\(\int \frac {\text {csch}^4(x)}{a+b \cosh ^2(x)} \, dx\) [17]

Optimal result

Integrand size = 15, antiderivative size = 59 \[ \int \frac {\text {csch}^4(x)}{a+b \cosh ^2(x)} \, dx=\frac {b^2 \text {arctanh}\left (\frac {\sqrt {a} \tanh (x)}{\sqrt {a+b}}\right )}{\sqrt {a} (a+b)^{5/2}}+\frac {(a+2 b) \coth (x)}{(a+b)^2}-\frac {\coth ^3(x)}{3 (a+b)} \] Output:

b^2*arctanh(a^(1/2)*tanh(x)/(a+b)^(1/2))/a^(1/2)/(a+b)^(5/2)+(a+2*b)*coth( 
x)/(a+b)^2-coth(x)^3/(3*a+3*b)
 

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00 \[ \int \frac {\text {csch}^4(x)}{a+b \cosh ^2(x)} \, dx=\frac {b^2 \text {arctanh}\left (\frac {\sqrt {a} \tanh (x)}{\sqrt {a+b}}\right )}{\sqrt {a} (a+b)^{5/2}}-\frac {\coth (x) \left (-2 a-5 b+(a+b) \text {csch}^2(x)\right )}{3 (a+b)^2} \] Input:

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

Output:

(b^2*ArcTanh[(Sqrt[a]*Tanh[x])/Sqrt[a + b]])/(Sqrt[a]*(a + b)^(5/2)) - (Co 
th[x]*(-2*a - 5*b + (a + b)*Csch[x]^2))/(3*(a + b)^2)
 

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {3042, 3670, 300, 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.

\(\displaystyle \int \frac {\text {csch}^4(x)}{a+b \cosh ^2(x)} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{\cos \left (\frac {\pi }{2}+i x\right )^4 \left (a+b \sin \left (\frac {\pi }{2}+i x\right )^2\right )}dx\)

\(\Big \downarrow \) 3670

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

\(\Big \downarrow \) 300

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {b^2 \text {arctanh}\left (\frac {\sqrt {a+b} \coth (x)}{\sqrt {a}}\right )}{\sqrt {a} (a+b)^{5/2}}-\frac {\coth ^3(x)}{3 (a+b)}+\frac {(a+2 b) \coth (x)}{(a+b)^2}\)

Input:

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

Output:

(b^2*ArcTanh[(Sqrt[a + b]*Coth[x])/Sqrt[a]])/(Sqrt[a]*(a + b)^(5/2)) + ((a 
 + 2*b)*Coth[x])/(a + b)^2 - Coth[x]^3/(3*(a + b))
 

Defintions of rubi rules used

rule 300

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Int 
[PolynomialDivide[(a + b*x^2)^p, (c + d*x^2)^(-q), x], x] /; FreeQ[{a, b, c 
, d}, x] && NeQ[b*c - a*d, 0] && IGtQ[p, 0] && ILtQ[q, 0] && GeQ[p, -q]
 

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 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. \(161\) vs. \(2(53)=106\).

Time = 24.96 (sec) , antiderivative size = 162, normalized size of antiderivative = 2.75

method	result	size
default	\(-\frac {\frac {a \tanh \left (\frac {x}{2}\right )^{3}}{3}+\frac {b \tanh \left (\frac {x}{2}\right )^{3}}{3}-3 a \tanh \left (\frac {x}{2}\right )-7 b \tanh \left (\frac {x}{2}\right )}{8 \left (a +b \right )^{2}}-\frac {1}{24 \left (a +b \right ) \tanh \left (\frac {x}{2}\right )^{3}}-\frac {-3 a -7 b}{8 \left (a +b \right )^{2} \tanh \left (\frac {x}{2}\right )}-\frac {2 b^{2} \left (-\frac {\ln \left (\sqrt {a +b}\, \tanh \left (\frac {x}{2}\right )^{2}+2 \tanh \left (\frac {x}{2}\right ) \sqrt {a}+\sqrt {a +b}\right )}{4 \sqrt {a}\, \sqrt {a +b}}+\frac {\ln \left (\sqrt {a +b}\, \tanh \left (\frac {x}{2}\right )^{2}-2 \tanh \left (\frac {x}{2}\right ) \sqrt {a}+\sqrt {a +b}\right )}{4 \sqrt {a}\, \sqrt {a +b}}\right )}{\left (a +b \right )^{2}}\)	\(162\)
risch	\(-\frac {2 \left (-3 \,{\mathrm e}^{4 x} b +6 \,{\mathrm e}^{2 x} a +12 \,{\mathrm e}^{2 x} b -2 a -5 b \right )}{3 \left (a +b \right )^{2} \left ({\mathrm e}^{2 x}-1\right )^{3}}+\frac {b^{2} \ln \left ({\mathrm e}^{2 x}+\frac {2 a \sqrt {a^{2}+a b}+b \sqrt {a^{2}+a b}-2 a^{2}-2 a b}{b \sqrt {a^{2}+a b}}\right )}{2 \sqrt {a^{2}+a b}\, \left (a +b \right )^{2}}-\frac {b^{2} \ln \left ({\mathrm e}^{2 x}+\frac {2 a \sqrt {a^{2}+a b}+b \sqrt {a^{2}+a b}+2 a^{2}+2 a b}{b \sqrt {a^{2}+a b}}\right )}{2 \sqrt {a^{2}+a b}\, \left (a +b \right )^{2}}\)	\(187\)

Input:

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

Output:

-1/8/(a+b)^2*(1/3*a*tanh(1/2*x)^3+1/3*b*tanh(1/2*x)^3-3*a*tanh(1/2*x)-7*b* 
tanh(1/2*x))-1/24/(a+b)/tanh(1/2*x)^3-1/8/(a+b)^2*(-3*a-7*b)/tanh(1/2*x)-2 
/(a+b)^2*b^2*(-1/4/a^(1/2)/(a+b)^(1/2)*ln((a+b)^(1/2)*tanh(1/2*x)^2+2*tanh 
(1/2*x)*a^(1/2)+(a+b)^(1/2))+1/4/a^(1/2)/(a+b)^(1/2)*ln((a+b)^(1/2)*tanh(1 
/2*x)^2-2*tanh(1/2*x)*a^(1/2)+(a+b)^(1/2)))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 857 vs. \(2 (49) = 98\).

Time = 0.12 (sec) , antiderivative size = 1875, normalized size of antiderivative = 31.78 \[ \int \frac {\text {csch}^4(x)}{a+b \cosh ^2(x)} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/6*(12*(a^2*b + a*b^2)*cosh(x)^4 + 48*(a^2*b + a*b^2)*cosh(x)*sinh(x)^3 
+ 12*(a^2*b + a*b^2)*sinh(x)^4 + 8*a^3 + 28*a^2*b + 20*a*b^2 - 24*(a^3 + 3 
*a^2*b + 2*a*b^2)*cosh(x)^2 - 24*(a^3 + 3*a^2*b + 2*a*b^2 - 3*(a^2*b + a*b 
^2)*cosh(x)^2)*sinh(x)^2 + 3*(b^2*cosh(x)^6 + 6*b^2*cosh(x)*sinh(x)^5 + b^ 
2*sinh(x)^6 - 3*b^2*cosh(x)^4 + 3*(5*b^2*cosh(x)^2 - b^2)*sinh(x)^4 + 3*b^ 
2*cosh(x)^2 + 4*(5*b^2*cosh(x)^3 - 3*b^2*cosh(x))*sinh(x)^3 + 3*(5*b^2*cos 
h(x)^4 - 6*b^2*cosh(x)^2 + b^2)*sinh(x)^2 - b^2 + 6*(b^2*cosh(x)^5 - 2*b^2 
*cosh(x)^3 + b^2*cosh(x))*sinh(x))*sqrt(a^2 + a*b)*log((b^2*cosh(x)^4 + 4* 
b^2*cosh(x)*sinh(x)^3 + b^2*sinh(x)^4 + 2*(2*a*b + b^2)*cosh(x)^2 + 2*(3*b 
^2*cosh(x)^2 + 2*a*b + b^2)*sinh(x)^2 + 8*a^2 + 8*a*b + b^2 + 4*(b^2*cosh( 
x)^3 + (2*a*b + b^2)*cosh(x))*sinh(x) - 4*(b*cosh(x)^2 + 2*b*cosh(x)*sinh( 
x) + b*sinh(x)^2 + 2*a + b)*sqrt(a^2 + a*b))/(b*cosh(x)^4 + 4*b*cosh(x)*si 
nh(x)^3 + b*sinh(x)^4 + 2*(2*a + b)*cosh(x)^2 + 2*(3*b*cosh(x)^2 + 2*a + b 
)*sinh(x)^2 + 4*(b*cosh(x)^3 + (2*a + b)*cosh(x))*sinh(x) + b)) + 48*((a^2 
*b + a*b^2)*cosh(x)^3 - (a^3 + 3*a^2*b + 2*a*b^2)*cosh(x))*sinh(x))/((a^4 
+ 3*a^3*b + 3*a^2*b^2 + a*b^3)*cosh(x)^6 + 6*(a^4 + 3*a^3*b + 3*a^2*b^2 + 
a*b^3)*cosh(x)*sinh(x)^5 + (a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*sinh(x)^6 - 
 3*(a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*cosh(x)^4 - 3*(a^4 + 3*a^3*b + 3*a^ 
2*b^2 + a*b^3 - 5*(a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*cosh(x)^2)*sinh(x)^4 
 - a^4 - 3*a^3*b - 3*a^2*b^2 - a*b^3 + 4*(5*(a^4 + 3*a^3*b + 3*a^2*b^2 ...
 

Sympy [F]

\[ \int \frac {\text {csch}^4(x)}{a+b \cosh ^2(x)} \, dx=\int \frac {\operatorname {csch}^{4}{\left (x \right )}}{a + b \cosh ^{2}{\left (x \right )}}\, dx \] Input:

integrate(csch(x)**4/(a+b*cosh(x)**2),x)
 

Output:

Integral(csch(x)**4/(a + b*cosh(x)**2), x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 161 vs. \(2 (49) = 98\).

Time = 0.15 (sec) , antiderivative size = 161, normalized size of antiderivative = 2.73 \[ \int \frac {\text {csch}^4(x)}{a+b \cosh ^2(x)} \, dx=-\frac {b^{2} \log \left (\frac {b e^{\left (-2 \, x\right )} + 2 \, a + b - 2 \, \sqrt {{\left (a + b\right )} a}}{b e^{\left (-2 \, x\right )} + 2 \, a + b + 2 \, \sqrt {{\left (a + b\right )} a}}\right )}{2 \, \sqrt {{\left (a + b\right )} a} {\left (a^{2} + 2 \, a b + b^{2}\right )}} - \frac {2 \, {\left (6 \, {\left (a + 2 \, b\right )} e^{\left (-2 \, x\right )} - 3 \, b e^{\left (-4 \, x\right )} - 2 \, a - 5 \, b\right )}}{3 \, {\left (a^{2} + 2 \, a b + b^{2} - 3 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} e^{\left (-2 \, x\right )} + 3 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} e^{\left (-4 \, x\right )} - {\left (a^{2} + 2 \, a b + b^{2}\right )} e^{\left (-6 \, x\right )}\right )}} \] Input:

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

Output:

-1/2*b^2*log((b*e^(-2*x) + 2*a + b - 2*sqrt((a + b)*a))/(b*e^(-2*x) + 2*a 
+ b + 2*sqrt((a + b)*a)))/(sqrt((a + b)*a)*(a^2 + 2*a*b + b^2)) - 2/3*(6*( 
a + 2*b)*e^(-2*x) - 3*b*e^(-4*x) - 2*a - 5*b)/(a^2 + 2*a*b + b^2 - 3*(a^2 
+ 2*a*b + b^2)*e^(-2*x) + 3*(a^2 + 2*a*b + b^2)*e^(-4*x) - (a^2 + 2*a*b + 
b^2)*e^(-6*x))
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 107 vs. \(2 (49) = 98\).

Time = 0.19 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.81 \[ \int \frac {\text {csch}^4(x)}{a+b \cosh ^2(x)} \, dx=\frac {b^{2} \arctan \left (\frac {b e^{\left (2 \, x\right )} + 2 \, a + b}{2 \, \sqrt {-a^{2} - a b}}\right )}{{\left (a^{2} + 2 \, a b + b^{2}\right )} \sqrt {-a^{2} - a b}} + \frac {2 \, {\left (3 \, b e^{\left (4 \, x\right )} - 6 \, a e^{\left (2 \, x\right )} - 12 \, b e^{\left (2 \, x\right )} + 2 \, a + 5 \, b\right )}}{3 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} {\left (e^{\left (2 \, x\right )} - 1\right )}^{3}} \] Input:

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

Output:

b^2*arctan(1/2*(b*e^(2*x) + 2*a + b)/sqrt(-a^2 - a*b))/((a^2 + 2*a*b + b^2 
)*sqrt(-a^2 - a*b)) + 2/3*(3*b*e^(4*x) - 6*a*e^(2*x) - 12*b*e^(2*x) + 2*a 
+ 5*b)/((a^2 + 2*a*b + b^2)*(e^(2*x) - 1)^3)
 

Mupad [B] (verification not implemented)

Time = 2.73 (sec) , antiderivative size = 245, normalized size of antiderivative = 4.15 \[ \int \frac {\text {csch}^4(x)}{a+b \cosh ^2(x)} \, dx=\frac {2\,b}{{\left (a+b\right )}^2\,\left ({\mathrm {e}}^{2\,x}-1\right )}-\frac {4}{\left (a+b\right )\,\left ({\mathrm {e}}^{4\,x}-2\,{\mathrm {e}}^{2\,x}+1\right )}-\frac {8}{3\,\left (a+b\right )\,\left (3\,{\mathrm {e}}^{2\,x}-3\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}-1\right )}-\frac {b^2\,\ln \left (\frac {4\,b^2\,\left (2\,a\,b+8\,a^2\,{\mathrm {e}}^{2\,x}+b^2\,{\mathrm {e}}^{2\,x}+b^2+8\,a\,b\,{\mathrm {e}}^{2\,x}\right )}{a\,{\left (a+b\right )}^5}-\frac {8\,b^2\,\left (b+4\,a\,{\mathrm {e}}^{2\,x}+2\,b\,{\mathrm {e}}^{2\,x}\right )}{\sqrt {a}\,{\left (a+b\right )}^{9/2}}\right )}{2\,\sqrt {a}\,{\left (a+b\right )}^{5/2}}+\frac {b^2\,\ln \left (\frac {8\,b^2\,\left (b+4\,a\,{\mathrm {e}}^{2\,x}+2\,b\,{\mathrm {e}}^{2\,x}\right )}{\sqrt {a}\,{\left (a+b\right )}^{9/2}}+\frac {4\,b^2\,\left (2\,a\,b+8\,a^2\,{\mathrm {e}}^{2\,x}+b^2\,{\mathrm {e}}^{2\,x}+b^2+8\,a\,b\,{\mathrm {e}}^{2\,x}\right )}{a\,{\left (a+b\right )}^5}\right )}{2\,\sqrt {a}\,{\left (a+b\right )}^{5/2}} \] Input:

int(1/(sinh(x)^4*(a + b*cosh(x)^2)),x)
 

Output:

(2*b)/((a + b)^2*(exp(2*x) - 1)) - 4/((a + b)*(exp(4*x) - 2*exp(2*x) + 1)) 
 - 8/(3*(a + b)*(3*exp(2*x) - 3*exp(4*x) + exp(6*x) - 1)) - (b^2*log((4*b^ 
2*(2*a*b + 8*a^2*exp(2*x) + b^2*exp(2*x) + b^2 + 8*a*b*exp(2*x)))/(a*(a + 
b)^5) - (8*b^2*(b + 4*a*exp(2*x) + 2*b*exp(2*x)))/(a^(1/2)*(a + b)^(9/2))) 
)/(2*a^(1/2)*(a + b)^(5/2)) + (b^2*log((8*b^2*(b + 4*a*exp(2*x) + 2*b*exp( 
2*x)))/(a^(1/2)*(a + b)^(9/2)) + (4*b^2*(2*a*b + 8*a^2*exp(2*x) + b^2*exp( 
2*x) + b^2 + 8*a*b*exp(2*x)))/(a*(a + b)^5)))/(2*a^(1/2)*(a + b)^(5/2))
 

Reduce [B] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 687, normalized size of antiderivative = 11.64 \[ \int \frac {\text {csch}^4(x)}{a+b \cosh ^2(x)} \, dx=\frac {3 e^{6 x} \sqrt {a}\, \sqrt {a +b}\, \mathrm {log}\left (-\sqrt {2 \sqrt {a}\, \sqrt {a +b}-2 a -b}+e^{x} \sqrt {b}\right ) b^{2}+3 e^{6 x} \sqrt {a}\, \sqrt {a +b}\, \mathrm {log}\left (\sqrt {2 \sqrt {a}\, \sqrt {a +b}-2 a -b}+e^{x} \sqrt {b}\right ) b^{2}-3 e^{6 x} \sqrt {a}\, \sqrt {a +b}\, \mathrm {log}\left (2 \sqrt {a}\, \sqrt {a +b}+e^{2 x} b +2 a +b \right ) b^{2}-9 e^{4 x} \sqrt {a}\, \sqrt {a +b}\, \mathrm {log}\left (-\sqrt {2 \sqrt {a}\, \sqrt {a +b}-2 a -b}+e^{x} \sqrt {b}\right ) b^{2}-9 e^{4 x} \sqrt {a}\, \sqrt {a +b}\, \mathrm {log}\left (\sqrt {2 \sqrt {a}\, \sqrt {a +b}-2 a -b}+e^{x} \sqrt {b}\right ) b^{2}+9 e^{4 x} \sqrt {a}\, \sqrt {a +b}\, \mathrm {log}\left (2 \sqrt {a}\, \sqrt {a +b}+e^{2 x} b +2 a +b \right ) b^{2}+9 e^{2 x} \sqrt {a}\, \sqrt {a +b}\, \mathrm {log}\left (-\sqrt {2 \sqrt {a}\, \sqrt {a +b}-2 a -b}+e^{x} \sqrt {b}\right ) b^{2}+9 e^{2 x} \sqrt {a}\, \sqrt {a +b}\, \mathrm {log}\left (\sqrt {2 \sqrt {a}\, \sqrt {a +b}-2 a -b}+e^{x} \sqrt {b}\right ) b^{2}-9 e^{2 x} \sqrt {a}\, \sqrt {a +b}\, \mathrm {log}\left (2 \sqrt {a}\, \sqrt {a +b}+e^{2 x} b +2 a +b \right ) b^{2}-3 \sqrt {a}\, \sqrt {a +b}\, \mathrm {log}\left (-\sqrt {2 \sqrt {a}\, \sqrt {a +b}-2 a -b}+e^{x} \sqrt {b}\right ) b^{2}-3 \sqrt {a}\, \sqrt {a +b}\, \mathrm {log}\left (\sqrt {2 \sqrt {a}\, \sqrt {a +b}-2 a -b}+e^{x} \sqrt {b}\right ) b^{2}+3 \sqrt {a}\, \sqrt {a +b}\, \mathrm {log}\left (2 \sqrt {a}\, \sqrt {a +b}+e^{2 x} b +2 a +b \right ) b^{2}+4 e^{6 x} a^{2} b +4 e^{6 x} a \,b^{2}-24 e^{2 x} a^{3}-60 e^{2 x} a^{2} b -36 e^{2 x} a \,b^{2}+8 a^{3}+24 a^{2} b +16 a \,b^{2}}{6 a \left (e^{6 x} a^{3}+3 e^{6 x} a^{2} b +3 e^{6 x} a \,b^{2}+e^{6 x} b^{3}-3 e^{4 x} a^{3}-9 e^{4 x} a^{2} b -9 e^{4 x} a \,b^{2}-3 e^{4 x} b^{3}+3 e^{2 x} a^{3}+9 e^{2 x} a^{2} b +9 e^{2 x} a \,b^{2}+3 e^{2 x} b^{3}-a^{3}-3 a^{2} b -3 a \,b^{2}-b^{3}\right )} \] Input:

int(csch(x)^4/(a+b*cosh(x)^2),x)
 

Output:

(3*e**(6*x)*sqrt(a)*sqrt(a + b)*log( - sqrt(2*sqrt(a)*sqrt(a + b) - 2*a - 
b) + e**x*sqrt(b))*b**2 + 3*e**(6*x)*sqrt(a)*sqrt(a + b)*log(sqrt(2*sqrt(a 
)*sqrt(a + b) - 2*a - b) + e**x*sqrt(b))*b**2 - 3*e**(6*x)*sqrt(a)*sqrt(a 
+ b)*log(2*sqrt(a)*sqrt(a + b) + e**(2*x)*b + 2*a + b)*b**2 - 9*e**(4*x)*s 
qrt(a)*sqrt(a + b)*log( - sqrt(2*sqrt(a)*sqrt(a + b) - 2*a - b) + e**x*sqr 
t(b))*b**2 - 9*e**(4*x)*sqrt(a)*sqrt(a + b)*log(sqrt(2*sqrt(a)*sqrt(a + b) 
 - 2*a - b) + e**x*sqrt(b))*b**2 + 9*e**(4*x)*sqrt(a)*sqrt(a + b)*log(2*sq 
rt(a)*sqrt(a + b) + e**(2*x)*b + 2*a + b)*b**2 + 9*e**(2*x)*sqrt(a)*sqrt(a 
 + b)*log( - sqrt(2*sqrt(a)*sqrt(a + b) - 2*a - b) + e**x*sqrt(b))*b**2 + 
9*e**(2*x)*sqrt(a)*sqrt(a + b)*log(sqrt(2*sqrt(a)*sqrt(a + b) - 2*a - b) + 
 e**x*sqrt(b))*b**2 - 9*e**(2*x)*sqrt(a)*sqrt(a + b)*log(2*sqrt(a)*sqrt(a 
+ b) + e**(2*x)*b + 2*a + b)*b**2 - 3*sqrt(a)*sqrt(a + b)*log( - sqrt(2*sq 
rt(a)*sqrt(a + b) - 2*a - b) + e**x*sqrt(b))*b**2 - 3*sqrt(a)*sqrt(a + b)* 
log(sqrt(2*sqrt(a)*sqrt(a + b) - 2*a - b) + e**x*sqrt(b))*b**2 + 3*sqrt(a) 
*sqrt(a + b)*log(2*sqrt(a)*sqrt(a + b) + e**(2*x)*b + 2*a + b)*b**2 + 4*e* 
*(6*x)*a**2*b + 4*e**(6*x)*a*b**2 - 24*e**(2*x)*a**3 - 60*e**(2*x)*a**2*b 
- 36*e**(2*x)*a*b**2 + 8*a**3 + 24*a**2*b + 16*a*b**2)/(6*a*(e**(6*x)*a**3 
 + 3*e**(6*x)*a**2*b + 3*e**(6*x)*a*b**2 + e**(6*x)*b**3 - 3*e**(4*x)*a**3 
 - 9*e**(4*x)*a**2*b - 9*e**(4*x)*a*b**2 - 3*e**(4*x)*b**3 + 3*e**(2*x)*a* 
*3 + 9*e**(2*x)*a**2*b + 9*e**(2*x)*a*b**2 + 3*e**(2*x)*b**3 - a**3 - 3...