\(\int \frac {(a+b \text {arctanh}(c x))^2}{x^2 (d+c d x)^3} \, dx\) [117]

Optimal result

Integrand size = 22, antiderivative size = 448 \[ \int \frac {(a+b \text {arctanh}(c x))^2}{x^2 (d+c d x)^3} \, dx=-\frac {b^2 c}{16 d^3 (1+c x)^2}-\frac {19 b^2 c}{16 d^3 (1+c x)}+\frac {19 b^2 c \text {arctanh}(c x)}{16 d^3}-\frac {b c (a+b \text {arctanh}(c x))}{4 d^3 (1+c x)^2}-\frac {9 b c (a+b \text {arctanh}(c x))}{4 d^3 (1+c x)}+\frac {17 c (a+b \text {arctanh}(c x))^2}{8 d^3}-\frac {(a+b \text {arctanh}(c x))^2}{d^3 x}-\frac {c (a+b \text {arctanh}(c x))^2}{2 d^3 (1+c x)^2}-\frac {2 c (a+b \text {arctanh}(c x))^2}{d^3 (1+c x)}-\frac {6 c (a+b \text {arctanh}(c x))^2 \text {arctanh}\left (1-\frac {2}{1-c x}\right )}{d^3}-\frac {3 c (a+b \text {arctanh}(c x))^2 \log \left (\frac {2}{1+c x}\right )}{d^3}+\frac {2 b c (a+b \text {arctanh}(c x)) \log \left (2-\frac {2}{1+c x}\right )}{d^3}+\frac {3 b c (a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{d^3}-\frac {3 b c (a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-c x}\right )}{d^3}+\frac {3 b c (a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+c x}\right )}{d^3}-\frac {b^2 c \operatorname {PolyLog}\left (2,-1+\frac {2}{1+c x}\right )}{d^3}-\frac {3 b^2 c \operatorname {PolyLog}\left (3,1-\frac {2}{1-c x}\right )}{2 d^3}+\frac {3 b^2 c \operatorname {PolyLog}\left (3,-1+\frac {2}{1-c x}\right )}{2 d^3}+\frac {3 b^2 c \operatorname {PolyLog}\left (3,1-\frac {2}{1+c x}\right )}{2 d^3} \] Output:

-1/16*b^2*c/d^3/(c*x+1)^2-19/16*b^2*c/d^3/(c*x+1)+19/16*b^2*c*arctanh(c*x) 
/d^3-1/4*b*c*(a+b*arctanh(c*x))/d^3/(c*x+1)^2-9/4*b*c*(a+b*arctanh(c*x))/d 
^3/(c*x+1)+17/8*c*(a+b*arctanh(c*x))^2/d^3-(a+b*arctanh(c*x))^2/d^3/x-1/2* 
c*(a+b*arctanh(c*x))^2/d^3/(c*x+1)^2-2*c*(a+b*arctanh(c*x))^2/d^3/(c*x+1)+ 
6*c*(a+b*arctanh(c*x))^2*arctanh(-1+2/(-c*x+1))/d^3-3*c*(a+b*arctanh(c*x)) 
^2*ln(2/(c*x+1))/d^3+2*b*c*(a+b*arctanh(c*x))*ln(2-2/(c*x+1))/d^3+3*b*c*(a 
+b*arctanh(c*x))*polylog(2,1-2/(-c*x+1))/d^3-3*b*c*(a+b*arctanh(c*x))*poly 
log(2,-1+2/(-c*x+1))/d^3+3*b*c*(a+b*arctanh(c*x))*polylog(2,1-2/(c*x+1))/d 
^3-b^2*c*polylog(2,-1+2/(c*x+1))/d^3-3/2*b^2*c*polylog(3,1-2/(-c*x+1))/d^3 
+3/2*b^2*c*polylog(3,-1+2/(-c*x+1))/d^3+3/2*b^2*c*polylog(3,1-2/(c*x+1))/d 
^3
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 1.57 (sec) , antiderivative size = 479, normalized size of antiderivative = 1.07 \[ \int \frac {(a+b \text {arctanh}(c x))^2}{x^2 (d+c d x)^3} \, dx=\frac {-\frac {64 a^2}{x}-\frac {32 a^2 c}{(1+c x)^2}-\frac {128 a^2 c}{1+c x}-192 a^2 c \log (x)+192 a^2 c \log (1+c x)+b^2 c \left (-8 i \pi ^3+64 \text {arctanh}(c x)^2-\frac {64 \text {arctanh}(c x)^2}{c x}+128 \text {arctanh}(c x)^3-40 \cosh (2 \text {arctanh}(c x))-80 \text {arctanh}(c x) \cosh (2 \text {arctanh}(c x))-80 \text {arctanh}(c x)^2 \cosh (2 \text {arctanh}(c x))-\cosh (4 \text {arctanh}(c x))-4 \text {arctanh}(c x) \cosh (4 \text {arctanh}(c x))-8 \text {arctanh}(c x)^2 \cosh (4 \text {arctanh}(c x))+128 \text {arctanh}(c x) \log \left (1-e^{-2 \text {arctanh}(c x)}\right )-192 \text {arctanh}(c x)^2 \log \left (1-e^{2 \text {arctanh}(c x)}\right )-64 \operatorname {PolyLog}\left (2,e^{-2 \text {arctanh}(c x)}\right )-192 \text {arctanh}(c x) \operatorname {PolyLog}\left (2,e^{2 \text {arctanh}(c x)}\right )+96 \operatorname {PolyLog}\left (3,e^{2 \text {arctanh}(c x)}\right )+40 \sinh (2 \text {arctanh}(c x))+80 \text {arctanh}(c x) \sinh (2 \text {arctanh}(c x))+80 \text {arctanh}(c x)^2 \sinh (2 \text {arctanh}(c x))+\sinh (4 \text {arctanh}(c x))+4 \text {arctanh}(c x) \sinh (4 \text {arctanh}(c x))+8 \text {arctanh}(c x)^2 \sinh (4 \text {arctanh}(c x))\right )+\frac {4 a b \left (48 c x \operatorname {PolyLog}\left (2,e^{-2 \text {arctanh}(c x)}\right )+c x \left (-20 \cosh (2 \text {arctanh}(c x))-\cosh (4 \text {arctanh}(c x))+32 \log \left (\frac {c x}{\sqrt {1-c^2 x^2}}\right )+20 \sinh (2 \text {arctanh}(c x))+\sinh (4 \text {arctanh}(c x))\right )-4 \text {arctanh}(c x) \left (8+10 c x \cosh (2 \text {arctanh}(c x))+c x \cosh (4 \text {arctanh}(c x))+24 c x \log \left (1-e^{-2 \text {arctanh}(c x)}\right )-10 c x \sinh (2 \text {arctanh}(c x))-c x \sinh (4 \text {arctanh}(c x))\right )\right )}{x}}{64 d^3} \] Input:

Integrate[(a + b*ArcTanh[c*x])^2/(x^2*(d + c*d*x)^3),x]
 

Output:

((-64*a^2)/x - (32*a^2*c)/(1 + c*x)^2 - (128*a^2*c)/(1 + c*x) - 192*a^2*c* 
Log[x] + 192*a^2*c*Log[1 + c*x] + b^2*c*((-8*I)*Pi^3 + 64*ArcTanh[c*x]^2 - 
 (64*ArcTanh[c*x]^2)/(c*x) + 128*ArcTanh[c*x]^3 - 40*Cosh[2*ArcTanh[c*x]] 
- 80*ArcTanh[c*x]*Cosh[2*ArcTanh[c*x]] - 80*ArcTanh[c*x]^2*Cosh[2*ArcTanh[ 
c*x]] - Cosh[4*ArcTanh[c*x]] - 4*ArcTanh[c*x]*Cosh[4*ArcTanh[c*x]] - 8*Arc 
Tanh[c*x]^2*Cosh[4*ArcTanh[c*x]] + 128*ArcTanh[c*x]*Log[1 - E^(-2*ArcTanh[ 
c*x])] - 192*ArcTanh[c*x]^2*Log[1 - E^(2*ArcTanh[c*x])] - 64*PolyLog[2, E^ 
(-2*ArcTanh[c*x])] - 192*ArcTanh[c*x]*PolyLog[2, E^(2*ArcTanh[c*x])] + 96* 
PolyLog[3, E^(2*ArcTanh[c*x])] + 40*Sinh[2*ArcTanh[c*x]] + 80*ArcTanh[c*x] 
*Sinh[2*ArcTanh[c*x]] + 80*ArcTanh[c*x]^2*Sinh[2*ArcTanh[c*x]] + Sinh[4*Ar 
cTanh[c*x]] + 4*ArcTanh[c*x]*Sinh[4*ArcTanh[c*x]] + 8*ArcTanh[c*x]^2*Sinh[ 
4*ArcTanh[c*x]]) + (4*a*b*(48*c*x*PolyLog[2, E^(-2*ArcTanh[c*x])] + c*x*(- 
20*Cosh[2*ArcTanh[c*x]] - Cosh[4*ArcTanh[c*x]] + 32*Log[(c*x)/Sqrt[1 - c^2 
*x^2]] + 20*Sinh[2*ArcTanh[c*x]] + Sinh[4*ArcTanh[c*x]]) - 4*ArcTanh[c*x]* 
(8 + 10*c*x*Cosh[2*ArcTanh[c*x]] + c*x*Cosh[4*ArcTanh[c*x]] + 24*c*x*Log[1 
 - E^(-2*ArcTanh[c*x])] - 10*c*x*Sinh[2*ArcTanh[c*x]] - c*x*Sinh[4*ArcTanh 
[c*x]])))/x)/(64*d^3)
 

Rubi [A] (verified)

Time = 1.33 (sec) , antiderivative size = 448, 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.

\(\displaystyle \int \frac {(a+b \text {arctanh}(c x))^2}{x^2 (c d x+d)^3} \, dx\)

\(\Big \downarrow \) 6502

\(\displaystyle \int \left (\frac {3 c^2 (a+b \text {arctanh}(c x))^2}{d^3 (c x+1)}+\frac {2 c^2 (a+b \text {arctanh}(c x))^2}{d^3 (c x+1)^2}+\frac {c^2 (a+b \text {arctanh}(c x))^2}{d^3 (c x+1)^3}+\frac {(a+b \text {arctanh}(c x))^2}{d^3 x^2}-\frac {3 c (a+b \text {arctanh}(c x))^2}{d^3 x}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {3 b c \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))}{d^3}-\frac {3 b c \operatorname {PolyLog}\left (2,\frac {2}{1-c x}-1\right ) (a+b \text {arctanh}(c x))}{d^3}+\frac {3 b c \operatorname {PolyLog}\left (2,1-\frac {2}{c x+1}\right ) (a+b \text {arctanh}(c x))}{d^3}-\frac {9 b c (a+b \text {arctanh}(c x))}{4 d^3 (c x+1)}-\frac {b c (a+b \text {arctanh}(c x))}{4 d^3 (c x+1)^2}-\frac {(a+b \text {arctanh}(c x))^2}{d^3 x}-\frac {2 c (a+b \text {arctanh}(c x))^2}{d^3 (c x+1)}-\frac {c (a+b \text {arctanh}(c x))^2}{2 d^3 (c x+1)^2}+\frac {17 c (a+b \text {arctanh}(c x))^2}{8 d^3}-\frac {6 c \text {arctanh}\left (1-\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))^2}{d^3}+\frac {2 b c \log \left (2-\frac {2}{c x+1}\right ) (a+b \text {arctanh}(c x))}{d^3}-\frac {3 c \log \left (\frac {2}{c x+1}\right ) (a+b \text {arctanh}(c x))^2}{d^3}+\frac {19 b^2 c \text {arctanh}(c x)}{16 d^3}-\frac {b^2 c \operatorname {PolyLog}\left (2,\frac {2}{c x+1}-1\right )}{d^3}-\frac {3 b^2 c \operatorname {PolyLog}\left (3,1-\frac {2}{1-c x}\right )}{2 d^3}+\frac {3 b^2 c \operatorname {PolyLog}\left (3,\frac {2}{1-c x}-1\right )}{2 d^3}+\frac {3 b^2 c \operatorname {PolyLog}\left (3,1-\frac {2}{c x+1}\right )}{2 d^3}-\frac {19 b^2 c}{16 d^3 (c x+1)}-\frac {b^2 c}{16 d^3 (c x+1)^2}\)

Input:

Int[(a + b*ArcTanh[c*x])^2/(x^2*(d + c*d*x)^3),x]
 

Output:

-1/16*(b^2*c)/(d^3*(1 + c*x)^2) - (19*b^2*c)/(16*d^3*(1 + c*x)) + (19*b^2* 
c*ArcTanh[c*x])/(16*d^3) - (b*c*(a + b*ArcTanh[c*x]))/(4*d^3*(1 + c*x)^2) 
- (9*b*c*(a + b*ArcTanh[c*x]))/(4*d^3*(1 + c*x)) + (17*c*(a + b*ArcTanh[c* 
x])^2)/(8*d^3) - (a + b*ArcTanh[c*x])^2/(d^3*x) - (c*(a + b*ArcTanh[c*x])^ 
2)/(2*d^3*(1 + c*x)^2) - (2*c*(a + b*ArcTanh[c*x])^2)/(d^3*(1 + c*x)) - (6 
*c*(a + b*ArcTanh[c*x])^2*ArcTanh[1 - 2/(1 - c*x)])/d^3 - (3*c*(a + b*ArcT 
anh[c*x])^2*Log[2/(1 + c*x)])/d^3 + (2*b*c*(a + b*ArcTanh[c*x])*Log[2 - 2/ 
(1 + c*x)])/d^3 + (3*b*c*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 - c*x)]) 
/d^3 - (3*b*c*(a + b*ArcTanh[c*x])*PolyLog[2, -1 + 2/(1 - c*x)])/d^3 + (3* 
b*c*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 + c*x)])/d^3 - (b^2*c*PolyLog 
[2, -1 + 2/(1 + c*x)])/d^3 - (3*b^2*c*PolyLog[3, 1 - 2/(1 - c*x)])/(2*d^3) 
 + (3*b^2*c*PolyLog[3, -1 + 2/(1 - c*x)])/(2*d^3) + (3*b^2*c*PolyLog[3, 1 
- 2/(1 + c*x)])/(2*d^3)
 

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 = 3.03 (sec) , antiderivative size = 4301, normalized size of antiderivative = 9.60

method	result	size
parts	\(\text {Expression too large to display}\)	\(4301\)
derivativedivides	\(\text {Expression too large to display}\)	\(4346\)
default	\(\text {Expression too large to display}\)	\(4346\)

Input:

int((a+b*arctanh(c*x))^2/x^2/(c*d*x+d)^3,x,method=_RETURNVERBOSE)
 

Output:

a^2/d^3*(-1/2/(c*x+1)^2*c-2*c/(c*x+1)+3*c*ln(c*x+1)-1/x-3*c*ln(x))+b^2/d^3 
*c*(-3*arctanh(c*x)^2*ln(1-(c*x+1)/(-c^2*x^2+1)^(1/2))-6*arctanh(c*x)*poly 
log(2,(c*x+1)/(-c^2*x^2+1)^(1/2))-3*arctanh(c*x)^2*ln(1+(c*x+1)/(-c^2*x^2+ 
1)^(1/2))-6*arctanh(c*x)*polylog(2,-(c*x+1)/(-c^2*x^2+1)^(1/2))+3*arctanh( 
c*x)^2*ln((c*x+1)^2/(-c^2*x^2+1)-1)+3/2*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)))*(arctanh(c*x)*ln(1+(c*x+1)/(-c^2*x^2+1)^(1/2))+dilog(1+(c 
*x+1)/(-c^2*x^2+1)^(1/2))-dilog((c*x+1)/(-c^2*x^2+1)^(1/2)))-3*arctanh(c*x 
)^2*ln(c*x)+3*ln(2)*dilog((c*x+1)/(-c^2*x^2+1)^(1/2))-3*ln(2)*dilog(1+(c*x 
+1)/(-c^2*x^2+1)^(1/2))+3*ln(2)*polylog(2,(c*x+1)/(-c^2*x^2+1)^(1/2))+3*ln 
(2)*polylog(2,-(c*x+1)/(-c^2*x^2+1)^(1/2))+3*arctanh(c*x)^2*ln(c*x+1)-1/8* 
arctanh(c*x)*ln(1-(c*x+1)/(-c^2*x^2+1)^(1/2))+2*arctanh(c*x)*ln(1+(c*x+1)/ 
(-c^2*x^2+1)^(1/2))-3/2*I*Pi*csgn(I*(-(c*x+1)^2/(c^2*x^2-1)-1)/(1-(c*x+1)^ 
2/(c^2*x^2-1)))^2*csgn(I/(1-(c*x+1)^2/(c^2*x^2-1)))*(arctanh(c*x)*ln(1-(c* 
x+1)/(-c^2*x^2+1)^(1/2))+arctanh(c*x)*ln(1+(c*x+1)/(-c^2*x^2+1)^(1/2))-arc 
tanh(c*x)^2+polylog(2,(c*x+1)/(-c^2*x^2+1)^(1/2))+polylog(2,-(c*x+1)/(-c^2 
*x^2+1)^(1/2)))+3/2*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)))*(arctanh(c*x)*ln(1-(c*x+1)/(-c^ 
2*x^2+1)^(1/2))+arctanh(c*x)*ln(1+(c*x+1)/(-c^2*x^2+1)^(1/2))-arctanh(c*x) 
^2+polylog(2,(c*x+1)/(-c^2*x^2+1)^(1/2))+polylog(2,-(c*x+1)/(-c^2*x^2+1...
 

Fricas [F]

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

integrate((a+b*arctanh(c*x))^2/x^2/(c*d*x+d)^3,x, algorithm="fricas")
 

Output:

integral((b^2*arctanh(c*x)^2 + 2*a*b*arctanh(c*x) + a^2)/(c^3*d^3*x^5 + 3* 
c^2*d^3*x^4 + 3*c*d^3*x^3 + d^3*x^2), x)
 

Sympy [F]

\[ \int \frac {(a+b \text {arctanh}(c x))^2}{x^2 (d+c d x)^3} \, dx=\frac {\int \frac {a^{2}}{c^{3} x^{5} + 3 c^{2} x^{4} + 3 c x^{3} + x^{2}}\, dx + \int \frac {b^{2} \operatorname {atanh}^{2}{\left (c x \right )}}{c^{3} x^{5} + 3 c^{2} x^{4} + 3 c x^{3} + x^{2}}\, dx + \int \frac {2 a b \operatorname {atanh}{\left (c x \right )}}{c^{3} x^{5} + 3 c^{2} x^{4} + 3 c x^{3} + x^{2}}\, dx}{d^{3}} \] Input:

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

Output:

(Integral(a**2/(c**3*x**5 + 3*c**2*x**4 + 3*c*x**3 + x**2), x) + Integral( 
b**2*atanh(c*x)**2/(c**3*x**5 + 3*c**2*x**4 + 3*c*x**3 + x**2), x) + Integ 
ral(2*a*b*atanh(c*x)/(c**3*x**5 + 3*c**2*x**4 + 3*c*x**3 + x**2), x))/d**3
 

Maxima [F]

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

integrate((a+b*arctanh(c*x))^2/x^2/(c*d*x+d)^3,x, algorithm="maxima")
 

Output:

-1/2*a^2*((6*c^2*x^2 + 9*c*x + 2)/(c^2*d^3*x^3 + 2*c*d^3*x^2 + d^3*x) - 6* 
c*log(c*x + 1)/d^3 + 6*c*log(x)/d^3) - 1/8*(6*b^2*c^2*x^2 + 9*b^2*c*x + 2* 
b^2 - 6*(b^2*c^3*x^3 + 2*b^2*c^2*x^2 + b^2*c*x)*log(c*x + 1))*log(-c*x + 1 
)^2/(c^2*d^3*x^3 + 2*c*d^3*x^2 + d^3*x) - 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) + (6*b^2*c^4*x^4 + 15*b^2* 
c^3*x^3 + 11*b^2*c^2*x^2 + 4*a*b - 2*(2*a*b*c - b^2*c)*x - 2*(3*b^2*c^5*x^ 
5 + 9*b^2*c^4*x^4 + 9*b^2*c^3*x^3 + 3*b^2*c^2*x^2 + b^2*c*x - b^2)*log(c*x 
 + 1))*log(-c*x + 1))/(c^4*d^3*x^6 + 2*c^3*d^3*x^5 - 2*c*d^3*x^3 - d^3*x^2 
), x)
 

Giac [F]

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

integrate((a+b*arctanh(c*x))^2/x^2/(c*d*x+d)^3,x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {(a+b \text {arctanh}(c x))^2}{x^2 (d+c d x)^3} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 16*atanh(c*x)**3*b**2*c**3*x**3 - 32*atanh(c*x)**3*b**2*c**2*x**2 - 16 
*atanh(c*x)**3*b**2*c*x - 48*atanh(c*x)**2*a*b*c**3*x**3 - 96*atanh(c*x)** 
2*a*b*c**2*x**2 - 48*atanh(c*x)**2*a*b*c*x - 60*atanh(c*x)**2*b**2*c**3*x* 
*3 - 72*atanh(c*x)**2*b**2*c**2*x**2 + 36*atanh(c*x)**2*b**2*c*x + 64*atan 
h(c*x)**2*b**2 - 48*atanh(c*x)*a*b*c**3*x**3 + 144*atanh(c*x)*a*b*c*x + 12 
8*atanh(c*x)*a*b - 60*atanh(c*x)*b**2*c**3*x**3 + 132*atanh(c*x)*b**2*c*x 
+ 64*atanh(c*x)*b**2 - 384*int(atanh(c*x)/(c**4*x**6 + 2*c**3*x**5 - 2*c*x 
**3 - x**2),x)*a*b*c**2*x**3 - 768*int(atanh(c*x)/(c**4*x**6 + 2*c**3*x**5 
 - 2*c*x**3 - x**2),x)*a*b*c*x**2 - 384*int(atanh(c*x)/(c**4*x**6 + 2*c**3 
*x**5 - 2*c*x**3 - x**2),x)*a*b*x - 64*int(atanh(c*x)/(c**4*x**6 + 2*c**3* 
x**5 - 2*c*x**3 - x**2),x)*b**2*c**2*x**3 - 128*int(atanh(c*x)/(c**4*x**6 
+ 2*c**3*x**5 - 2*c*x**3 - x**2),x)*b**2*c*x**2 - 64*int(atanh(c*x)/(c**4* 
x**6 + 2*c**3*x**5 - 2*c*x**3 - x**2),x)*b**2*x - 192*int(atanh(c*x)**2/(c 
**4*x**6 + 2*c**3*x**5 - 2*c*x**3 - x**2),x)*b**2*c**2*x**3 - 384*int(atan 
h(c*x)**2/(c**4*x**6 + 2*c**3*x**5 - 2*c*x**3 - x**2),x)*b**2*c*x**2 - 192 
*int(atanh(c*x)**2/(c**4*x**6 + 2*c**3*x**5 - 2*c*x**3 - x**2),x)*b**2*x + 
 28*log(c*x - 1)*a*b*c**3*x**3 + 56*log(c*x - 1)*a*b*c**2*x**2 + 28*log(c* 
x - 1)*a*b*c*x + 17*log(c*x - 1)*b**2*c**3*x**3 + 34*log(c*x - 1)*b**2*c** 
2*x**2 + 17*log(c*x - 1)*b**2*c*x + 384*log(c*x + 1)*a**2*c**3*x**3 + 768* 
log(c*x + 1)*a**2*c**2*x**2 + 384*log(c*x + 1)*a**2*c*x + 100*log(c*x +...