\(\int \frac {x^4 (a+b \text {arctanh}(c x))^2}{(d+c d x)^3} \, dx\) [111]

Optimal result

Integrand size = 22, antiderivative size = 408 \[ \int \frac {x^4 (a+b \text {arctanh}(c x))^2}{(d+c d x)^3} \, dx=\frac {a b x}{c^4 d^3}-\frac {b^2}{16 c^5 d^3 (1+c x)^2}+\frac {29 b^2}{16 c^5 d^3 (1+c x)}-\frac {29 b^2 \text {arctanh}(c x)}{16 c^5 d^3}+\frac {b^2 x \text {arctanh}(c x)}{c^4 d^3}-\frac {b (a+b \text {arctanh}(c x))}{4 c^5 d^3 (1+c x)^2}+\frac {15 b (a+b \text {arctanh}(c x))}{4 c^5 d^3 (1+c x)}-\frac {43 (a+b \text {arctanh}(c x))^2}{8 c^5 d^3}-\frac {3 x (a+b \text {arctanh}(c x))^2}{c^4 d^3}+\frac {x^2 (a+b \text {arctanh}(c x))^2}{2 c^3 d^3}-\frac {(a+b \text {arctanh}(c x))^2}{2 c^5 d^3 (1+c x)^2}+\frac {4 (a+b \text {arctanh}(c x))^2}{c^5 d^3 (1+c x)}+\frac {6 b (a+b \text {arctanh}(c x)) \log \left (\frac {2}{1-c x}\right )}{c^5 d^3}-\frac {6 (a+b \text {arctanh}(c x))^2 \log \left (\frac {2}{1+c x}\right )}{c^5 d^3}+\frac {b^2 \log \left (1-c^2 x^2\right )}{2 c^5 d^3}+\frac {3 b^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{c^5 d^3}+\frac {6 b (a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+c x}\right )}{c^5 d^3}+\frac {3 b^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1+c x}\right )}{c^5 d^3} \] Output:

a*b*x/c^4/d^3-1/16*b^2/c^5/d^3/(c*x+1)^2+29/16*b^2/c^5/d^3/(c*x+1)-29/16*b 
^2*arctanh(c*x)/c^5/d^3+b^2*x*arctanh(c*x)/c^4/d^3-1/4*b*(a+b*arctanh(c*x) 
)/c^5/d^3/(c*x+1)^2+15/4*b*(a+b*arctanh(c*x))/c^5/d^3/(c*x+1)-43/8*(a+b*ar 
ctanh(c*x))^2/c^5/d^3-3*x*(a+b*arctanh(c*x))^2/c^4/d^3+1/2*x^2*(a+b*arctan 
h(c*x))^2/c^3/d^3-1/2*(a+b*arctanh(c*x))^2/c^5/d^3/(c*x+1)^2+4*(a+b*arctan 
h(c*x))^2/c^5/d^3/(c*x+1)+6*b*(a+b*arctanh(c*x))*ln(2/(-c*x+1))/c^5/d^3-6* 
(a+b*arctanh(c*x))^2*ln(2/(c*x+1))/c^5/d^3+1/2*b^2*ln(-c^2*x^2+1)/c^5/d^3+ 
3*b^2*polylog(2,1-2/(-c*x+1))/c^5/d^3+6*b*(a+b*arctanh(c*x))*polylog(2,1-2 
/(c*x+1))/c^5/d^3+3*b^2*polylog(3,1-2/(c*x+1))/c^5/d^3
 

Mathematica [A] (verified)

Time = 1.40 (sec) , antiderivative size = 420, normalized size of antiderivative = 1.03 \[ \int \frac {x^4 (a+b \text {arctanh}(c x))^2}{(d+c d x)^3} \, dx=\frac {-48 a^2 c x+8 a^2 c^2 x^2-\frac {8 a^2}{(1+c x)^2}+\frac {64 a^2}{1+c x}+96 a^2 \log (1+c x)+a b \left (16 c x+28 \cosh (2 \text {arctanh}(c x))-\cosh (4 \text {arctanh}(c x))-48 \log \left (1-c^2 x^2\right )+96 \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(c x)}\right )-28 \sinh (2 \text {arctanh}(c x))+\sinh (4 \text {arctanh}(c x))+4 \text {arctanh}(c x) \left (-4-24 c x+4 c^2 x^2+14 \cosh (2 \text {arctanh}(c x))-\cosh (4 \text {arctanh}(c x))-48 \log \left (1+e^{-2 \text {arctanh}(c x)}\right )-14 \sinh (2 \text {arctanh}(c x))+\sinh (4 \text {arctanh}(c x))\right )\right )+16 b^2 \left ((-3+6 \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(c x)}\right )+\frac {1}{64} \left (56 \cosh (2 \text {arctanh}(c x))-\cosh (4 \text {arctanh}(c x))+32 \log \left (1-c^2 x^2\right )+192 \operatorname {PolyLog}\left (3,-e^{-2 \text {arctanh}(c x)}\right )-56 \sinh (2 \text {arctanh}(c x))+\sinh (4 \text {arctanh}(c x))+4 \text {arctanh}(c x) \left (16 c x+28 \cosh (2 \text {arctanh}(c x))-\cosh (4 \text {arctanh}(c x))+96 \log \left (1+e^{-2 \text {arctanh}(c x)}\right )-28 \sinh (2 \text {arctanh}(c x))+\sinh (4 \text {arctanh}(c x))\right )+8 \text {arctanh}(c x)^2 \left (20-24 c x+4 c^2 x^2+14 \cosh (2 \text {arctanh}(c x))-\cosh (4 \text {arctanh}(c x))-48 \log \left (1+e^{-2 \text {arctanh}(c x)}\right )-14 \sinh (2 \text {arctanh}(c x))+\sinh (4 \text {arctanh}(c x))\right )\right )\right )}{16 c^5 d^3} \] Input:

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

Output:

(-48*a^2*c*x + 8*a^2*c^2*x^2 - (8*a^2)/(1 + c*x)^2 + (64*a^2)/(1 + c*x) + 
96*a^2*Log[1 + c*x] + a*b*(16*c*x + 28*Cosh[2*ArcTanh[c*x]] - Cosh[4*ArcTa 
nh[c*x]] - 48*Log[1 - c^2*x^2] + 96*PolyLog[2, -E^(-2*ArcTanh[c*x])] - 28* 
Sinh[2*ArcTanh[c*x]] + Sinh[4*ArcTanh[c*x]] + 4*ArcTanh[c*x]*(-4 - 24*c*x 
+ 4*c^2*x^2 + 14*Cosh[2*ArcTanh[c*x]] - Cosh[4*ArcTanh[c*x]] - 48*Log[1 + 
E^(-2*ArcTanh[c*x])] - 14*Sinh[2*ArcTanh[c*x]] + Sinh[4*ArcTanh[c*x]])) + 
16*b^2*((-3 + 6*ArcTanh[c*x])*PolyLog[2, -E^(-2*ArcTanh[c*x])] + (56*Cosh[ 
2*ArcTanh[c*x]] - Cosh[4*ArcTanh[c*x]] + 32*Log[1 - c^2*x^2] + 192*PolyLog 
[3, -E^(-2*ArcTanh[c*x])] - 56*Sinh[2*ArcTanh[c*x]] + Sinh[4*ArcTanh[c*x]] 
 + 4*ArcTanh[c*x]*(16*c*x + 28*Cosh[2*ArcTanh[c*x]] - Cosh[4*ArcTanh[c*x]] 
 + 96*Log[1 + E^(-2*ArcTanh[c*x])] - 28*Sinh[2*ArcTanh[c*x]] + Sinh[4*ArcT 
anh[c*x]]) + 8*ArcTanh[c*x]^2*(20 - 24*c*x + 4*c^2*x^2 + 14*Cosh[2*ArcTanh 
[c*x]] - Cosh[4*ArcTanh[c*x]] - 48*Log[1 + E^(-2*ArcTanh[c*x])] - 14*Sinh[ 
2*ArcTanh[c*x]] + Sinh[4*ArcTanh[c*x]]))/64))/(16*c^5*d^3)
 

Rubi [A] (verified)

Time = 1.15 (sec) , antiderivative size = 408, 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.

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

\(\Big \downarrow \) 6502

\(\displaystyle \int \left (\frac {6 (a+b \text {arctanh}(c x))^2}{c^4 d^3 (c x+1)}-\frac {4 (a+b \text {arctanh}(c x))^2}{c^4 d^3 (c x+1)^2}-\frac {3 (a+b \text {arctanh}(c x))^2}{c^4 d^3}+\frac {(a+b \text {arctanh}(c x))^2}{c^4 d^3 (c x+1)^3}+\frac {x (a+b \text {arctanh}(c x))^2}{c^3 d^3}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {6 b \operatorname {PolyLog}\left (2,1-\frac {2}{c x+1}\right ) (a+b \text {arctanh}(c x))}{c^5 d^3}+\frac {15 b (a+b \text {arctanh}(c x))}{4 c^5 d^3 (c x+1)}-\frac {b (a+b \text {arctanh}(c x))}{4 c^5 d^3 (c x+1)^2}+\frac {4 (a+b \text {arctanh}(c x))^2}{c^5 d^3 (c x+1)}-\frac {(a+b \text {arctanh}(c x))^2}{2 c^5 d^3 (c x+1)^2}-\frac {43 (a+b \text {arctanh}(c x))^2}{8 c^5 d^3}+\frac {6 b \log \left (\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))}{c^5 d^3}-\frac {6 \log \left (\frac {2}{c x+1}\right ) (a+b \text {arctanh}(c x))^2}{c^5 d^3}-\frac {3 x (a+b \text {arctanh}(c x))^2}{c^4 d^3}+\frac {x^2 (a+b \text {arctanh}(c x))^2}{2 c^3 d^3}+\frac {a b x}{c^4 d^3}-\frac {29 b^2 \text {arctanh}(c x)}{16 c^5 d^3}+\frac {b^2 x \text {arctanh}(c x)}{c^4 d^3}+\frac {3 b^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{c^5 d^3}+\frac {3 b^2 \operatorname {PolyLog}\left (3,1-\frac {2}{c x+1}\right )}{c^5 d^3}+\frac {29 b^2}{16 c^5 d^3 (c x+1)}-\frac {b^2}{16 c^5 d^3 (c x+1)^2}+\frac {b^2 \log \left (1-c^2 x^2\right )}{2 c^5 d^3}\)

Input:

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

Output:

(a*b*x)/(c^4*d^3) - b^2/(16*c^5*d^3*(1 + c*x)^2) + (29*b^2)/(16*c^5*d^3*(1 
 + c*x)) - (29*b^2*ArcTanh[c*x])/(16*c^5*d^3) + (b^2*x*ArcTanh[c*x])/(c^4* 
d^3) - (b*(a + b*ArcTanh[c*x]))/(4*c^5*d^3*(1 + c*x)^2) + (15*b*(a + b*Arc 
Tanh[c*x]))/(4*c^5*d^3*(1 + c*x)) - (43*(a + b*ArcTanh[c*x])^2)/(8*c^5*d^3 
) - (3*x*(a + b*ArcTanh[c*x])^2)/(c^4*d^3) + (x^2*(a + b*ArcTanh[c*x])^2)/ 
(2*c^3*d^3) - (a + b*ArcTanh[c*x])^2/(2*c^5*d^3*(1 + c*x)^2) + (4*(a + b*A 
rcTanh[c*x])^2)/(c^5*d^3*(1 + c*x)) + (6*b*(a + b*ArcTanh[c*x])*Log[2/(1 - 
 c*x)])/(c^5*d^3) - (6*(a + b*ArcTanh[c*x])^2*Log[2/(1 + c*x)])/(c^5*d^3) 
+ (b^2*Log[1 - c^2*x^2])/(2*c^5*d^3) + (3*b^2*PolyLog[2, 1 - 2/(1 - c*x)]) 
/(c^5*d^3) + (6*b*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 + c*x)])/(c^5*d 
^3) + (3*b^2*PolyLog[3, 1 - 2/(1 + c*x)])/(c^5*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 = 5.48 (sec) , antiderivative size = 1069, normalized size of antiderivative = 2.62

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1069\)
default	\(\text {Expression too large to display}\)	\(1069\)
parts	\(\text {Expression too large to display}\)	\(1082\)

Input:

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

Output:

1/c^5*(a^2/d^3*(1/2*c^2*x^2-3*c*x-1/2/(c*x+1)^2+4/(c*x+1)+6*ln(c*x+1))+b^2 
/d^3*(3*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)^2+ 
6*arctanh(c*x)^2*ln(c*x+1)+6*arctanh(c*x)*ln(1+I*(c*x+1)/(-c^2*x^2+1)^(1/2 
))+6*arctanh(c*x)*ln(1-I*(c*x+1)/(-c^2*x^2+1)^(1/2))+6*dilog(1+I*(c*x+1)/( 
-c^2*x^2+1)^(1/2))+6*dilog(1-I*(c*x+1)/(-c^2*x^2+1)^(1/2))+3*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*arctanh(c*x)^2-3*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))*arctanh(c*x)^2-3*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)))*arctanh(c*x)^ 
2-6*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*arctanh(c*x)^2-1/64/(c*x+1)^2*(c*x-1)^2+4*arctanh(c*x)^3+1/2*arctanh(c*x 
)^2*c^2*x^2-3*arctanh(c*x)^2*c*x-1/16*arctanh(c*x)*(c*x-1)^2/(c*x+1)^2-7/8 
/(c*x+1)*(c*x-1)-43/8*arctanh(c*x)^2+3*polylog(3,-(c*x+1)^2/(-c^2*x^2+1))+ 
(c*x+1)*arctanh(c*x)-ln(1+(c*x+1)^2/(-c^2*x^2+1))-6*arctanh(c*x)*polylog(2 
,-(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)))^3*arctanh(c*x)^2-3*I*Pi*csgn(I*(c*x+1)^2/(c^2*x^2-1))^3*arc 
tanh(c*x)^2-12*arctanh(c*x)^2*ln((c*x+1)/(-c^2*x^2+1)^(1/2))-6*ln(2)*arcta 
nh(c*x)^2-7/4*arctanh(c*x)*(c*x-1)/(c*x+1)+4/(c*x+1)*arctanh(c*x)^2-1/2/(c 
*x+1)^2*arctanh(c*x)^2)+2*b*a/d^3*(1/2*arctanh(c*x)*c^2*x^2-3*arctanh(c...
 

Fricas [F]

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

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

Output:

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

Sympy [F]

\[ \int \frac {x^4 (a+b \text {arctanh}(c x))^2}{(d+c d x)^3} \, dx=\frac {\int \frac {a^{2} x^{4}}{c^{3} x^{3} + 3 c^{2} x^{2} + 3 c x + 1}\, dx + \int \frac {b^{2} x^{4} \operatorname {atanh}^{2}{\left (c x \right )}}{c^{3} x^{3} + 3 c^{2} x^{2} + 3 c x + 1}\, dx + \int \frac {2 a b x^{4} \operatorname {atanh}{\left (c x \right )}}{c^{3} x^{3} + 3 c^{2} x^{2} + 3 c x + 1}\, dx}{d^{3}} \] Input:

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

Output:

(Integral(a**2*x**4/(c**3*x**3 + 3*c**2*x**2 + 3*c*x + 1), x) + Integral(b 
**2*x**4*atanh(c*x)**2/(c**3*x**3 + 3*c**2*x**2 + 3*c*x + 1), x) + Integra 
l(2*a*b*x**4*atanh(c*x)/(c**3*x**3 + 3*c**2*x**2 + 3*c*x + 1), x))/d**3
 

Maxima [F]

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

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

Output:

1/2*a^2*((8*c*x + 7)/(c^7*d^3*x^2 + 2*c^6*d^3*x + c^5*d^3) + (c*x^2 - 6*x) 
/(c^4*d^3) + 12*log(c*x + 1)/(c^5*d^3)) + 1/8*(b^2*c^4*x^4 - 4*b^2*c^3*x^3 
 - 11*b^2*c^2*x^2 + 2*b^2*c*x + 7*b^2 + 12*(b^2*c^2*x^2 + 2*b^2*c*x + b^2) 
*log(c*x + 1))*log(-c*x + 1)^2/(c^7*d^3*x^2 + 2*c^6*d^3*x + c^5*d^3) - int 
egrate(-1/4*((b^2*c^5*x^5 - b^2*c^4*x^4)*log(c*x + 1)^2 + 4*(a*b*c^5*x^5 - 
 a*b*c^4*x^4)*log(c*x + 1) + (15*b^2*c^3*x^3 + 9*b^2*c^2*x^2 - (4*a*b*c^5 
+ b^2*c^5)*x^5 + (4*a*b*c^4 + 3*b^2*c^4)*x^4 - 9*b^2*c*x - 7*b^2 - 2*(b^2* 
c^5*x^5 - b^2*c^4*x^4 + 6*b^2*c^3*x^3 + 18*b^2*c^2*x^2 + 18*b^2*c*x + 6*b^ 
2)*log(c*x + 1))*log(-c*x + 1))/(c^8*d^3*x^4 + 2*c^7*d^3*x^3 - 2*c^5*d^3*x 
 - c^4*d^3), x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {x^4 (a+b \text {arctanh}(c x))^2}{(d+c d x)^3} \, dx=\frac {4 \left (\int \frac {\mathit {atanh} \left (c x \right ) x^{4}}{c^{3} x^{3}+3 c^{2} x^{2}+3 c x +1}d x \right ) a b \,c^{7} x^{2}+8 \left (\int \frac {\mathit {atanh} \left (c x \right ) x^{4}}{c^{3} x^{3}+3 c^{2} x^{2}+3 c x +1}d x \right ) a b \,c^{6} x +4 \left (\int \frac {\mathit {atanh} \left (c x \right ) x^{4}}{c^{3} x^{3}+3 c^{2} x^{2}+3 c x +1}d x \right ) a b \,c^{5}+2 \left (\int \frac {\mathit {atanh} \left (c x \right )^{2} x^{4}}{c^{3} x^{3}+3 c^{2} x^{2}+3 c x +1}d x \right ) b^{2} c^{7} x^{2}+4 \left (\int \frac {\mathit {atanh} \left (c x \right )^{2} x^{4}}{c^{3} x^{3}+3 c^{2} x^{2}+3 c x +1}d x \right ) b^{2} c^{6} x +2 \left (\int \frac {\mathit {atanh} \left (c x \right )^{2} x^{4}}{c^{3} x^{3}+3 c^{2} x^{2}+3 c x +1}d x \right ) b^{2} c^{5}+12 \,\mathrm {log}\left (c x +1\right ) a^{2} c^{2} x^{2}+24 \,\mathrm {log}\left (c x +1\right ) a^{2} c x +12 \,\mathrm {log}\left (c x +1\right ) a^{2}+a^{2} c^{4} x^{4}-4 a^{2} c^{3} x^{3}-12 a^{2} c^{2} x^{2}+6 a^{2}}{2 c^{5} d^{3} \left (c^{2} x^{2}+2 c x +1\right )} \] Input:

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

Output:

(4*int((atanh(c*x)*x**4)/(c**3*x**3 + 3*c**2*x**2 + 3*c*x + 1),x)*a*b*c**7 
*x**2 + 8*int((atanh(c*x)*x**4)/(c**3*x**3 + 3*c**2*x**2 + 3*c*x + 1),x)*a 
*b*c**6*x + 4*int((atanh(c*x)*x**4)/(c**3*x**3 + 3*c**2*x**2 + 3*c*x + 1), 
x)*a*b*c**5 + 2*int((atanh(c*x)**2*x**4)/(c**3*x**3 + 3*c**2*x**2 + 3*c*x 
+ 1),x)*b**2*c**7*x**2 + 4*int((atanh(c*x)**2*x**4)/(c**3*x**3 + 3*c**2*x* 
*2 + 3*c*x + 1),x)*b**2*c**6*x + 2*int((atanh(c*x)**2*x**4)/(c**3*x**3 + 3 
*c**2*x**2 + 3*c*x + 1),x)*b**2*c**5 + 12*log(c*x + 1)*a**2*c**2*x**2 + 24 
*log(c*x + 1)*a**2*c*x + 12*log(c*x + 1)*a**2 + a**2*c**4*x**4 - 4*a**2*c* 
*3*x**3 - 12*a**2*c**2*x**2 + 6*a**2)/(2*c**5*d**3*(c**2*x**2 + 2*c*x + 1) 
)