\(\int \frac {\text {csch}(c+d x)}{(a+b \tanh ^2(c+d x))^2} \, dx\) [37]

Optimal result

Integrand size = 21, antiderivative size = 103 \[ \int \frac {\text {csch}(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx=-\frac {\text {arctanh}(\cosh (c+d x))}{a^2 d}+\frac {\sqrt {b} (3 a+2 b) \text {arctanh}\left (\frac {\sqrt {b} \text {sech}(c+d x)}{\sqrt {a+b}}\right )}{2 a^2 (a+b)^{3/2} d}+\frac {b \text {sech}(c+d x)}{2 a (a+b) d \left (a+b-b \text {sech}^2(c+d x)\right )} \] Output:

-arctanh(cosh(d*x+c))/a^2/d+1/2*b^(1/2)*(3*a+2*b)*arctanh(b^(1/2)*sech(d*x 
+c)/(a+b)^(1/2))/a^2/(a+b)^(3/2)/d+1/2*b*sech(d*x+c)/a/(a+b)/d/(a+b-b*sech 
(d*x+c)^2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.94 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.83 \[ \int \frac {\text {csch}(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx=\frac {\frac {i \sqrt {b} (3 a+2 b) \arctan \left (\frac {-i \sqrt {a+b}-\sqrt {a} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {b}}\right )}{(a+b)^{3/2}}+\frac {i \sqrt {b} (3 a+2 b) \arctan \left (\frac {-i \sqrt {a+b}+\sqrt {a} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {b}}\right )}{(a+b)^{3/2}}+\frac {2 a b \cosh (c+d x)}{(a+b) (a-b+(a+b) \cosh (2 (c+d x)))}-2 \log \left (\cosh \left (\frac {1}{2} (c+d x)\right )\right )+2 \log \left (\sinh \left (\frac {1}{2} (c+d x)\right )\right )}{2 a^2 d} \] Input:

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

Output:

((I*Sqrt[b]*(3*a + 2*b)*ArcTan[((-I)*Sqrt[a + b] - Sqrt[a]*Tanh[(c + d*x)/ 
2])/Sqrt[b]])/(a + b)^(3/2) + (I*Sqrt[b]*(3*a + 2*b)*ArcTan[((-I)*Sqrt[a + 
 b] + Sqrt[a]*Tanh[(c + d*x)/2])/Sqrt[b]])/(a + b)^(3/2) + (2*a*b*Cosh[c + 
 d*x])/((a + b)*(a - b + (a + b)*Cosh[2*(c + d*x)])) - 2*Log[Cosh[(c + d*x 
)/2]] + 2*Log[Sinh[(c + d*x)/2]])/(2*a^2*d)
 

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.09, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {3042, 26, 4147, 25, 316, 25, 397, 219, 221}

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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

\(\displaystyle i \int \frac {1}{\sin (i c+i d x) \left (a-b \tan (i c+i d x)^2\right )^2}dx\)

\(\Big \downarrow \) 4147

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 316

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 397

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

(-1/2*((2*(a + b)*ArcTanh[Sech[c + d*x]])/a - (Sqrt[b]*(3*a + 2*b)*ArcTanh 
[(Sqrt[b]*Sech[c + d*x])/Sqrt[a + b]])/(a*Sqrt[a + b]))/(a*(a + b)) + (b*S 
ech[c + d*x])/(2*a*(a + b)*(a + b - b*Sech[c + d*x]^2)))/d
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

rule 316

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim 
p[(-b)*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(2*a*(p + 1)*(b*c - a*d)) 
), x] + Simp[1/(2*a*(p + 1)*(b*c - a*d))   Int[(a + b*x^2)^(p + 1)*(c + d*x 
^2)^q*Simp[b*c + 2*(p + 1)*(b*c - a*d) + d*b*(2*(p + q + 2) + 1)*x^2, x], x 
], x] /; FreeQ[{a, b, c, d, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] &&  ! 
( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomialQ[a, b, c, d, 2, 
 p, q, x]
 

rule 397

Int[((e_) + (f_.)*(x_)^2)/(((a_) + (b_.)*(x_)^2)*((c_) + (d_.)*(x_)^2)), x_ 
Symbol] :> Simp[(b*e - a*f)/(b*c - a*d)   Int[1/(a + b*x^2), x], x] - Simp[ 
(d*e - c*f)/(b*c - a*d)   Int[1/(c + d*x^2), x], x] /; FreeQ[{a, b, c, d, e 
, f}, x]
 

rule 3042

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

rule 4147

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

Maple [A] (verified)

Time = 2.08 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.56

method	result	size
derivativedivides	\(\frac {\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{2}}-\frac {4 b \left (\frac {-\frac {\left (a +2 b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{4 \left (a +b \right )}-\frac {a}{4 \left (a +b \right )}}{\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +4 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+a}-\frac {\left (3 a +2 b \right ) \operatorname {arctanh}\left (\frac {2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +2 a +4 b}{4 \sqrt {a b +b^{2}}}\right )}{8 \left (a +b \right ) \sqrt {a b +b^{2}}}\right )}{a^{2}}}{d}\)	\(161\)
default	\(\frac {\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{2}}-\frac {4 b \left (\frac {-\frac {\left (a +2 b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{4 \left (a +b \right )}-\frac {a}{4 \left (a +b \right )}}{\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +4 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+a}-\frac {\left (3 a +2 b \right ) \operatorname {arctanh}\left (\frac {2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +2 a +4 b}{4 \sqrt {a b +b^{2}}}\right )}{8 \left (a +b \right ) \sqrt {a b +b^{2}}}\right )}{a^{2}}}{d}\)	\(161\)
risch	\(\frac {b \,{\mathrm e}^{d x +c} \left ({\mathrm e}^{2 d x +2 c}+1\right )}{a d \left (a +b \right ) \left ({\mathrm e}^{4 d x +4 c} a +b \,{\mathrm e}^{4 d x +4 c}+2 \,{\mathrm e}^{2 d x +2 c} a -2 \,{\mathrm e}^{2 d x +2 c} b +a +b \right )}-\frac {\ln \left ({\mathrm e}^{d x +c}+1\right )}{a^{2} d}+\frac {\ln \left ({\mathrm e}^{d x +c}-1\right )}{a^{2} d}+\frac {3 \sqrt {\left (a +b \right ) b}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {\left (a +b \right ) b}\, {\mathrm e}^{d x +c}}{a +b}+1\right )}{4 \left (a +b \right )^{2} d a}+\frac {\sqrt {\left (a +b \right ) b}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {\left (a +b \right ) b}\, {\mathrm e}^{d x +c}}{a +b}+1\right ) b}{2 \left (a +b \right )^{2} d \,a^{2}}-\frac {3 \sqrt {\left (a +b \right ) b}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {\left (a +b \right ) b}\, {\mathrm e}^{d x +c}}{a +b}+1\right )}{4 \left (a +b \right )^{2} d a}-\frac {\sqrt {\left (a +b \right ) b}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {\left (a +b \right ) b}\, {\mathrm e}^{d x +c}}{a +b}+1\right ) b}{2 \left (a +b \right )^{2} d \,a^{2}}\)	\(326\)

Input:

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

Output:

1/d*(1/a^2*ln(tanh(1/2*d*x+1/2*c))-4*b/a^2*((-1/4*(a+2*b)/(a+b)*tanh(1/2*d 
*x+1/2*c)^2-1/4*a/(a+b))/(tanh(1/2*d*x+1/2*c)^4*a+2*tanh(1/2*d*x+1/2*c)^2* 
a+4*b*tanh(1/2*d*x+1/2*c)^2+a)-1/8*(3*a+2*b)/(a+b)/(a*b+b^2)^(1/2)*arctanh 
(1/4*(2*tanh(1/2*d*x+1/2*c)^2*a+2*a+4*b)/(a*b+b^2)^(1/2))))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1324 vs. \(2 (94) = 188\).

Time = 0.16 (sec) , antiderivative size = 2614, normalized size of antiderivative = 25.38 \[ \int \frac {\text {csch}(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/4*(4*a*b*cosh(d*x + c)^3 + 12*a*b*cosh(d*x + c)*sinh(d*x + c)^2 + 4*a*b 
*sinh(d*x + c)^3 + 4*a*b*cosh(d*x + c) + ((3*a^2 + 5*a*b + 2*b^2)*cosh(d*x 
 + c)^4 + 4*(3*a^2 + 5*a*b + 2*b^2)*cosh(d*x + c)*sinh(d*x + c)^3 + (3*a^2 
 + 5*a*b + 2*b^2)*sinh(d*x + c)^4 + 2*(3*a^2 - a*b - 2*b^2)*cosh(d*x + c)^ 
2 + 2*(3*(3*a^2 + 5*a*b + 2*b^2)*cosh(d*x + c)^2 + 3*a^2 - a*b - 2*b^2)*si 
nh(d*x + c)^2 + 3*a^2 + 5*a*b + 2*b^2 + 4*((3*a^2 + 5*a*b + 2*b^2)*cosh(d* 
x + c)^3 + (3*a^2 - a*b - 2*b^2)*cosh(d*x + c))*sinh(d*x + c))*sqrt(b/(a + 
 b))*log(((a + b)*cosh(d*x + c)^4 + 4*(a + b)*cosh(d*x + c)*sinh(d*x + c)^ 
3 + (a + b)*sinh(d*x + c)^4 + 2*(a + 3*b)*cosh(d*x + c)^2 + 2*(3*(a + b)*c 
osh(d*x + c)^2 + a + 3*b)*sinh(d*x + c)^2 + 4*((a + b)*cosh(d*x + c)^3 + ( 
a + 3*b)*cosh(d*x + c))*sinh(d*x + c) + 4*((a + b)*cosh(d*x + c)^3 + 3*(a 
+ b)*cosh(d*x + c)*sinh(d*x + c)^2 + (a + b)*sinh(d*x + c)^3 + (a + b)*cos 
h(d*x + c) + (3*(a + b)*cosh(d*x + c)^2 + a + b)*sinh(d*x + c))*sqrt(b/(a 
+ b)) + a + b)/((a + b)*cosh(d*x + c)^4 + 4*(a + b)*cosh(d*x + c)*sinh(d*x 
 + c)^3 + (a + b)*sinh(d*x + c)^4 + 2*(a - b)*cosh(d*x + c)^2 + 2*(3*(a + 
b)*cosh(d*x + c)^2 + a - b)*sinh(d*x + c)^2 + 4*((a + b)*cosh(d*x + c)^3 + 
 (a - b)*cosh(d*x + c))*sinh(d*x + c) + a + b)) - 4*((a^2 + 2*a*b + b^2)*c 
osh(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 - b^2)*cosh(d*x + c)^2 + 2*(3*(a 
^2 + 2*a*b + b^2)*cosh(d*x + c)^2 + a^2 - b^2)*sinh(d*x + c)^2 + a^2 + ...
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

(b*e^(3*d*x + 3*c) + b*e^(d*x + c))/(a^3*d + 2*a^2*b*d + a*b^2*d + (a^3*d* 
e^(4*c) + 2*a^2*b*d*e^(4*c) + a*b^2*d*e^(4*c))*e^(4*d*x) + 2*(a^3*d*e^(2*c 
) - a*b^2*d*e^(2*c))*e^(2*d*x)) - log((e^(d*x + c) + 1)*e^(-c))/(a^2*d) + 
log((e^(d*x + c) - 1)*e^(-c))/(a^2*d) - 2*integrate(1/2*((3*a*b*e^(3*c) + 
2*b^2*e^(3*c))*e^(3*d*x) - (3*a*b*e^c + 2*b^2*e^c)*e^(d*x))/(a^4 + 2*a^3*b 
 + a^2*b^2 + (a^4*e^(4*c) + 2*a^3*b*e^(4*c) + a^2*b^2*e^(4*c))*e^(4*d*x) + 
 2*(a^4*e^(2*c) - a^2*b^2*e^(2*c))*e^(2*d*x)), x)
 

Giac [F(-2)]

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

integrate(csch(d*x+c)/(a+b*tanh(d*x+c)^2)^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 [F(-1)]

Timed out. \[ \int \frac {\text {csch}(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx=\int \frac {1}{\mathrm {sinh}\left (c+d\,x\right )\,{\left (b\,{\mathrm {tanh}\left (c+d\,x\right )}^2+a\right )}^2} \,d x \] Input:

int(1/(sinh(c + d*x)*(a + b*tanh(c + d*x)^2)^2),x)
 

Output:

int(1/(sinh(c + d*x)*(a + b*tanh(c + d*x)^2)^2), x)
 

Reduce [B] (verification not implemented)

Time = 0.32 (sec) , antiderivative size = 1630, normalized size of antiderivative = 15.83 \[ \int \frac {\text {csch}(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 3*e**(4*c + 4*d*x)*sqrt(b)*sqrt(a + b)*log(e**(2*c + 2*d*x)*sqrt(a + b 
) + sqrt(a + b) - 2*e**(c + d*x)*sqrt(b))*a**2 - 5*e**(4*c + 4*d*x)*sqrt(b 
)*sqrt(a + b)*log(e**(2*c + 2*d*x)*sqrt(a + b) + sqrt(a + b) - 2*e**(c + d 
*x)*sqrt(b))*a*b - 2*e**(4*c + 4*d*x)*sqrt(b)*sqrt(a + b)*log(e**(2*c + 2* 
d*x)*sqrt(a + b) + sqrt(a + b) - 2*e**(c + d*x)*sqrt(b))*b**2 + 3*e**(4*c 
+ 4*d*x)*sqrt(b)*sqrt(a + b)*log(e**(2*c + 2*d*x)*sqrt(a + b) + sqrt(a + b 
) + 2*e**(c + d*x)*sqrt(b))*a**2 + 5*e**(4*c + 4*d*x)*sqrt(b)*sqrt(a + b)* 
log(e**(2*c + 2*d*x)*sqrt(a + b) + sqrt(a + b) + 2*e**(c + d*x)*sqrt(b))*a 
*b + 2*e**(4*c + 4*d*x)*sqrt(b)*sqrt(a + b)*log(e**(2*c + 2*d*x)*sqrt(a + 
b) + sqrt(a + b) + 2*e**(c + d*x)*sqrt(b))*b**2 - 6*e**(2*c + 2*d*x)*sqrt( 
b)*sqrt(a + b)*log(e**(2*c + 2*d*x)*sqrt(a + b) + sqrt(a + b) - 2*e**(c + 
d*x)*sqrt(b))*a**2 + 2*e**(2*c + 2*d*x)*sqrt(b)*sqrt(a + b)*log(e**(2*c + 
2*d*x)*sqrt(a + b) + sqrt(a + b) - 2*e**(c + d*x)*sqrt(b))*a*b + 4*e**(2*c 
 + 2*d*x)*sqrt(b)*sqrt(a + b)*log(e**(2*c + 2*d*x)*sqrt(a + b) + sqrt(a + 
b) - 2*e**(c + d*x)*sqrt(b))*b**2 + 6*e**(2*c + 2*d*x)*sqrt(b)*sqrt(a + b) 
*log(e**(2*c + 2*d*x)*sqrt(a + b) + sqrt(a + b) + 2*e**(c + d*x)*sqrt(b))* 
a**2 - 2*e**(2*c + 2*d*x)*sqrt(b)*sqrt(a + b)*log(e**(2*c + 2*d*x)*sqrt(a 
+ b) + sqrt(a + b) + 2*e**(c + d*x)*sqrt(b))*a*b - 4*e**(2*c + 2*d*x)*sqrt 
(b)*sqrt(a + b)*log(e**(2*c + 2*d*x)*sqrt(a + b) + sqrt(a + b) + 2*e**(c + 
 d*x)*sqrt(b))*b**2 - 3*sqrt(b)*sqrt(a + b)*log(e**(2*c + 2*d*x)*sqrt(a...