\(\int \frac {(a+b \text {arctanh}(c x))^2}{x (d+c d x)^2} \, dx\) [108]

Optimal result

Integrand size = 22, antiderivative size = 295 \[ \int \frac {(a+b \text {arctanh}(c x))^2}{x (d+c d x)^2} \, dx=\frac {b^2}{2 d^2 (1+c x)}-\frac {b^2 \text {arctanh}(c x)}{2 d^2}+\frac {b (a+b \text {arctanh}(c x))}{d^2 (1+c x)}-\frac {(a+b \text {arctanh}(c x))^2}{2 d^2}+\frac {(a+b \text {arctanh}(c x))^2}{d^2 (1+c x)}+\frac {2 (a+b \text {arctanh}(c x))^2 \text {arctanh}\left (1-\frac {2}{1-c x}\right )}{d^2}+\frac {(a+b \text {arctanh}(c x))^2 \log \left (\frac {2}{1+c x}\right )}{d^2}-\frac {b (a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{d^2}+\frac {b (a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-c x}\right )}{d^2}-\frac {b (a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+c x}\right )}{d^2}+\frac {b^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1-c x}\right )}{2 d^2}-\frac {b^2 \operatorname {PolyLog}\left (3,-1+\frac {2}{1-c x}\right )}{2 d^2}-\frac {b^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1+c x}\right )}{2 d^2} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.91 (sec) , antiderivative size = 254, normalized size of antiderivative = 0.86 \[ \int \frac {(a+b \text {arctanh}(c x))^2}{x (d+c d x)^2} \, dx=\frac {\frac {24 a^2}{1+c x}+24 a^2 \log (c x)-24 a^2 \log (1+c x)+12 a b \left (\cosh (2 \text {arctanh}(c x))-2 \operatorname {PolyLog}\left (2,e^{-2 \text {arctanh}(c x)}\right )+2 \text {arctanh}(c x) \left (\cosh (2 \text {arctanh}(c x))+2 \log \left (1-e^{-2 \text {arctanh}(c x)}\right )-\sinh (2 \text {arctanh}(c x))\right )-\sinh (2 \text {arctanh}(c x))\right )+b^2 \left (i \pi ^3-16 \text {arctanh}(c x)^3+6 \cosh (2 \text {arctanh}(c x))+12 \text {arctanh}(c x) \cosh (2 \text {arctanh}(c x))+12 \text {arctanh}(c x)^2 \cosh (2 \text {arctanh}(c x))+24 \text {arctanh}(c x)^2 \log \left (1-e^{2 \text {arctanh}(c x)}\right )+24 \text {arctanh}(c x) \operatorname {PolyLog}\left (2,e^{2 \text {arctanh}(c x)}\right )-12 \operatorname {PolyLog}\left (3,e^{2 \text {arctanh}(c x)}\right )-6 \sinh (2 \text {arctanh}(c x))-12 \text {arctanh}(c x) \sinh (2 \text {arctanh}(c x))-12 \text {arctanh}(c x)^2 \sinh (2 \text {arctanh}(c x))\right )}{24 d^2} \] Input:

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

Output:

((24*a^2)/(1 + c*x) + 24*a^2*Log[c*x] - 24*a^2*Log[1 + c*x] + 12*a*b*(Cosh 
[2*ArcTanh[c*x]] - 2*PolyLog[2, E^(-2*ArcTanh[c*x])] + 2*ArcTanh[c*x]*(Cos 
h[2*ArcTanh[c*x]] + 2*Log[1 - E^(-2*ArcTanh[c*x])] - Sinh[2*ArcTanh[c*x]]) 
 - Sinh[2*ArcTanh[c*x]]) + b^2*(I*Pi^3 - 16*ArcTanh[c*x]^3 + 6*Cosh[2*ArcT 
anh[c*x]] + 12*ArcTanh[c*x]*Cosh[2*ArcTanh[c*x]] + 12*ArcTanh[c*x]^2*Cosh[ 
2*ArcTanh[c*x]] + 24*ArcTanh[c*x]^2*Log[1 - E^(2*ArcTanh[c*x])] + 24*ArcTa 
nh[c*x]*PolyLog[2, E^(2*ArcTanh[c*x])] - 12*PolyLog[3, E^(2*ArcTanh[c*x])] 
 - 6*Sinh[2*ArcTanh[c*x]] - 12*ArcTanh[c*x]*Sinh[2*ArcTanh[c*x]] - 12*ArcT 
anh[c*x]^2*Sinh[2*ArcTanh[c*x]]))/(24*d^2)
 

Rubi [A] (verified)

Time = 0.97 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, 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.

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

\(\Big \downarrow \) 6502

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

b^2/(2*d^2*(1 + c*x)) - (b^2*ArcTanh[c*x])/(2*d^2) + (b*(a + b*ArcTanh[c*x 
]))/(d^2*(1 + c*x)) - (a + b*ArcTanh[c*x])^2/(2*d^2) + (a + b*ArcTanh[c*x] 
)^2/(d^2*(1 + c*x)) + (2*(a + b*ArcTanh[c*x])^2*ArcTanh[1 - 2/(1 - c*x)])/ 
d^2 + ((a + b*ArcTanh[c*x])^2*Log[2/(1 + c*x)])/d^2 - (b*(a + b*ArcTanh[c* 
x])*PolyLog[2, 1 - 2/(1 - c*x)])/d^2 + (b*(a + b*ArcTanh[c*x])*PolyLog[2, 
-1 + 2/(1 - c*x)])/d^2 - (b*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 + c*x 
)])/d^2 + (b^2*PolyLog[3, 1 - 2/(1 - c*x)])/(2*d^2) - (b^2*PolyLog[3, -1 + 
 2/(1 - c*x)])/(2*d^2) - (b^2*PolyLog[3, 1 - 2/(1 + c*x)])/(2*d^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 = 1.20 (sec) , antiderivative size = 1239, normalized size of antiderivative = 4.20

method	result	size
parts	\(\text {Expression too large to display}\)	\(1239\)
derivativedivides	\(\text {Expression too large to display}\)	\(1241\)
default	\(\text {Expression too large to display}\)	\(1241\)

Input:

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

Output:

a^2/d^2*(1/(c*x+1)-ln(c*x+1)+ln(x))+b^2/d^2*(arctanh(c*x)^2*ln(c*x)+1/(c*x 
+1)*arctanh(c*x)^2-arctanh(c*x)^2*ln(c*x+1)+2*arctanh(c*x)^2*ln((c*x+1)/(- 
c^2*x^2+1)^(1/2))-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*csgn(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)))*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/(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-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+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+2*ln(2)-1) 
-1/2*arctanh(c*x)*(c*x-1)/(c*x+1)-1/4/(c*x+1)*(c*x-1)-arctanh(c*x)^2*ln((c 
*x+1)^2/(-c^2*x^2+1)-1)+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))-2*polylog(3,(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*arct...
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {(a+b \text {arctanh}(c x))^2}{x (d+c d x)^2} \, dx=\frac {-48 \left (\int \frac {\mathit {atanh} \left (c x \right )}{c^{3} x^{4}+c^{2} x^{3}-c \,x^{2}-x}d x \right ) a b -4 \mathit {atanh} \left (c x \right )^{3} b^{2}-24 \,\mathrm {log}\left (c x +1\right ) a^{2}-6 b^{2} c x -12 \mathit {atanh} \left (c x \right )^{2} a b c x -6 \,\mathrm {log}\left (c x -1\right ) a b +6 \,\mathrm {log}\left (c x +1\right ) a b -12 \mathit {atanh} \left (c x \right ) b^{2} c x -12 a b c x -6 \mathit {atanh} \left (c x \right )^{2} b^{2} c x +24 \,\mathrm {log}\left (x \right ) a^{2} c x -24 \,\mathrm {log}\left (c x +1\right ) a^{2} c x -24 \mathit {atanh} \left (c x \right ) a b c x -6 \,\mathrm {log}\left (c x -1\right ) a b c x +6 \,\mathrm {log}\left (c x +1\right ) a b c x -24 \left (\int \frac {\mathit {atanh} \left (c x \right )^{2}}{c^{3} x^{4}+c^{2} x^{3}-c \,x^{2}-x}d x \right ) b^{2} c x -3 \,\mathrm {log}\left (c x -1\right ) b^{2}+3 \,\mathrm {log}\left (c x +1\right ) b^{2}-4 \mathit {atanh} \left (c x \right )^{3} b^{2} c x -24 a^{2} c x +6 \mathit {atanh} \left (c x \right )^{2} b^{2}+24 \,\mathrm {log}\left (x \right ) a^{2}-3 \,\mathrm {log}\left (c x -1\right ) b^{2} c x +3 \,\mathrm {log}\left (c x +1\right ) b^{2} c x -48 \left (\int \frac {\mathit {atanh} \left (c x \right )}{c^{3} x^{4}+c^{2} x^{3}-c \,x^{2}-x}d x \right ) a b c x -24 \left (\int \frac {\mathit {atanh} \left (c x \right )^{2}}{c^{3} x^{4}+c^{2} x^{3}-c \,x^{2}-x}d x \right ) b^{2}-12 \mathit {atanh} \left (c x \right )^{2} a b}{24 d^{2} \left (c x +1\right )} \] Input:

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

Output:

( - 4*atanh(c*x)**3*b**2*c*x - 4*atanh(c*x)**3*b**2 - 12*atanh(c*x)**2*a*b 
*c*x - 12*atanh(c*x)**2*a*b - 6*atanh(c*x)**2*b**2*c*x + 6*atanh(c*x)**2*b 
**2 - 24*atanh(c*x)*a*b*c*x - 12*atanh(c*x)*b**2*c*x - 48*int(atanh(c*x)/( 
c**3*x**4 + c**2*x**3 - c*x**2 - x),x)*a*b*c*x - 48*int(atanh(c*x)/(c**3*x 
**4 + c**2*x**3 - c*x**2 - x),x)*a*b - 24*int(atanh(c*x)**2/(c**3*x**4 + c 
**2*x**3 - c*x**2 - x),x)*b**2*c*x - 24*int(atanh(c*x)**2/(c**3*x**4 + c** 
2*x**3 - c*x**2 - x),x)*b**2 - 6*log(c*x - 1)*a*b*c*x - 6*log(c*x - 1)*a*b 
 - 3*log(c*x - 1)*b**2*c*x - 3*log(c*x - 1)*b**2 - 24*log(c*x + 1)*a**2*c* 
x - 24*log(c*x + 1)*a**2 + 6*log(c*x + 1)*a*b*c*x + 6*log(c*x + 1)*a*b + 3 
*log(c*x + 1)*b**2*c*x + 3*log(c*x + 1)*b**2 + 24*log(x)*a**2*c*x + 24*log 
(x)*a**2 - 24*a**2*c*x - 12*a*b*c*x - 6*b**2*c*x)/(24*d**2*(c*x + 1))