\(\int (e+f x)^2 (a+b \coth ^{-1}(c+d x))^3 \, dx\) [33]

Optimal result

Integrand size = 20, antiderivative size = 546 \[ \int (e+f x)^2 \left (a+b \coth ^{-1}(c+d x)\right )^3 \, dx=\frac {a b^2 f^2 x}{d^2}+\frac {b^3 f^2 (c+d x) \coth ^{-1}(c+d x)}{d^3}-\frac {b f^2 \left (a+b \coth ^{-1}(c+d x)\right )^2}{2 d^3}+\frac {3 b f (d e-c f) \left (a+b \coth ^{-1}(c+d x)\right )^2}{d^3}+\frac {3 b f (d e-c f) (c+d x) \left (a+b \coth ^{-1}(c+d x)\right )^2}{d^3}+\frac {b f^2 (c+d x)^2 \left (a+b \coth ^{-1}(c+d x)\right )^2}{2 d^3}-\frac {(d e-c f) \left (d^2 e^2-2 c d e f+\left (3+c^2\right ) f^2\right ) \left (a+b \coth ^{-1}(c+d x)\right )^3}{3 d^3 f}+\frac {\left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right ) \left (a+b \coth ^{-1}(c+d x)\right )^3}{3 d^3}+\frac {(e+f x)^3 \left (a+b \coth ^{-1}(c+d x)\right )^3}{3 f}-\frac {6 b^2 f (d e-c f) \left (a+b \coth ^{-1}(c+d x)\right ) \log \left (\frac {2}{1-c-d x}\right )}{d^3}-\frac {b \left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right ) \left (a+b \coth ^{-1}(c+d x)\right )^2 \log \left (\frac {2}{1-c-d x}\right )}{d^3}+\frac {b^3 f^2 \log \left (1-(c+d x)^2\right )}{2 d^3}-\frac {3 b^3 f (d e-c f) \operatorname {PolyLog}\left (2,-\frac {1+c+d x}{1-c-d x}\right )}{d^3}-\frac {b^2 \left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right ) \left (a+b \coth ^{-1}(c+d x)\right ) \operatorname {PolyLog}\left (2,1-\frac {2}{1-c-d x}\right )}{d^3}+\frac {b^3 \left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right ) \operatorname {PolyLog}\left (3,1-\frac {2}{1-c-d x}\right )}{2 d^3} \] Output:

a*b^2*f^2*x/d^2+b^3*f^2*(d*x+c)*arccoth(d*x+c)/d^3-1/2*b*f^2*(a+b*arccoth( 
d*x+c))^2/d^3+3*b*f*(-c*f+d*e)*(a+b*arccoth(d*x+c))^2/d^3+3*b*f*(-c*f+d*e) 
*(d*x+c)*(a+b*arccoth(d*x+c))^2/d^3+1/2*b*f^2*(d*x+c)^2*(a+b*arccoth(d*x+c 
))^2/d^3-1/3*(-c*f+d*e)*(d^2*e^2-2*c*d*e*f+(c^2+3)*f^2)*(a+b*arccoth(d*x+c 
))^3/d^3/f+1/3*(3*d^2*e^2-6*c*d*e*f+(3*c^2+1)*f^2)*(a+b*arccoth(d*x+c))^3/ 
d^3+1/3*(f*x+e)^3*(a+b*arccoth(d*x+c))^3/f-6*b^2*f*(-c*f+d*e)*(a+b*arccoth 
(d*x+c))*ln(2/(-d*x-c+1))/d^3-b*(3*d^2*e^2-6*c*d*e*f+(3*c^2+1)*f^2)*(a+b*a 
rccoth(d*x+c))^2*ln(2/(-d*x-c+1))/d^3+1/2*b^3*f^2*ln(1-(d*x+c)^2)/d^3-3*b^ 
3*f*(-c*f+d*e)*polylog(2,-(d*x+c+1)/(-d*x-c+1))/d^3-b^2*(3*d^2*e^2-6*c*d*e 
*f+(3*c^2+1)*f^2)*(a+b*arccoth(d*x+c))*polylog(2,1-2/(-d*x-c+1))/d^3+1/2*b 
^3*(3*d^2*e^2-6*c*d*e*f+(3*c^2+1)*f^2)*polylog(3,1-2/(-d*x-c+1))/d^3
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 8.79 (sec) , antiderivative size = 2574, normalized size of antiderivative = 4.71 \[ \int (e+f x)^2 \left (a+b \coth ^{-1}(c+d x)\right )^3 \, dx=\text {Result too large to show} \] Input:

Integrate[(e + f*x)^2*(a + b*ArcCoth[c + d*x])^3,x]
 

Output:

(a^2*(a*d^2*e^2 + 3*b*d*e*f - 2*b*c*f^2)*x)/d^2 + (a^2*f*(2*a*d*e + b*f)*x 
^2)/(2*d) + (a^3*f^2*x^3)/3 + a^2*b*x*(3*e^2 + 3*e*f*x + f^2*x^2)*ArcCoth[ 
c + d*x] + ((3*a^2*b*d^2*e^2 - 3*a^2*b*c*d^2*e^2 + 3*a^2*b*d*e*f - 6*a^2*b 
*c*d*e*f + 3*a^2*b*c^2*d*e*f + a^2*b*f^2 - 3*a^2*b*c*f^2 + 3*a^2*b*c^2*f^2 
 - a^2*b*c^3*f^2)*Log[1 - c - d*x])/(2*d^3) + ((3*a^2*b*d^2*e^2 + 3*a^2*b* 
c*d^2*e^2 - 3*a^2*b*d*e*f - 6*a^2*b*c*d*e*f - 3*a^2*b*c^2*d*e*f + a^2*b*f^ 
2 + 3*a^2*b*c*f^2 + 3*a^2*b*c^2*f^2 + a^2*b*c^3*f^2)*Log[1 + c + d*x])/(2* 
d^3) - (6*a*b^2*e*f*(1 - (c + d*x)^2)*(((c + d*x)*ArcCoth[c + d*x])/d^2 - 
(c*(c + d*x)*ArcCoth[c + d*x]^2)/d^2 + ((c + d*x)^2*(1 - (c + d*x)^(-2))*A 
rcCoth[c + d*x]^2)/(2*d^2) - Log[1/((c + d*x)*Sqrt[1 - (c + d*x)^(-2)])]/d 
^2 + (2*c*(ArcCoth[c + d*x]^2/2 + ArcCoth[c + d*x]*Log[1 - E^(-2*ArcCoth[c 
 + d*x])] - PolyLog[2, E^(-2*ArcCoth[c + d*x])]/2))/d^2))/((c + d*x)^2*(1 
- (c + d*x)^(-2))) + (3*a*b^2*e^2*(1 - (c + d*x)^2)*(ArcCoth[c + d*x]*(Arc 
Coth[c + d*x] - (c + d*x)*ArcCoth[c + d*x] + 2*Log[1 - E^(-2*ArcCoth[c + d 
*x])]) - PolyLog[2, E^(-2*ArcCoth[c + d*x])]))/(d*(c + d*x)^2*(1 - (c + d* 
x)^(-2))) + (b^3*e^2*(1 - (c + d*x)^2)*(ArcCoth[c + d*x]^2*(ArcCoth[c + d* 
x] - (c + d*x)*ArcCoth[c + d*x] + 3*Log[1 - E^(-2*ArcCoth[c + d*x])]) - 3* 
ArcCoth[c + d*x]*PolyLog[2, E^(-2*ArcCoth[c + d*x])] - (3*PolyLog[3, E^(-2 
*ArcCoth[c + d*x])])/2))/(d*(c + d*x)^2*(1 - (c + d*x)^(-2))) - (b^3*e*f*( 
1 - (c + d*x)^2)*(ArcCoth[c + d*x]*(-3*ArcCoth[c + d*x] + 2*c*ArcCoth[c...
 

Rubi [A] (verified)

Time = 1.20 (sec) , antiderivative size = 533, normalized size of antiderivative = 0.98, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6662, 27, 6481, 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.

\(\displaystyle \int (e+f x)^2 \left (a+b \coth ^{-1}(c+d x)\right )^3 \, dx\)

\(\Big \downarrow \) 6662

\(\displaystyle \frac {\int \frac {\left (d \left (e-\frac {c f}{d}\right )+f (c+d x)\right )^2 \left (a+b \coth ^{-1}(c+d x)\right )^3}{d^2}d(c+d x)}{d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int (d e-c f+f (c+d x))^2 \left (a+b \coth ^{-1}(c+d x)\right )^3d(c+d x)}{d^3}\)

\(\Big \downarrow \) 6481

\(\displaystyle \frac {\frac {(f (c+d x)-c f+d e)^3 \left (a+b \coth ^{-1}(c+d x)\right )^3}{3 f}-\frac {b \int \left (-\left ((c+d x) \left (a+b \coth ^{-1}(c+d x)\right )^2 f^3\right )-3 (d e-c f) \left (a+b \coth ^{-1}(c+d x)\right )^2 f^2+\frac {\left ((d e-c f) \left (d^2 e^2-2 c d f e+\left (c^2+3\right ) f^2\right )+f \left (3 d^2 e^2-6 c d f e+\left (3 c^2+1\right ) f^2\right ) (c+d x)\right ) \left (a+b \coth ^{-1}(c+d x)\right )^2}{1-(c+d x)^2}\right )d(c+d x)}{f}}{d^3}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\frac {(f (c+d x)-c f+d e)^3 \left (a+b \coth ^{-1}(c+d x)\right )^3}{3 f}-\frac {b \left (b f \left (\left (3 c^2+1\right ) f^2-6 c d e f+3 d^2 e^2\right ) \operatorname {PolyLog}\left (2,1-\frac {2}{-c-d x+1}\right ) \left (a+b \coth ^{-1}(c+d x)\right )-\frac {f \left (\left (3 c^2+1\right ) f^2-6 c d e f+3 d^2 e^2\right ) \left (a+b \coth ^{-1}(c+d x)\right )^3}{3 b}+\frac {(d e-c f) \left (\left (c^2+3\right ) f^2-2 c d e f+d^2 e^2\right ) \left (a+b \coth ^{-1}(c+d x)\right )^3}{3 b}+f \left (\left (3 c^2+1\right ) f^2-6 c d e f+3 d^2 e^2\right ) \log \left (\frac {2}{-c-d x+1}\right ) \left (a+b \coth ^{-1}(c+d x)\right )^2-3 f^2 (d e-c f) \left (a+b \coth ^{-1}(c+d x)\right )^2-3 f^2 (c+d x) (d e-c f) \left (a+b \coth ^{-1}(c+d x)\right )^2+6 b f^2 (d e-c f) \log \left (\frac {2}{-c-d x+1}\right ) \left (a+b \coth ^{-1}(c+d x)\right )-a b f^3 (c+d x)-\frac {1}{2} f^3 (c+d x)^2 \left (a+b \coth ^{-1}(c+d x)\right )^2+\frac {1}{2} f^3 \left (a+b \coth ^{-1}(c+d x)\right )^2-\frac {1}{2} b^2 f \left (\left (3 c^2+1\right ) f^2-6 c d e f+3 d^2 e^2\right ) \operatorname {PolyLog}\left (3,1-\frac {2}{-c-d x+1}\right )+3 b^2 f^2 (d e-c f) \operatorname {PolyLog}\left (2,-\frac {c+d x+1}{-c-d x+1}\right )-\frac {1}{2} b^2 f^3 \log \left (1-(c+d x)^2\right )-b^2 f^3 (c+d x) \coth ^{-1}(c+d x)\right )}{f}}{d^3}\)

Input:

Int[(e + f*x)^2*(a + b*ArcCoth[c + d*x])^3,x]
 

Output:

(((d*e - c*f + f*(c + d*x))^3*(a + b*ArcCoth[c + d*x])^3)/(3*f) - (b*(-(a* 
b*f^3*(c + d*x)) - b^2*f^3*(c + d*x)*ArcCoth[c + d*x] + (f^3*(a + b*ArcCot 
h[c + d*x])^2)/2 - 3*f^2*(d*e - c*f)*(a + b*ArcCoth[c + d*x])^2 - 3*f^2*(d 
*e - c*f)*(c + d*x)*(a + b*ArcCoth[c + d*x])^2 - (f^3*(c + d*x)^2*(a + b*A 
rcCoth[c + d*x])^2)/2 + ((d*e - c*f)*(d^2*e^2 - 2*c*d*e*f + (3 + c^2)*f^2) 
*(a + b*ArcCoth[c + d*x])^3)/(3*b) - (f*(3*d^2*e^2 - 6*c*d*e*f + (1 + 3*c^ 
2)*f^2)*(a + b*ArcCoth[c + d*x])^3)/(3*b) + 6*b*f^2*(d*e - c*f)*(a + b*Arc 
Coth[c + d*x])*Log[2/(1 - c - d*x)] + f*(3*d^2*e^2 - 6*c*d*e*f + (1 + 3*c^ 
2)*f^2)*(a + b*ArcCoth[c + d*x])^2*Log[2/(1 - c - d*x)] - (b^2*f^3*Log[1 - 
 (c + d*x)^2])/2 + 3*b^2*f^2*(d*e - c*f)*PolyLog[2, -((1 + c + d*x)/(1 - c 
 - d*x))] + b*f*(3*d^2*e^2 - 6*c*d*e*f + (1 + 3*c^2)*f^2)*(a + b*ArcCoth[c 
 + d*x])*PolyLog[2, 1 - 2/(1 - c - d*x)] - (b^2*f*(3*d^2*e^2 - 6*c*d*e*f + 
 (1 + 3*c^2)*f^2)*PolyLog[3, 1 - 2/(1 - c - d*x)])/2))/f)/d^3
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2009

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

rule 6481

Int[((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_)*((d_) + (e_.)*(x_))^(q_.), x_S 
ymbol] :> Simp[(d + e*x)^(q + 1)*((a + b*ArcCoth[c*x])^p/(e*(q + 1))), x] - 
 Simp[b*c*(p/(e*(q + 1)))   Int[ExpandIntegrand[(a + b*ArcCoth[c*x])^(p - 1 
), (d + e*x)^(q + 1)/(1 - c^2*x^2), x], x], x] /; FreeQ[{a, b, c, d, e}, x] 
 && IGtQ[p, 1] && IntegerQ[q] && NeQ[q, -1]
 

rule 6662

Int[((a_.) + ArcCoth[(c_) + (d_.)*(x_)]*(b_.))^(p_.)*((e_.) + (f_.)*(x_))^( 
m_.), x_Symbol] :> Simp[1/d   Subst[Int[((d*e - c*f)/d + f*(x/d))^m*(a + b* 
ArcCoth[x])^p, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IG 
tQ[p, 0]
 

Maple [C] (warning: unable to verify)

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

Time = 26.44 (sec) , antiderivative size = 8597, normalized size of antiderivative = 15.75

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(8597\)
default	\(\text {Expression too large to display}\)	\(8597\)
parts	\(\text {Expression too large to display}\)	\(8610\)

Input:

int((f*x+e)^2*(a+b*arccoth(d*x+c))^3,x,method=_RETURNVERBOSE)
 

Output:

result too large to display
 

Fricas [F]

\[ \int (e+f x)^2 \left (a+b \coth ^{-1}(c+d x)\right )^3 \, dx=\int { {\left (f x + e\right )}^{2} {\left (b \operatorname {arcoth}\left (d x + c\right ) + a\right )}^{3} \,d x } \] Input:

integrate((f*x+e)^2*(a+b*arccoth(d*x+c))^3,x, algorithm="fricas")
 

Output:

integral(a^3*f^2*x^2 + 2*a^3*e*f*x + a^3*e^2 + (b^3*f^2*x^2 + 2*b^3*e*f*x 
+ b^3*e^2)*arccoth(d*x + c)^3 + 3*(a*b^2*f^2*x^2 + 2*a*b^2*e*f*x + a*b^2*e 
^2)*arccoth(d*x + c)^2 + 3*(a^2*b*f^2*x^2 + 2*a^2*b*e*f*x + a^2*b*e^2)*arc 
coth(d*x + c), x)
 

Sympy [F]

\[ \int (e+f x)^2 \left (a+b \coth ^{-1}(c+d x)\right )^3 \, dx=\int \left (a + b \operatorname {acoth}{\left (c + d x \right )}\right )^{3} \left (e + f x\right )^{2}\, dx \] Input:

integrate((f*x+e)**2*(a+b*acoth(d*x+c))**3,x)
 

Output:

Integral((a + b*acoth(c + d*x))**3*(e + f*x)**2, x)
 

Maxima [F]

\[ \int (e+f x)^2 \left (a+b \coth ^{-1}(c+d x)\right )^3 \, dx=\int { {\left (f x + e\right )}^{2} {\left (b \operatorname {arcoth}\left (d x + c\right ) + a\right )}^{3} \,d x } \] Input:

integrate((f*x+e)^2*(a+b*arccoth(d*x+c))^3,x, algorithm="maxima")
 

Output:

1/3*a^3*f^2*x^3 + a^3*e*f*x^2 + 3/2*(2*x^2*arccoth(d*x + c) + d*(2*x/d^2 - 
 (c^2 + 2*c + 1)*log(d*x + c + 1)/d^3 + (c^2 - 2*c + 1)*log(d*x + c - 1)/d 
^3))*a^2*b*e*f + 1/2*(2*x^3*arccoth(d*x + c) + d*((d*x^2 - 4*c*x)/d^3 + (c 
^3 + 3*c^2 + 3*c + 1)*log(d*x + c + 1)/d^4 - (c^3 - 3*c^2 + 3*c - 1)*log(d 
*x + c - 1)/d^4))*a^2*b*f^2 + a^3*e^2*x + 3/2*(2*(d*x + c)*arccoth(d*x + c 
) + log(-(d*x + c)^2 + 1))*a^2*b*e^2/d + 1/24*((b^3*d^3*f^2*x^3 + 3*b^3*d^ 
3*e*f*x^2 + 3*b^3*d^3*e^2*x + (c^3*f^2 + 3*d^2*e^2 - 3*(d*e*f - f^2)*c^2 - 
 3*d*e*f + 3*(d^2*e^2 - 2*d*e*f + f^2)*c + f^2)*b^3)*log(d*x + c + 1)^3 + 
3*(2*a*b^2*d^3*f^2*x^3 + (6*a*b^2*d^3*e*f + b^3*d^2*f^2)*x^2 + 2*(3*a*b^2* 
d^3*e^2 + (3*d^2*e*f - 2*c*d*f^2)*b^3)*x - (b^3*d^3*f^2*x^3 + 3*b^3*d^3*e* 
f*x^2 + 3*b^3*d^3*e^2*x + (c^3*f^2 - 3*d^2*e^2 - 3*(d*e*f + f^2)*c^2 - 3*d 
*e*f + 3*(d^2*e^2 + 2*d*e*f + f^2)*c - f^2)*b^3)*log(d*x + c - 1))*log(d*x 
 + c + 1)^2)/d^3 + integrate(-1/8*((b^3*d^3*f^2*x^3 + (2*d^3*e*f + c*d^2*f 
^2 + d^2*f^2)*b^3*x^2 + (d^3*e^2 + 2*c*d^2*e*f + 2*d^2*e*f)*b^3*x + (c*d^2 
*e^2 + d^2*e^2)*b^3)*log(d*x + c - 1)^3 - 6*(a*b^2*d^3*f^2*x^3 + (2*d^3*e* 
f + c*d^2*f^2 + d^2*f^2)*a*b^2*x^2 + (d^3*e^2 + 2*c*d^2*e*f + 2*d^2*e*f)*a 
*b^2*x + (c*d^2*e^2 + d^2*e^2)*a*b^2)*log(d*x + c - 1)^2 + (4*a*b^2*d^3*f^ 
2*x^3 + 2*(6*a*b^2*d^3*e*f + b^3*d^2*f^2)*x^2 - 3*(b^3*d^3*f^2*x^3 + (2*d^ 
3*e*f + c*d^2*f^2 + d^2*f^2)*b^3*x^2 + (d^3*e^2 + 2*c*d^2*e*f + 2*d^2*e*f) 
*b^3*x + (c*d^2*e^2 + d^2*e^2)*b^3)*log(d*x + c - 1)^2 + 4*(3*a*b^2*d^3...
 

Giac [F]

\[ \int (e+f x)^2 \left (a+b \coth ^{-1}(c+d x)\right )^3 \, dx=\int { {\left (f x + e\right )}^{2} {\left (b \operatorname {arcoth}\left (d x + c\right ) + a\right )}^{3} \,d x } \] Input:

integrate((f*x+e)^2*(a+b*arccoth(d*x+c))^3,x, algorithm="giac")
 

Output:

integrate((f*x + e)^2*(b*arccoth(d*x + c) + a)^3, x)
 

Mupad [F(-1)]

Timed out. \[ \int (e+f x)^2 \left (a+b \coth ^{-1}(c+d x)\right )^3 \, dx=\int {\left (e+f\,x\right )}^2\,{\left (a+b\,\mathrm {acoth}\left (c+d\,x\right )\right )}^3 \,d x \] Input:

int((e + f*x)^2*(a + b*acoth(c + d*x))^3,x)
 

Output:

int((e + f*x)^2*(a + b*acoth(c + d*x))^3, x)
 

Reduce [F]

\[ \int (e+f x)^2 \left (a+b \coth ^{-1}(c+d x)\right )^3 \, dx =\text {Too large to display} \] Input:

int((f*x+e)^2*(a+b*acoth(d*x+c))^3,x)
                                                                                    
                                                                                    
 

Output:

( - 4*acoth(c + d*x)**3*b**3*c**3*f**2 + 6*acoth(c + d*x)**3*b**3*c**2*d*e 
*f + 4*acoth(c + d*x)**3*b**3*c*f**2 + 6*acoth(c + d*x)**3*b**3*d**3*e**2* 
x + 6*acoth(c + d*x)**3*b**3*d**3*e*f*x**2 + 2*acoth(c + d*x)**3*b**3*d**3 
*f**2*x**3 - 6*acoth(c + d*x)**3*b**3*d*e*f - 12*acoth(c + d*x)**2*a*b**2* 
c**3*f**2 + 18*acoth(c + d*x)**2*a*b**2*c**2*d*e*f + 12*acoth(c + d*x)**2* 
a*b**2*c*f**2 + 18*acoth(c + d*x)**2*a*b**2*d**3*e**2*x + 18*acoth(c + d*x 
)**2*a*b**2*d**3*e*f*x**2 + 6*acoth(c + d*x)**2*a*b**2*d**3*f**2*x**3 - 18 
*acoth(c + d*x)**2*a*b**2*d*e*f - 3*acoth(c + d*x)**2*b**3*c**2*f**2 + 12* 
acoth(c + d*x)**2*b**3*c*d*f**2*x - 18*acoth(c + d*x)**2*b**3*d**2*e*f*x - 
 3*acoth(c + d*x)**2*b**3*d**2*f**2*x**2 + 3*acoth(c + d*x)**2*b**3*f**2 + 
 6*acoth(c + d*x)*a**2*b*c**3*f**2 - 18*acoth(c + d*x)*a**2*b*c**2*d*e*f + 
 18*acoth(c + d*x)*a**2*b*c**2*f**2 + 18*acoth(c + d*x)*a**2*b*c*d**2*e**2 
 - 36*acoth(c + d*x)*a**2*b*c*d*e*f + 18*acoth(c + d*x)*a**2*b*c*f**2 + 18 
*acoth(c + d*x)*a**2*b*d**3*e**2*x + 18*acoth(c + d*x)*a**2*b*d**3*e*f*x** 
2 + 6*acoth(c + d*x)*a**2*b*d**3*f**2*x**3 + 18*acoth(c + d*x)*a**2*b*d**2 
*e**2 - 18*acoth(c + d*x)*a**2*b*d*e*f + 6*acoth(c + d*x)*a**2*b*f**2 + 30 
*acoth(c + d*x)*a*b**2*c**2*f**2 - 36*acoth(c + d*x)*a*b**2*c*d*e*f + 24*a 
coth(c + d*x)*a*b**2*c*d*f**2*x + 36*acoth(c + d*x)*a*b**2*c*f**2 - 36*aco 
th(c + d*x)*a*b**2*d**2*e*f*x - 6*acoth(c + d*x)*a*b**2*d**2*f**2*x**2 - 3 
6*acoth(c + d*x)*a*b**2*d*e*f + 6*acoth(c + d*x)*a*b**2*f**2 + 6*acoth(...