\(\int (e+f x)^2 (a+b \coth ^{-1}(c+d x))^2 \, dx\) [28]

Optimal result

Integrand size = 20, antiderivative size = 374 \[ \int (e+f x)^2 \left (a+b \coth ^{-1}(c+d x)\right )^2 \, dx=\frac {b^2 f^2 x}{3 d^2}+\frac {2 a b f (d e-c f) x}{d^2}+\frac {2 b^2 f (d e-c f) (c+d x) \coth ^{-1}(c+d x)}{d^3}+\frac {b f^2 (c+d x)^2 \left (a+b \coth ^{-1}(c+d x)\right )}{3 d^3}-\frac {(d e-c f) \left (d^2 e^2-2 c d e f+\left (3+c^2\right ) f^2\right ) \left (a+b \coth ^{-1}(c+d x)\right )^2}{3 d^3 f}+\frac {\left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right ) \left (a+b \coth ^{-1}(c+d x)\right )^2}{3 d^3}+\frac {(e+f x)^3 \left (a+b \coth ^{-1}(c+d x)\right )^2}{3 f}-\frac {b^2 f^2 \text {arctanh}(c+d x)}{3 d^3}-\frac {2 b \left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right ) \left (a+b \coth ^{-1}(c+d x)\right ) \log \left (\frac {2}{1-c-d x}\right )}{3 d^3}+\frac {b^2 f (d e-c f) \log \left (1-(c+d x)^2\right )}{d^3}-\frac {b^2 \left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right ) \operatorname {PolyLog}\left (2,-\frac {1+c+d x}{1-c-d x}\right )}{3 d^3} \] Output:

1/3*b^2*f^2*x/d^2+2*a*b*f*(-c*f+d*e)*x/d^2+2*b^2*f*(-c*f+d*e)*(d*x+c)*arcc 
oth(d*x+c)/d^3+1/3*b*f^2*(d*x+c)^2*(a+b*arccoth(d*x+c))/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))^2/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))^2/d^3+1/3*(f*x+e)^3*(a+b* 
arccoth(d*x+c))^2/f-1/3*b^2*f^2*arctanh(d*x+c)/d^3-2/3*b*(3*d^2*e^2-6*c*d* 
e*f+(3*c^2+1)*f^2)*(a+b*arccoth(d*x+c))*ln(2/(-d*x-c+1))/d^3+b^2*f*(-c*f+d 
*e)*ln(1-(d*x+c)^2)/d^3-1/3*b^2*(3*d^2*e^2-6*c*d*e*f+(3*c^2+1)*f^2)*polylo 
g(2,-(d*x+c+1)/(-d*x-c+1))/d^3
 

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(1078\) vs. \(2(374)=748\).

Time = 6.91 (sec) , antiderivative size = 1078, normalized size of antiderivative = 2.88 \[ \int (e+f x)^2 \left (a+b \coth ^{-1}(c+d x)\right )^2 \, dx =\text {Too large to display} \] Input:

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

Output:

a^2*e^2*x + a^2*e*f*x^2 + (a^2*f^2*x^3)/3 + (a*b*(2*x*(3*e^2 + 3*e*f*x + f 
^2*x^2)*ArcCoth[c + d*x] + (d*f*x*(6*d*e - 4*c*f + d*f*x) - (-1 + c)*(3*d^ 
2*e^2 - 3*(-1 + c)*d*e*f + (-1 + c)^2*f^2)*Log[1 - c - d*x] + (1 + c)*(3*d 
^2*e^2 - 3*(1 + c)*d*e*f + (1 + c)^2*f^2)*Log[1 + c + d*x])/d^3))/3 - (2*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))*ArcCoth[c + d* 
x]^2)/(2*d^2) - Log[1/((c + d*x)*Sqrt[1 - (c + d*x)^(-2)])]/d^2 + (2*c*(Ar 
cCoth[c + d*x]^2/2 + ArcCoth[c + d*x]*Log[1 - E^(-2*ArcCoth[c + d*x])] - P 
olyLog[2, E^(-2*ArcCoth[c + d*x])]/2))/d^2))/((c + d*x)^2*(1 - (c + d*x)^( 
-2))) + (b^2*e^2*(1 - (c + d*x)^2)*(ArcCoth[c + d*x]*(ArcCoth[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^2* 
f^2*(c + d*x)*Sqrt[1 - (c + d*x)^(-2)]*(1 - (c + d*x)^2)*((4*ArcCoth[c + d 
*x])/((c + d*x)*Sqrt[1 - (c + d*x)^(-2)]) + (3*ArcCoth[c + d*x]^2)/((c + d 
*x)*Sqrt[1 - (c + d*x)^(-2)]) - (12*c*ArcCoth[c + d*x]^2)/((c + d*x)*Sqrt[ 
1 - (c + d*x)^(-2)]) + (9*c^2*ArcCoth[c + d*x]^2)/((c + d*x)*Sqrt[1 - (c + 
 d*x)^(-2)]) + (-1 + 6*c*ArcCoth[c + d*x] + 3*ArcCoth[c + d*x]^2 - 3*c^2*A 
rcCoth[c + d*x]^2)/Sqrt[1 - (c + d*x)^(-2)] + Cosh[3*ArcCoth[c + d*x]] - 6 
*c*ArcCoth[c + d*x]*Cosh[3*ArcCoth[c + d*x]] + ArcCoth[c + d*x]^2*Cosh[3*A 
rcCoth[c + d*x]] + 3*c^2*ArcCoth[c + d*x]^2*Cosh[3*ArcCoth[c + d*x]] + ...
 

Rubi [A] (verified)

Time = 0.82 (sec) , antiderivative size = 374, normalized size of antiderivative = 1.00, 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.

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

\(\Big \downarrow \) 6662

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 6481

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(((d*e - c*f + f*(c + d*x))^3*(a + b*ArcCoth[c + d*x])^2)/(3*f) - (2*b*(-1 
/2*(b*f^3*(c + d*x)) - 3*a*f^2*(d*e - c*f)*(c + d*x) - 3*b*f^2*(d*e - c*f) 
*(c + d*x)*ArcCoth[c + d*x] - (f^3*(c + d*x)^2*(a + b*ArcCoth[c + d*x]))/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])^2)/(2*b) - (f*(3*d^2*e^2 - 6*c*d*e*f + (1 + 3*c^2)*f^2)*(a + b*ArcCot 
h[c + d*x])^2)/(2*b) + (b*f^3*ArcTanh[c + d*x])/2 + f*(3*d^2*e^2 - 6*c*d*e 
*f + (1 + 3*c^2)*f^2)*(a + b*ArcCoth[c + d*x])*Log[2/(1 - c - d*x)] - (3*b 
*f^2*(d*e - c*f)*Log[1 - (c + d*x)^2])/2 + (b*f*(3*d^2*e^2 - 6*c*d*e*f + ( 
1 + 3*c^2)*f^2)*PolyLog[2, -((1 + c + d*x)/(1 - c - d*x))])/2))/(3*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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1410\) vs. \(2(358)=716\).

Time = 0.85 (sec) , antiderivative size = 1411, normalized size of antiderivative = 3.77

method	result	size
parts	\(\text {Expression too large to display}\)	\(1411\)
derivativedivides	\(\text {Expression too large to display}\)	\(1412\)
default	\(\text {Expression too large to display}\)	\(1412\)
risch	\(\text {Expression too large to display}\)	\(1685\)

Input:

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

Output:

1/3*a^2*(f*x+e)^3/f+b^2/d*(1/3/d^2*f^2*arccoth(d*x+c)^2*(d*x+c)^3-1/d^2*f^ 
2*arccoth(d*x+c)^2*(d*x+c)^2*c+1/d*f*arccoth(d*x+c)^2*(d*x+c)^2*e+1/d^2*f^ 
2*arccoth(d*x+c)^2*(d*x+c)*c^2-2/d*f*arccoth(d*x+c)^2*(d*x+c)*c*e+arccoth( 
d*x+c)^2*(d*x+c)*e^2-1/3/d^2*f^2*arccoth(d*x+c)^2*c^3+1/d*f*arccoth(d*x+c) 
^2*c^2*e-arccoth(d*x+c)^2*c*e^2+1/3*d/f*arccoth(d*x+c)^2*e^3+2/3/d^2/f*(1/ 
2*arccoth(d*x+c)*f^3*(d*x+c)^2+1/2*arccoth(d*x+c)*ln(d*x+c-1)*f^3+1/2*arcc 
oth(d*x+c)*ln(d*x+c+1)*f^3-3*arccoth(d*x+c)*c*f^3*(d*x+c)-1/2*arccoth(d*x+ 
c)*ln(d*x+c-1)*c^3*f^3+1/2*arccoth(d*x+c)*ln(d*x+c-1)*d^3*e^3+3/2*arccoth( 
d*x+c)*ln(d*x+c-1)*c^2*f^3-3/2*arccoth(d*x+c)*ln(d*x+c-1)*c*f^3+1/2*arccot 
h(d*x+c)*ln(d*x+c+1)*c^3*f^3-1/2*arccoth(d*x+c)*ln(d*x+c+1)*d^3*e^3+3/2*ar 
ccoth(d*x+c)*ln(d*x+c+1)*c^2*f^3+3/2*arccoth(d*x+c)*ln(d*x+c+1)*c*f^3+3/2* 
arccoth(d*x+c)*ln(d*x+c+1)*c*d^2*e^2*f-3*arccoth(d*x+c)*ln(d*x+c+1)*c*d*e* 
f^2+3/2*arccoth(d*x+c)*ln(d*x+c-1)*c^2*d*e*f^2-3/2*arccoth(d*x+c)*ln(d*x+c 
-1)*c*d^2*e^2*f-3*arccoth(d*x+c)*ln(d*x+c-1)*c*d*e*f^2-3/2*arccoth(d*x+c)* 
ln(d*x+c+1)*c^2*d*e*f^2+3*arccoth(d*x+c)*d*e*f^2*(d*x+c)+3/2*arccoth(d*x+c 
)*ln(d*x+c-1)*d^2*e^2*f+3/2*arccoth(d*x+c)*ln(d*x+c-1)*d*e*f^2+3/2*arccoth 
(d*x+c)*ln(d*x+c+1)*d^2*e^2*f-3/2*arccoth(d*x+c)*ln(d*x+c+1)*d*e*f^2+1/2*f 
^2*(f*(d*x+c)+1/2*(-6*c*f+6*d*e+f)*ln(d*x+c-1)-1/2*(6*c*f-6*d*e+f)*ln(d*x+ 
c+1))+1/2*(-c^3*f^3+3*c^2*d*e*f^2-3*c*d^2*e^2*f+d^3*e^3+3*c^2*f^3-6*c*d*e* 
f^2+3*d^2*e^2*f-3*c*f^3+3*d*e*f^2+f^3)*(1/4*ln(d*x+c-1)^2-1/2*dilog(1/2...
 

Fricas [F]

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

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

Output:

integral(a^2*f^2*x^2 + 2*a^2*e*f*x + a^2*e^2 + (b^2*f^2*x^2 + 2*b^2*e*f*x 
+ b^2*e^2)*arccoth(d*x + c)^2 + 2*(a*b*f^2*x^2 + 2*a*b*e*f*x + a*b*e^2)*ar 
ccoth(d*x + c), x)
 

Sympy [F]

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

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

Output:

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 791 vs. \(2 (350) = 700\).

Time = 0.24 (sec) , antiderivative size = 791, normalized size of antiderivative = 2.11 \[ \int (e+f x)^2 \left (a+b \coth ^{-1}(c+d x)\right )^2 \, dx =\text {Too large to display} \] Input:

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

Output:

1/3*a^2*f^2*x^3 + a^2*e*f*x^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*b*e*f + 1/3*(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*b*f^2 + a^2*e^2*x + (2*(d*x + c)*arccoth(d*x + c) + log(-(d* 
x + c)^2 + 1))*a*b*e^2/d - 1/3*(3*d^2*e^2 - 6*c*d*e*f + 3*c^2*f^2 + f^2)*( 
log(d*x + c - 1)*log(1/2*d*x + 1/2*c + 1/2) + dilog(-1/2*d*x - 1/2*c + 1/2 
))*b^2/d^3 - 1/6*(5*c^2*f^2 - 6*d*e*f - 6*(d*e*f - f^2)*c + f^2)*b^2*log(d 
*x + c + 1)/d^3 + 1/12*(4*b^2*d*f^2*x + (b^2*d^3*f^2*x^3 + 3*b^2*d^3*e*f*x 
^2 + 3*b^2*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^2)*log(d*x + c + 1)^2 + (b^2*d^ 
3*f^2*x^3 + 3*b^2*d^3*e*f*x^2 + 3*b^2*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^2)*l 
og(d*x + c - 1)^2 + 2*(b^2*d^2*f^2*x^2 + 2*(3*d^2*e*f - 2*c*d*f^2)*b^2*x - 
 (b^2*d^3*f^2*x^3 + 3*b^2*d^3*e*f*x^2 + 3*b^2*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^2)*log(d*x + c - 1))*log(d*x + c + 1) - 2*(b^2*d^2*f^2*x^2 + 2*(3*d^2* 
e*f - 2*c*d*f^2)*b^2*x - (5*c^2*f^2 + 6*d*e*f - 6*(d*e*f + f^2)*c + f^2)*b 
^2)*log(d*x + c - 1))/d^3
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

( - 2*acoth(c + d*x)**2*b**2*c**3*f**2 + 3*acoth(c + d*x)**2*b**2*c**2*d*e 
*f + 2*acoth(c + d*x)**2*b**2*c*f**2 + 3*acoth(c + d*x)**2*b**2*d**3*e**2* 
x + 3*acoth(c + d*x)**2*b**2*d**3*e*f*x**2 + acoth(c + d*x)**2*b**2*d**3*f 
**2*x**3 - 3*acoth(c + d*x)**2*b**2*d*e*f + 2*acoth(c + d*x)*a*b*c**3*f**2 
 - 6*acoth(c + d*x)*a*b*c**2*d*e*f + 6*acoth(c + d*x)*a*b*c**2*f**2 + 6*ac 
oth(c + d*x)*a*b*c*d**2*e**2 - 12*acoth(c + d*x)*a*b*c*d*e*f + 6*acoth(c + 
 d*x)*a*b*c*f**2 + 6*acoth(c + d*x)*a*b*d**3*e**2*x + 6*acoth(c + d*x)*a*b 
*d**3*e*f*x**2 + 2*acoth(c + d*x)*a*b*d**3*f**2*x**3 + 6*acoth(c + d*x)*a* 
b*d**2*e**2 - 6*acoth(c + d*x)*a*b*d*e*f + 2*acoth(c + d*x)*a*b*f**2 + 5*a 
coth(c + d*x)*b**2*c**2*f**2 - 6*acoth(c + d*x)*b**2*c*d*e*f + 4*acoth(c + 
 d*x)*b**2*c*d*f**2*x + 6*acoth(c + d*x)*b**2*c*f**2 - 6*acoth(c + d*x)*b* 
*2*d**2*e*f*x - acoth(c + d*x)*b**2*d**2*f**2*x**2 - 6*acoth(c + d*x)*b**2 
*d*e*f + acoth(c + d*x)*b**2*f**2 - 6*int((acoth(c + d*x)*x)/(c**2 + 2*c*d 
*x + d**2*x**2 - 1),x)*b**2*c**2*d**2*f**2 + 12*int((acoth(c + d*x)*x)/(c* 
*2 + 2*c*d*x + d**2*x**2 - 1),x)*b**2*c*d**3*e*f - 6*int((acoth(c + d*x)*x 
)/(c**2 + 2*c*d*x + d**2*x**2 - 1),x)*b**2*d**4*e**2 - 2*int((acoth(c + d* 
x)*x)/(c**2 + 2*c*d*x + d**2*x**2 - 1),x)*b**2*d**2*f**2 - 6*log(c + d*x - 
 1)*a*b*c**2*f**2 + 12*log(c + d*x - 1)*a*b*c*d*e*f - 6*log(c + d*x - 1)*a 
*b*d**2*e**2 - 2*log(c + d*x - 1)*a*b*f**2 - 6*log(c + d*x - 1)*b**2*c*f** 
2 + 6*log(c + d*x - 1)*b**2*d*e*f + 3*a**2*d**3*e**2*x + 3*a**2*d**3*e*...