\(\int e^{2 (a+b x)} \text {csch}^3(d+b x) \text {sech}^2(d+b x) \, dx\) [77]

Optimal result

Integrand size = 26, antiderivative size = 137 \[ \int e^{2 (a+b x)} \text {csch}^3(d+b x) \text {sech}^2(d+b x) \, dx=-\frac {4 e^{2 a+d+3 b x}}{b \left (1-e^{2 d+2 b x}\right )^2 \left (1+e^{2 d+2 b x}\right )}+\frac {e^{2 a-d+b x} \left (5+3 e^{2 d+2 b x}\right )}{b \left (1-e^{4 d+4 b x}\right )}-\frac {4 e^{2 a-2 d} \arctan \left (e^{d+b x}\right )}{b}-\frac {e^{2 a-2 d} \text {arctanh}\left (e^{d+b x}\right )}{b} \] Output:

-4*exp(3*b*x+2*a+d)/b/(1-exp(2*b*x+2*d))^2/(1+exp(2*b*x+2*d))+exp(b*x+2*a- 
d)*(5+3*exp(2*b*x+2*d))/b/(1-exp(4*b*x+4*d))-4*exp(2*a-2*d)*arctan(exp(b*x 
+d))/b-exp(2*a-2*d)*arctanh(exp(b*x+d))/b
 

Mathematica [A] (verified)

Time = 0.58 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.85 \[ \int e^{2 (a+b x)} \text {csch}^3(d+b x) \text {sech}^2(d+b x) \, dx=\frac {e^{2 a-2 d} \left (-\frac {4 e^{d+b x}}{\left (-1+e^{2 (d+b x)}\right )^2}-\frac {10 e^{d+b x}}{-1+e^{2 (d+b x)}}+\frac {4 e^{d+b x}}{1+e^{2 (d+b x)}}-8 \arctan \left (e^{d+b x}\right )+\log \left (1-e^{d+b x}\right )-\log \left (1+e^{d+b x}\right )\right )}{2 b} \] Input:

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

Output:

(E^(2*a - 2*d)*((-4*E^(d + b*x))/(-1 + E^(2*(d + b*x)))^2 - (10*E^(d + b*x 
))/(-1 + E^(2*(d + b*x))) + (4*E^(d + b*x))/(1 + E^(2*(d + b*x))) - 8*ArcT 
an[E^(d + b*x)] + Log[1 - E^(d + b*x)] - Log[1 + E^(d + b*x)]))/(2*b)
 

Rubi [A] (warning: unable to verify)

Time = 0.32 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.84, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {2720, 27, 372, 440, 27, 402, 25, 397, 216, 219}

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.

\(\displaystyle \int e^{2 (a+b x)} \text {csch}^3(b x+d) \text {sech}^2(b x+d) \, dx\)

\(\Big \downarrow \) 2720

\(\displaystyle \frac {\int -\frac {32 e^{2 a+6 b x}}{\left (1-e^{2 b x}\right )^3 \left (1+e^{2 b x}\right )^2}de^{b x}}{b}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {32 e^{2 a} \int \frac {e^{6 b x}}{\left (1-e^{2 b x}\right )^3 \left (1+e^{2 b x}\right )^2}de^{b x}}{b}\)

\(\Big \downarrow \) 372

\(\displaystyle -\frac {32 e^{2 a} \left (\frac {e^{3 b x}}{8 \left (1-e^{2 b x}\right )^2 \left (e^{2 b x}+1\right )}-\frac {1}{8} \int \frac {e^{2 b x} \left (3+5 e^{2 b x}\right )}{\left (1-e^{2 b x}\right )^2 \left (1+e^{2 b x}\right )^2}de^{b x}\right )}{b}\)

\(\Big \downarrow \) 440

\(\displaystyle -\frac {32 e^{2 a} \left (\frac {1}{8} \left (-\frac {1}{4} \int -\frac {4 \left (2-e^{2 b x}\right )}{\left (1-e^{2 b x}\right ) \left (1+e^{2 b x}\right )^2}de^{b x}-\frac {2 e^{b x}}{\left (1-e^{2 b x}\right ) \left (e^{2 b x}+1\right )}\right )+\frac {e^{3 b x}}{8 \left (1-e^{2 b x}\right )^2 \left (e^{2 b x}+1\right )}\right )}{b}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {32 e^{2 a} \left (\frac {1}{8} \left (\int \frac {2-e^{2 b x}}{\left (1-e^{2 b x}\right ) \left (1+e^{2 b x}\right )^2}de^{b x}-\frac {2 e^{b x}}{\left (1-e^{2 b x}\right ) \left (e^{2 b x}+1\right )}\right )+\frac {e^{3 b x}}{8 \left (1-e^{2 b x}\right )^2 \left (e^{2 b x}+1\right )}\right )}{b}\)

\(\Big \downarrow \) 402

\(\displaystyle -\frac {32 e^{2 a} \left (\frac {1}{8} \left (-\frac {1}{4} \int -\frac {5-3 e^{2 b x}}{\left (1-e^{2 b x}\right ) \left (1+e^{2 b x}\right )}de^{b x}+\frac {3 e^{b x}}{4 \left (e^{2 b x}+1\right )}-\frac {2 e^{b x}}{\left (1-e^{2 b x}\right ) \left (e^{2 b x}+1\right )}\right )+\frac {e^{3 b x}}{8 \left (1-e^{2 b x}\right )^2 \left (e^{2 b x}+1\right )}\right )}{b}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {32 e^{2 a} \left (\frac {1}{8} \left (\frac {1}{4} \int \frac {5-3 e^{2 b x}}{\left (1-e^{2 b x}\right ) \left (1+e^{2 b x}\right )}de^{b x}+\frac {3 e^{b x}}{4 \left (e^{2 b x}+1\right )}-\frac {2 e^{b x}}{\left (1-e^{2 b x}\right ) \left (e^{2 b x}+1\right )}\right )+\frac {e^{3 b x}}{8 \left (1-e^{2 b x}\right )^2 \left (e^{2 b x}+1\right )}\right )}{b}\)

\(\Big \downarrow \) 397

\(\displaystyle -\frac {32 e^{2 a} \left (\frac {1}{8} \left (\frac {1}{4} \left (\int \frac {1}{1-e^{2 b x}}de^{b x}+4 \int \frac {1}{1+e^{2 b x}}de^{b x}\right )+\frac {3 e^{b x}}{4 \left (e^{2 b x}+1\right )}-\frac {2 e^{b x}}{\left (1-e^{2 b x}\right ) \left (e^{2 b x}+1\right )}\right )+\frac {e^{3 b x}}{8 \left (1-e^{2 b x}\right )^2 \left (e^{2 b x}+1\right )}\right )}{b}\)

\(\Big \downarrow \) 216

\(\displaystyle -\frac {32 e^{2 a} \left (\frac {1}{8} \left (\frac {1}{4} \left (\int \frac {1}{1-e^{2 b x}}de^{b x}+4 \arctan \left (e^{b x}\right )\right )+\frac {3 e^{b x}}{4 \left (e^{2 b x}+1\right )}-\frac {2 e^{b x}}{\left (1-e^{2 b x}\right ) \left (e^{2 b x}+1\right )}\right )+\frac {e^{3 b x}}{8 \left (1-e^{2 b x}\right )^2 \left (e^{2 b x}+1\right )}\right )}{b}\)

\(\Big \downarrow \) 219

\(\displaystyle -\frac {32 e^{2 a} \left (\frac {1}{8} \left (\frac {1}{4} \left (4 \arctan \left (e^{b x}\right )+\text {arctanh}\left (e^{b x}\right )\right )+\frac {3 e^{b x}}{4 \left (e^{2 b x}+1\right )}-\frac {2 e^{b x}}{\left (1-e^{2 b x}\right ) \left (e^{2 b x}+1\right )}\right )+\frac {e^{3 b x}}{8 \left (1-e^{2 b x}\right )^2 \left (e^{2 b x}+1\right )}\right )}{b}\)

Input:

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

Output:

(-32*E^(2*a)*(E^(3*b*x)/(8*(1 - E^(2*b*x))^2*(1 + E^(2*b*x))) + ((3*E^(b*x 
))/(4*(1 + E^(2*b*x))) - (2*E^(b*x))/((1 - E^(2*b*x))*(1 + E^(2*b*x))) + ( 
4*ArcTan[E^(b*x)] + ArcTanh[E^(b*x)])/4)/8))/b
 

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 216

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A 
rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a 
, 0] || GtQ[b, 0])
 

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 372

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ 
), x_Symbol] :> Simp[(-a)*e^3*(e*x)^(m - 3)*(a + b*x^2)^(p + 1)*((c + d*x^2 
)^(q + 1)/(2*b*(b*c - a*d)*(p + 1))), x] + Simp[e^4/(2*b*(b*c - a*d)*(p + 1 
))   Int[(e*x)^(m - 4)*(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[a*c*(m - 3) + 
 (a*d*(m + 2*q - 1) + 2*b*c*(p + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, 
e, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && GtQ[m, 3] && 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 402

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

rule 440

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

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]]]
 

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 182.69 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.53

method	result	size
risch	\(\frac {\left (-3 \,{\mathrm e}^{4 b x +4 a +4 d}-6 \,{\mathrm e}^{2 b x +4 a +2 d}+5 \,{\mathrm e}^{4 a}\right ) {\mathrm e}^{b x +4 a -d}}{\left ({\mathrm e}^{2 b x +2 a +2 d}+{\mathrm e}^{2 a}\right ) \left (-{\mathrm e}^{2 b x +2 a +2 d}+{\mathrm e}^{2 a}\right )^{2} b}+\frac {\ln \left ({\mathrm e}^{b x +a}-{\mathrm e}^{a -d}\right ) {\mathrm e}^{2 a -2 d}}{2 b}-\frac {\ln \left ({\mathrm e}^{b x +a}+{\mathrm e}^{a -d}\right ) {\mathrm e}^{2 a -2 d}}{2 b}+\frac {2 i \ln \left ({\mathrm e}^{b x +a}-i {\mathrm e}^{a -d}\right ) {\mathrm e}^{2 a -2 d}}{b}-\frac {2 i \ln \left ({\mathrm e}^{b x +a}+i {\mathrm e}^{a -d}\right ) {\mathrm e}^{2 a -2 d}}{b}\)	\(210\)

Input:

int(exp(2*b*x+2*a)*csch(b*x+d)^3*sech(b*x+d)^2,x,method=_RETURNVERBOSE)
 

Output:

1/(exp(2*b*x+2*a+2*d)+exp(2*a))/(-exp(2*b*x+2*a+2*d)+exp(2*a))^2/b*(-3*exp 
(4*b*x+4*a+4*d)-6*exp(2*b*x+4*a+2*d)+5*exp(4*a))*exp(b*x+4*a-d)+1/2*ln(exp 
(b*x+a)-exp(a-d))/b*exp(2*a-2*d)-1/2*ln(exp(b*x+a)+exp(a-d))/b*exp(2*a-2*d 
)+2*I*ln(exp(b*x+a)-I*exp(a-d))/b*exp(2*a-2*d)-2*I*ln(exp(b*x+a)+I*exp(a-d 
))/b*exp(2*a-2*d)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2109 vs. \(2 (124) = 248\).

Time = 0.10 (sec) , antiderivative size = 2109, normalized size of antiderivative = 15.39 \[ \int e^{2 (a+b x)} \text {csch}^3(d+b x) \text {sech}^2(d+b x) \, dx=\text {Too large to display} \] Input:

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

Output:

-1/2*(6*cosh(b*x + d)^5*cosh(-2*a + 2*d) + 6*(cosh(-2*a + 2*d) - sinh(-2*a 
 + 2*d))*sinh(b*x + d)^5 + 30*(cosh(b*x + d)*cosh(-2*a + 2*d) - cosh(b*x + 
 d)*sinh(-2*a + 2*d))*sinh(b*x + d)^4 + 12*cosh(b*x + d)^3*cosh(-2*a + 2*d 
) + 12*(5*cosh(b*x + d)^2*cosh(-2*a + 2*d) - (5*cosh(b*x + d)^2 + 1)*sinh( 
-2*a + 2*d) + cosh(-2*a + 2*d))*sinh(b*x + d)^3 + 12*(5*cosh(b*x + d)^3*co 
sh(-2*a + 2*d) + 3*cosh(b*x + d)*cosh(-2*a + 2*d) - (5*cosh(b*x + d)^3 + 3 
*cosh(b*x + d))*sinh(-2*a + 2*d))*sinh(b*x + d)^2 + 8*(cosh(b*x + d)^6*cos 
h(-2*a + 2*d) + (cosh(-2*a + 2*d) - sinh(-2*a + 2*d))*sinh(b*x + d)^6 + 6* 
(cosh(b*x + d)*cosh(-2*a + 2*d) - cosh(b*x + d)*sinh(-2*a + 2*d))*sinh(b*x 
 + d)^5 - cosh(b*x + d)^4*cosh(-2*a + 2*d) + (15*cosh(b*x + d)^2*cosh(-2*a 
 + 2*d) - (15*cosh(b*x + d)^2 - 1)*sinh(-2*a + 2*d) - cosh(-2*a + 2*d))*si 
nh(b*x + d)^4 + 4*(5*cosh(b*x + d)^3*cosh(-2*a + 2*d) - cosh(b*x + d)*cosh 
(-2*a + 2*d) - (5*cosh(b*x + d)^3 - cosh(b*x + d))*sinh(-2*a + 2*d))*sinh( 
b*x + d)^3 - cosh(b*x + d)^2*cosh(-2*a + 2*d) + (15*cosh(b*x + d)^4*cosh(- 
2*a + 2*d) - 6*cosh(b*x + d)^2*cosh(-2*a + 2*d) - (15*cosh(b*x + d)^4 - 6* 
cosh(b*x + d)^2 - 1)*sinh(-2*a + 2*d) - cosh(-2*a + 2*d))*sinh(b*x + d)^2 
+ 2*(3*cosh(b*x + d)^5*cosh(-2*a + 2*d) - 2*cosh(b*x + d)^3*cosh(-2*a + 2* 
d) - cosh(b*x + d)*cosh(-2*a + 2*d) - (3*cosh(b*x + d)^5 - 2*cosh(b*x + d) 
^3 - cosh(b*x + d))*sinh(-2*a + 2*d))*sinh(b*x + d) - (cosh(b*x + d)^6 - c 
osh(b*x + d)^4 - cosh(b*x + d)^2 + 1)*sinh(-2*a + 2*d) + cosh(-2*a + 2*...
 

Sympy [F]

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

integrate(exp(2*b*x+2*a)*csch(b*x+d)**3*sech(b*x+d)**2,x)
 

Output:

exp(2*a)*Integral(exp(2*b*x)*csch(b*x + d)**3*sech(b*x + d)**2, x)
                                                                                    
                                                                                    
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.12 \[ \int e^{2 (a+b x)} \text {csch}^3(d+b x) \text {sech}^2(d+b x) \, dx=\frac {4 \, \arctan \left (e^{\left (-b x - d\right )}\right ) e^{\left (2 \, a - 2 \, d\right )}}{b} - \frac {e^{\left (2 \, a - 2 \, d\right )} \log \left (e^{\left (-b x - d\right )} + 1\right )}{2 \, b} + \frac {e^{\left (2 \, a - 2 \, d\right )} \log \left (e^{\left (-b x - d\right )} - 1\right )}{2 \, b} + \frac {{\left (3 \, e^{\left (-b x - d\right )} + 6 \, e^{\left (-3 \, b x - 3 \, d\right )} - 5 \, e^{\left (-5 \, b x - 5 \, d\right )}\right )} e^{\left (2 \, a - 2 \, d\right )}}{b {\left (e^{\left (-2 \, b x - 2 \, d\right )} + e^{\left (-4 \, b x - 4 \, d\right )} - e^{\left (-6 \, b x - 6 \, d\right )} - 1\right )}} \] Input:

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.84 \[ \int e^{2 (a+b x)} \text {csch}^3(d+b x) \text {sech}^2(d+b x) \, dx=-\frac {{\left (8 \, \arctan \left (e^{\left (b x + d\right )}\right ) e^{\left (-2 \, d\right )} + e^{\left (-2 \, d\right )} \log \left (e^{\left (b x + d\right )} + 1\right ) - e^{\left (-2 \, d\right )} \log \left ({\left | e^{\left (b x + d\right )} - 1 \right |}\right ) - \frac {4 \, e^{\left (b x - d\right )}}{e^{\left (2 \, b x + 2 \, d\right )} + 1} + \frac {2 \, {\left (5 \, e^{\left (3 \, b x + 3 \, d\right )} - 3 \, e^{\left (b x + d\right )}\right )} e^{\left (-2 \, d\right )}}{{\left (e^{\left (2 \, b x + 2 \, d\right )} - 1\right )}^{2}}\right )} e^{\left (2 \, a\right )}}{2 \, b} \] Input:

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

Output:

-1/2*(8*arctan(e^(b*x + d))*e^(-2*d) + e^(-2*d)*log(e^(b*x + d) + 1) - e^( 
-2*d)*log(abs(e^(b*x + d) - 1)) - 4*e^(b*x - d)/(e^(2*b*x + 2*d) + 1) + 2* 
(5*e^(3*b*x + 3*d) - 3*e^(b*x + d))*e^(-2*d)/(e^(2*b*x + 2*d) - 1)^2)*e^(2 
*a)/b
 

Mupad [B] (verification not implemented)

Time = 3.62 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.82 \[ \int e^{2 (a+b x)} \text {csch}^3(d+b x) \text {sech}^2(d+b x) \, dx=\frac {2\,{\mathrm {e}}^{2\,a-d+b\,x}}{b+b\,{\mathrm {e}}^{2\,d+2\,b\,x}}-\frac {2\,{\mathrm {e}}^{2\,a-d+b\,x}}{b-2\,b\,{\mathrm {e}}^{2\,d+2\,b\,x}+b\,{\mathrm {e}}^{4\,d+4\,b\,x}}+\frac {5\,{\mathrm {e}}^{2\,a-d+b\,x}}{b-b\,{\mathrm {e}}^{2\,d+2\,b\,x}}-\frac {2\,{\mathrm {e}}^{2\,a-2\,d}\,\left (\ln \left (-272\,{\mathrm {e}}^{6\,a}\,{\mathrm {e}}^{-5\,d}\,{\mathrm {e}}^{b\,x}-{\mathrm {e}}^{6\,a}\,{\mathrm {e}}^{-6\,d}\,272{}\mathrm {i}\right )\,1{}\mathrm {i}-\ln \left (-272\,{\mathrm {e}}^{6\,a}\,{\mathrm {e}}^{-5\,d}\,{\mathrm {e}}^{b\,x}+{\mathrm {e}}^{6\,a}\,{\mathrm {e}}^{-6\,d}\,272{}\mathrm {i}\right )\,1{}\mathrm {i}\right )}{b}-\frac {{\mathrm {e}}^{2\,a-2\,d}\,\ln \left (-68\,{\mathrm {e}}^{6\,a}\,{\mathrm {e}}^{-6\,d}-68\,{\mathrm {e}}^{6\,a}\,{\mathrm {e}}^{-5\,d}\,{\mathrm {e}}^{b\,x}\right )}{2\,b}+\frac {{\mathrm {e}}^{2\,a-2\,d}\,\ln \left (68\,{\mathrm {e}}^{6\,a}\,{\mathrm {e}}^{-6\,d}-68\,{\mathrm {e}}^{6\,a}\,{\mathrm {e}}^{-5\,d}\,{\mathrm {e}}^{b\,x}\right )}{2\,b} \] Input:

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

Output:

(2*exp(2*a - d + b*x))/(b + b*exp(2*d + 2*b*x)) - (2*exp(2*a - d + b*x))/( 
b - 2*b*exp(2*d + 2*b*x) + b*exp(4*d + 4*b*x)) + (5*exp(2*a - d + b*x))/(b 
 - b*exp(2*d + 2*b*x)) - (2*exp(2*a - 2*d)*(log(- exp(6*a)*exp(-6*d)*272i 
- 272*exp(6*a)*exp(-5*d)*exp(b*x))*1i - log(exp(6*a)*exp(-6*d)*272i - 272* 
exp(6*a)*exp(-5*d)*exp(b*x))*1i))/b - (exp(2*a - 2*d)*log(- 68*exp(6*a)*ex 
p(-6*d) - 68*exp(6*a)*exp(-5*d)*exp(b*x)))/(2*b) + (exp(2*a - 2*d)*log(68* 
exp(6*a)*exp(-6*d) - 68*exp(6*a)*exp(-5*d)*exp(b*x)))/(2*b)
 

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 310, normalized size of antiderivative = 2.26 \[ \int e^{2 (a+b x)} \text {csch}^3(d+b x) \text {sech}^2(d+b x) \, dx=\frac {e^{2 a} \left (-8 e^{6 b x +6 d} \mathit {atan} \left (e^{b x +d}\right )+8 e^{4 b x +4 d} \mathit {atan} \left (e^{b x +d}\right )+8 e^{2 b x +2 d} \mathit {atan} \left (e^{b x +d}\right )-8 \mathit {atan} \left (e^{b x +d}\right )+e^{6 b x +6 d} \mathrm {log}\left (e^{b x +d}-1\right )-e^{6 b x +6 d} \mathrm {log}\left (e^{b x +d}+1\right )-6 e^{5 b x +5 d}-e^{4 b x +4 d} \mathrm {log}\left (e^{b x +d}-1\right )+e^{4 b x +4 d} \mathrm {log}\left (e^{b x +d}+1\right )-12 e^{3 b x +3 d}-e^{2 b x +2 d} \mathrm {log}\left (e^{b x +d}-1\right )+e^{2 b x +2 d} \mathrm {log}\left (e^{b x +d}+1\right )+10 e^{b x +d}+\mathrm {log}\left (e^{b x +d}-1\right )-\mathrm {log}\left (e^{b x +d}+1\right )\right )}{2 e^{2 d} b \left (e^{6 b x +6 d}-e^{4 b x +4 d}-e^{2 b x +2 d}+1\right )} \] Input:

int(exp(2*b*x+2*a)*csch(b*x+d)^3*sech(b*x+d)^2,x)
 

Output:

(e**(2*a)*( - 8*e**(6*b*x + 6*d)*atan(e**(b*x + d)) + 8*e**(4*b*x + 4*d)*a 
tan(e**(b*x + d)) + 8*e**(2*b*x + 2*d)*atan(e**(b*x + d)) - 8*atan(e**(b*x 
 + d)) + e**(6*b*x + 6*d)*log(e**(b*x + d) - 1) - e**(6*b*x + 6*d)*log(e** 
(b*x + d) + 1) - 6*e**(5*b*x + 5*d) - e**(4*b*x + 4*d)*log(e**(b*x + d) - 
1) + e**(4*b*x + 4*d)*log(e**(b*x + d) + 1) - 12*e**(3*b*x + 3*d) - e**(2* 
b*x + 2*d)*log(e**(b*x + d) - 1) + e**(2*b*x + 2*d)*log(e**(b*x + d) + 1) 
+ 10*e**(b*x + d) + log(e**(b*x + d) - 1) - log(e**(b*x + d) + 1)))/(2*e** 
(2*d)*b*(e**(6*b*x + 6*d) - e**(4*b*x + 4*d) - e**(2*b*x + 2*d) + 1))