\(\int \frac {(d+c d x) (a+b \text {arctanh}(c x))^2}{x} \, dx\) [72]

Optimal result

Integrand size = 20, antiderivative size = 191 \[ \int \frac {(d+c d x) (a+b \text {arctanh}(c x))^2}{x} \, dx=d (a+b \text {arctanh}(c x))^2+c d x (a+b \text {arctanh}(c x))^2+2 d (a+b \text {arctanh}(c x))^2 \text {arctanh}\left (1-\frac {2}{1-c x}\right )-2 b d (a+b \text {arctanh}(c x)) \log \left (\frac {2}{1-c x}\right )-b^2 d \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )-b d (a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )+b d (a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-c x}\right )+\frac {1}{2} b^2 d \operatorname {PolyLog}\left (3,1-\frac {2}{1-c x}\right )-\frac {1}{2} b^2 d \operatorname {PolyLog}\left (3,-1+\frac {2}{1-c x}\right ) \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.31 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.19 \[ \int \frac {(d+c d x) (a+b \text {arctanh}(c x))^2}{x} \, dx=d \left (a^2 c x+a^2 \log (c x)+a b \left (2 c x \text {arctanh}(c x)+\log \left (1-c^2 x^2\right )\right )+b^2 \left (\text {arctanh}(c x) \left ((-1+c x) \text {arctanh}(c x)-2 \log \left (1+e^{-2 \text {arctanh}(c x)}\right )\right )+\operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(c x)}\right )\right )+a b (-\operatorname {PolyLog}(2,-c x)+\operatorname {PolyLog}(2,c x))+b^2 \left (\frac {i \pi ^3}{24}-\frac {2}{3} \text {arctanh}(c x)^3-\text {arctanh}(c x)^2 \log \left (1+e^{-2 \text {arctanh}(c x)}\right )+\text {arctanh}(c x)^2 \log \left (1-e^{2 \text {arctanh}(c x)}\right )+\text {arctanh}(c x) \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(c x)}\right )+\text {arctanh}(c x) \operatorname {PolyLog}\left (2,e^{2 \text {arctanh}(c x)}\right )+\frac {1}{2} \operatorname {PolyLog}\left (3,-e^{-2 \text {arctanh}(c x)}\right )-\frac {1}{2} \operatorname {PolyLog}\left (3,e^{2 \text {arctanh}(c x)}\right )\right )\right ) \] Input:

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

Output:

d*(a^2*c*x + a^2*Log[c*x] + a*b*(2*c*x*ArcTanh[c*x] + Log[1 - c^2*x^2]) + 
b^2*(ArcTanh[c*x]*((-1 + c*x)*ArcTanh[c*x] - 2*Log[1 + E^(-2*ArcTanh[c*x]) 
]) + PolyLog[2, -E^(-2*ArcTanh[c*x])]) + a*b*(-PolyLog[2, -(c*x)] + PolyLo 
g[2, c*x]) + b^2*((I/24)*Pi^3 - (2*ArcTanh[c*x]^3)/3 - ArcTanh[c*x]^2*Log[ 
1 + E^(-2*ArcTanh[c*x])] + ArcTanh[c*x]^2*Log[1 - E^(2*ArcTanh[c*x])] + Ar 
cTanh[c*x]*PolyLog[2, -E^(-2*ArcTanh[c*x])] + ArcTanh[c*x]*PolyLog[2, E^(2 
*ArcTanh[c*x])] + PolyLog[3, -E^(-2*ArcTanh[c*x])]/2 - PolyLog[3, E^(2*Arc 
Tanh[c*x])]/2))
 

Rubi [A] (verified)

Time = 0.79 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {6502, 2009}

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.

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

\(\Big \downarrow \) 6502

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 6502

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e 
_.)*(x_))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*ArcTanh[c*x])^p, ( 
f*x)^m*(d + e*x)^q, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[p, 0] 
 && IntegerQ[q] && (GtQ[q, 0] || NeQ[a, 0] || IntegerQ[m])
 

Maple [C] (warning: unable to verify)

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

Time = 2.62 (sec) , antiderivative size = 1827, normalized size of antiderivative = 9.57

method	result	size
parts	\(\text {Expression too large to display}\)	\(1827\)
derivativedivides	\(\text {Expression too large to display}\)	\(1837\)
default	\(\text {Expression too large to display}\)	\(1837\)

Input:

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

Output:

d*a^2*(c*x+ln(x))+d*b^2*(arctanh(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))+arctanh(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))-arctanh(c*x)^2*ln((c*x+1)^2/(-c^2*x^2+1)-1)+arctanh(c*x)^2*ln(c*x)-1 
/2*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*(arctanh(c*x)*ln(1+I*(c*x+1)/(-c^2*x^2+1)^ 
(1/2))+arctanh(c*x)*ln(1-I*(c*x+1)/(-c^2*x^2+1)^(1/2))+dilog(1+I*(c*x+1)/( 
-c^2*x^2+1)^(1/2))+dilog(1-I*(c*x+1)/(-c^2*x^2+1)^(1/2)))-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)/(1-(c*x+1)^2/ 
(c^2*x^2-1)))^2*(arctanh(c*x)*ln(1+I*(c*x+1)/(-c^2*x^2+1)^(1/2))+arctanh(c 
*x)*ln(1-I*(c*x+1)/(-c^2*x^2+1)^(1/2))+dilog(1+I*(c*x+1)/(-c^2*x^2+1)^(1/2 
))+dilog(1-I*(c*x+1)/(-c^2*x^2+1)^(1/2)))-arctanh(c*x)*ln(1+(c*x+1)^2/(-c^ 
2*x^2+1))-arctanh(c*x)*ln(1+I*(c*x+1)/(-c^2*x^2+1)^(1/2))-arctanh(c*x)*ln( 
1-I*(c*x+1)/(-c^2*x^2+1)^(1/2))-dilog(1+I*(c*x+1)/(-c^2*x^2+1)^(1/2))-dilo 
g(1-I*(c*x+1)/(-c^2*x^2+1)^(1/2))-1/4*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)))*csgn(I/(1 
-(c*x+1)^2/(c^2*x^2-1)))*(2*arctanh(c*x)*ln(1+(c*x+1)^2/(-c^2*x^2+1))-2*ar 
ctanh(c*x)^2+polylog(2,-(c*x+1)^2/(-c^2*x^2+1)))+arctanh(c*x)^2*c*x+arctan 
h(c*x)^2-1/2*polylog(2,-(c*x+1)^2/(-c^2*x^2+1))+1/2*polylog(3,-(c*x+1)^2/( 
-c^2*x^2+1))-arctanh(c*x)*polylog(2,-(c*x+1)^2/(-c^2*x^2+1))-2*polylog(...
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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