\(\int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))^2}{x^4} \, dx\) [91]

Optimal result

Integrand size = 22, antiderivative size = 396 \[ \int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))^2}{x^4} \, dx=-\frac {b^2 c^2 d^3}{3 x}+\frac {1}{3} b^2 c^3 d^3 \text {arctanh}(c x)-\frac {b c d^3 (a+b \text {arctanh}(c x))}{3 x^2}-\frac {3 b c^2 d^3 (a+b \text {arctanh}(c x))}{x}+\frac {29}{6} c^3 d^3 (a+b \text {arctanh}(c x))^2-\frac {d^3 (a+b \text {arctanh}(c x))^2}{3 x^3}-\frac {3 c d^3 (a+b \text {arctanh}(c x))^2}{2 x^2}-\frac {3 c^2 d^3 (a+b \text {arctanh}(c x))^2}{x}+2 c^3 d^3 (a+b \text {arctanh}(c x))^2 \text {arctanh}\left (1-\frac {2}{1-c x}\right )+3 b^2 c^3 d^3 \log (x)-\frac {3}{2} b^2 c^3 d^3 \log \left (1-c^2 x^2\right )+\frac {20}{3} b c^3 d^3 (a+b \text {arctanh}(c x)) \log \left (2-\frac {2}{1+c x}\right )-b c^3 d^3 (a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )+b c^3 d^3 (a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-c x}\right )-\frac {10}{3} b^2 c^3 d^3 \operatorname {PolyLog}\left (2,-1+\frac {2}{1+c x}\right )+\frac {1}{2} b^2 c^3 d^3 \operatorname {PolyLog}\left (3,1-\frac {2}{1-c x}\right )-\frac {1}{2} b^2 c^3 d^3 \operatorname {PolyLog}\left (3,-1+\frac {2}{1-c x}\right ) \] Output:

-1/3*b^2*c^2*d^3/x+1/3*b^2*c^3*d^3*arctanh(c*x)-1/3*b*c*d^3*(a+b*arctanh(c 
*x))/x^2-3*b*c^2*d^3*(a+b*arctanh(c*x))/x+29/6*c^3*d^3*(a+b*arctanh(c*x))^ 
2-1/3*d^3*(a+b*arctanh(c*x))^2/x^3-3/2*c*d^3*(a+b*arctanh(c*x))^2/x^2-3*c^ 
2*d^3*(a+b*arctanh(c*x))^2/x-2*c^3*d^3*(a+b*arctanh(c*x))^2*arctanh(-1+2/( 
-c*x+1))+3*b^2*c^3*d^3*ln(x)-3/2*b^2*c^3*d^3*ln(-c^2*x^2+1)+20/3*b*c^3*d^3 
*(a+b*arctanh(c*x))*ln(2-2/(c*x+1))-b*c^3*d^3*(a+b*arctanh(c*x))*polylog(2 
,1-2/(-c*x+1))+b*c^3*d^3*(a+b*arctanh(c*x))*polylog(2,-1+2/(-c*x+1))-10/3* 
b^2*c^3*d^3*polylog(2,-1+2/(c*x+1))+1/2*b^2*c^3*d^3*polylog(3,1-2/(-c*x+1) 
)-1/2*b^2*c^3*d^3*polylog(3,-1+2/(-c*x+1))
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.46 (sec) , antiderivative size = 569, normalized size of antiderivative = 1.44 \[ \int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))^2}{x^4} \, dx=\frac {d^3 \left (-8 a^2-36 a^2 c x-8 a b c x-72 a^2 c^2 x^2-72 a b c^2 x^2-8 b^2 c^2 x^2+i b^2 c^3 \pi ^3 x^3-16 a b \text {arctanh}(c x)-72 a b c x \text {arctanh}(c x)-8 b^2 c x \text {arctanh}(c x)-144 a b c^2 x^2 \text {arctanh}(c x)-72 b^2 c^2 x^2 \text {arctanh}(c x)+8 b^2 c^3 x^3 \text {arctanh}(c x)-8 b^2 \text {arctanh}(c x)^2-36 b^2 c x \text {arctanh}(c x)^2-72 b^2 c^2 x^2 \text {arctanh}(c x)^2+116 b^2 c^3 x^3 \text {arctanh}(c x)^2-16 b^2 c^3 x^3 \text {arctanh}(c x)^3+160 b^2 c^3 x^3 \text {arctanh}(c x) \log \left (1-e^{-2 \text {arctanh}(c x)}\right )-24 b^2 c^3 x^3 \text {arctanh}(c x)^2 \log \left (1+e^{-2 \text {arctanh}(c x)}\right )+24 b^2 c^3 x^3 \text {arctanh}(c x)^2 \log \left (1-e^{2 \text {arctanh}(c x)}\right )+24 a^2 c^3 x^3 \log (x)+160 a b c^3 x^3 \log (c x)-36 a b c^3 x^3 \log (1-c x)+36 a b c^3 x^3 \log (1+c x)+72 b^2 c^3 x^3 \log \left (\frac {c x}{\sqrt {1-c^2 x^2}}\right )-80 a b c^3 x^3 \log \left (1-c^2 x^2\right )+24 b^2 c^3 x^3 \text {arctanh}(c x) \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(c x)}\right )-80 b^2 c^3 x^3 \operatorname {PolyLog}\left (2,e^{-2 \text {arctanh}(c x)}\right )+24 b^2 c^3 x^3 \text {arctanh}(c x) \operatorname {PolyLog}\left (2,e^{2 \text {arctanh}(c x)}\right )-24 a b c^3 x^3 \operatorname {PolyLog}(2,-c x)+24 a b c^3 x^3 \operatorname {PolyLog}(2,c x)+12 b^2 c^3 x^3 \operatorname {PolyLog}\left (3,-e^{-2 \text {arctanh}(c x)}\right )-12 b^2 c^3 x^3 \operatorname {PolyLog}\left (3,e^{2 \text {arctanh}(c x)}\right )\right )}{24 x^3} \] Input:

Integrate[((d + c*d*x)^3*(a + b*ArcTanh[c*x])^2)/x^4,x]
 

Output:

(d^3*(-8*a^2 - 36*a^2*c*x - 8*a*b*c*x - 72*a^2*c^2*x^2 - 72*a*b*c^2*x^2 - 
8*b^2*c^2*x^2 + I*b^2*c^3*Pi^3*x^3 - 16*a*b*ArcTanh[c*x] - 72*a*b*c*x*ArcT 
anh[c*x] - 8*b^2*c*x*ArcTanh[c*x] - 144*a*b*c^2*x^2*ArcTanh[c*x] - 72*b^2* 
c^2*x^2*ArcTanh[c*x] + 8*b^2*c^3*x^3*ArcTanh[c*x] - 8*b^2*ArcTanh[c*x]^2 - 
 36*b^2*c*x*ArcTanh[c*x]^2 - 72*b^2*c^2*x^2*ArcTanh[c*x]^2 + 116*b^2*c^3*x 
^3*ArcTanh[c*x]^2 - 16*b^2*c^3*x^3*ArcTanh[c*x]^3 + 160*b^2*c^3*x^3*ArcTan 
h[c*x]*Log[1 - E^(-2*ArcTanh[c*x])] - 24*b^2*c^3*x^3*ArcTanh[c*x]^2*Log[1 
+ E^(-2*ArcTanh[c*x])] + 24*b^2*c^3*x^3*ArcTanh[c*x]^2*Log[1 - E^(2*ArcTan 
h[c*x])] + 24*a^2*c^3*x^3*Log[x] + 160*a*b*c^3*x^3*Log[c*x] - 36*a*b*c^3*x 
^3*Log[1 - c*x] + 36*a*b*c^3*x^3*Log[1 + c*x] + 72*b^2*c^3*x^3*Log[(c*x)/S 
qrt[1 - c^2*x^2]] - 80*a*b*c^3*x^3*Log[1 - c^2*x^2] + 24*b^2*c^3*x^3*ArcTa 
nh[c*x]*PolyLog[2, -E^(-2*ArcTanh[c*x])] - 80*b^2*c^3*x^3*PolyLog[2, E^(-2 
*ArcTanh[c*x])] + 24*b^2*c^3*x^3*ArcTanh[c*x]*PolyLog[2, E^(2*ArcTanh[c*x] 
)] - 24*a*b*c^3*x^3*PolyLog[2, -(c*x)] + 24*a*b*c^3*x^3*PolyLog[2, c*x] + 
12*b^2*c^3*x^3*PolyLog[3, -E^(-2*ArcTanh[c*x])] - 12*b^2*c^3*x^3*PolyLog[3 
, E^(2*ArcTanh[c*x])]))/(24*x^3)
 

Rubi [A] (verified)

Time = 1.80 (sec) , antiderivative size = 396, 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.

\(\displaystyle \int \frac {(c d x+d)^3 (a+b \text {arctanh}(c x))^2}{x^4} \, dx\)

\(\Big \downarrow \) 6502

\(\displaystyle \int \left (\frac {c^3 d^3 (a+b \text {arctanh}(c x))^2}{x}+\frac {3 c^2 d^3 (a+b \text {arctanh}(c x))^2}{x^2}+\frac {d^3 (a+b \text {arctanh}(c x))^2}{x^4}+\frac {3 c d^3 (a+b \text {arctanh}(c x))^2}{x^3}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -b c^3 d^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))+b c^3 d^3 \operatorname {PolyLog}\left (2,\frac {2}{1-c x}-1\right ) (a+b \text {arctanh}(c x))+\frac {29}{6} c^3 d^3 (a+b \text {arctanh}(c x))^2+2 c^3 d^3 \text {arctanh}\left (1-\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))^2+\frac {20}{3} b c^3 d^3 \log \left (2-\frac {2}{c x+1}\right ) (a+b \text {arctanh}(c x))-\frac {3 c^2 d^3 (a+b \text {arctanh}(c x))^2}{x}-\frac {3 b c^2 d^3 (a+b \text {arctanh}(c x))}{x}-\frac {d^3 (a+b \text {arctanh}(c x))^2}{3 x^3}-\frac {3 c d^3 (a+b \text {arctanh}(c x))^2}{2 x^2}-\frac {b c d^3 (a+b \text {arctanh}(c x))}{3 x^2}+\frac {1}{3} b^2 c^3 d^3 \text {arctanh}(c x)-\frac {10}{3} b^2 c^3 d^3 \operatorname {PolyLog}\left (2,\frac {2}{c x+1}-1\right )+\frac {1}{2} b^2 c^3 d^3 \operatorname {PolyLog}\left (3,1-\frac {2}{1-c x}\right )-\frac {1}{2} b^2 c^3 d^3 \operatorname {PolyLog}\left (3,\frac {2}{1-c x}-1\right )+3 b^2 c^3 d^3 \log (x)-\frac {b^2 c^2 d^3}{3 x}-\frac {3}{2} b^2 c^3 d^3 \log \left (1-c^2 x^2\right )\)

Input:

Int[((d + c*d*x)^3*(a + b*ArcTanh[c*x])^2)/x^4,x]
 

Output:

-1/3*(b^2*c^2*d^3)/x + (b^2*c^3*d^3*ArcTanh[c*x])/3 - (b*c*d^3*(a + b*ArcT 
anh[c*x]))/(3*x^2) - (3*b*c^2*d^3*(a + b*ArcTanh[c*x]))/x + (29*c^3*d^3*(a 
 + b*ArcTanh[c*x])^2)/6 - (d^3*(a + b*ArcTanh[c*x])^2)/(3*x^3) - (3*c*d^3* 
(a + b*ArcTanh[c*x])^2)/(2*x^2) - (3*c^2*d^3*(a + b*ArcTanh[c*x])^2)/x + 2 
*c^3*d^3*(a + b*ArcTanh[c*x])^2*ArcTanh[1 - 2/(1 - c*x)] + 3*b^2*c^3*d^3*L 
og[x] - (3*b^2*c^3*d^3*Log[1 - c^2*x^2])/2 + (20*b*c^3*d^3*(a + b*ArcTanh[ 
c*x])*Log[2 - 2/(1 + c*x)])/3 - b*c^3*d^3*(a + b*ArcTanh[c*x])*PolyLog[2, 
1 - 2/(1 - c*x)] + b*c^3*d^3*(a + b*ArcTanh[c*x])*PolyLog[2, -1 + 2/(1 - c 
*x)] - (10*b^2*c^3*d^3*PolyLog[2, -1 + 2/(1 + c*x)])/3 + (b^2*c^3*d^3*Poly 
Log[3, 1 - 2/(1 - c*x)])/2 - (b^2*c^3*d^3*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 = 5.59 (sec) , antiderivative size = 1197, normalized size of antiderivative = 3.02

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1197\)
default	\(\text {Expression too large to display}\)	\(1197\)
parts	\(\text {Expression too large to display}\)	\(1255\)

Input:

int((c*d*x+d)^3*(a+b*arctanh(c*x))^2/x^4,x,method=_RETURNVERBOSE)
 

Output:

c^3*(1/2*I*d^3*b^2*Pi*csgn(I*(-(c*x+1)^2/(c^2*x^2-1)-1)/(1-(c*x+1)^2/(c^2* 
x^2-1)))^3*arctanh(c*x)^2+1/2*I*d^3*b^2*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)))*arctanh(c*x)^2+d^3*b^2*arctanh(c*x)^2*ln(c*x)+d^3 
*b^2*arctanh(c*x)^2*ln(1+(c*x+1)/(-c^2*x^2+1)^(1/2))+1/2*d^3*b^2*polylog(3 
,-(c*x+1)^2/(-c^2*x^2+1))-8/3*b^2*d^3*arctanh(c*x)-11/6*d^3*b^2*arctanh(c* 
x)^2+d^3*b^2*arctanh(c*x)^2*ln(1-(c*x+1)/(-c^2*x^2+1)^(1/2))+2*d^3*b^2*arc 
tanh(c*x)*polylog(2,(c*x+1)/(-c^2*x^2+1)^(1/2))+2*d^3*b^2*arctanh(c*x)*pol 
ylog(2,-(c*x+1)/(-c^2*x^2+1)^(1/2))-d^3*b^2*arctanh(c*x)*polylog(2,-(c*x+1 
)^2/(-c^2*x^2+1))-1/3*d^3*b^2/(c*x-(-c^2*x^2+1)^(1/2)+1)*(-c^2*x^2+1)^(1/2 
)+1/3*d^3*b^2/(c*x+(-c^2*x^2+1)^(1/2)+1)*(-c^2*x^2+1)^(1/2)+20/3*d^3*b^2*a 
rctanh(c*x)*ln(1+(c*x+1)/(-c^2*x^2+1)^(1/2))-d^3*b^2*arctanh(c*x)^2*ln((c* 
x+1)^2/(-c^2*x^2+1)-1)+2*d^3*b*a*(-1/3*arctanh(c*x)/c^3/x^3-3/2*arctanh(c* 
x)/c^2/x^2+arctanh(c*x)*ln(c*x)-3*arctanh(c*x)/c/x-29/12*ln(c*x-1)-1/6/c^2 
/x^2-3/2/c/x+10/3*ln(c*x)-11/12*ln(c*x+1)-1/2*dilog(c*x)-1/2*dilog(c*x+1)- 
1/2*ln(c*x)*ln(c*x+1))-1/2*I*d^3*b^2*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)^2-1/2*I*d^3*b^2*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)^2-1/3*d^3*b^2*ar 
ctanh(c*x)^2/c^3/x^3-3/2*d^3*b^2*arctanh(c*x)^2/c^2/x^2-3*d^3*b^2*arcta...
 

Fricas [F]

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

integrate((c*d*x+d)^3*(a+b*arctanh(c*x))^2/x^4,x, algorithm="fricas")
 

Output:

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

Sympy [F]

\[ \int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))^2}{x^4} \, dx=d^{3} \left (\int \frac {a^{2}}{x^{4}}\, dx + \int \frac {3 a^{2} c}{x^{3}}\, dx + \int \frac {3 a^{2} c^{2}}{x^{2}}\, dx + \int \frac {a^{2} c^{3}}{x}\, dx + \int \frac {b^{2} \operatorname {atanh}^{2}{\left (c x \right )}}{x^{4}}\, dx + \int \frac {2 a b \operatorname {atanh}{\left (c x \right )}}{x^{4}}\, dx + \int \frac {3 b^{2} c \operatorname {atanh}^{2}{\left (c x \right )}}{x^{3}}\, dx + \int \frac {3 b^{2} c^{2} \operatorname {atanh}^{2}{\left (c x \right )}}{x^{2}}\, dx + \int \frac {b^{2} c^{3} \operatorname {atanh}^{2}{\left (c x \right )}}{x}\, dx + \int \frac {6 a b c \operatorname {atanh}{\left (c x \right )}}{x^{3}}\, dx + \int \frac {6 a b c^{2} \operatorname {atanh}{\left (c x \right )}}{x^{2}}\, dx + \int \frac {2 a b c^{3} \operatorname {atanh}{\left (c x \right )}}{x}\, dx\right ) \] Input:

integrate((c*d*x+d)**3*(a+b*atanh(c*x))**2/x**4,x)
 

Output:

d**3*(Integral(a**2/x**4, x) + Integral(3*a**2*c/x**3, x) + Integral(3*a** 
2*c**2/x**2, x) + Integral(a**2*c**3/x, x) + Integral(b**2*atanh(c*x)**2/x 
**4, x) + Integral(2*a*b*atanh(c*x)/x**4, x) + Integral(3*b**2*c*atanh(c*x 
)**2/x**3, x) + Integral(3*b**2*c**2*atanh(c*x)**2/x**2, x) + Integral(b** 
2*c**3*atanh(c*x)**2/x, x) + Integral(6*a*b*c*atanh(c*x)/x**3, x) + Integr 
al(6*a*b*c**2*atanh(c*x)/x**2, x) + Integral(2*a*b*c**3*atanh(c*x)/x, x))
 

Maxima [F]

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

integrate((c*d*x+d)^3*(a+b*arctanh(c*x))^2/x^4,x, algorithm="maxima")
 

Output:

a^2*c^3*d^3*log(x) - 3*(c*(log(c^2*x^2 - 1) - log(x^2)) + 2*arctanh(c*x)/x 
)*a*b*c^2*d^3 + 3/2*((c*log(c*x + 1) - c*log(c*x - 1) - 2/x)*c - 2*arctanh 
(c*x)/x^2)*a*b*c*d^3 - 1/3*((c^2*log(c^2*x^2 - 1) - c^2*log(x^2) + 1/x^2)* 
c + 2*arctanh(c*x)/x^3)*a*b*d^3 - 3*a^2*c^2*d^3/x - 3/2*a^2*c*d^3/x^2 - 1/ 
3*a^2*d^3/x^3 - 1/24*(18*b^2*c^2*d^3*x^2 + 9*b^2*c*d^3*x + 2*b^2*d^3)*log( 
-c*x + 1)^2/x^3 - integrate(-1/12*(3*(b^2*c^4*d^3*x^4 + 2*b^2*c^3*d^3*x^3 
- 2*b^2*c*d^3*x - b^2*d^3)*log(c*x + 1)^2 + 12*(a*b*c^4*d^3*x^4 - a*b*c^3* 
d^3*x^3)*log(c*x + 1) - (12*a*b*c^4*d^3*x^4 - 9*b^2*c^2*d^3*x^2 - 2*b^2*c* 
d^3*x - 6*(2*a*b*c^3*d^3 + 3*b^2*c^3*d^3)*x^3 + 6*(b^2*c^4*d^3*x^4 + 2*b^2 
*c^3*d^3*x^3 - 2*b^2*c*d^3*x - b^2*d^3)*log(c*x + 1))*log(-c*x + 1))/(c*x^ 
5 - x^4), x)
 

Giac [F]

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

integrate((c*d*x+d)^3*(a+b*arctanh(c*x))^2/x^4,x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

int(((a + b*atanh(c*x))^2*(d + c*d*x)^3)/x^4,x)
 

Output:

int(((a + b*atanh(c*x))^2*(d + c*d*x)^3)/x^4, x)
 

Reduce [F]

\[ \int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))^2}{x^4} \, dx=\frac {d^{3} \left (6 \,\mathrm {log}\left (x \right ) a^{2} c^{3} x^{3}-4 \mathit {atanh} \left (c x \right ) a b -2 b^{2} c^{2} x^{2}-16 \mathit {atanh} \left (c x \right ) b^{2} c^{3} x^{3}-2 \mathit {atanh} \left (c x \right ) b^{2} c x -2 a b c x -9 \mathit {atanh} \left (c x \right )^{2} b^{2} c x -18 \mathit {atanh} \left (c x \right ) b^{2} c^{2} x^{2}-18 a b \,c^{2} x^{2}-18 \,\mathrm {log}\left (c^{2} x -c \right ) b^{2} c^{3} x^{3}+18 \,\mathrm {log}\left (x \right ) b^{2} c^{3} x^{3}-2 a^{2}+9 \mathit {atanh} \left (c x \right )^{2} b^{2} c^{3} x^{3}-40 \left (\int \frac {\mathit {atanh} \left (c x \right )}{c^{2} x^{3}-x}d x \right ) b^{2} c^{3} x^{3}-18 \mathit {atanh} \left (c x \right ) a b c x -40 \,\mathrm {log}\left (c^{2} x -c \right ) a b \,c^{3} x^{3}+40 \,\mathrm {log}\left (x \right ) a b \,c^{3} x^{3}-18 \mathit {atanh} \left (c x \right )^{2} b^{2} c^{2} x^{2}-22 \mathit {atanh} \left (c x \right ) a b \,c^{3} x^{3}-9 a^{2} c x -2 \mathit {atanh} \left (c x \right )^{2} b^{2}-36 \mathit {atanh} \left (c x \right ) a b \,c^{2} x^{2}+12 \left (\int \frac {\mathit {atanh} \left (c x \right )}{x}d x \right ) a b \,c^{3} x^{3}-18 a^{2} c^{2} x^{2}+6 \left (\int \frac {\mathit {atanh} \left (c x \right )^{2}}{x}d x \right ) b^{2} c^{3} x^{3}\right )}{6 x^{3}} \] Input:

int((c*d*x+d)^3*(a+b*atanh(c*x))^2/x^4,x)
 

Output:

(d**3*(9*atanh(c*x)**2*b**2*c**3*x**3 - 18*atanh(c*x)**2*b**2*c**2*x**2 - 
9*atanh(c*x)**2*b**2*c*x - 2*atanh(c*x)**2*b**2 - 22*atanh(c*x)*a*b*c**3*x 
**3 - 36*atanh(c*x)*a*b*c**2*x**2 - 18*atanh(c*x)*a*b*c*x - 4*atanh(c*x)*a 
*b - 16*atanh(c*x)*b**2*c**3*x**3 - 18*atanh(c*x)*b**2*c**2*x**2 - 2*atanh 
(c*x)*b**2*c*x - 40*int(atanh(c*x)/(c**2*x**3 - x),x)*b**2*c**3*x**3 + 12* 
int(atanh(c*x)/x,x)*a*b*c**3*x**3 + 6*int(atanh(c*x)**2/x,x)*b**2*c**3*x** 
3 - 40*log(c**2*x - c)*a*b*c**3*x**3 - 18*log(c**2*x - c)*b**2*c**3*x**3 + 
 6*log(x)*a**2*c**3*x**3 + 40*log(x)*a*b*c**3*x**3 + 18*log(x)*b**2*c**3*x 
**3 - 18*a**2*c**2*x**2 - 9*a**2*c*x - 2*a**2 - 18*a*b*c**2*x**2 - 2*a*b*c 
*x - 2*b**2*c**2*x**2))/(6*x**3)