\(\int \frac {\tanh ^5(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx\) [138]

Optimal result

Integrand size = 23, antiderivative size = 70 \[ \int \frac {\tanh ^5(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx=-\frac {(a+2 b) \log (\cosh (c+d x))}{b^2 d}+\frac {(a+b)^2 \log \left (b+a \cosh ^2(c+d x)\right )}{2 a b^2 d}-\frac {\text {sech}^2(c+d x)}{2 b d} \] Output:

-(a+2*b)*ln(cosh(d*x+c))/b^2/d+1/2*(a+b)^2*ln(b+a*cosh(d*x+c)^2)/a/b^2/d-1 
/2*sech(d*x+c)^2/b/d
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.40 \[ \int \frac {\tanh ^5(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx=-\frac {(a+2 b+a \cosh (2 (c+d x))) \text {sech}^2(c+d x) \left (2 a (a+2 b) \log (\cosh (c+d x))-(a+b)^2 \log \left (a+b+a \sinh ^2(c+d x)\right )+a b \text {sech}^2(c+d x)\right )}{4 a b^2 d \left (a+b \text {sech}^2(c+d x)\right )} \] Input:

Integrate[Tanh[c + d*x]^5/(a + b*Sech[c + d*x]^2),x]
 

Output:

-1/4*((a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d*x]^2*(2*a*(a + 2*b)*Log[C 
osh[c + d*x]] - (a + b)^2*Log[a + b + a*Sinh[c + d*x]^2] + a*b*Sech[c + d* 
x]^2))/(a*b^2*d*(a + b*Sech[c + d*x]^2))
 

Rubi [A] (warning: unable to verify)

Time = 0.31 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.90, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3042, 26, 4626, 354, 99, 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.

\(\displaystyle \int \frac {\tanh ^5(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int -\frac {i \tan (i c+i d x)^5}{a+b \sec (i c+i d x)^2}dx\)

\(\Big \downarrow \) 26

\(\displaystyle -i \int \frac {\tan (i c+i d x)^5}{b \sec (i c+i d x)^2+a}dx\)

\(\Big \downarrow \) 4626

\(\displaystyle \frac {\int \frac {\left (1-\cosh ^2(c+d x)\right )^2 \text {sech}^3(c+d x)}{a \cosh ^2(c+d x)+b}d\cosh (c+d x)}{d}\)

\(\Big \downarrow \) 354

\(\displaystyle \frac {\int \frac {\left (1-\cosh ^2(c+d x)\right )^2 \text {sech}^2(c+d x)}{a \cosh ^2(c+d x)+b}d\cosh ^2(c+d x)}{2 d}\)

\(\Big \downarrow \) 99

\(\displaystyle \frac {\int \left (\frac {(a+b)^2}{b^2 \left (a \cosh ^2(c+d x)+b\right )}+\frac {\text {sech}^2(c+d x)}{b}+\frac {(-a-2 b) \text {sech}(c+d x)}{b^2}\right )d\cosh ^2(c+d x)}{2 d}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\frac {(a+b)^2 \log \left (a \cosh ^2(c+d x)+b\right )}{a b^2}-\frac {(a+2 b) \log \left (\cosh ^2(c+d x)\right )}{b^2}-\frac {\text {sech}(c+d x)}{b}}{2 d}\)

Input:

Int[Tanh[c + d*x]^5/(a + b*Sech[c + d*x]^2),x]
 

Output:

(-(((a + 2*b)*Log[Cosh[c + d*x]^2])/b^2) + ((a + b)^2*Log[b + a*Cosh[c + d 
*x]^2])/(a*b^2) - Sech[c + d*x]/b)/(2*d)
 

Defintions of rubi rules used

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 99

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], 
 x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] && (IntegerQ[p] | 
| (GtQ[m, 0] && GeQ[n, -1]))
 

rule 354

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_S 
ymbol] :> Simp[1/2   Subst[Int[x^((m - 1)/2)*(a + b*x)^p*(c + d*x)^q, x], x 
, x^2], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IntegerQ 
[(m - 1)/2]
 

rule 2009

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

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4626

Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_.)*tan[(e_.) + (f_.)*(x_ 
)]^(m_.), x_Symbol] :> Module[{ff = FreeFactors[Cos[e + f*x], x]}, Simp[-(f 
*ff^(m + n*p - 1))^(-1)   Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*((b + a*(ff* 
x)^n)^p/x^(m + n*p)), x], x, Cos[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, n} 
, x] && IntegerQ[(m - 1)/2] && IntegerQ[n] && IntegerQ[p]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(182\) vs. \(2(66)=132\).

Time = 4.06 (sec) , antiderivative size = 183, normalized size of antiderivative = 2.61

method	result	size
derivativedivides	\(\frac {-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a}-\frac {\left (a +2 b \right ) \ln \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )-\frac {2 b}{1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {2 b}{\left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2}}}{b^{2}}+\frac {2 \left (\frac {1}{4} a^{2}+\frac {1}{2} a b +\frac {1}{4} b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} b +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a +b \right )}{a \,b^{2}}}{d}\)	\(183\)
default	\(\frac {-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a}-\frac {\left (a +2 b \right ) \ln \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )-\frac {2 b}{1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {2 b}{\left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2}}}{b^{2}}+\frac {2 \left (\frac {1}{4} a^{2}+\frac {1}{2} a b +\frac {1}{4} b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} b +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a +b \right )}{a \,b^{2}}}{d}\)	\(183\)
risch	\(-\frac {x}{a}-\frac {2 c}{d a}-\frac {2 \,{\mathrm e}^{2 d x +2 c}}{b d \left ({\mathrm e}^{2 d x +2 c}+1\right )^{2}}-\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+1\right ) a}{b^{2} d}-\frac {2 \ln \left ({\mathrm e}^{2 d x +2 c}+1\right )}{b d}+\frac {a \ln \left ({\mathrm e}^{4 d x +4 c}+\frac {2 \left (a +2 b \right ) {\mathrm e}^{2 d x +2 c}}{a}+1\right )}{2 b^{2} d}+\frac {\ln \left ({\mathrm e}^{4 d x +4 c}+\frac {2 \left (a +2 b \right ) {\mathrm e}^{2 d x +2 c}}{a}+1\right )}{b d}+\frac {\ln \left ({\mathrm e}^{4 d x +4 c}+\frac {2 \left (a +2 b \right ) {\mathrm e}^{2 d x +2 c}}{a}+1\right )}{2 a d}\)	\(205\)

Input:

int(tanh(d*x+c)^5/(a+b*sech(d*x+c)^2),x,method=_RETURNVERBOSE)
 

Output:

1/d*(-1/a*ln(tanh(1/2*d*x+1/2*c)-1)-1/a*ln(tanh(1/2*d*x+1/2*c)+1)-1/b^2*(( 
a+2*b)*ln(1+tanh(1/2*d*x+1/2*c)^2)-2*b/(1+tanh(1/2*d*x+1/2*c)^2)+2*b/(1+ta 
nh(1/2*d*x+1/2*c)^2)^2)+2/a/b^2*(1/4*a^2+1/2*a*b+1/4*b^2)*ln(tanh(1/2*d*x+ 
1/2*c)^4*a+tanh(1/2*d*x+1/2*c)^4*b+2*tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d* 
x+1/2*c)^2*b+a+b))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 736 vs. \(2 (66) = 132\).

Time = 0.32 (sec) , antiderivative size = 736, normalized size of antiderivative = 10.51 \[ \int \frac {\tanh ^5(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx =\text {Too large to display} \] Input:

integrate(tanh(d*x+c)^5/(a+b*sech(d*x+c)^2),x, algorithm="fricas")
 

Output:

-1/2*(2*b^2*d*x*cosh(d*x + c)^4 + 8*b^2*d*x*cosh(d*x + c)*sinh(d*x + c)^3 
+ 2*b^2*d*x*sinh(d*x + c)^4 + 2*b^2*d*x + 4*(b^2*d*x + a*b)*cosh(d*x + c)^ 
2 + 4*(3*b^2*d*x*cosh(d*x + c)^2 + b^2*d*x + a*b)*sinh(d*x + c)^2 - ((a^2 
+ 2*a*b + b^2)*cosh(d*x + c)^4 + 4*(a^2 + 2*a*b + b^2)*cosh(d*x + c)*sinh( 
d*x + c)^3 + (a^2 + 2*a*b + b^2)*sinh(d*x + c)^4 + 2*(a^2 + 2*a*b + b^2)*c 
osh(d*x + c)^2 + 2*(3*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^2 + a^2 + 2*a*b + 
b^2)*sinh(d*x + c)^2 + a^2 + 2*a*b + b^2 + 4*((a^2 + 2*a*b + b^2)*cosh(d*x 
 + c)^3 + (a^2 + 2*a*b + b^2)*cosh(d*x + c))*sinh(d*x + c))*log(2*(a*cosh( 
d*x + c)^2 + a*sinh(d*x + c)^2 + a + 2*b)/(cosh(d*x + c)^2 - 2*cosh(d*x + 
c)*sinh(d*x + c) + sinh(d*x + c)^2)) + 2*((a^2 + 2*a*b)*cosh(d*x + c)^4 + 
4*(a^2 + 2*a*b)*cosh(d*x + c)*sinh(d*x + c)^3 + (a^2 + 2*a*b)*sinh(d*x + c 
)^4 + 2*(a^2 + 2*a*b)*cosh(d*x + c)^2 + 2*(3*(a^2 + 2*a*b)*cosh(d*x + c)^2 
 + a^2 + 2*a*b)*sinh(d*x + c)^2 + a^2 + 2*a*b + 4*((a^2 + 2*a*b)*cosh(d*x 
+ c)^3 + (a^2 + 2*a*b)*cosh(d*x + c))*sinh(d*x + c))*log(2*cosh(d*x + c)/( 
cosh(d*x + c) - sinh(d*x + c))) + 8*(b^2*d*x*cosh(d*x + c)^3 + (b^2*d*x + 
a*b)*cosh(d*x + c))*sinh(d*x + c))/(a*b^2*d*cosh(d*x + c)^4 + 4*a*b^2*d*co 
sh(d*x + c)*sinh(d*x + c)^3 + a*b^2*d*sinh(d*x + c)^4 + 2*a*b^2*d*cosh(d*x 
 + c)^2 + a*b^2*d + 2*(3*a*b^2*d*cosh(d*x + c)^2 + a*b^2*d)*sinh(d*x + c)^ 
2 + 4*(a*b^2*d*cosh(d*x + c)^3 + a*b^2*d*cosh(d*x + c))*sinh(d*x + c))
 

Sympy [F]

\[ \int \frac {\tanh ^5(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx=\int \frac {\tanh ^{5}{\left (c + d x \right )}}{a + b \operatorname {sech}^{2}{\left (c + d x \right )}}\, dx \] Input:

integrate(tanh(d*x+c)**5/(a+b*sech(d*x+c)**2),x)
 

Output:

Integral(tanh(c + d*x)**5/(a + b*sech(c + d*x)**2), x)
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.87 \[ \int \frac {\tanh ^5(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx=\frac {d x + c}{a d} - \frac {2 \, e^{\left (-2 \, d x - 2 \, c\right )}}{{\left (2 \, b e^{\left (-2 \, d x - 2 \, c\right )} + b e^{\left (-4 \, d x - 4 \, c\right )} + b\right )} d} - \frac {{\left (a + 2 \, b\right )} \log \left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}{b^{2} d} + \frac {{\left (a^{2} + 2 \, a b + b^{2}\right )} \log \left (2 \, {\left (a + 2 \, b\right )} e^{\left (-2 \, d x - 2 \, c\right )} + a e^{\left (-4 \, d x - 4 \, c\right )} + a\right )}{2 \, a b^{2} d} \] Input:

integrate(tanh(d*x+c)^5/(a+b*sech(d*x+c)^2),x, algorithm="maxima")
 

Output:

(d*x + c)/(a*d) - 2*e^(-2*d*x - 2*c)/((2*b*e^(-2*d*x - 2*c) + b*e^(-4*d*x 
- 4*c) + b)*d) - (a + 2*b)*log(e^(-2*d*x - 2*c) + 1)/(b^2*d) + 1/2*(a^2 + 
2*a*b + b^2)*log(2*(a + 2*b)*e^(-2*d*x - 2*c) + a*e^(-4*d*x - 4*c) + a)/(a 
*b^2*d)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {\tanh ^5(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(tanh(d*x+c)^5/(a+b*sech(d*x+c)^2),x, algorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Limit: Max order reached or unable 
to make series expansion Error: Bad Argument Value
 

Mupad [B] (verification not implemented)

Time = 2.62 (sec) , antiderivative size = 421, normalized size of antiderivative = 6.01 \[ \int \frac {\tanh ^5(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx=\frac {2}{b\,d\,\left (2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1\right )}-\frac {2}{b\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}-\frac {x}{a}-\frac {\ln \left (39\,a\,b^7+243\,a^7\,b+27\,a^8+2\,b^8+289\,a^2\,b^6+1017\,a^3\,b^5+1791\,a^4\,b^4+1701\,a^5\,b^3+891\,a^6\,b^2+27\,a^8\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+2\,b^8\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+39\,a\,b^7\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+243\,a^7\,b\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+289\,a^2\,b^6\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+1017\,a^3\,b^5\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+1791\,a^4\,b^4\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+1701\,a^5\,b^3\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+891\,a^6\,b^2\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}\right )\,\left (a+2\,b\right )}{b^2\,d}+\frac {\ln \left (a\,b^2+6\,a^2\,b+3\,a^3+6\,a^3\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+3\,a^3\,{\mathrm {e}}^{4\,c}\,{\mathrm {e}}^{4\,d\,x}+4\,b^3\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+26\,a\,b^2\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+24\,a^2\,b\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+a\,b^2\,{\mathrm {e}}^{4\,c}\,{\mathrm {e}}^{4\,d\,x}+6\,a^2\,b\,{\mathrm {e}}^{4\,c}\,{\mathrm {e}}^{4\,d\,x}\right )\,\left (a^2+2\,a\,b+b^2\right )}{2\,a\,b^2\,d} \] Input:

int(tanh(c + d*x)^5/(a + b/cosh(c + d*x)^2),x)
 

Output:

2/(b*d*(2*exp(2*c + 2*d*x) + exp(4*c + 4*d*x) + 1)) - 2/(b*d*(exp(2*c + 2* 
d*x) + 1)) - x/a - (log(39*a*b^7 + 243*a^7*b + 27*a^8 + 2*b^8 + 289*a^2*b^ 
6 + 1017*a^3*b^5 + 1791*a^4*b^4 + 1701*a^5*b^3 + 891*a^6*b^2 + 27*a^8*exp( 
2*c)*exp(2*d*x) + 2*b^8*exp(2*c)*exp(2*d*x) + 39*a*b^7*exp(2*c)*exp(2*d*x) 
 + 243*a^7*b*exp(2*c)*exp(2*d*x) + 289*a^2*b^6*exp(2*c)*exp(2*d*x) + 1017* 
a^3*b^5*exp(2*c)*exp(2*d*x) + 1791*a^4*b^4*exp(2*c)*exp(2*d*x) + 1701*a^5* 
b^3*exp(2*c)*exp(2*d*x) + 891*a^6*b^2*exp(2*c)*exp(2*d*x))*(a + 2*b))/(b^2 
*d) + (log(a*b^2 + 6*a^2*b + 3*a^3 + 6*a^3*exp(2*c)*exp(2*d*x) + 3*a^3*exp 
(4*c)*exp(4*d*x) + 4*b^3*exp(2*c)*exp(2*d*x) + 26*a*b^2*exp(2*c)*exp(2*d*x 
) + 24*a^2*b*exp(2*c)*exp(2*d*x) + a*b^2*exp(4*c)*exp(4*d*x) + 6*a^2*b*exp 
(4*c)*exp(4*d*x))*(2*a*b + a^2 + b^2))/(2*a*b^2*d)
 

Reduce [B] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 1292, normalized size of antiderivative = 18.46 \[ \int \frac {\tanh ^5(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx =\text {Too large to display} \] Input:

int(tanh(d*x+c)^5/(a+b*sech(d*x+c)^2),x)
 

Output:

( - 2*e**(4*c + 4*d*x)*log(e**(2*c + 2*d*x) + 1)*a**2 - 4*e**(4*c + 4*d*x) 
*log(e**(2*c + 2*d*x) + 1)*a*b + e**(4*c + 4*d*x)*log( - sqrt(2*sqrt(b)*sq 
rt(a + b) - a - 2*b) + e**(c + d*x)*sqrt(a))*a**2 + 2*e**(4*c + 4*d*x)*log 
( - sqrt(2*sqrt(b)*sqrt(a + b) - a - 2*b) + e**(c + d*x)*sqrt(a))*a*b + e* 
*(4*c + 4*d*x)*log( - sqrt(2*sqrt(b)*sqrt(a + b) - a - 2*b) + e**(c + d*x) 
*sqrt(a))*b**2 + e**(4*c + 4*d*x)*log(sqrt(2*sqrt(b)*sqrt(a + b) - a - 2*b 
) + e**(c + d*x)*sqrt(a))*a**2 + 2*e**(4*c + 4*d*x)*log(sqrt(2*sqrt(b)*sqr 
t(a + b) - a - 2*b) + e**(c + d*x)*sqrt(a))*a*b + e**(4*c + 4*d*x)*log(sqr 
t(2*sqrt(b)*sqrt(a + b) - a - 2*b) + e**(c + d*x)*sqrt(a))*b**2 + e**(4*c 
+ 4*d*x)*log(2*sqrt(b)*sqrt(a + b) + e**(2*c + 2*d*x)*a + a + 2*b)*a**2 + 
2*e**(4*c + 4*d*x)*log(2*sqrt(b)*sqrt(a + b) + e**(2*c + 2*d*x)*a + a + 2* 
b)*a*b + e**(4*c + 4*d*x)*log(2*sqrt(b)*sqrt(a + b) + e**(2*c + 2*d*x)*a + 
 a + 2*b)*b**2 + 2*e**(4*c + 4*d*x)*a*b - 2*e**(4*c + 4*d*x)*b**2*d*x - 4* 
e**(2*c + 2*d*x)*log(e**(2*c + 2*d*x) + 1)*a**2 - 8*e**(2*c + 2*d*x)*log(e 
**(2*c + 2*d*x) + 1)*a*b + 2*e**(2*c + 2*d*x)*log( - sqrt(2*sqrt(b)*sqrt(a 
 + b) - a - 2*b) + e**(c + d*x)*sqrt(a))*a**2 + 4*e**(2*c + 2*d*x)*log( - 
sqrt(2*sqrt(b)*sqrt(a + b) - a - 2*b) + e**(c + d*x)*sqrt(a))*a*b + 2*e**( 
2*c + 2*d*x)*log( - sqrt(2*sqrt(b)*sqrt(a + b) - a - 2*b) + e**(c + d*x)*s 
qrt(a))*b**2 + 2*e**(2*c + 2*d*x)*log(sqrt(2*sqrt(b)*sqrt(a + b) - a - 2*b 
) + e**(c + d*x)*sqrt(a))*a**2 + 4*e**(2*c + 2*d*x)*log(sqrt(2*sqrt(b)*...