\(\int \frac {\coth ^4(c+d x)}{(a+b \text {sech}^2(c+d x))^3} \, dx\) [168]

Optimal result

Integrand size = 23, antiderivative size = 232 \[ \int \frac {\coth ^4(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\frac {x}{a^3}-\frac {b^{5/2} \left (63 a^2+36 a b+8 b^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{8 a^3 (a+b)^{9/2} d}-\frac {\left (8 a^3+32 a^2 b-15 a b^2-4 b^3\right ) \coth (c+d x)}{8 a^2 (a+b)^4 d}-\frac {\left (8 a^2-39 a b-12 b^2\right ) \coth ^3(c+d x)}{24 a^2 (a+b)^3 d}-\frac {b \coth ^3(c+d x)}{4 a (a+b) d \left (a+b-b \tanh ^2(c+d x)\right )^2}-\frac {b (11 a+4 b) \coth ^3(c+d x)}{8 a^2 (a+b)^2 d \left (a+b-b \tanh ^2(c+d x)\right )} \] Output:

x/a^3-1/8*b^(5/2)*(63*a^2+36*a*b+8*b^2)*arctanh(b^(1/2)*tanh(d*x+c)/(a+b)^ 
(1/2))/a^3/(a+b)^(9/2)/d-1/8*(8*a^3+32*a^2*b-15*a*b^2-4*b^3)*coth(d*x+c)/a 
^2/(a+b)^4/d-1/24*(8*a^2-39*a*b-12*b^2)*coth(d*x+c)^3/a^2/(a+b)^3/d-1/4*b* 
coth(d*x+c)^3/a/(a+b)/d/(a+b-b*tanh(d*x+c)^2)^2-1/8*b*(11*a+4*b)*coth(d*x+ 
c)^3/a^2/(a+b)^2/d/(a+b-b*tanh(d*x+c)^2)
                                                                                    
                                                                                    
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 8.32 (sec) , antiderivative size = 3334, normalized size of antiderivative = 14.37 \[ \int \frac {\coth ^4(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\text {Result too large to show} \] Input:

Integrate[Coth[c + d*x]^4/(a + b*Sech[c + d*x]^2)^3,x]
 

Output:

((63*a^2 + 36*a*b + 8*b^2)*(a + 2*b + a*Cosh[2*c + 2*d*x])^3*Sech[c + d*x] 
^6*(((I/64)*b^3*ArcTan[Sech[d*x]*(((-1/2*I)*Cosh[2*c])/(Sqrt[a + b]*Sqrt[b 
*Cosh[4*c] - b*Sinh[4*c]]) + ((I/2)*Sinh[2*c])/(Sqrt[a + b]*Sqrt[b*Cosh[4* 
c] - b*Sinh[4*c]]))*(-(a*Sinh[d*x]) - 2*b*Sinh[d*x] + a*Sinh[2*c + d*x])]* 
Cosh[2*c])/(a^3*Sqrt[a + b]*d*Sqrt[b*Cosh[4*c] - b*Sinh[4*c]]) - ((I/64)*b 
^3*ArcTan[Sech[d*x]*(((-1/2*I)*Cosh[2*c])/(Sqrt[a + b]*Sqrt[b*Cosh[4*c] - 
b*Sinh[4*c]]) + ((I/2)*Sinh[2*c])/(Sqrt[a + b]*Sqrt[b*Cosh[4*c] - b*Sinh[4 
*c]]))*(-(a*Sinh[d*x]) - 2*b*Sinh[d*x] + a*Sinh[2*c + d*x])]*Sinh[2*c])/(a 
^3*Sqrt[a + b]*d*Sqrt[b*Cosh[4*c] - b*Sinh[4*c]])))/((a + b)^4*(a + b*Sech 
[c + d*x]^2)^3) + ((a + 2*b + a*Cosh[2*c + 2*d*x])*Csch[c]*Csch[c + d*x]^3 
*Sech[2*c]*Sech[c + d*x]^6*(-36*a^6*d*x*Cosh[d*x] - 336*a^5*b*d*x*Cosh[d*x 
] - 1560*a^4*b^2*d*x*Cosh[d*x] - 3600*a^3*b^3*d*x*Cosh[d*x] - 4260*a^2*b^4 
*d*x*Cosh[d*x] - 2496*a*b^5*d*x*Cosh[d*x] - 576*b^6*d*x*Cosh[d*x] + 36*a^6 
*d*x*Cosh[3*d*x] + 240*a^5*b*d*x*Cosh[3*d*x] + 408*a^4*b^2*d*x*Cosh[3*d*x] 
 - 48*a^3*b^3*d*x*Cosh[3*d*x] - 732*a^2*b^4*d*x*Cosh[3*d*x] - 672*a*b^5*d* 
x*Cosh[3*d*x] - 192*b^6*d*x*Cosh[3*d*x] + 36*a^6*d*x*Cosh[2*c - d*x] + 336 
*a^5*b*d*x*Cosh[2*c - d*x] + 1560*a^4*b^2*d*x*Cosh[2*c - d*x] + 3600*a^3*b 
^3*d*x*Cosh[2*c - d*x] + 4260*a^2*b^4*d*x*Cosh[2*c - d*x] + 2496*a*b^5*d*x 
*Cosh[2*c - d*x] + 576*b^6*d*x*Cosh[2*c - d*x] + 36*a^6*d*x*Cosh[2*c + d*x 
] + 336*a^5*b*d*x*Cosh[2*c + d*x] + 1560*a^4*b^2*d*x*Cosh[2*c + d*x] + ...
 

Rubi [A] (verified)

Time = 0.64 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.13, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.609, Rules used = {3042, 4629, 2075, 374, 25, 441, 25, 445, 27, 445, 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.

\(\displaystyle \int \frac {\coth ^4(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{\tan (i c+i d x)^4 \left (a+b \sec (i c+i d x)^2\right )^3}dx\)

\(\Big \downarrow \) 4629

\(\displaystyle \frac {\int \frac {\coth ^4(c+d x)}{\left (1-\tanh ^2(c+d x)\right ) \left (a+b \left (1-\tanh ^2(c+d x)\right )\right )^3}d\tanh (c+d x)}{d}\)

\(\Big \downarrow \) 2075

\(\displaystyle \frac {\int \frac {\coth ^4(c+d x)}{\left (1-\tanh ^2(c+d x)\right ) \left (-b \tanh ^2(c+d x)+a+b\right )^3}d\tanh (c+d x)}{d}\)

\(\Big \downarrow \) 374

\(\displaystyle \frac {-\frac {\int -\frac {\coth ^4(c+d x) \left (7 b \tanh ^2(c+d x)+4 a-3 b\right )}{\left (1-\tanh ^2(c+d x)\right ) \left (-b \tanh ^2(c+d x)+a+b\right )^2}d\tanh (c+d x)}{4 a (a+b)}-\frac {b \coth ^3(c+d x)}{4 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )^2}}{d}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\int \frac {\coth ^4(c+d x) \left (7 b \tanh ^2(c+d x)+4 a-3 b\right )}{\left (1-\tanh ^2(c+d x)\right ) \left (-b \tanh ^2(c+d x)+a+b\right )^2}d\tanh (c+d x)}{4 a (a+b)}-\frac {b \coth ^3(c+d x)}{4 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )^2}}{d}\)

\(\Big \downarrow \) 441

\(\displaystyle \frac {\frac {-\frac {\int -\frac {\coth ^4(c+d x) \left (8 a^2-39 b a-12 b^2+5 b (11 a+4 b) \tanh ^2(c+d x)\right )}{\left (1-\tanh ^2(c+d x)\right ) \left (-b \tanh ^2(c+d x)+a+b\right )}d\tanh (c+d x)}{2 a (a+b)}-\frac {b (11 a+4 b) \coth ^3(c+d x)}{2 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )}}{4 a (a+b)}-\frac {b \coth ^3(c+d x)}{4 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )^2}}{d}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\frac {\int \frac {\coth ^4(c+d x) \left (8 a^2-39 b a-12 b^2+5 b (11 a+4 b) \tanh ^2(c+d x)\right )}{\left (1-\tanh ^2(c+d x)\right ) \left (-b \tanh ^2(c+d x)+a+b\right )}d\tanh (c+d x)}{2 a (a+b)}-\frac {b (11 a+4 b) \coth ^3(c+d x)}{2 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )}}{4 a (a+b)}-\frac {b \coth ^3(c+d x)}{4 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )^2}}{d}\)

\(\Big \downarrow \) 445

\(\displaystyle \frac {\frac {\frac {-\frac {\int -\frac {3 \coth ^2(c+d x) \left (8 a^3+32 b a^2-15 b^2 a-4 b^3-b \left (8 a^2-39 b a-12 b^2\right ) \tanh ^2(c+d x)\right )}{\left (1-\tanh ^2(c+d x)\right ) \left (-b \tanh ^2(c+d x)+a+b\right )}d\tanh (c+d x)}{3 (a+b)}-\frac {\left (8 a^2-39 a b-12 b^2\right ) \coth ^3(c+d x)}{3 (a+b)}}{2 a (a+b)}-\frac {b (11 a+4 b) \coth ^3(c+d x)}{2 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )}}{4 a (a+b)}-\frac {b \coth ^3(c+d x)}{4 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )^2}}{d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {\frac {\int \frac {\coth ^2(c+d x) \left (8 a^3+32 b a^2-15 b^2 a-4 b^3-b \left (8 a^2-39 b a-12 b^2\right ) \tanh ^2(c+d x)\right )}{\left (1-\tanh ^2(c+d x)\right ) \left (-b \tanh ^2(c+d x)+a+b\right )}d\tanh (c+d x)}{a+b}-\frac {\left (8 a^2-39 a b-12 b^2\right ) \coth ^3(c+d x)}{3 (a+b)}}{2 a (a+b)}-\frac {b (11 a+4 b) \coth ^3(c+d x)}{2 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )}}{4 a (a+b)}-\frac {b \coth ^3(c+d x)}{4 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )^2}}{d}\)

\(\Big \downarrow \) 445

\(\displaystyle \frac {\frac {\frac {\frac {-\frac {\int -\frac {8 a^4+40 b a^3+80 b^2 a^2+17 b^3 a+4 b^4-b \left (8 a^3+32 b a^2-15 b^2 a-4 b^3\right ) \tanh ^2(c+d x)}{\left (1-\tanh ^2(c+d x)\right ) \left (-b \tanh ^2(c+d x)+a+b\right )}d\tanh (c+d x)}{a+b}-\frac {\left (8 a^3+32 a^2 b-15 a b^2-4 b^3\right ) \coth (c+d x)}{a+b}}{a+b}-\frac {\left (8 a^2-39 a b-12 b^2\right ) \coth ^3(c+d x)}{3 (a+b)}}{2 a (a+b)}-\frac {b (11 a+4 b) \coth ^3(c+d x)}{2 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )}}{4 a (a+b)}-\frac {b \coth ^3(c+d x)}{4 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )^2}}{d}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\frac {\frac {\frac {\int \frac {8 a^4+40 b a^3+80 b^2 a^2+17 b^3 a+4 b^4-b \left (8 a^3+32 b a^2-15 b^2 a-4 b^3\right ) \tanh ^2(c+d x)}{\left (1-\tanh ^2(c+d x)\right ) \left (-b \tanh ^2(c+d x)+a+b\right )}d\tanh (c+d x)}{a+b}-\frac {\left (8 a^3+32 a^2 b-15 a b^2-4 b^3\right ) \coth (c+d x)}{a+b}}{a+b}-\frac {\left (8 a^2-39 a b-12 b^2\right ) \coth ^3(c+d x)}{3 (a+b)}}{2 a (a+b)}-\frac {b (11 a+4 b) \coth ^3(c+d x)}{2 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )}}{4 a (a+b)}-\frac {b \coth ^3(c+d x)}{4 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )^2}}{d}\)

\(\Big \downarrow \) 397

\(\displaystyle \frac {\frac {\frac {\frac {\frac {\frac {8 (a+b)^4 \int \frac {1}{1-\tanh ^2(c+d x)}d\tanh (c+d x)}{a}-\frac {b^3 \left (63 a^2+36 a b+8 b^2\right ) \int \frac {1}{-b \tanh ^2(c+d x)+a+b}d\tanh (c+d x)}{a}}{a+b}-\frac {\left (8 a^3+32 a^2 b-15 a b^2-4 b^3\right ) \coth (c+d x)}{a+b}}{a+b}-\frac {\left (8 a^2-39 a b-12 b^2\right ) \coth ^3(c+d x)}{3 (a+b)}}{2 a (a+b)}-\frac {b (11 a+4 b) \coth ^3(c+d x)}{2 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )}}{4 a (a+b)}-\frac {b \coth ^3(c+d x)}{4 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )^2}}{d}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {\frac {\frac {\frac {\frac {8 (a+b)^4 \text {arctanh}(\tanh (c+d x))}{a}-\frac {b^3 \left (63 a^2+36 a b+8 b^2\right ) \int \frac {1}{-b \tanh ^2(c+d x)+a+b}d\tanh (c+d x)}{a}}{a+b}-\frac {\left (8 a^3+32 a^2 b-15 a b^2-4 b^3\right ) \coth (c+d x)}{a+b}}{a+b}-\frac {\left (8 a^2-39 a b-12 b^2\right ) \coth ^3(c+d x)}{3 (a+b)}}{2 a (a+b)}-\frac {b (11 a+4 b) \coth ^3(c+d x)}{2 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )}}{4 a (a+b)}-\frac {b \coth ^3(c+d x)}{4 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )^2}}{d}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {\frac {\frac {\frac {\frac {8 (a+b)^4 \text {arctanh}(\tanh (c+d x))}{a}-\frac {b^{5/2} \left (63 a^2+36 a b+8 b^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{a \sqrt {a+b}}}{a+b}-\frac {\left (8 a^3+32 a^2 b-15 a b^2-4 b^3\right ) \coth (c+d x)}{a+b}}{a+b}-\frac {\left (8 a^2-39 a b-12 b^2\right ) \coth ^3(c+d x)}{3 (a+b)}}{2 a (a+b)}-\frac {b (11 a+4 b) \coth ^3(c+d x)}{2 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )}}{4 a (a+b)}-\frac {b \coth ^3(c+d x)}{4 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )^2}}{d}\)

Input:

Int[Coth[c + d*x]^4/(a + b*Sech[c + d*x]^2)^3,x]
 

Output:

(-1/4*(b*Coth[c + d*x]^3)/(a*(a + b)*(a + b - b*Tanh[c + d*x]^2)^2) + ((-1 
/3*((8*a^2 - 39*a*b - 12*b^2)*Coth[c + d*x]^3)/(a + b) + (((8*(a + b)^4*Ar 
cTanh[Tanh[c + d*x]])/a - (b^(5/2)*(63*a^2 + 36*a*b + 8*b^2)*ArcTanh[(Sqrt 
[b]*Tanh[c + d*x])/Sqrt[a + b]])/(a*Sqrt[a + b]))/(a + b) - ((8*a^3 + 32*a 
^2*b - 15*a*b^2 - 4*b^3)*Coth[c + d*x])/(a + b))/(a + b))/(2*a*(a + b)) - 
(b*(11*a + 4*b)*Coth[c + d*x]^3)/(2*a*(a + b)*(a + b - b*Tanh[c + d*x]^2)) 
)/(4*a*(a + b)))/d
 

Defintions of rubi rules used

rule 25

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

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, 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 374

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ 
), x_Symbol] :> Simp[(-b)*(e*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q 
 + 1)/(a*e*2*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*2*(b*c - a*d)*(p + 1)) 
 Int[(e*x)^m*(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[b*c*(m + 1) + 2*(b*c - 
a*d)*(p + 1) + d*b*(m + 2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b, 
c, d, e, m, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && IntBinomialQ[a, b, 
 c, d, e, m, 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 441

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ 
)*((e_) + (f_.)*(x_)^2), x_Symbol] :> Simp[(-(b*e - a*f))*(g*x)^(m + 1)*(a 
+ b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(a*g*2*(b*c - a*d)*(p + 1))), x] + Si 
mp[1/(a*2*(b*c - a*d)*(p + 1))   Int[(g*x)^m*(a + b*x^2)^(p + 1)*(c + d*x^2 
)^q*Simp[c*(b*e - a*f)*(m + 1) + e*2*(b*c - a*d)*(p + 1) + d*(b*e - a*f)*(m 
 + 2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m, q}, 
 x] && LtQ[p, -1]
 

rule 445

Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_ 
.)*((e_) + (f_.)*(x_)^2), x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b*x^2)^(p 
+ 1)*((c + d*x^2)^(q + 1)/(a*c*g*(m + 1))), x] + Simp[1/(a*c*g^2*(m + 1)) 
 Int[(g*x)^(m + 2)*(a + b*x^2)^p*(c + d*x^2)^q*Simp[a*f*c*(m + 1) - e*(b*c 
+ a*d)*(m + 2 + 1) - e*2*(b*c*p + a*d*q) - b*e*d*(m + 2*(p + q + 2) + 1)*x^ 
2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] && LtQ[m, -1]
 

rule 2075

Int[(u_)^(p_.)*(v_)^(q_.)*((e_.)*(x_))^(m_.), x_Symbol] :> Int[(e*x)^m*Expa 
ndToSum[u, x]^p*ExpandToSum[v, x]^q, x] /; FreeQ[{e, m, p, q}, x] && Binomi 
alQ[{u, v}, x] && EqQ[BinomialDegree[u, x] - BinomialDegree[v, x], 0] &&  ! 
BinomialMatchQ[{u, v}, x]
 

rule 3042

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

rule 4629

Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_.)*((d_.)*tan[(e_.) + (f 
_.)*(x_)])^(m_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Sim 
p[ff/f   Subst[Int[(d*ff*x)^m*((a + b*(1 + ff^2*x^2)^(n/2))^p/(1 + ff^2*x^2 
)), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, d, e, f, m, p}, x] && Inte 
gerQ[n/2] && (IntegerQ[m/2] || EqQ[n, 2])
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(467\) vs. \(2(214)=428\).

Time = 255.44 (sec) , antiderivative size = 468, normalized size of antiderivative = 2.02

method	result	size
derivativedivides	\(\frac {-\frac {\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} a}{3}+\frac {b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3}+5 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+17 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right ) \left (a +b \right )}-\frac {1}{24 \left (a +b \right )^{3} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {5 a +17 b}{8 \left (a +b \right )^{4} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a^{3}}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a^{3}}+\frac {2 b^{3} \left (\frac {\left (-\frac {17}{8} a^{3}-\frac {21}{8} a^{2} b -\frac {1}{2} a \,b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}-\frac {\left (51 a^{2}+3 a b -4 b^{2}\right ) a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{8}-\frac {\left (51 a^{2}+3 a b -4 b^{2}\right ) a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{8}+\left (-\frac {17}{8} a^{3}-\frac {21}{8} a^{2} b -\frac {1}{2} a \,b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} b +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a +b \right )^{2}}+\frac {\left (63 a^{2}+36 a b +8 b^{2}\right ) \left (-\frac {\ln \left (\sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}+\frac {\ln \left (\sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}\right )}{8}\right )}{\left (a +b \right )^{4} a^{3}}}{d}\)	\(468\)
default	\(\frac {-\frac {\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} a}{3}+\frac {b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3}+5 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+17 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right ) \left (a +b \right )}-\frac {1}{24 \left (a +b \right )^{3} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {5 a +17 b}{8 \left (a +b \right )^{4} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a^{3}}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a^{3}}+\frac {2 b^{3} \left (\frac {\left (-\frac {17}{8} a^{3}-\frac {21}{8} a^{2} b -\frac {1}{2} a \,b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}-\frac {\left (51 a^{2}+3 a b -4 b^{2}\right ) a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{8}-\frac {\left (51 a^{2}+3 a b -4 b^{2}\right ) a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{8}+\left (-\frac {17}{8} a^{3}-\frac {21}{8} a^{2} b -\frac {1}{2} a \,b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} b +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a +b \right )^{2}}+\frac {\left (63 a^{2}+36 a b +8 b^{2}\right ) \left (-\frac {\ln \left (\sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}+\frac {\ln \left (\sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}\right )}{8}\right )}{\left (a +b \right )^{4} a^{3}}}{d}\)	\(468\)
risch	\(\frac {x}{a^{3}}-\frac {104 a^{5} b +80 \,{\mathrm e}^{2 d x +2 c} a^{6}+48 \,{\mathrm e}^{4 d x +4 c} a^{6}+144 b^{6} {\mathrm e}^{4 d x +4 c}+18 a^{2} b^{4}+32 a^{6}+432 b^{6} {\mathrm e}^{8 d x +8 c}+32 a^{6} {\mathrm e}^{6 d x +6 c}-432 b^{6} {\mathrm e}^{6 d x +6 c}+51 a^{3} b^{3}+48 a^{6} {\mathrm e}^{12 d x +12 c}+144 a^{6} {\mathrm e}^{10 d x +10 c}-144 b^{6} {\mathrm e}^{10 d x +10 c}+128 a^{6} {\mathrm e}^{8 d x +8 c}+960 a^{4} b^{2} {\mathrm e}^{10 d x +10 c}-66 a^{2} b^{4} {\mathrm e}^{10 d x +10 c}-408 a \,b^{5} {\mathrm e}^{10 d x +10 c}+440 a^{5} b \,{\mathrm e}^{8 d x +8 c}+1152 a^{4} b^{2} {\mathrm e}^{8 d x +8 c}+2073 a^{3} b^{3} {\mathrm e}^{8 d x +8 c}+642 a^{2} b^{4} {\mathrm e}^{8 d x +8 c}+1416 a \,b^{5} {\mathrm e}^{8 d x +8 c}+120 a^{5} b \,{\mathrm e}^{12 d x +12 c}-51 a^{3} b^{3} {\mathrm e}^{12 d x +12 c}-132 a^{2} b^{4} {\mathrm e}^{12 d x +12 c}-48 a \,b^{5} {\mathrm e}^{12 d x +12 c}+672 a^{5} b \,{\mathrm e}^{10 d x +10 c}+480 \,{\mathrm e}^{2 d x +2 c} a^{5} b +294 a^{2} b^{4} {\mathrm e}^{2 d x +2 c}+96 a \,b^{5} {\mathrm e}^{2 d x +2 c}-1320 a \,b^{5} {\mathrm e}^{6 d x +6 c}+104 a^{5} b \,{\mathrm e}^{4 d x +4 c}+640 a^{4} b^{2} {\mathrm e}^{4 d x +4 c}+1511 a^{3} b^{3} {\mathrm e}^{4 d x +4 c}-528 a^{2} b^{4} {\mathrm e}^{4 d x +4 c}+264 a \,b^{5} {\mathrm e}^{4 d x +4 c}+832 a^{4} b^{2} {\mathrm e}^{2 d x +2 c}-228 a^{2} b^{4} {\mathrm e}^{6 d x +6 c}-384 a^{5} b \,{\mathrm e}^{6 d x +6 c}-2048 a^{4} b^{2} {\mathrm e}^{6 d x +6 c}-3072 a^{3} b^{3} {\mathrm e}^{6 d x +6 c}}{12 a^{3} d \left (a +b \right )^{4} \left ({\mathrm e}^{2 d x +2 c}-1\right )^{3} \left ({\mathrm e}^{4 d x +4 c} a +2 a \,{\mathrm e}^{2 d x +2 c}+4 b \,{\mathrm e}^{2 d x +2 c}+a \right )^{2}}+\frac {63 \sqrt {b \left (a +b \right )}\, b^{2} \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {a +2 \sqrt {b \left (a +b \right )}+2 b}{a}\right )}{16 \left (a +b \right )^{5} d a}+\frac {9 \sqrt {b \left (a +b \right )}\, b^{3} \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {a +2 \sqrt {b \left (a +b \right )}+2 b}{a}\right )}{4 \left (a +b \right )^{5} d \,a^{2}}+\frac {\sqrt {b \left (a +b \right )}\, b^{4} \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {a +2 \sqrt {b \left (a +b \right )}+2 b}{a}\right )}{2 \left (a +b \right )^{5} d \,a^{3}}-\frac {63 \sqrt {b \left (a +b \right )}\, b^{2} \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {-a +2 \sqrt {b \left (a +b \right )}-2 b}{a}\right )}{16 \left (a +b \right )^{5} d a}-\frac {9 \sqrt {b \left (a +b \right )}\, b^{3} \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {-a +2 \sqrt {b \left (a +b \right )}-2 b}{a}\right )}{4 \left (a +b \right )^{5} d \,a^{2}}-\frac {\sqrt {b \left (a +b \right )}\, b^{4} \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {-a +2 \sqrt {b \left (a +b \right )}-2 b}{a}\right )}{2 \left (a +b \right )^{5} d \,a^{3}}\)	\(996\)

Input:

int(coth(d*x+c)^4/(a+b*sech(d*x+c)^2)^3,x,method=_RETURNVERBOSE)
 

Output:

1/d*(-1/8/(a^3+3*a^2*b+3*a*b^2+b^3)/(a+b)*(1/3*tanh(1/2*d*x+1/2*c)^3*a+1/3 
*b*tanh(1/2*d*x+1/2*c)^3+5*a*tanh(1/2*d*x+1/2*c)+17*b*tanh(1/2*d*x+1/2*c)) 
-1/24/(a+b)^3/tanh(1/2*d*x+1/2*c)^3-1/8*(5*a+17*b)/(a+b)^4/tanh(1/2*d*x+1/ 
2*c)-1/a^3*ln(tanh(1/2*d*x+1/2*c)-1)+1/a^3*ln(tanh(1/2*d*x+1/2*c)+1)+2*b^3 
/(a+b)^4/a^3*(((-17/8*a^3-21/8*a^2*b-1/2*a*b^2)*tanh(1/2*d*x+1/2*c)^7-1/8* 
(51*a^2+3*a*b-4*b^2)*a*tanh(1/2*d*x+1/2*c)^5-1/8*(51*a^2+3*a*b-4*b^2)*a*ta 
nh(1/2*d*x+1/2*c)^3+(-17/8*a^3-21/8*a^2*b-1/2*a*b^2)*tanh(1/2*d*x+1/2*c))/ 
(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)^2+1/8*(63*a^2+36*a*b+8*b^2)*(-1/4/b^(1/2)/ 
(a+b)^(1/2)*ln((a+b)^(1/2)*tanh(1/2*d*x+1/2*c)^2+2*tanh(1/2*d*x+1/2*c)*b^( 
1/2)+(a+b)^(1/2))+1/4/b^(1/2)/(a+b)^(1/2)*ln((a+b)^(1/2)*tanh(1/2*d*x+1/2* 
c)^2-2*tanh(1/2*d*x+1/2*c)*b^(1/2)+(a+b)^(1/2)))))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 11993 vs. \(2 (220) = 440\).

Time = 0.86 (sec) , antiderivative size = 24263, normalized size of antiderivative = 104.58 \[ \int \frac {\coth ^4(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\text {Too large to display} \] Input:

integrate(coth(d*x+c)^4/(a+b*sech(d*x+c)^2)^3,x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F]

\[ \int \frac {\coth ^4(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\int \frac {\coth ^{4}{\left (c + d x \right )}}{\left (a + b \operatorname {sech}^{2}{\left (c + d x \right )}\right )^{3}}\, dx \] Input:

integrate(coth(d*x+c)**4/(a+b*sech(d*x+c)**2)**3,x)
 

Output:

Integral(coth(c + d*x)**4/(a + b*sech(c + d*x)**2)**3, x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4920 vs. \(2 (220) = 440\).

Time = 0.92 (sec) , antiderivative size = 4920, normalized size of antiderivative = 21.21 \[ \int \frac {\coth ^4(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\text {Too large to display} \] Input:

integrate(coth(d*x+c)^4/(a+b*sech(d*x+c)^2)^3,x, algorithm="maxima")
 

Output:

1/8*(3*a^3*b + 12*a^2*b^2 + 8*a*b^3 + 2*b^4)*log(a*e^(4*d*x + 4*c) + 2*(a 
+ 2*b)*e^(2*d*x + 2*c) + a)/((a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3* 
b^4)*d) - 3/4*b*log(a*e^(4*d*x + 4*c) + 2*(a + 2*b)*e^(2*d*x + 2*c) + a)/( 
(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*d) - 1/8*(3*a^3*b + 12*a^2*b^2 
 + 8*a*b^3 + 2*b^4)*log(2*(a + 2*b)*e^(-2*d*x - 2*c) + a*e^(-4*d*x - 4*c) 
+ a)/((a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*d) + 3/4*b*log(2*( 
a + 2*b)*e^(-2*d*x - 2*c) + a*e^(-4*d*x - 4*c) + a)/((a^4 + 4*a^3*b + 6*a^ 
2*b^2 + 4*a*b^3 + b^4)*d) + 1/4*(2*a + 5*b)*log(e^(2*d*x + 2*c) - 1)/((a^4 
 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*d) + 3/2*b*log(e^(2*d*x + 2*c) - 1 
)/((a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*d) - 1/4*(2*a + 5*b)*log(e^ 
(-2*d*x - 2*c) - 1)/((a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*d) - 3/2* 
b*log(e^(-2*d*x - 2*c) - 1)/((a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*d 
) - 1/256*(15*a^4*b + 260*a^3*b^2 + 504*a^2*b^3 + 288*a*b^4 + 64*b^5)*log( 
(a*e^(2*d*x + 2*c) + a + 2*b - 2*sqrt((a + b)*b))/(a*e^(2*d*x + 2*c) + a + 
 2*b + 2*sqrt((a + b)*b)))/((a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b 
^4)*sqrt((a + b)*b)*d) + 5/64*(3*a*b + 10*b^2)*log((a*e^(2*d*x + 2*c) + a 
+ 2*b - 2*sqrt((a + b)*b))/(a*e^(2*d*x + 2*c) + a + 2*b + 2*sqrt((a + b)*b 
)))/((a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*sqrt((a + b)*b)*d) + 1/25 
6*(15*a^4*b + 260*a^3*b^2 + 504*a^2*b^3 + 288*a*b^4 + 64*b^5)*log((a*e^(-2 
*d*x - 2*c) + a + 2*b - 2*sqrt((a + b)*b))/(a*e^(-2*d*x - 2*c) + a + 2*...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 469 vs. \(2 (220) = 440\).

Time = 1.21 (sec) , antiderivative size = 469, normalized size of antiderivative = 2.02 \[ \int \frac {\coth ^4(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=-\frac {\frac {3 \, {\left (63 \, a^{2} b^{3} + 36 \, a b^{4} + 8 \, b^{5}\right )} \arctan \left (\frac {a e^{\left (2 \, d x + 2 \, c\right )} + a + 2 \, b}{2 \, \sqrt {-a b - b^{2}}}\right )}{{\left (a^{7} + 4 \, a^{6} b + 6 \, a^{5} b^{2} + 4 \, a^{4} b^{3} + a^{3} b^{4}\right )} \sqrt {-a b - b^{2}}} - \frac {6 \, {\left (17 \, a^{3} b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 44 \, a^{2} b^{4} e^{\left (6 \, d x + 6 \, c\right )} + 16 \, a b^{5} e^{\left (6 \, d x + 6 \, c\right )} + 51 \, a^{3} b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 154 \, a^{2} b^{4} e^{\left (4 \, d x + 4 \, c\right )} + 184 \, a b^{5} e^{\left (4 \, d x + 4 \, c\right )} + 48 \, b^{6} e^{\left (4 \, d x + 4 \, c\right )} + 51 \, a^{3} b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 116 \, a^{2} b^{4} e^{\left (2 \, d x + 2 \, c\right )} + 32 \, a b^{5} e^{\left (2 \, d x + 2 \, c\right )} + 17 \, a^{3} b^{3} + 6 \, a^{2} b^{4}\right )}}{{\left (a^{7} + 4 \, a^{6} b + 6 \, a^{5} b^{2} + 4 \, a^{4} b^{3} + a^{3} b^{4}\right )} {\left (a e^{\left (4 \, d x + 4 \, c\right )} + 2 \, a e^{\left (2 \, d x + 2 \, c\right )} + 4 \, b e^{\left (2 \, d x + 2 \, c\right )} + a\right )}^{2}} - \frac {24 \, {\left (d x + c\right )}}{a^{3}} + \frac {16 \, {\left (6 \, a e^{\left (4 \, d x + 4 \, c\right )} + 15 \, b e^{\left (4 \, d x + 4 \, c\right )} - 6 \, a e^{\left (2 \, d x + 2 \, c\right )} - 24 \, b e^{\left (2 \, d x + 2 \, c\right )} + 4 \, a + 13 \, b\right )}}{{\left (a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right )} {\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{3}}}{24 \, d} \] Input:

integrate(coth(d*x+c)^4/(a+b*sech(d*x+c)^2)^3,x, algorithm="giac")
 

Output:

-1/24*(3*(63*a^2*b^3 + 36*a*b^4 + 8*b^5)*arctan(1/2*(a*e^(2*d*x + 2*c) + a 
 + 2*b)/sqrt(-a*b - b^2))/((a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^ 
4)*sqrt(-a*b - b^2)) - 6*(17*a^3*b^3*e^(6*d*x + 6*c) + 44*a^2*b^4*e^(6*d*x 
 + 6*c) + 16*a*b^5*e^(6*d*x + 6*c) + 51*a^3*b^3*e^(4*d*x + 4*c) + 154*a^2* 
b^4*e^(4*d*x + 4*c) + 184*a*b^5*e^(4*d*x + 4*c) + 48*b^6*e^(4*d*x + 4*c) + 
 51*a^3*b^3*e^(2*d*x + 2*c) + 116*a^2*b^4*e^(2*d*x + 2*c) + 32*a*b^5*e^(2* 
d*x + 2*c) + 17*a^3*b^3 + 6*a^2*b^4)/((a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b 
^3 + a^3*b^4)*(a*e^(4*d*x + 4*c) + 2*a*e^(2*d*x + 2*c) + 4*b*e^(2*d*x + 2* 
c) + a)^2) - 24*(d*x + c)/a^3 + 16*(6*a*e^(4*d*x + 4*c) + 15*b*e^(4*d*x + 
4*c) - 6*a*e^(2*d*x + 2*c) - 24*b*e^(2*d*x + 2*c) + 4*a + 13*b)/((a^4 + 4* 
a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*(e^(2*d*x + 2*c) - 1)^3))/d
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\coth ^4(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^6\,{\mathrm {coth}\left (c+d\,x\right )}^4}{{\left (a\,{\mathrm {cosh}\left (c+d\,x\right )}^2+b\right )}^3} \,d x \] Input:

int(coth(c + d*x)^4/(a + b/cosh(c + d*x)^2)^3,x)
 

Output:

int((cosh(c + d*x)^6*coth(c + d*x)^4)/(b + a*cosh(c + d*x)^2)^3, x)
 

Reduce [B] (verification not implemented)

Time = 25.08 (sec) , antiderivative size = 9527, normalized size of antiderivative = 41.06 \[ \int \frac {\coth ^4(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx =\text {Too large to display} \] Input:

int(coth(d*x+c)^4/(a+b*sech(d*x+c)^2)^3,x)
 

Output:

( - 189*e**(14*c + 14*d*x)*sqrt(b)*sqrt(a + b)*log( - sqrt(2*sqrt(b)*sqrt( 
a + b) - a - 2*b) + e**(c + d*x)*sqrt(a))*a**5*b**2 - 1620*e**(14*c + 14*d 
*x)*sqrt(b)*sqrt(a + b)*log( - sqrt(2*sqrt(b)*sqrt(a + b) - a - 2*b) + e** 
(c + d*x)*sqrt(a))*a**4*b**3 - 888*e**(14*c + 14*d*x)*sqrt(b)*sqrt(a + b)* 
log( - sqrt(2*sqrt(b)*sqrt(a + b) - a - 2*b) + e**(c + d*x)*sqrt(a))*a**3* 
b**4 - 192*e**(14*c + 14*d*x)*sqrt(b)*sqrt(a + b)*log( - sqrt(2*sqrt(b)*sq 
rt(a + b) - a - 2*b) + e**(c + d*x)*sqrt(a))*a**2*b**5 - 189*e**(14*c + 14 
*d*x)*sqrt(b)*sqrt(a + b)*log(sqrt(2*sqrt(b)*sqrt(a + b) - a - 2*b) + e**( 
c + d*x)*sqrt(a))*a**5*b**2 - 1620*e**(14*c + 14*d*x)*sqrt(b)*sqrt(a + b)* 
log(sqrt(2*sqrt(b)*sqrt(a + b) - a - 2*b) + e**(c + d*x)*sqrt(a))*a**4*b** 
3 - 888*e**(14*c + 14*d*x)*sqrt(b)*sqrt(a + b)*log(sqrt(2*sqrt(b)*sqrt(a + 
 b) - a - 2*b) + e**(c + d*x)*sqrt(a))*a**3*b**4 - 192*e**(14*c + 14*d*x)* 
sqrt(b)*sqrt(a + b)*log(sqrt(2*sqrt(b)*sqrt(a + b) - a - 2*b) + e**(c + d* 
x)*sqrt(a))*a**2*b**5 + 189*e**(14*c + 14*d*x)*sqrt(b)*sqrt(a + b)*log(2*s 
qrt(b)*sqrt(a + b) + e**(2*c + 2*d*x)*a + a + 2*b)*a**5*b**2 + 1620*e**(14 
*c + 14*d*x)*sqrt(b)*sqrt(a + b)*log(2*sqrt(b)*sqrt(a + b) + e**(2*c + 2*d 
*x)*a + a + 2*b)*a**4*b**3 + 888*e**(14*c + 14*d*x)*sqrt(b)*sqrt(a + b)*lo 
g(2*sqrt(b)*sqrt(a + b) + e**(2*c + 2*d*x)*a + a + 2*b)*a**3*b**4 + 192*e* 
*(14*c + 14*d*x)*sqrt(b)*sqrt(a + b)*log(2*sqrt(b)*sqrt(a + b) + e**(2*c + 
 2*d*x)*a + a + 2*b)*a**2*b**5 - 189*e**(12*c + 12*d*x)*sqrt(b)*sqrt(a ...