*x^2-1)))^2+I*Pi*csgn(I/(1 
-(c*x+1)^2/(c^2*x^2-1)))*csgn(I*(-(c*x+1)^2/(c^2*x^2-1)-1))*csgn(I*(-(c*x+ 
1)^2/(c^2*x^2-1)-1)/(1-(c*x+1)^2/(c^2*x^2-1)))+I*Pi*csgn(I*(-(c*x+1)^2/(c^ 
2*x^2-1)-1)/(1-(c*x+1)^2/(c^2*x^2-1)))^3-I*Pi*csgn(I*(-(c*x+1)^2/(c^2*x^2- 
1)-1))*csgn(I*(-(c*x+1)^2/(c^2*x^2-1)-1)/(1-(c*x+1)^2/(c^2*x^2-1)))^2+2*ln 
(2))+arctanh(c*x)^2*ln(1-(c*x+1)/(-c^2*x^2+1)^(1/2))+2*arctanh(c*x)*polylo 
g(2,(c*x+1)/(-c^2*x^2+1)^(1/2))-2*polylog(3,(c*x+1)/(-c^2*x^2+1)^(1/2))+ar 
ctanh(c*x)^2*ln(1+(c*x+1)/(-c^2*x^2+1)^(1/2))+2*arctanh(c*x)*polylog(2,-(c 
*x+1)/(-c^2*x^2+1)^(1/2))-2*polylog(3,-(c*x+1)/(-c^2*x^2+1)^(1/2)))+2*b...
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {(a+b \text {arctanh}(c x))^2}{x (d+c d x)} \, dx=\frac {-\mathit {atanh} \left (c x \right )^{3} b^{2}-3 \mathit {atanh} \left (c x \right )^{2} a b -6 \left (\int \frac {\mathit {atanh} \left (c x \right )}{c^{2} x^{3}-x}d x \right ) a b -3 \left (\int \frac {\mathit {atanh} \left (c x \right )^{2}}{c^{2} x^{3}-x}d x \right ) b^{2}-3 \,\mathrm {log}\left (c x +1\right ) a^{2}+3 \,\mathrm {log}\left (x \right ) a^{2}}{3 d} \] Input:

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

Output:

( - atanh(c*x)**3*b**2 - 3*atanh(c*x)**2*a*b - 6*int(atanh(c*x)/(c**2*x**3 
 - x),x)*a*b - 3*int(atanh(c*x)**2/(c**2*x**3 - x),x)*b**2 - 3*log(c*x + 1 
)*a**2 + 3*log(x)*a**2)/(3*d)