Integrand size = 12, antiderivative size = 85 \[ \int \frac {1}{(a+b \tanh (c+d x))^2} \, dx=\frac {\left (a^2+b^2\right ) x}{\left (a^2-b^2\right )^2}-\frac {2 a b \log (a \cosh (c+d x)+b \sinh (c+d x))}{\left (a^2-b^2\right )^2 d}+\frac {b}{\left (a^2-b^2\right ) d (a+b \tanh (c+d x))} \] Output:
(a^2+b^2)*x/(a^2-b^2)^2-2*a*b*ln(a*cosh(d*x+c)+b*sinh(d*x+c))/(a^2-b^2)^2/ d+b/(a^2-b^2)/d/(a+b*tanh(d*x+c))
Time = 0.71 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.11 \[ \int \frac {1}{(a+b \tanh (c+d x))^2} \, dx=\frac {-\frac {\log (1-\tanh (c+d x))}{(a+b)^2}+\frac {\log (1+\tanh (c+d x))}{(a-b)^2}+\frac {2 b \left (-2 a \log (a+b \tanh (c+d x))+\frac {a^2-b^2}{a+b \tanh (c+d x)}\right )}{\left (a^2-b^2\right )^2}}{2 d} \] Input:
Integrate[(a + b*Tanh[c + d*x])^(-2),x]
Output:
(-(Log[1 - Tanh[c + d*x]]/(a + b)^2) + Log[1 + Tanh[c + d*x]]/(a - b)^2 + (2*b*(-2*a*Log[a + b*Tanh[c + d*x]] + (a^2 - b^2)/(a + b*Tanh[c + d*x])))/ (a^2 - b^2)^2)/(2*d)
Time = 0.50 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.15, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {3042, 3964, 3042, 4014, 26, 3042, 4013}
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.
| \(\displaystyle \int \frac {1}{(a+b \tanh (c+d x))^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(a-i b \tan (i c+i d x))^2}dx\) |
| \(\Big \downarrow \) 3964 |
| \(\displaystyle \frac {\int \frac {a-b \tanh (c+d x)}{a+b \tanh (c+d x)}dx}{a^2-b^2}+\frac {b}{d \left (a^2-b^2\right ) (a+b \tanh (c+d x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {b}{d \left (a^2-b^2\right ) (a+b \tanh (c+d x))}+\frac {\int \frac {a+i b \tan (i c+i d x)}{a-i b \tan (i c+i d x)}dx}{a^2-b^2}\) |
| \(\Big \downarrow \) 4014 |
| \(\displaystyle \frac {b}{d \left (a^2-b^2\right ) (a+b \tanh (c+d x))}+\frac {\frac {x \left (a^2+b^2\right )}{a^2-b^2}-\frac {2 i a b \int -\frac {i (b+a \tanh (c+d x))}{a+b \tanh (c+d x)}dx}{a^2-b^2}}{a^2-b^2}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {\frac {x \left (a^2+b^2\right )}{a^2-b^2}-\frac {2 a b \int \frac {b+a \tanh (c+d x)}{a+b \tanh (c+d x)}dx}{a^2-b^2}}{a^2-b^2}+\frac {b}{d \left (a^2-b^2\right ) (a+b \tanh (c+d x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {b}{d \left (a^2-b^2\right ) (a+b \tanh (c+d x))}+\frac {\frac {x \left (a^2+b^2\right )}{a^2-b^2}-\frac {2 a b \int \frac {b-i a \tan (i c+i d x)}{a-i b \tan (i c+i d x)}dx}{a^2-b^2}}{a^2-b^2}\) |
| \(\Big \downarrow \) 4013 |
| \(\displaystyle \frac {b}{d \left (a^2-b^2\right ) (a+b \tanh (c+d x))}+\frac {\frac {x \left (a^2+b^2\right )}{a^2-b^2}-\frac {2 a b \log (a \cosh (c+d x)+b \sinh (c+d x))}{d \left (a^2-b^2\right )}}{a^2-b^2}\) |
Input:
Int[(a + b*Tanh[c + d*x])^(-2),x]
Output:
(((a^2 + b^2)*x)/(a^2 - b^2) - (2*a*b*Log[a*Cosh[c + d*x] + b*Sinh[c + d*x ]])/((a^2 - b^2)*d))/(a^2 - b^2) + b/((a^2 - b^2)*d*(a + b*Tanh[c + d*x]))
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*((a + b*Tan[c + d*x])^(n + 1)/(d*(n + 1)*(a^2 + b^2))), x] + Simp[1/(a^2 + b^2) Int[(a - b*Tan[c + d*x])*(a + b*Tan[c + d*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && LtQ[n, -1]
Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/((a_) + (b_.)*tan[(e_.) + (f_.)* (x_)]), x_Symbol] :> Simp[(c/(b*f))*Log[RemoveContent[a*Cos[e + f*x] + b*Si n[e + f*x], x]], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && EqQ[a*c + b*d, 0]
Int[((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])/((a_.) + (b_.)*tan[(e_.) + (f_. )*(x_)]), x_Symbol] :> Simp[(a*c + b*d)*(x/(a^2 + b^2)), x] + Simp[(b*c - a *d)/(a^2 + b^2) Int[(b - a*Tan[e + f*x])/(a + b*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && N eQ[a*c + b*d, 0]
Time = 0.23 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.09
| method | result | size |
| derivativedivides | \(\frac {-\frac {\ln \left (\tanh \left (d x +c \right )-1\right )}{2 \left (a +b \right )^{2}}+\frac {\ln \left (\tanh \left (d x +c \right )+1\right )}{2 \left (a -b \right )^{2}}+\frac {b}{\left (a -b \right ) \left (a +b \right ) \left (a +b \tanh \left (d x +c \right )\right )}-\frac {2 a b \ln \left (a +b \tanh \left (d x +c \right )\right )}{\left (a +b \right )^{2} \left (a -b \right )^{2}}}{d}\) | \(93\) |
| default | \(\frac {-\frac {\ln \left (\tanh \left (d x +c \right )-1\right )}{2 \left (a +b \right )^{2}}+\frac {\ln \left (\tanh \left (d x +c \right )+1\right )}{2 \left (a -b \right )^{2}}+\frac {b}{\left (a -b \right ) \left (a +b \right ) \left (a +b \tanh \left (d x +c \right )\right )}-\frac {2 a b \ln \left (a +b \tanh \left (d x +c \right )\right )}{\left (a +b \right )^{2} \left (a -b \right )^{2}}}{d}\) | \(93\) |
| risch | \(\frac {x}{a^{2}+2 a b +b^{2}}+\frac {4 a b x}{a^{4}-2 a^{2} b^{2}+b^{4}}+\frac {4 a b c}{d \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}+\frac {2 b^{2}}{\left (a -b \right ) d \left (a^{2}+2 a b +b^{2}\right ) \left ({\mathrm e}^{2 d x +2 c} a +{\mathrm e}^{2 d x +2 c} b +a -b \right )}-\frac {2 a b \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {a -b}{a +b}\right )}{d \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}\) | \(167\) |
| parallelrisch | \(-\frac {-2 \ln \left (1-\tanh \left (d x +c \right )\right ) \tanh \left (d x +c \right ) a^{2} b^{2}+2 \ln \left (a +b \tanh \left (d x +c \right )\right ) \tanh \left (d x +c \right ) a^{2} b^{2}-b^{4} \tanh \left (d x +c \right )-a^{2} b^{2} d x +a^{2} b^{2} \tanh \left (d x +c \right )-x \tanh \left (d x +c \right ) a \,b^{3} d -2 a^{3} b d x -a^{4} d x -2 \ln \left (1-\tanh \left (d x +c \right )\right ) a^{3} b +2 \ln \left (a +b \tanh \left (d x +c \right )\right ) a^{3} b -2 x \tanh \left (d x +c \right ) a^{2} b^{2} d -x \tanh \left (d x +c \right ) a^{3} b d}{\left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \left (a +b \tanh \left (d x +c \right )\right ) a d}\) | \(216\) |
Input:
int(1/(a+b*tanh(d*x+c))^2,x,method=_RETURNVERBOSE)
Output:
1/d*(-1/2/(a+b)^2*ln(tanh(d*x+c)-1)+1/2/(a-b)^2*ln(tanh(d*x+c)+1)+b/(a-b)/ (a+b)/(a+b*tanh(d*x+c))-2*a*b/(a+b)^2/(a-b)^2*ln(a+b*tanh(d*x+c)))
Leaf count of result is larger than twice the leaf count of optimal. 422 vs. \(2 (85) = 170\).
Time = 0.09 (sec) , antiderivative size = 422, normalized size of antiderivative = 4.96 \[ \int \frac {1}{(a+b \tanh (c+d x))^2} \, dx=\frac {{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} d x \cosh \left (d x + c\right )^{2} + 2 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} d x \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} d x \sinh \left (d x + c\right )^{2} + 2 \, a b^{2} - 2 \, b^{3} + {\left (a^{3} + a^{2} b - a b^{2} - b^{3}\right )} d x - 2 \, {\left (a^{2} b - a b^{2} + {\left (a^{2} b + a b^{2}\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (a^{2} b + a b^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + {\left (a^{2} b + a b^{2}\right )} \sinh \left (d x + c\right )^{2}\right )} \log \left (\frac {2 \, {\left (a \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right )\right )}}{\cosh \left (d x + c\right ) - \sinh \left (d x + c\right )}\right )}{{\left (a^{5} + a^{4} b - 2 \, a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4} + b^{5}\right )} d \cosh \left (d x + c\right )^{2} + 2 \, {\left (a^{5} + a^{4} b - 2 \, a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4} + b^{5}\right )} d \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + {\left (a^{5} + a^{4} b - 2 \, a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4} + b^{5}\right )} d \sinh \left (d x + c\right )^{2} + {\left (a^{5} - a^{4} b - 2 \, a^{3} b^{2} + 2 \, a^{2} b^{3} + a b^{4} - b^{5}\right )} d} \] Input:
integrate(1/(a+b*tanh(d*x+c))^2,x, algorithm="fricas")
Output:
((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*x*cosh(d*x + c)^2 + 2*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*x*cosh(d*x + c)*sinh(d*x + c) + (a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*x*sinh(d*x + c)^2 + 2*a*b^2 - 2*b^3 + (a^3 + a^2*b - a*b^2 - b^3) *d*x - 2*(a^2*b - a*b^2 + (a^2*b + a*b^2)*cosh(d*x + c)^2 + 2*(a^2*b + a*b ^2)*cosh(d*x + c)*sinh(d*x + c) + (a^2*b + a*b^2)*sinh(d*x + c)^2)*log(2*( a*cosh(d*x + c) + b*sinh(d*x + c))/(cosh(d*x + c) - sinh(d*x + c))))/((a^5 + a^4*b - 2*a^3*b^2 - 2*a^2*b^3 + a*b^4 + b^5)*d*cosh(d*x + c)^2 + 2*(a^5 + a^4*b - 2*a^3*b^2 - 2*a^2*b^3 + a*b^4 + b^5)*d*cosh(d*x + c)*sinh(d*x + c) + (a^5 + a^4*b - 2*a^3*b^2 - 2*a^2*b^3 + a*b^4 + b^5)*d*sinh(d*x + c)^ 2 + (a^5 - a^4*b - 2*a^3*b^2 + 2*a^2*b^3 + a*b^4 - b^5)*d)
Leaf count of result is larger than twice the leaf count of optimal. 1389 vs. \(2 (70) = 140\).
Time = 9.21 (sec) , antiderivative size = 1389, normalized size of antiderivative = 16.34 \[ \int \frac {1}{(a+b \tanh (c+d x))^2} \, dx=\text {Too large to display} \] Input:
integrate(1/(a+b*tanh(d*x+c))**2,x)
Output:
Piecewise((zoo*x/tanh(c)**2, Eq(a, 0) & Eq(b, 0) & Eq(d, 0)), (x/a**2, Eq( b, 0)), (d*x*tanh(c + d*x)**2/(4*b**2*d*tanh(c + d*x)**2 - 8*b**2*d*tanh(c + d*x) + 4*b**2*d) - 2*d*x*tanh(c + d*x)/(4*b**2*d*tanh(c + d*x)**2 - 8*b **2*d*tanh(c + d*x) + 4*b**2*d) + d*x/(4*b**2*d*tanh(c + d*x)**2 - 8*b**2* d*tanh(c + d*x) + 4*b**2*d) - tanh(c + d*x)/(4*b**2*d*tanh(c + d*x)**2 - 8 *b**2*d*tanh(c + d*x) + 4*b**2*d) + 2/(4*b**2*d*tanh(c + d*x)**2 - 8*b**2* d*tanh(c + d*x) + 4*b**2*d), Eq(a, -b)), (d*x*tanh(c + d*x)**2/(4*b**2*d*t anh(c + d*x)**2 + 8*b**2*d*tanh(c + d*x) + 4*b**2*d) + 2*d*x*tanh(c + d*x) /(4*b**2*d*tanh(c + d*x)**2 + 8*b**2*d*tanh(c + d*x) + 4*b**2*d) + d*x/(4* b**2*d*tanh(c + d*x)**2 + 8*b**2*d*tanh(c + d*x) + 4*b**2*d) - tanh(c + d* x)/(4*b**2*d*tanh(c + d*x)**2 + 8*b**2*d*tanh(c + d*x) + 4*b**2*d) - 2/(4* b**2*d*tanh(c + d*x)**2 + 8*b**2*d*tanh(c + d*x) + 4*b**2*d), Eq(a, b)), ( x/(a + b*tanh(c))**2, Eq(d, 0)), (a**3*d*x/(a**5*d + a**4*b*d*tanh(c + d*x ) - 2*a**3*b**2*d - 2*a**2*b**3*d*tanh(c + d*x) + a*b**4*d + b**5*d*tanh(c + d*x)) + a**2*b*d*x*tanh(c + d*x)/(a**5*d + a**4*b*d*tanh(c + d*x) - 2*a **3*b**2*d - 2*a**2*b**3*d*tanh(c + d*x) + a*b**4*d + b**5*d*tanh(c + d*x) ) - 2*a**2*b*d*x/(a**5*d + a**4*b*d*tanh(c + d*x) - 2*a**3*b**2*d - 2*a**2 *b**3*d*tanh(c + d*x) + a*b**4*d + b**5*d*tanh(c + d*x)) - 2*a**2*b*log(a/ b + tanh(c + d*x))/(a**5*d + a**4*b*d*tanh(c + d*x) - 2*a**3*b**2*d - 2*a* *2*b**3*d*tanh(c + d*x) + a*b**4*d + b**5*d*tanh(c + d*x)) + 2*a**2*b*l...
Time = 0.05 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.49 \[ \int \frac {1}{(a+b \tanh (c+d x))^2} \, dx=-\frac {2 \, a b \log \left (-{\left (a - b\right )} e^{\left (-2 \, d x - 2 \, c\right )} - a - b\right )}{{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} d} - \frac {2 \, b^{2}}{{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4} + {\left (a^{4} - 2 \, a^{3} b + 2 \, a b^{3} - b^{4}\right )} e^{\left (-2 \, d x - 2 \, c\right )}\right )} d} + \frac {d x + c}{{\left (a^{2} + 2 \, a b + b^{2}\right )} d} \] Input:
integrate(1/(a+b*tanh(d*x+c))^2,x, algorithm="maxima")
Output:
-2*a*b*log(-(a - b)*e^(-2*d*x - 2*c) - a - b)/((a^4 - 2*a^2*b^2 + b^4)*d) - 2*b^2/((a^4 - 2*a^2*b^2 + b^4 + (a^4 - 2*a^3*b + 2*a*b^3 - b^4)*e^(-2*d* x - 2*c))*d) + (d*x + c)/((a^2 + 2*a*b + b^2)*d)
Time = 0.13 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.55 \[ \int \frac {1}{(a+b \tanh (c+d x))^2} \, dx=-\frac {\frac {2 \, a b \log \left ({\left | -a e^{\left (2 \, d x + 2 \, c\right )} - b e^{\left (2 \, d x + 2 \, c\right )} - a + b \right |}\right )}{a^{4} - 2 \, a^{2} b^{2} + b^{4}} - \frac {d x + c}{a^{2} - 2 \, a b + b^{2}} - \frac {2 \, {\left (a b^{2} - b^{3}\right )}}{{\left (a e^{\left (2 \, d x + 2 \, c\right )} + b e^{\left (2 \, d x + 2 \, c\right )} + a - b\right )} {\left (a + b\right )}^{2} {\left (a - b\right )}^{2}}}{d} \] Input:
integrate(1/(a+b*tanh(d*x+c))^2,x, algorithm="giac")
Output:
-(2*a*b*log(abs(-a*e^(2*d*x + 2*c) - b*e^(2*d*x + 2*c) - a + b))/(a^4 - 2* a^2*b^2 + b^4) - (d*x + c)/(a^2 - 2*a*b + b^2) - 2*(a*b^2 - b^3)/((a*e^(2* d*x + 2*c) + b*e^(2*d*x + 2*c) + a - b)*(a + b)^2*(a - b)^2))/d
Time = 2.38 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.49 \[ \int \frac {1}{(a+b \tanh (c+d x))^2} \, dx=\frac {\frac {a\,x}{{\left (a+b\right )}^2}+\frac {b\,x\,\mathrm {tanh}\left (c+d\,x\right )}{{\left (a+b\right )}^2}-\frac {b^2\,\mathrm {tanh}\left (c+d\,x\right )}{a\,d\,\left (a^2-b^2\right )}}{a+b\,\mathrm {tanh}\left (c+d\,x\right )}-\frac {2\,a\,b\,\ln \left (a+b\,\mathrm {tanh}\left (c+d\,x\right )\right )}{d\,\left (a^4-2\,a^2\,b^2+b^4\right )}+\frac {2\,a\,b\,\ln \left (\mathrm {tanh}\left (c+d\,x\right )+1\right )}{d\,{\left (a^2-b^2\right )}^2} \] Input:
int(1/(a + b*tanh(c + d*x))^2,x)
Output:
((a*x)/(a + b)^2 + (b*x*tanh(c + d*x))/(a + b)^2 - (b^2*tanh(c + d*x))/(a* d*(a^2 - b^2)))/(a + b*tanh(c + d*x)) - (2*a*b*log(a + b*tanh(c + d*x)))/( d*(a^4 + b^4 - 2*a^2*b^2)) + (2*a*b*log(tanh(c + d*x) + 1))/(d*(a^2 - b^2) ^2)
Time = 0.23 (sec) , antiderivative size = 428, normalized size of antiderivative = 5.04 \[ \int \frac {1}{(a+b \tanh (c+d x))^2} \, dx=\frac {-2 e^{2 d x +2 c} \mathrm {log}\left (e^{2 d x +2 c} a +e^{2 d x +2 c} b +a -b \right ) a^{2} b -2 e^{2 d x +2 c} \mathrm {log}\left (e^{2 d x +2 c} a +e^{2 d x +2 c} b +a -b \right ) a \,b^{2}+e^{2 d x +2 c} a^{3} d x +3 e^{2 d x +2 c} a^{2} b d x +3 e^{2 d x +2 c} a \,b^{2} d x -2 e^{2 d x +2 c} a \,b^{2}+e^{2 d x +2 c} b^{3} d x -2 e^{2 d x +2 c} b^{3}-2 \,\mathrm {log}\left (e^{2 d x +2 c} a +e^{2 d x +2 c} b +a -b \right ) a^{2} b +2 \,\mathrm {log}\left (e^{2 d x +2 c} a +e^{2 d x +2 c} b +a -b \right ) a \,b^{2}+a^{3} d x +a^{2} b d x -a \,b^{2} d x -b^{3} d x}{d \left (e^{2 d x +2 c} a^{5}+e^{2 d x +2 c} a^{4} b -2 e^{2 d x +2 c} a^{3} b^{2}-2 e^{2 d x +2 c} a^{2} b^{3}+e^{2 d x +2 c} a \,b^{4}+e^{2 d x +2 c} b^{5}+a^{5}-a^{4} b -2 a^{3} b^{2}+2 a^{2} b^{3}+a \,b^{4}-b^{5}\right )} \] Input:
int(1/(a+b*tanh(d*x+c))^2,x)
Output:
( - 2*e**(2*c + 2*d*x)*log(e**(2*c + 2*d*x)*a + e**(2*c + 2*d*x)*b + a - b )*a**2*b - 2*e**(2*c + 2*d*x)*log(e**(2*c + 2*d*x)*a + e**(2*c + 2*d*x)*b + a - b)*a*b**2 + e**(2*c + 2*d*x)*a**3*d*x + 3*e**(2*c + 2*d*x)*a**2*b*d* x + 3*e**(2*c + 2*d*x)*a*b**2*d*x - 2*e**(2*c + 2*d*x)*a*b**2 + e**(2*c + 2*d*x)*b**3*d*x - 2*e**(2*c + 2*d*x)*b**3 - 2*log(e**(2*c + 2*d*x)*a + e** (2*c + 2*d*x)*b + a - b)*a**2*b + 2*log(e**(2*c + 2*d*x)*a + e**(2*c + 2*d *x)*b + a - b)*a*b**2 + a**3*d*x + a**2*b*d*x - a*b**2*d*x - b**3*d*x)/(d* (e**(2*c + 2*d*x)*a**5 + e**(2*c + 2*d*x)*a**4*b - 2*e**(2*c + 2*d*x)*a**3 *b**2 - 2*e**(2*c + 2*d*x)*a**2*b**3 + e**(2*c + 2*d*x)*a*b**4 + e**(2*c + 2*d*x)*b**5 + a**5 - a**4*b - 2*a**3*b**2 + 2*a**2*b**3 + a*b**4 - b**5))