\(\int e^{c (a+b x)} \text {csch}^2(a c+b c x)^{7/2} \, dx\) [125]

Optimal result

Integrand size = 25, antiderivative size = 199 \[ \int e^{c (a+b x)} \text {csch}^2(a c+b c x)^{7/2} \, dx=-\frac {32 \sqrt {\text {csch}^2(a c+b c x)} \sinh (a c+b c x)}{3 b c \left (1-e^{2 c (a+b x)}\right )^6}+\frac {192 \sqrt {\text {csch}^2(a c+b c x)} \sinh (a c+b c x)}{5 b c \left (1-e^{2 c (a+b x)}\right )^5}-\frac {48 \sqrt {\text {csch}^2(a c+b c x)} \sinh (a c+b c x)}{b c \left (1-e^{2 c (a+b x)}\right )^4}+\frac {64 \sqrt {\text {csch}^2(a c+b c x)} \sinh (a c+b c x)}{3 b c \left (1-e^{2 c (a+b x)}\right )^3} \] Output:

-32/3*(csch(b*c*x+a*c)^2)^(1/2)*sinh(b*c*x+a*c)/b/c/(1-exp(2*c*(b*x+a)))^6 
+192/5*(csch(b*c*x+a*c)^2)^(1/2)*sinh(b*c*x+a*c)/b/c/(1-exp(2*c*(b*x+a)))^ 
5-48*(csch(b*c*x+a*c)^2)^(1/2)*sinh(b*c*x+a*c)/b/c/(1-exp(2*c*(b*x+a)))^4+ 
64/3*(csch(b*c*x+a*c)^2)^(1/2)*sinh(b*c*x+a*c)/b/c/(1-exp(2*c*(b*x+a)))^3
 

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.42 \[ \int e^{c (a+b x)} \text {csch}^2(a c+b c x)^{7/2} \, dx=-\frac {16 \left (-1+6 e^{2 c (a+b x)}-15 e^{4 c (a+b x)}+20 e^{6 c (a+b x)}\right ) \sqrt {\text {csch}^2(c (a+b x))} \sinh (c (a+b x))}{15 b c \left (-1+e^{2 c (a+b x)}\right )^6} \] Input:

Integrate[E^(c*(a + b*x))*(Csch[a*c + b*c*x]^2)^(7/2),x]
 

Output:

(-16*(-1 + 6*E^(2*c*(a + b*x)) - 15*E^(4*c*(a + b*x)) + 20*E^(6*c*(a + b*x 
)))*Sqrt[Csch[c*(a + b*x)]^2]*Sinh[c*(a + b*x)])/(15*b*c*(-1 + E^(2*c*(a + 
 b*x)))^6)
 

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.57, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {7271, 2720, 27, 243, 53, 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.

\(\displaystyle \int e^{c (a+b x)} \text {csch}^2(a c+b c x)^{7/2} \, dx\)

\(\Big \downarrow \) 7271

\(\displaystyle \sinh (a c+b c x) \sqrt {\text {csch}^2(a c+b c x)} \int e^{c (a+b x)} \text {csch}^7(a c+b x c)dx\)

\(\Big \downarrow \) 2720

\(\displaystyle \frac {\sinh (a c+b c x) \sqrt {\text {csch}^2(a c+b c x)} \int -\frac {128 e^{7 c (a+b x)}}{\left (1-e^{2 c (a+b x)}\right )^7}de^{c (a+b x)}}{b c}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {128 \sinh (a c+b c x) \sqrt {\text {csch}^2(a c+b c x)} \int \frac {e^{7 c (a+b x)}}{\left (1-e^{2 c (a+b x)}\right )^7}de^{c (a+b x)}}{b c}\)

\(\Big \downarrow \) 243

\(\displaystyle -\frac {64 \sinh (a c+b c x) \sqrt {\text {csch}^2(a c+b c x)} \int \frac {e^{3 c (a+b x)}}{\left (1-e^{2 c (a+b x)}\right )^7}de^{2 c (a+b x)}}{b c}\)

\(\Big \downarrow \) 53

\(\displaystyle -\frac {64 \sinh (a c+b c x) \sqrt {\text {csch}^2(a c+b c x)} \int \left (-\frac {1}{\left (-1+e^{2 c (a+b x)}\right )^4}-\frac {3}{\left (-1+e^{2 c (a+b x)}\right )^5}-\frac {3}{\left (-1+e^{2 c (a+b x)}\right )^6}-\frac {1}{\left (-1+e^{2 c (a+b x)}\right )^7}\right )de^{2 c (a+b x)}}{b c}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {64 \left (-\frac {1}{3 \left (1-e^{2 c (a+b x)}\right )^3}+\frac {3}{4 \left (1-e^{2 c (a+b x)}\right )^4}-\frac {3}{5 \left (1-e^{2 c (a+b x)}\right )^5}+\frac {1}{6 \left (1-e^{2 c (a+b x)}\right )^6}\right ) \sinh (a c+b c x) \sqrt {\text {csch}^2(a c+b c x)}}{b c}\)

Input:

Int[E^(c*(a + b*x))*(Csch[a*c + b*c*x]^2)^(7/2),x]
 

Output:

(-64*(1/(6*(1 - E^(2*c*(a + b*x)))^6) - 3/(5*(1 - E^(2*c*(a + b*x)))^5) + 
3/(4*(1 - E^(2*c*(a + b*x)))^4) - 1/(3*(1 - E^(2*c*(a + b*x)))^3))*Sqrt[Cs 
ch[a*c + b*c*x]^2]*Sinh[a*c + b*c*x])/(b*c)
 

Defintions of rubi rules used

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 53

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int 
[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, 
x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) 
|| LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
 

rule 243

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2   Subst[In 
t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I 
ntegerQ[(m - 1)/2]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2720

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] 
   Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct 
ionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ 
[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) 
*(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
 

rule 7271

Int[(u_.)*((a_.)*(v_)^(m_.))^(p_), x_Symbol] :> Simp[a^IntPart[p]*((a*v^m)^ 
FracPart[p]/v^(m*FracPart[p]))   Int[u*v^(m*p), x], x] /; FreeQ[{a, m, p}, 
x] &&  !IntegerQ[p] &&  !FreeQ[v, x] &&  !(EqQ[a, 1] && EqQ[m, 1]) &&  !(Eq 
Q[v, x] && EqQ[m, 1])
 

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 1.35 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.44

method	result	size
default	\(-\frac {\operatorname {csgn}\left (\operatorname {csch}\left (c \left (b x +a \right )\right )\right ) \left (\frac {\coth \left (c \left (b x +a \right )\right )^{6}}{6}+\frac {\coth \left (c \left (b x +a \right )\right )^{5}}{5}-\frac {\coth \left (c \left (b x +a \right )\right )^{4}}{2}-\frac {2 \coth \left (c \left (b x +a \right )\right )^{3}}{3}+\frac {\coth \left (c \left (b x +a \right )\right )^{2}}{2}+\coth \left (c \left (b x +a \right )\right )\right )}{b c}\)	\(87\)
risch	\(-\frac {16 \left (20 \,{\mathrm e}^{6 c \left (b x +a \right )}-15 \,{\mathrm e}^{4 c \left (b x +a \right )}+6 \,{\mathrm e}^{2 c \left (b x +a \right )}-1\right ) \sqrt {\frac {{\mathrm e}^{2 c \left (b x +a \right )}}{\left ({\mathrm e}^{2 c \left (b x +a \right )}-1\right )^{2}}}\, {\mathrm e}^{-c \left (b x +a \right )}}{15 b c \left ({\mathrm e}^{2 c \left (b x +a \right )}-1\right )^{5}}\)	\(91\)

Input:

int(exp(c*(b*x+a))*(csch(b*c*x+a*c)^2)^(7/2),x,method=_RETURNVERBOSE)
 

Output:

-csgn(csch(c*(b*x+a)))/b/c*(1/6*coth(c*(b*x+a))^6+1/5*coth(c*(b*x+a))^5-1/ 
2*coth(c*(b*x+a))^4-2/3*coth(c*(b*x+a))^3+1/2*coth(c*(b*x+a))^2+coth(c*(b* 
x+a)))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 592 vs. \(2 (173) = 346\).

Time = 0.08 (sec) , antiderivative size = 592, normalized size of antiderivative = 2.97 \[ \int e^{c (a+b x)} \text {csch}^2(a c+b c x)^{7/2} \, dx=-\frac {16 \, {\left (19 \, \cosh \left (b c x + a c\right )^{3} + 57 \, \cosh \left (b c x + a c\right ) \sinh \left (b c x + a c\right )^{2} + 21 \, \sinh \left (b c x + a c\right )^{3} + 21 \, {\left (3 \, \cosh \left (b c x + a c\right )^{2} - 1\right )} \sinh \left (b c x + a c\right ) - 9 \, \cosh \left (b c x + a c\right )\right )}}{15 \, {\left (b c \cosh \left (b c x + a c\right )^{9} + 9 \, b c \cosh \left (b c x + a c\right ) \sinh \left (b c x + a c\right )^{8} + b c \sinh \left (b c x + a c\right )^{9} - 6 \, b c \cosh \left (b c x + a c\right )^{7} + 6 \, {\left (6 \, b c \cosh \left (b c x + a c\right )^{2} - b c\right )} \sinh \left (b c x + a c\right )^{7} + 15 \, b c \cosh \left (b c x + a c\right )^{5} + 42 \, {\left (2 \, b c \cosh \left (b c x + a c\right )^{3} - b c \cosh \left (b c x + a c\right )\right )} \sinh \left (b c x + a c\right )^{6} + 3 \, {\left (42 \, b c \cosh \left (b c x + a c\right )^{4} - 42 \, b c \cosh \left (b c x + a c\right )^{2} + 5 \, b c\right )} \sinh \left (b c x + a c\right )^{5} - 19 \, b c \cosh \left (b c x + a c\right )^{3} + 3 \, {\left (42 \, b c \cosh \left (b c x + a c\right )^{5} - 70 \, b c \cosh \left (b c x + a c\right )^{3} + 25 \, b c \cosh \left (b c x + a c\right )\right )} \sinh \left (b c x + a c\right )^{4} + 3 \, {\left (28 \, b c \cosh \left (b c x + a c\right )^{6} - 70 \, b c \cosh \left (b c x + a c\right )^{4} + 50 \, b c \cosh \left (b c x + a c\right )^{2} - 7 \, b c\right )} \sinh \left (b c x + a c\right )^{3} + 9 \, b c \cosh \left (b c x + a c\right ) + 3 \, {\left (12 \, b c \cosh \left (b c x + a c\right )^{7} - 42 \, b c \cosh \left (b c x + a c\right )^{5} + 50 \, b c \cosh \left (b c x + a c\right )^{3} - 19 \, b c \cosh \left (b c x + a c\right )\right )} \sinh \left (b c x + a c\right )^{2} + 3 \, {\left (3 \, b c \cosh \left (b c x + a c\right )^{8} - 14 \, b c \cosh \left (b c x + a c\right )^{6} + 25 \, b c \cosh \left (b c x + a c\right )^{4} - 21 \, b c \cosh \left (b c x + a c\right )^{2} + 7 \, b c\right )} \sinh \left (b c x + a c\right )\right )}} \] Input:

integrate(exp(c*(b*x+a))*(csch(b*c*x+a*c)^2)^(7/2),x, algorithm="fricas")
 

Output:

-16/15*(19*cosh(b*c*x + a*c)^3 + 57*cosh(b*c*x + a*c)*sinh(b*c*x + a*c)^2 
+ 21*sinh(b*c*x + a*c)^3 + 21*(3*cosh(b*c*x + a*c)^2 - 1)*sinh(b*c*x + a*c 
) - 9*cosh(b*c*x + a*c))/(b*c*cosh(b*c*x + a*c)^9 + 9*b*c*cosh(b*c*x + a*c 
)*sinh(b*c*x + a*c)^8 + b*c*sinh(b*c*x + a*c)^9 - 6*b*c*cosh(b*c*x + a*c)^ 
7 + 6*(6*b*c*cosh(b*c*x + a*c)^2 - b*c)*sinh(b*c*x + a*c)^7 + 15*b*c*cosh( 
b*c*x + a*c)^5 + 42*(2*b*c*cosh(b*c*x + a*c)^3 - b*c*cosh(b*c*x + a*c))*si 
nh(b*c*x + a*c)^6 + 3*(42*b*c*cosh(b*c*x + a*c)^4 - 42*b*c*cosh(b*c*x + a* 
c)^2 + 5*b*c)*sinh(b*c*x + a*c)^5 - 19*b*c*cosh(b*c*x + a*c)^3 + 3*(42*b*c 
*cosh(b*c*x + a*c)^5 - 70*b*c*cosh(b*c*x + a*c)^3 + 25*b*c*cosh(b*c*x + a* 
c))*sinh(b*c*x + a*c)^4 + 3*(28*b*c*cosh(b*c*x + a*c)^6 - 70*b*c*cosh(b*c* 
x + a*c)^4 + 50*b*c*cosh(b*c*x + a*c)^2 - 7*b*c)*sinh(b*c*x + a*c)^3 + 9*b 
*c*cosh(b*c*x + a*c) + 3*(12*b*c*cosh(b*c*x + a*c)^7 - 42*b*c*cosh(b*c*x + 
 a*c)^5 + 50*b*c*cosh(b*c*x + a*c)^3 - 19*b*c*cosh(b*c*x + a*c))*sinh(b*c* 
x + a*c)^2 + 3*(3*b*c*cosh(b*c*x + a*c)^8 - 14*b*c*cosh(b*c*x + a*c)^6 + 2 
5*b*c*cosh(b*c*x + a*c)^4 - 21*b*c*cosh(b*c*x + a*c)^2 + 7*b*c)*sinh(b*c*x 
 + a*c))
 

Sympy [F]

\[ \int e^{c (a+b x)} \text {csch}^2(a c+b c x)^{7/2} \, dx=e^{a c} \int \left (\operatorname {csch}^{2}{\left (a c + b c x \right )}\right )^{\frac {7}{2}} e^{b c x}\, dx \] Input:

integrate(exp(c*(b*x+a))*(csch(b*c*x+a*c)**2)**(7/2),x)
 

Output:

exp(a*c)*Integral((csch(a*c + b*c*x)**2)**(7/2)*exp(b*c*x), x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 386 vs. \(2 (173) = 346\).

Time = 0.14 (sec) , antiderivative size = 386, normalized size of antiderivative = 1.94 \[ \int e^{c (a+b x)} \text {csch}^2(a c+b c x)^{7/2} \, dx=-\frac {64 \, e^{\left (6 \, b c x + 6 \, a c\right )}}{3 \, b c {\left (e^{\left (12 \, b c x + 12 \, a c\right )} - 6 \, e^{\left (10 \, b c x + 10 \, a c\right )} + 15 \, e^{\left (8 \, b c x + 8 \, a c\right )} - 20 \, e^{\left (6 \, b c x + 6 \, a c\right )} + 15 \, e^{\left (4 \, b c x + 4 \, a c\right )} - 6 \, e^{\left (2 \, b c x + 2 \, a c\right )} + 1\right )}} + \frac {16 \, e^{\left (4 \, b c x + 4 \, a c\right )}}{b c {\left (e^{\left (12 \, b c x + 12 \, a c\right )} - 6 \, e^{\left (10 \, b c x + 10 \, a c\right )} + 15 \, e^{\left (8 \, b c x + 8 \, a c\right )} - 20 \, e^{\left (6 \, b c x + 6 \, a c\right )} + 15 \, e^{\left (4 \, b c x + 4 \, a c\right )} - 6 \, e^{\left (2 \, b c x + 2 \, a c\right )} + 1\right )}} - \frac {32 \, e^{\left (2 \, b c x + 2 \, a c\right )}}{5 \, b c {\left (e^{\left (12 \, b c x + 12 \, a c\right )} - 6 \, e^{\left (10 \, b c x + 10 \, a c\right )} + 15 \, e^{\left (8 \, b c x + 8 \, a c\right )} - 20 \, e^{\left (6 \, b c x + 6 \, a c\right )} + 15 \, e^{\left (4 \, b c x + 4 \, a c\right )} - 6 \, e^{\left (2 \, b c x + 2 \, a c\right )} + 1\right )}} + \frac {16}{15 \, b c {\left (e^{\left (12 \, b c x + 12 \, a c\right )} - 6 \, e^{\left (10 \, b c x + 10 \, a c\right )} + 15 \, e^{\left (8 \, b c x + 8 \, a c\right )} - 20 \, e^{\left (6 \, b c x + 6 \, a c\right )} + 15 \, e^{\left (4 \, b c x + 4 \, a c\right )} - 6 \, e^{\left (2 \, b c x + 2 \, a c\right )} + 1\right )}} \] Input:

integrate(exp(c*(b*x+a))*(csch(b*c*x+a*c)^2)^(7/2),x, algorithm="maxima")
 

Output:

-64/3*e^(6*b*c*x + 6*a*c)/(b*c*(e^(12*b*c*x + 12*a*c) - 6*e^(10*b*c*x + 10 
*a*c) + 15*e^(8*b*c*x + 8*a*c) - 20*e^(6*b*c*x + 6*a*c) + 15*e^(4*b*c*x + 
4*a*c) - 6*e^(2*b*c*x + 2*a*c) + 1)) + 16*e^(4*b*c*x + 4*a*c)/(b*c*(e^(12* 
b*c*x + 12*a*c) - 6*e^(10*b*c*x + 10*a*c) + 15*e^(8*b*c*x + 8*a*c) - 20*e^ 
(6*b*c*x + 6*a*c) + 15*e^(4*b*c*x + 4*a*c) - 6*e^(2*b*c*x + 2*a*c) + 1)) - 
 32/5*e^(2*b*c*x + 2*a*c)/(b*c*(e^(12*b*c*x + 12*a*c) - 6*e^(10*b*c*x + 10 
*a*c) + 15*e^(8*b*c*x + 8*a*c) - 20*e^(6*b*c*x + 6*a*c) + 15*e^(4*b*c*x + 
4*a*c) - 6*e^(2*b*c*x + 2*a*c) + 1)) + 16/15/(b*c*(e^(12*b*c*x + 12*a*c) - 
 6*e^(10*b*c*x + 10*a*c) + 15*e^(8*b*c*x + 8*a*c) - 20*e^(6*b*c*x + 6*a*c) 
 + 15*e^(4*b*c*x + 4*a*c) - 6*e^(2*b*c*x + 2*a*c) + 1))
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.45 \[ \int e^{c (a+b x)} \text {csch}^2(a c+b c x)^{7/2} \, dx=-\frac {16 \, {\left (20 \, e^{\left (6 \, b c x + 6 \, a c\right )} - 15 \, e^{\left (4 \, b c x + 4 \, a c\right )} + 6 \, e^{\left (2 \, b c x + 2 \, a c\right )} - 1\right )}}{15 \, b c {\left (e^{\left (2 \, b c x + 2 \, a c\right )} - 1\right )}^{6} \mathrm {sgn}\left (e^{\left (b c x + a c\right )} - e^{\left (-b c x - a c\right )}\right )} \] Input:

integrate(exp(c*(b*x+a))*(csch(b*c*x+a*c)^2)^(7/2),x, algorithm="giac")
 

Output:

-16/15*(20*e^(6*b*c*x + 6*a*c) - 15*e^(4*b*c*x + 4*a*c) + 6*e^(2*b*c*x + 2 
*a*c) - 1)/(b*c*(e^(2*b*c*x + 2*a*c) - 1)^6*sgn(e^(b*c*x + a*c) - e^(-b*c* 
x - a*c)))
 

Mupad [B] (verification not implemented)

Time = 2.95 (sec) , antiderivative size = 413, normalized size of antiderivative = 2.08 \[ \int e^{c (a+b x)} \text {csch}^2(a c+b c x)^{7/2} \, dx=\frac {32\,\sqrt {\frac {1}{{\left (\frac {{\mathrm {e}}^{a\,c+b\,c\,x}}{2}-\frac {{\mathrm {e}}^{-a\,c-b\,c\,x}}{2}\right )}^2}}\,\left ({\mathrm {e}}^{4\,a\,c+4\,b\,c\,x}-2\,{\mathrm {e}}^{2\,a\,c+2\,b\,c\,x}+1\right )}{3\,b\,c\,\left ({\mathrm {e}}^{a\,c+b\,c\,x}-{\mathrm {e}}^{3\,a\,c+3\,b\,c\,x}\right )\,{\left ({\mathrm {e}}^{2\,a\,c+2\,b\,c\,x}-1\right )}^3}+\frac {24\,\sqrt {\frac {1}{{\left (\frac {{\mathrm {e}}^{a\,c+b\,c\,x}}{2}-\frac {{\mathrm {e}}^{-a\,c-b\,c\,x}}{2}\right )}^2}}\,\left ({\mathrm {e}}^{4\,a\,c+4\,b\,c\,x}-2\,{\mathrm {e}}^{2\,a\,c+2\,b\,c\,x}+1\right )}{b\,c\,\left ({\mathrm {e}}^{a\,c+b\,c\,x}-{\mathrm {e}}^{3\,a\,c+3\,b\,c\,x}\right )\,{\left ({\mathrm {e}}^{2\,a\,c+2\,b\,c\,x}-1\right )}^4}+\frac {96\,\sqrt {\frac {1}{{\left (\frac {{\mathrm {e}}^{a\,c+b\,c\,x}}{2}-\frac {{\mathrm {e}}^{-a\,c-b\,c\,x}}{2}\right )}^2}}\,\left ({\mathrm {e}}^{4\,a\,c+4\,b\,c\,x}-2\,{\mathrm {e}}^{2\,a\,c+2\,b\,c\,x}+1\right )}{5\,b\,c\,\left ({\mathrm {e}}^{a\,c+b\,c\,x}-{\mathrm {e}}^{3\,a\,c+3\,b\,c\,x}\right )\,{\left ({\mathrm {e}}^{2\,a\,c+2\,b\,c\,x}-1\right )}^5}+\frac {16\,\sqrt {\frac {1}{{\left (\frac {{\mathrm {e}}^{a\,c+b\,c\,x}}{2}-\frac {{\mathrm {e}}^{-a\,c-b\,c\,x}}{2}\right )}^2}}\,\left ({\mathrm {e}}^{4\,a\,c+4\,b\,c\,x}-2\,{\mathrm {e}}^{2\,a\,c+2\,b\,c\,x}+1\right )}{3\,b\,c\,\left ({\mathrm {e}}^{a\,c+b\,c\,x}-{\mathrm {e}}^{3\,a\,c+3\,b\,c\,x}\right )\,{\left ({\mathrm {e}}^{2\,a\,c+2\,b\,c\,x}-1\right )}^6} \] Input:

int(exp(c*(a + b*x))*(1/sinh(a*c + b*c*x)^2)^(7/2),x)
                                                                                    
                                                                                    
 

Output:

(32*(1/(exp(a*c + b*c*x)/2 - exp(- a*c - b*c*x)/2)^2)^(1/2)*(exp(4*a*c + 4 
*b*c*x) - 2*exp(2*a*c + 2*b*c*x) + 1))/(3*b*c*(exp(a*c + b*c*x) - exp(3*a* 
c + 3*b*c*x))*(exp(2*a*c + 2*b*c*x) - 1)^3) + (24*(1/(exp(a*c + b*c*x)/2 - 
 exp(- a*c - b*c*x)/2)^2)^(1/2)*(exp(4*a*c + 4*b*c*x) - 2*exp(2*a*c + 2*b* 
c*x) + 1))/(b*c*(exp(a*c + b*c*x) - exp(3*a*c + 3*b*c*x))*(exp(2*a*c + 2*b 
*c*x) - 1)^4) + (96*(1/(exp(a*c + b*c*x)/2 - exp(- a*c - b*c*x)/2)^2)^(1/2 
)*(exp(4*a*c + 4*b*c*x) - 2*exp(2*a*c + 2*b*c*x) + 1))/(5*b*c*(exp(a*c + b 
*c*x) - exp(3*a*c + 3*b*c*x))*(exp(2*a*c + 2*b*c*x) - 1)^5) + (16*(1/(exp( 
a*c + b*c*x)/2 - exp(- a*c - b*c*x)/2)^2)^(1/2)*(exp(4*a*c + 4*b*c*x) - 2* 
exp(2*a*c + 2*b*c*x) + 1))/(3*b*c*(exp(a*c + b*c*x) - exp(3*a*c + 3*b*c*x) 
)*(exp(2*a*c + 2*b*c*x) - 1)^6)
 

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.69 \[ \int e^{c (a+b x)} \text {csch}^2(a c+b c x)^{7/2} \, dx=\frac {-\frac {64 e^{6 b c x +6 a c}}{3}+16 e^{4 b c x +4 a c}-\frac {32 e^{2 b c x +2 a c}}{5}+\frac {16}{15}}{b c \left (e^{12 b c x +12 a c}-6 e^{10 b c x +10 a c}+15 e^{8 b c x +8 a c}-20 e^{6 b c x +6 a c}+15 e^{4 b c x +4 a c}-6 e^{2 b c x +2 a c}+1\right )} \] Input:

int(exp(c*(b*x+a))*(csch(b*c*x+a*c)^2)^(7/2),x)
 

Output:

(16*( - 20*e**(6*a*c + 6*b*c*x) + 15*e**(4*a*c + 4*b*c*x) - 6*e**(2*a*c + 
2*b*c*x) + 1))/(15*b*c*(e**(12*a*c + 12*b*c*x) - 6*e**(10*a*c + 10*b*c*x) 
+ 15*e**(8*a*c + 8*b*c*x) - 20*e**(6*a*c + 6*b*c*x) + 15*e**(4*a*c + 4*b*c 
*x) - 6*e**(2*a*c + 2*b*c*x) + 1))