\(\int \frac {\text {csch}^4(c+d x)}{(a+b \text {sech}^2(c+d x))^3} \, dx\) [48]

Optimal result

Integrand size = 23, antiderivative size = 165 \[ \int \frac {\text {csch}^4(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=-\frac {5 (3 a-4 b) \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{8 (a+b)^{9/2} d}+\frac {(a-2 b) \coth (c+d x)}{(a+b)^4 d}-\frac {\coth ^3(c+d x)}{3 (a+b)^3 d}-\frac {a b \tanh (c+d x)}{4 (a+b)^3 d \left (a+b-b \tanh ^2(c+d x)\right )^2}-\frac {(7 a-4 b) b \tanh (c+d x)}{8 (a+b)^4 d \left (a+b-b \tanh ^2(c+d x)\right )} \] Output:

-5/8*(3*a-4*b)*b^(1/2)*arctanh(b^(1/2)*tanh(d*x+c)/(a+b)^(1/2))/(a+b)^(9/2 
)/d+(a-2*b)*coth(d*x+c)/(a+b)^4/d-1/3*coth(d*x+c)^3/(a+b)^3/d-1/4*a*b*tanh 
(d*x+c)/(a+b)^3/d/(a+b-b*tanh(d*x+c)^2)^2-1/8*(7*a-4*b)*b*tanh(d*x+c)/(a+b 
)^4/d/(a+b-b*tanh(d*x+c)^2)
                                                                                    
                                                                                    
 

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(985\) vs. \(2(165)=330\).

Time = 3.93 (sec) , antiderivative size = 985, normalized size of antiderivative = 5.97 \[ \int \frac {\text {csch}^4(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=-\frac {(a+2 b+a \cosh (2 (c+d x))) \text {sech}^6(c+d x) \left (\frac {480 (3 a-4 b) b \text {arctanh}\left (\frac {\text {sech}(d x) (\cosh (2 c)-\sinh (2 c)) ((a+2 b) \sinh (d x)-a \sinh (2 c+d x))}{2 \sqrt {a+b} \sqrt {b (\cosh (c)-\sinh (c))^4}}\right ) (a+2 b+a \cosh (2 (c+d x)))^2 (\cosh (2 c)-\sinh (2 c))}{\sqrt {a+b} \sqrt {b (\cosh (c)-\sinh (c))^4}}+\frac {\text {csch}(c) \text {csch}^3(c+d x) \text {sech}(2 c) \left (4 \left (44 a^4+122 a^3 b+63 a^2 b^2+126 a b^3+36 b^4\right ) \sinh (d x)+\left (-96 a^4-71 a^3 b+344 a^2 b^2-1208 a b^3+48 b^4\right ) \sinh (3 d x)+224 a^4 \sinh (2 c-d x)+576 a^3 b \sinh (2 c-d x)+124 a^2 b^2 \sinh (2 c-d x)-2184 a b^3 \sinh (2 c-d x)+144 b^4 \sinh (2 c-d x)-224 a^4 \sinh (2 c+d x)-657 a^3 b \sinh (2 c+d x)-538 a^2 b^2 \sinh (2 c+d x)+984 a b^3 \sinh (2 c+d x)+144 b^4 \sinh (2 c+d x)+176 a^4 \sinh (4 c+d x)+569 a^3 b \sinh (4 c+d x)+666 a^2 b^2 \sinh (4 c+d x)+1704 a b^3 \sinh (4 c+d x)-144 b^4 \sinh (4 c+d x)+48 a^4 \sinh (2 c+3 d x)+111 a^3 b \sinh (2 c+3 d x)+360 a^2 b^2 \sinh (2 c+3 d x)+312 a b^3 \sinh (2 c+3 d x)-48 b^4 \sinh (2 c+3 d x)-96 a^4 \sinh (4 c+3 d x)-152 a^3 b \sinh (4 c+3 d x)+146 a^2 b^2 \sinh (4 c+3 d x)-728 a b^3 \sinh (4 c+3 d x)-48 b^4 \sinh (4 c+3 d x)+48 a^4 \sinh (6 c+3 d x)+192 a^3 b \sinh (6 c+3 d x)+558 a^2 b^2 \sinh (6 c+3 d x)-168 a b^3 \sinh (6 c+3 d x)+48 b^4 \sinh (6 c+3 d x)+16 a^4 \sinh (2 c+5 d x)-598 a^2 b^2 \sinh (2 c+5 d x)+48 a b^3 \sinh (2 c+5 d x)+72 a^3 b \sinh (4 c+5 d x)+150 a^2 b^2 \sinh (4 c+5 d x)-48 a b^3 \sinh (4 c+5 d x)+16 a^4 \sinh (6 c+5 d x)+27 a^3 b \sinh (6 c+5 d x)-388 a^2 b^2 \sinh (6 c+5 d x)+45 a^3 b \sinh (8 c+5 d x)-60 a^2 b^2 \sinh (8 c+5 d x)+16 a^4 \sinh (4 c+7 d x)-83 a^3 b \sinh (4 c+7 d x)+6 a^2 b^2 \sinh (4 c+7 d x)+27 a^3 b \sinh (6 c+7 d x)-6 a^2 b^2 \sinh (6 c+7 d x)+16 a^4 \sinh (8 c+7 d x)-56 a^3 b \sinh (8 c+7 d x)\right )}{a}\right )}{6144 (a+b)^4 d \left (a+b \text {sech}^2(c+d x)\right )^3} \] Input:

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

Output:

-1/6144*((a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d*x]^6*((480*(3*a - 4*b) 
*b*ArcTanh[(Sech[d*x]*(Cosh[2*c] - Sinh[2*c])*((a + 2*b)*Sinh[d*x] - a*Sin 
h[2*c + d*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cosh[c] - Sinh[c])^4])]*(a + 2*b + a 
*Cosh[2*(c + d*x)])^2*(Cosh[2*c] - Sinh[2*c]))/(Sqrt[a + b]*Sqrt[b*(Cosh[c 
] - Sinh[c])^4]) + (Csch[c]*Csch[c + d*x]^3*Sech[2*c]*(4*(44*a^4 + 122*a^3 
*b + 63*a^2*b^2 + 126*a*b^3 + 36*b^4)*Sinh[d*x] + (-96*a^4 - 71*a^3*b + 34 
4*a^2*b^2 - 1208*a*b^3 + 48*b^4)*Sinh[3*d*x] + 224*a^4*Sinh[2*c - d*x] + 5 
76*a^3*b*Sinh[2*c - d*x] + 124*a^2*b^2*Sinh[2*c - d*x] - 2184*a*b^3*Sinh[2 
*c - d*x] + 144*b^4*Sinh[2*c - d*x] - 224*a^4*Sinh[2*c + d*x] - 657*a^3*b* 
Sinh[2*c + d*x] - 538*a^2*b^2*Sinh[2*c + d*x] + 984*a*b^3*Sinh[2*c + d*x] 
+ 144*b^4*Sinh[2*c + d*x] + 176*a^4*Sinh[4*c + d*x] + 569*a^3*b*Sinh[4*c + 
 d*x] + 666*a^2*b^2*Sinh[4*c + d*x] + 1704*a*b^3*Sinh[4*c + d*x] - 144*b^4 
*Sinh[4*c + d*x] + 48*a^4*Sinh[2*c + 3*d*x] + 111*a^3*b*Sinh[2*c + 3*d*x] 
+ 360*a^2*b^2*Sinh[2*c + 3*d*x] + 312*a*b^3*Sinh[2*c + 3*d*x] - 48*b^4*Sin 
h[2*c + 3*d*x] - 96*a^4*Sinh[4*c + 3*d*x] - 152*a^3*b*Sinh[4*c + 3*d*x] + 
146*a^2*b^2*Sinh[4*c + 3*d*x] - 728*a*b^3*Sinh[4*c + 3*d*x] - 48*b^4*Sinh[ 
4*c + 3*d*x] + 48*a^4*Sinh[6*c + 3*d*x] + 192*a^3*b*Sinh[6*c + 3*d*x] + 55 
8*a^2*b^2*Sinh[6*c + 3*d*x] - 168*a*b^3*Sinh[6*c + 3*d*x] + 48*b^4*Sinh[6* 
c + 3*d*x] + 16*a^4*Sinh[2*c + 5*d*x] - 598*a^2*b^2*Sinh[2*c + 5*d*x] + 48 
*a*b^3*Sinh[2*c + 5*d*x] + 72*a^3*b*Sinh[4*c + 5*d*x] + 150*a^2*b^2*Sin...
 

Rubi [A] (verified)

Time = 0.49 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.02, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3042, 4620, 361, 1582, 1584, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4620

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

\(\Big \downarrow \) 361

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

\(\Big \downarrow \) 1582

\(\displaystyle \frac {\frac {1}{4} b \left (\frac {\int \frac {\coth ^4(c+d x) \left (-\frac {(7 a-4 b) b^2 \tanh ^4(c+d x)}{a+b}-8 (a-b) b \tanh ^2(c+d x)+8 b (a+b)\right )}{-b \tanh ^2(c+d x)+a+b}d\tanh (c+d x)}{2 b^2 (a+b)^3}-\frac {(7 a-4 b) \tanh (c+d x)}{2 (a+b)^4 \left (a-b \tanh ^2(c+d x)+b\right )}\right )-\frac {a b \tanh (c+d x)}{4 (a+b)^3 \left (a-b \tanh ^2(c+d x)+b\right )^2}}{d}\)

\(\Big \downarrow \) 1584

\(\displaystyle \frac {\frac {1}{4} b \left (\frac {\int \left (8 b \coth ^4(c+d x)+\frac {8 b (2 b-a) \coth ^2(c+d x)}{a+b}-\frac {5 b^2 (4 b-3 a)}{(a+b) \left (b \tanh ^2(c+d x)-a-b\right )}\right )d\tanh (c+d x)}{2 b^2 (a+b)^3}-\frac {(7 a-4 b) \tanh (c+d x)}{2 (a+b)^4 \left (a-b \tanh ^2(c+d x)+b\right )}\right )-\frac {a b \tanh (c+d x)}{4 (a+b)^3 \left (a-b \tanh ^2(c+d x)+b\right )^2}}{d}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\frac {1}{4} b \left (\frac {-\frac {5 b^{3/2} (3 a-4 b) \text {arctanh}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{(a+b)^{3/2}}+\frac {8 b (a-2 b) \coth (c+d x)}{a+b}-\frac {8}{3} b \coth ^3(c+d x)}{2 b^2 (a+b)^3}-\frac {(7 a-4 b) \tanh (c+d x)}{2 (a+b)^4 \left (a-b \tanh ^2(c+d x)+b\right )}\right )-\frac {a b \tanh (c+d x)}{4 (a+b)^3 \left (a-b \tanh ^2(c+d x)+b\right )^2}}{d}\)

Input:

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

Output:

(-1/4*(a*b*Tanh[c + d*x])/((a + b)^3*(a + b - b*Tanh[c + d*x]^2)^2) + (b*( 
((-5*(3*a - 4*b)*b^(3/2)*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(a 
+ b)^(3/2) + (8*(a - 2*b)*b*Coth[c + d*x])/(a + b) - (8*b*Coth[c + d*x]^3) 
/3)/(2*b^2*(a + b)^3) - ((7*a - 4*b)*Tanh[c + d*x])/(2*(a + b)^4*(a + b - 
b*Tanh[c + d*x]^2))))/4)/d
 

Defintions of rubi rules used

rule 361

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] : 
> Simp[(-a)^(m/2 - 1)*(b*c - a*d)*x*((a + b*x^2)^(p + 1)/(2*b^(m/2 + 1)*(p 
+ 1))), x] + Simp[1/(2*b^(m/2 + 1)*(p + 1))   Int[x^m*(a + b*x^2)^(p + 1)*E 
xpandToSum[2*b*(p + 1)*Together[(b^(m/2)*(c + d*x^2) - (-a)^(m/2 - 1)*(b*c 
- a*d)*x^(-m + 2))/(a + b*x^2)] - ((-a)^(m/2 - 1)*(b*c - a*d))/x^m, x], x], 
 x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && ILtQ[m/ 
2, 0] && (IntegerQ[p] || EqQ[m + 2*p + 1, 0])
 

rule 1582

Int[(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^ 
4)^(p_.), x_Symbol] :> Simp[(-d)^(m/2 - 1)*(c*d^2 - b*d*e + a*e^2)^p*x*((d 
+ e*x^2)^(q + 1)/(2*e^(2*p + m/2)*(q + 1))), x] + Simp[(-d)^(m/2 - 1)/(2*e^ 
(2*p)*(q + 1))   Int[x^m*(d + e*x^2)^(q + 1)*ExpandToSum[Together[(1/(d + e 
*x^2))*(2*(-d)^(-m/2 + 1)*e^(2*p)*(q + 1)*(a + b*x^2 + c*x^4)^p - ((c*d^2 - 
 b*d*e + a*e^2)^p/(e^(m/2)*x^m))*(d + e*(2*q + 3)*x^2))], x], x], x] /; Fre 
eQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && ILtQ[q, -1] 
&& ILtQ[m/2, 0]
 

rule 1584

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + ( 
c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(f*x)^m*(d + e*x^2)^q* 
(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && NeQ[ 
b^2 - 4*a*c, 0] && IGtQ[p, 0] && IGtQ[q, -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 4620

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1436\) vs. \(2(149)=298\).

Time = 0.25 (sec) , antiderivative size = 1437, normalized size of antiderivative = 8.71

\[\text {Expression too large to display}\]

Input:

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

Output:

-1/24/d/(a^3+3*a^2*b+3*a*b^2+b^3)/(a+b)*a*tanh(1/2*d*x+1/2*c)^3-1/24/d/(a^ 
3+3*a^2*b+3*a*b^2+b^3)/(a+b)*b*tanh(1/2*d*x+1/2*c)^3+3/8/d/(a^3+3*a^2*b+3* 
a*b^2+b^3)/(a+b)*tanh(1/2*d*x+1/2*c)*a-9/8/d/(a^3+3*a^2*b+3*a*b^2+b^3)/(a+ 
b)*tanh(1/2*d*x+1/2*c)*b-1/24/d/(a+b)^3/tanh(1/2*d*x+1/2*c)^3+3/8/d/(a+b)^ 
4/tanh(1/2*d*x+1/2*c)*a-9/8/d/(a+b)^4/tanh(1/2*d*x+1/2*c)*b-9/4/d*b/(a+b)^ 
4/(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*tanh(1/2*d*x+1/2*c)^7*a^2-5/4/d*b^2/(a 
+b)^4/(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*tanh(1/2*d*x+1/2*c)^7*a+1/d*b^3/(a 
+b)^4/(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*tanh(1/2*d*x+1/2*c)^7-27/4/d*b/(a+ 
b)^4/(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*tanh(1/2*d*x+1/2*c)^5*a^2+13/4/d*b^ 
2/(a+b)^4/(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*tanh(1/2*d*x+1/2*c)^5*a-1/d*b^ 
3/(a+b)^4/(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*tanh(1/2*d*x+1/2*c)^5-27/4/d*b 
/(a+b)^4/(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*tanh(1/2*d*x+1/2*c)^3*a^2+13/4/ 
d*b^2/(a+b)^4/(tanh(1/2*d*x+1/2*c)^4*a+tanh(1/2*d*x+1/2*c)^4*b+2*tanh(1...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 7442 vs. \(2 (155) = 310\).

Time = 0.86 (sec) , antiderivative size = 15161, normalized size of antiderivative = 91.88 \[ \int \frac {\text {csch}^4(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F]

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

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

Output:

Integral(csch(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. 782 vs. \(2 (155) = 310\).

Time = 0.29 (sec) , antiderivative size = 782, normalized size of antiderivative = 4.74 \[ \int \frac {\text {csch}^4(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

5/16*(3*a*b - 4*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) + 1/12*(16*a^4 - 83*a^3*b + 6* 
a^2*b^2 + 2*(8*a^4 - 299*a^2*b^2 + 24*a*b^3)*e^(-2*d*x - 2*c) - (96*a^4 + 
71*a^3*b - 344*a^2*b^2 + 1208*a*b^3 - 48*b^4)*e^(-4*d*x - 4*c) - 4*(56*a^4 
 + 144*a^3*b + 31*a^2*b^2 - 546*a*b^3 + 36*b^4)*e^(-6*d*x - 6*c) - (176*a^ 
4 + 569*a^3*b + 666*a^2*b^2 + 1704*a*b^3 - 144*b^4)*e^(-8*d*x - 8*c) - 6*( 
8*a^4 + 32*a^3*b + 93*a^2*b^2 - 28*a*b^3 + 8*b^4)*e^(-10*d*x - 10*c) - 15* 
(3*a^3*b - 4*a^2*b^2)*e^(-12*d*x - 12*c))/((a^7 + 4*a^6*b + 6*a^5*b^2 + 4* 
a^4*b^3 + a^3*b^4 + (a^7 + 12*a^6*b + 38*a^5*b^2 + 52*a^4*b^3 + 33*a^3*b^4 
 + 8*a^2*b^5)*e^(-2*d*x - 2*c) - (3*a^7 + 20*a^6*b + 34*a^5*b^2 - 4*a^4*b^ 
3 - 61*a^3*b^4 - 56*a^2*b^5 - 16*a*b^6)*e^(-4*d*x - 4*c) - (3*a^7 + 28*a^6 
*b + 130*a^5*b^2 + 300*a^4*b^3 + 355*a^3*b^4 + 208*a^2*b^5 + 48*a*b^6)*e^( 
-6*d*x - 6*c) + (3*a^7 + 28*a^6*b + 130*a^5*b^2 + 300*a^4*b^3 + 355*a^3*b^ 
4 + 208*a^2*b^5 + 48*a*b^6)*e^(-8*d*x - 8*c) + (3*a^7 + 20*a^6*b + 34*a^5* 
b^2 - 4*a^4*b^3 - 61*a^3*b^4 - 56*a^2*b^5 - 16*a*b^6)*e^(-10*d*x - 10*c) - 
 (a^7 + 12*a^6*b + 38*a^5*b^2 + 52*a^4*b^3 + 33*a^3*b^4 + 8*a^2*b^5)*e^(-1 
2*d*x - 12*c) - (a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*e^(-14*d 
*x - 14*c))*d)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 406 vs. \(2 (155) = 310\).

Time = 0.37 (sec) , antiderivative size = 406, normalized size of antiderivative = 2.46 \[ \int \frac {\text {csch}^4(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=-\frac {\frac {15 \, {\left (3 \, a b - 4 \, b^{2}\right )} \arctan \left (\frac {a e^{\left (2 \, d x + 2 \, c\right )} + a + 2 \, b}{2 \, \sqrt {-a b - b^{2}}}\right )}{{\left (a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right )} \sqrt {-a b - b^{2}}} - \frac {6 \, {\left (9 \, a^{3} b e^{\left (6 \, d x + 6 \, c\right )} + 20 \, a^{2} b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 27 \, a^{3} b e^{\left (4 \, d x + 4 \, c\right )} + 66 \, a^{2} b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 56 \, a b^{3} e^{\left (4 \, d x + 4 \, c\right )} - 16 \, b^{4} e^{\left (4 \, d x + 4 \, c\right )} + 27 \, a^{3} b e^{\left (2 \, d x + 2 \, c\right )} + 44 \, a^{2} b^{2} e^{\left (2 \, d x + 2 \, c\right )} - 16 \, a b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 9 \, a^{3} b - 2 \, a^{2} b^{2}\right )}}{{\left (a^{5} + 4 \, a^{4} b + 6 \, a^{3} b^{2} + 4 \, a^{2} b^{3} + a b^{4}\right )} {\left (a e^{\left (4 \, d x + 4 \, c\right )} + 2 \, a e^{\left (2 \, d x + 2 \, c\right )} + 4 \, b e^{\left (2 \, d x + 2 \, c\right )} + a\right )}^{2}} + \frac {16 \, {\left (9 \, b e^{\left (4 \, d x + 4 \, c\right )} + 6 \, a e^{\left (2 \, d x + 2 \, c\right )} - 12 \, b e^{\left (2 \, d x + 2 \, c\right )} - 2 \, a + 7 \, b\right )}}{{\left (a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right )} {\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{3}}}{24 \, d} \] Input:

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

Output:

-1/24*(15*(3*a*b - 4*b^2)*arctan(1/2*(a*e^(2*d*x + 2*c) + a + 2*b)/sqrt(-a 
*b - b^2))/((a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*sqrt(-a*b - b^2)) 
- 6*(9*a^3*b*e^(6*d*x + 6*c) + 20*a^2*b^2*e^(6*d*x + 6*c) + 27*a^3*b*e^(4* 
d*x + 4*c) + 66*a^2*b^2*e^(4*d*x + 4*c) + 56*a*b^3*e^(4*d*x + 4*c) - 16*b^ 
4*e^(4*d*x + 4*c) + 27*a^3*b*e^(2*d*x + 2*c) + 44*a^2*b^2*e^(2*d*x + 2*c) 
- 16*a*b^3*e^(2*d*x + 2*c) + 9*a^3*b - 2*a^2*b^2)/((a^5 + 4*a^4*b + 6*a^3* 
b^2 + 4*a^2*b^3 + a*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) + 16*(9*b*e^(4*d*x + 4*c) + 6*a*e^(2*d*x + 2*c) - 12 
*b*e^(2*d*x + 2*c) - 2*a + 7*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. \[ \int \frac {\text {csch}^4(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^6}{{\mathrm {sinh}\left (c+d\,x\right )}^4\,{\left (a\,{\mathrm {cosh}\left (c+d\,x\right )}^2+b\right )}^3} \,d x \] Input:

int(1/(sinh(c + d*x)^4*(a + b/cosh(c + d*x)^2)^3),x)
 

Output:

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

Reduce [B] (verification not implemented)

Time = 0.90 (sec) , antiderivative size = 6786, normalized size of antiderivative = 41.13 \[ \int \frac {\text {csch}^4(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 45*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 - 300*e**(14*c + 14*d*x)*sqr 
t(b)*sqrt(a + b)*log( - sqrt(2*sqrt(b)*sqrt(a + b) - a - 2*b) + e**(c + d* 
x)*sqrt(a))*a**4*b + 480*e**(14*c + 14*d*x)*sqrt(b)*sqrt(a + b)*log( - sqr 
t(2*sqrt(b)*sqrt(a + b) - a - 2*b) + e**(c + d*x)*sqrt(a))*a**3*b**2 - 45* 
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 - 300*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 + 480*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**2 + 45*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**5 + 300*e**(14*c + 14*d*x)*sqrt(b)*sqrt(a + b)*log(2*sqrt(b)*sqr 
t(a + b) + e**(2*c + 2*d*x)*a + a + 2*b)*a**4*b - 480*e**(14*c + 14*d*x)*s 
qrt(b)*sqrt(a + b)*log(2*sqrt(b)*sqrt(a + b) + e**(2*c + 2*d*x)*a + a + 2* 
b)*a**3*b**2 - 45*e**(12*c + 12*d*x)*sqrt(b)*sqrt(a + b)*log( - sqrt(2*sqr 
t(b)*sqrt(a + b) - a - 2*b) + e**(c + d*x)*sqrt(a))*a**5 - 660*e**(12*c + 
12*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 - 1920*e**(12*c + 12*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**2 + 3840*e**(12*c + 12*d*x)*sqrt(b)*sqrt(a + b)*log( - sqrt(2*sqrt...