\(\int \frac {\cosh ^4(c+d x)}{(a+b \sinh ^2(c+d x))^2} \, dx\) [295]

Optimal result

Integrand size = 23, antiderivative size = 100 \[ \int \frac {\cosh ^4(c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^2} \, dx=\frac {x}{b^2}-\frac {\sqrt {a-b} (2 a+b) \text {arctanh}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{2 a^{3/2} b^2 d}-\frac {(a-b) \tanh (c+d x)}{2 a b d \left (a-(a-b) \tanh ^2(c+d x)\right )} \] Output:

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

Mathematica [A] (verified)

Time = 0.66 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.08 \[ \int \frac {\cosh ^4(c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^2} \, dx=\frac {2 (c+d x)-\frac {\left (2 a^2-a b-b^2\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{a^{3/2} \sqrt {a-b}}+\frac {b (-a+b) \sinh (2 (c+d x))}{a (2 a-b+b \cosh (2 (c+d x)))}}{2 b^2 d} \] Input:

Integrate[Cosh[c + d*x]^4/(a + b*Sinh[c + d*x]^2)^2,x]
 

Output:

(2*(c + d*x) - ((2*a^2 - a*b - b^2)*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sq 
rt[a]])/(a^(3/2)*Sqrt[a - b]) + (b*(-a + b)*Sinh[2*(c + d*x)])/(a*(2*a - b 
 + b*Cosh[2*(c + d*x)])))/(2*b^2*d)
 

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.15, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {3042, 3670, 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.

\(\displaystyle \int \frac {\cosh ^4(c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^2} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\cos (i c+i d x)^4}{\left (a-b \sin (i c+i d x)^2\right )^2}dx\)

\(\Big \downarrow \) 3670

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

\(\Big \downarrow \) 316

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 397

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 221

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

Input:

Int[Cosh[c + d*x]^4/(a + b*Sinh[c + d*x]^2)^2,x]
 

Output:

(((2*a*ArcTanh[Tanh[c + d*x]])/b - (Sqrt[a - b]*(2*a + b)*ArcTanh[(Sqrt[a 
- b]*Tanh[c + d*x])/Sqrt[a]])/(Sqrt[a]*b))/(2*a*b) - ((a - b)*Tanh[c + d*x 
])/(2*a*b*(a - (a - b)*Tanh[c + d*x]^2)))/d
 

Defintions of rubi rules used

rule 25

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

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 3670

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(314\) vs. \(2(88)=176\).

Time = 58.01 (sec) , antiderivative size = 315, normalized size of antiderivative = 3.15

method	result	size
risch	\(\frac {x}{b^{2}}+\frac {2 \,{\mathrm e}^{2 d x +2 c} a^{2}-3 \,{\mathrm e}^{2 d x +2 c} b a +b^{2} {\mathrm e}^{2 d x +2 c}+a b -b^{2}}{d \,b^{2} a \left (b \,{\mathrm e}^{4 d x +4 c}+4 \,{\mathrm e}^{2 d x +2 c} a -2 \,{\mathrm e}^{2 d x +2 c} b +b \right )}+\frac {\sqrt {a \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a +2 \sqrt {a \left (a -b \right )}-b}{b}\right )}{2 a d \,b^{2}}+\frac {\sqrt {a \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a +2 \sqrt {a \left (a -b \right )}-b}{b}\right )}{4 a^{2} d b}-\frac {\sqrt {a \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {-2 a +2 \sqrt {a \left (a -b \right )}+b}{b}\right )}{2 a d \,b^{2}}-\frac {\sqrt {a \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {-2 a +2 \sqrt {a \left (a -b \right )}+b}{b}\right )}{4 a^{2} d b}\)	\(315\)
derivativedivides	\(\frac {\frac {\frac {2 \left (-\frac {b \left (a -b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{2 a}-\frac {b \left (a -b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a}\right )}{\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a -2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +4 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+a}+\left (2 a^{2}-a b -b^{2}\right ) \left (-\frac {\left (\sqrt {-b \left (a -b \right )}-b \right ) \operatorname {arctanh}\left (\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) a}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}+\frac {\left (\sqrt {-b \left (a -b \right )}+b \right ) \arctan \left (\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) a}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}\right )}{b^{2}}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{b^{2}}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{b^{2}}}{d}\)	\(320\)
default	\(\frac {\frac {\frac {2 \left (-\frac {b \left (a -b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{2 a}-\frac {b \left (a -b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a}\right )}{\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a -2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +4 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+a}+\left (2 a^{2}-a b -b^{2}\right ) \left (-\frac {\left (\sqrt {-b \left (a -b \right )}-b \right ) \operatorname {arctanh}\left (\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) a}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}+\frac {\left (\sqrt {-b \left (a -b \right )}+b \right ) \arctan \left (\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) a}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}\right )}{b^{2}}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{b^{2}}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{b^{2}}}{d}\)	\(320\)

Input:

int(cosh(d*x+c)^4/(a+b*sinh(d*x+c)^2)^2,x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 631 vs. \(2 (89) = 178\).

Time = 0.13 (sec) , antiderivative size = 1527, normalized size of antiderivative = 15.27 \[ \int \frac {\cosh ^4(c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^2} \, dx=\text {Too large to display} \] Input:

integrate(cosh(d*x+c)^4/(a+b*sinh(d*x+c)^2)^2,x, algorithm="fricas")
 

Output:

[1/4*(4*a*b*d*x*cosh(d*x + c)^4 + 16*a*b*d*x*cosh(d*x + c)*sinh(d*x + c)^3 
 + 4*a*b*d*x*sinh(d*x + c)^4 + 4*a*b*d*x + 4*(2*(2*a^2 - a*b)*d*x + 2*a^2 
- 3*a*b + b^2)*cosh(d*x + c)^2 + 4*(6*a*b*d*x*cosh(d*x + c)^2 + 2*(2*a^2 - 
 a*b)*d*x + 2*a^2 - 3*a*b + b^2)*sinh(d*x + c)^2 + ((2*a*b + b^2)*cosh(d*x 
 + c)^4 + 4*(2*a*b + b^2)*cosh(d*x + c)*sinh(d*x + c)^3 + (2*a*b + b^2)*si 
nh(d*x + c)^4 + 2*(4*a^2 - b^2)*cosh(d*x + c)^2 + 2*(3*(2*a*b + b^2)*cosh( 
d*x + c)^2 + 4*a^2 - b^2)*sinh(d*x + c)^2 + 2*a*b + b^2 + 4*((2*a*b + b^2) 
*cosh(d*x + c)^3 + (4*a^2 - b^2)*cosh(d*x + c))*sinh(d*x + c))*sqrt((a - b 
)/a)*log((b^2*cosh(d*x + c)^4 + 4*b^2*cosh(d*x + c)*sinh(d*x + c)^3 + b^2* 
sinh(d*x + c)^4 + 2*(2*a*b - b^2)*cosh(d*x + c)^2 + 2*(3*b^2*cosh(d*x + c) 
^2 + 2*a*b - b^2)*sinh(d*x + c)^2 + 8*a^2 - 8*a*b + b^2 + 4*(b^2*cosh(d*x 
+ c)^3 + (2*a*b - b^2)*cosh(d*x + c))*sinh(d*x + c) + 4*(a*b*cosh(d*x + c) 
^2 + 2*a*b*cosh(d*x + c)*sinh(d*x + c) + a*b*sinh(d*x + c)^2 + 2*a^2 - a*b 
)*sqrt((a - b)/a))/(b*cosh(d*x + c)^4 + 4*b*cosh(d*x + c)*sinh(d*x + c)^3 
+ b*sinh(d*x + c)^4 + 2*(2*a - b)*cosh(d*x + c)^2 + 2*(3*b*cosh(d*x + c)^2 
 + 2*a - b)*sinh(d*x + c)^2 + 4*(b*cosh(d*x + c)^3 + (2*a - b)*cosh(d*x + 
c))*sinh(d*x + c) + b)) + 4*a*b - 4*b^2 + 8*(2*a*b*d*x*cosh(d*x + c)^3 + ( 
2*(2*a^2 - a*b)*d*x + 2*a^2 - 3*a*b + b^2)*cosh(d*x + c))*sinh(d*x + c))/( 
a*b^3*d*cosh(d*x + c)^4 + 4*a*b^3*d*cosh(d*x + c)*sinh(d*x + c)^3 + a*b^3* 
d*sinh(d*x + c)^4 + a*b^3*d + 2*(2*a^2*b^2 - a*b^3)*d*cosh(d*x + c)^2 +...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\cosh ^4(c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^2} \, dx=\text {Timed out} \] Input:

integrate(cosh(d*x+c)**4/(a+b*sinh(d*x+c)**2)**2,x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\cosh ^4(c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^2} \, dx=\text {Exception raised: ValueError} \] Input:

integrate(cosh(d*x+c)^4/(a+b*sinh(d*x+c)^2)^2,x, algorithm="maxima")
 

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(b-a>0)', see `assume?` for more 
details)Is
 

Giac [A] (verification not implemented)

Time = 0.94 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.78 \[ \int \frac {\cosh ^4(c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^2} \, dx=\frac {\frac {2 \, {\left (d x + c\right )}}{b^{2}} - \frac {{\left (2 \, a^{2} - a b - b^{2}\right )} \arctan \left (\frac {b e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a - b}{2 \, \sqrt {-a^{2} + a b}}\right )}{\sqrt {-a^{2} + a b} a b^{2}} + \frac {2 \, {\left (2 \, a^{2} e^{\left (2 \, d x + 2 \, c\right )} - 3 \, a b e^{\left (2 \, d x + 2 \, c\right )} + b^{2} e^{\left (2 \, d x + 2 \, c\right )} + a b - b^{2}\right )}}{{\left (b e^{\left (4 \, d x + 4 \, c\right )} + 4 \, a e^{\left (2 \, d x + 2 \, c\right )} - 2 \, b e^{\left (2 \, d x + 2 \, c\right )} + b\right )} a b^{2}}}{2 \, d} \] Input:

integrate(cosh(d*x+c)^4/(a+b*sinh(d*x+c)^2)^2,x, algorithm="giac")
 

Output:

1/2*(2*(d*x + c)/b^2 - (2*a^2 - a*b - b^2)*arctan(1/2*(b*e^(2*d*x + 2*c) + 
 2*a - b)/sqrt(-a^2 + a*b))/(sqrt(-a^2 + a*b)*a*b^2) + 2*(2*a^2*e^(2*d*x + 
 2*c) - 3*a*b*e^(2*d*x + 2*c) + b^2*e^(2*d*x + 2*c) + a*b - b^2)/((b*e^(4* 
d*x + 4*c) + 4*a*e^(2*d*x + 2*c) - 2*b*e^(2*d*x + 2*c) + b)*a*b^2))/d
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\cosh ^4(c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^2} \, dx=\int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^4}{{\left (b\,{\mathrm {sinh}\left (c+d\,x\right )}^2+a\right )}^2} \,d x \] Input:

int(cosh(c + d*x)^4/(a + b*sinh(c + d*x)^2)^2,x)
 

Output:

int(cosh(c + d*x)^4/(a + b*sinh(c + d*x)^2)^2, x)
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 1033, normalized size of antiderivative = 10.33 \[ \int \frac {\cosh ^4(c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^2} \, dx =\text {Too large to display} \] Input:

int(cosh(d*x+c)^4/(a+b*sinh(d*x+c)^2)^2,x)
 

Output:

( - 2*e**(4*c + 4*d*x)*sqrt(a)*sqrt(a - b)*log( - sqrt(2*sqrt(a)*sqrt(a - 
b) - 2*a + b) + e**(c + d*x)*sqrt(b))*a*b - e**(4*c + 4*d*x)*sqrt(a)*sqrt( 
a - b)*log( - sqrt(2*sqrt(a)*sqrt(a - b) - 2*a + b) + e**(c + d*x)*sqrt(b) 
)*b**2 - 2*e**(4*c + 4*d*x)*sqrt(a)*sqrt(a - b)*log(sqrt(2*sqrt(a)*sqrt(a 
- b) - 2*a + b) + e**(c + d*x)*sqrt(b))*a*b - e**(4*c + 4*d*x)*sqrt(a)*sqr 
t(a - b)*log(sqrt(2*sqrt(a)*sqrt(a - b) - 2*a + b) + e**(c + d*x)*sqrt(b)) 
*b**2 + 2*e**(4*c + 4*d*x)*sqrt(a)*sqrt(a - b)*log(2*sqrt(a)*sqrt(a - b) + 
 e**(2*c + 2*d*x)*b + 2*a - b)*a*b + e**(4*c + 4*d*x)*sqrt(a)*sqrt(a - b)* 
log(2*sqrt(a)*sqrt(a - b) + e**(2*c + 2*d*x)*b + 2*a - b)*b**2 - 8*e**(2*c 
 + 2*d*x)*sqrt(a)*sqrt(a - b)*log( - sqrt(2*sqrt(a)*sqrt(a - b) - 2*a + b) 
 + e**(c + d*x)*sqrt(b))*a**2 + 2*e**(2*c + 2*d*x)*sqrt(a)*sqrt(a - b)*log 
( - sqrt(2*sqrt(a)*sqrt(a - b) - 2*a + b) + e**(c + d*x)*sqrt(b))*b**2 - 8 
*e**(2*c + 2*d*x)*sqrt(a)*sqrt(a - b)*log(sqrt(2*sqrt(a)*sqrt(a - b) - 2*a 
 + b) + e**(c + d*x)*sqrt(b))*a**2 + 2*e**(2*c + 2*d*x)*sqrt(a)*sqrt(a - b 
)*log(sqrt(2*sqrt(a)*sqrt(a - b) - 2*a + b) + e**(c + d*x)*sqrt(b))*b**2 + 
 8*e**(2*c + 2*d*x)*sqrt(a)*sqrt(a - b)*log(2*sqrt(a)*sqrt(a - b) + e**(2* 
c + 2*d*x)*b + 2*a - b)*a**2 - 2*e**(2*c + 2*d*x)*sqrt(a)*sqrt(a - b)*log( 
2*sqrt(a)*sqrt(a - b) + e**(2*c + 2*d*x)*b + 2*a - b)*b**2 - 2*sqrt(a)*sqr 
t(a - b)*log( - sqrt(2*sqrt(a)*sqrt(a - b) - 2*a + b) + e**(c + d*x)*sqrt( 
b))*a*b - sqrt(a)*sqrt(a - b)*log( - sqrt(2*sqrt(a)*sqrt(a - b) - 2*a +...