\(\int e^{2 (a+b x)} \text {csch}^4(d+b x) \text {sech}^3(d+b x) \, dx\) [85]

Optimal result

Integrand size = 26, antiderivative size = 193 \[ \int e^{2 (a+b x)} \text {csch}^4(d+b x) \text {sech}^3(d+b x) \, dx=\frac {32 e^{2 a+3 d+5 b x}}{3 b \left (1-e^{2 d+2 b x}\right )^3 \left (1+e^{2 d+2 b x}\right )^2}-\frac {4 e^{2 a-d+b x} \left (5+7 e^{2 d+2 b x}\right )}{3 b \left (1-e^{4 d+4 b x}\right )^2}+\frac {e^{2 a-d+b x} \left (5+21 e^{2 d+2 b x}\right )}{3 b \left (1-e^{4 d+4 b x}\right )}-\frac {e^{2 a-2 d} \arctan \left (e^{d+b x}\right )}{b}+\frac {6 e^{2 a-2 d} \text {arctanh}\left (e^{d+b x}\right )}{b} \] Output:

32/3*exp(5*b*x+2*a+3*d)/b/(1-exp(2*b*x+2*d))^3/(1+exp(2*b*x+2*d))^2-4/3*ex 
p(b*x+2*a-d)*(5+7*exp(2*b*x+2*d))/b/(1-exp(4*b*x+4*d))^2+1/3*exp(b*x+2*a-d 
)*(5+21*exp(2*b*x+2*d))/b/(1-exp(4*b*x+4*d))-exp(2*a-2*d)*arctan(exp(b*x+d 
))/b+6*exp(2*a-2*d)*arctanh(exp(b*x+d))/b
 

Mathematica [A] (verified)

Time = 0.71 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.84 \[ \int e^{2 (a+b x)} \text {csch}^4(d+b x) \text {sech}^3(d+b x) \, dx=\frac {e^{2 a-2 d} \left (-\frac {8 e^{d+b x}}{\left (-1+e^{2 (d+b x)}\right )^3}-\frac {20 e^{d+b x}}{\left (-1+e^{2 (d+b x)}\right )^2}-\frac {6 e^{d+b x}}{-1+e^{2 (d+b x)}}+\frac {6 e^{d+b x}}{\left (1+e^{2 (d+b x)}\right )^2}-\frac {15 e^{d+b x}}{1+e^{2 (d+b x)}}-3 \arctan \left (e^{d+b x}\right )-9 \log \left (1-e^{d+b x}\right )+9 \log \left (1+e^{d+b x}\right )\right )}{3 b} \] Input:

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

Output:

(E^(2*a - 2*d)*((-8*E^(d + b*x))/(-1 + E^(2*(d + b*x)))^3 - (20*E^(d + b*x 
))/(-1 + E^(2*(d + b*x)))^2 - (6*E^(d + b*x))/(-1 + E^(2*(d + b*x))) + (6* 
E^(d + b*x))/(1 + E^(2*(d + b*x)))^2 - (15*E^(d + b*x))/(1 + E^(2*(d + b*x 
))) - 3*ArcTan[E^(d + b*x)] - 9*Log[1 - E^(d + b*x)] + 9*Log[1 + E^(d + b* 
x)]))/(3*b)
 

Rubi [A] (warning: unable to verify)

Time = 0.38 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.91, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {2720, 27, 372, 440, 27, 440, 27, 402, 27, 402, 27, 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.

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

\(\Big \downarrow \) 2720

\(\displaystyle \frac {\int \frac {128 e^{2 a+8 b x}}{\left (1-e^{2 b x}\right )^4 \left (1+e^{2 b x}\right )^3}de^{b x}}{b}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {128 e^{2 a} \int \frac {e^{8 b x}}{\left (1-e^{2 b x}\right )^4 \left (1+e^{2 b x}\right )^3}de^{b x}}{b}\)

\(\Big \downarrow \) 372

\(\displaystyle \frac {128 e^{2 a} \left (\frac {e^{5 b x}}{12 \left (1-e^{2 b x}\right )^3 \left (e^{2 b x}+1\right )^2}-\frac {1}{12} \int \frac {e^{4 b x} \left (5+7 e^{2 b x}\right )}{\left (1-e^{2 b x}\right )^3 \left (1+e^{2 b x}\right )^3}de^{b x}\right )}{b}\)

\(\Big \downarrow \) 440

\(\displaystyle \frac {128 e^{2 a} \left (\frac {1}{12} \left (-\frac {1}{8} \int -\frac {4 e^{2 b x} \left (9-e^{2 b x}\right )}{\left (1-e^{2 b x}\right )^2 \left (1+e^{2 b x}\right )^3}de^{b x}-\frac {3 e^{3 b x}}{2 \left (1-e^{2 b x}\right )^2 \left (e^{2 b x}+1\right )^2}\right )+\frac {e^{5 b x}}{12 \left (1-e^{2 b x}\right )^3 \left (e^{2 b x}+1\right )^2}\right )}{b}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {128 e^{2 a} \left (\frac {1}{12} \left (\frac {1}{2} \int \frac {e^{2 b x} \left (9-e^{2 b x}\right )}{\left (1-e^{2 b x}\right )^2 \left (1+e^{2 b x}\right )^3}de^{b x}-\frac {3 e^{3 b x}}{2 \left (1-e^{2 b x}\right )^2 \left (e^{2 b x}+1\right )^2}\right )+\frac {e^{5 b x}}{12 \left (1-e^{2 b x}\right )^3 \left (e^{2 b x}+1\right )^2}\right )}{b}\)

\(\Big \downarrow \) 440

\(\displaystyle \frac {128 e^{2 a} \left (\frac {1}{12} \left (\frac {1}{2} \left (\frac {1}{4} \int -\frac {4 \left (2-11 e^{2 b x}\right )}{\left (1-e^{2 b x}\right ) \left (1+e^{2 b x}\right )^3}de^{b x}+\frac {2 e^{b x}}{\left (1-e^{2 b x}\right ) \left (e^{2 b x}+1\right )^2}\right )-\frac {3 e^{3 b x}}{2 \left (1-e^{2 b x}\right )^2 \left (e^{2 b x}+1\right )^2}\right )+\frac {e^{5 b x}}{12 \left (1-e^{2 b x}\right )^3 \left (e^{2 b x}+1\right )^2}\right )}{b}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {128 e^{2 a} \left (\frac {1}{12} \left (\frac {1}{2} \left (\frac {2 e^{b x}}{\left (1-e^{2 b x}\right ) \left (e^{2 b x}+1\right )^2}-\int \frac {2-11 e^{2 b x}}{\left (1-e^{2 b x}\right ) \left (1+e^{2 b x}\right )^3}de^{b x}\right )-\frac {3 e^{3 b x}}{2 \left (1-e^{2 b x}\right )^2 \left (e^{2 b x}+1\right )^2}\right )+\frac {e^{5 b x}}{12 \left (1-e^{2 b x}\right )^3 \left (e^{2 b x}+1\right )^2}\right )}{b}\)

\(\Big \downarrow \) 402

\(\displaystyle \frac {128 e^{2 a} \left (\frac {1}{12} \left (\frac {1}{2} \left (\frac {1}{8} \int -\frac {3 \left (1-13 e^{2 b x}\right )}{\left (1-e^{2 b x}\right ) \left (1+e^{2 b x}\right )^2}de^{b x}-\frac {13 e^{b x}}{8 \left (e^{2 b x}+1\right )^2}+\frac {2 e^{b x}}{\left (1-e^{2 b x}\right ) \left (e^{2 b x}+1\right )^2}\right )-\frac {3 e^{3 b x}}{2 \left (1-e^{2 b x}\right )^2 \left (e^{2 b x}+1\right )^2}\right )+\frac {e^{5 b x}}{12 \left (1-e^{2 b x}\right )^3 \left (e^{2 b x}+1\right )^2}\right )}{b}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {128 e^{2 a} \left (\frac {1}{12} \left (\frac {1}{2} \left (-\frac {3}{8} \int \frac {1-13 e^{2 b x}}{\left (1-e^{2 b x}\right ) \left (1+e^{2 b x}\right )^2}de^{b x}-\frac {13 e^{b x}}{8 \left (e^{2 b x}+1\right )^2}+\frac {2 e^{b x}}{\left (1-e^{2 b x}\right ) \left (e^{2 b x}+1\right )^2}\right )-\frac {3 e^{3 b x}}{2 \left (1-e^{2 b x}\right )^2 \left (e^{2 b x}+1\right )^2}\right )+\frac {e^{5 b x}}{12 \left (1-e^{2 b x}\right )^3 \left (e^{2 b x}+1\right )^2}\right )}{b}\)

\(\Big \downarrow \) 402

\(\displaystyle \frac {128 e^{2 a} \left (\frac {1}{12} \left (\frac {1}{2} \left (-\frac {3}{8} \left (\frac {7 e^{b x}}{2 \left (e^{2 b x}+1\right )}-\frac {1}{4} \int \frac {2 \left (5+7 e^{2 b x}\right )}{\left (1-e^{2 b x}\right ) \left (1+e^{2 b x}\right )}de^{b x}\right )-\frac {13 e^{b x}}{8 \left (e^{2 b x}+1\right )^2}+\frac {2 e^{b x}}{\left (1-e^{2 b x}\right ) \left (e^{2 b x}+1\right )^2}\right )-\frac {3 e^{3 b x}}{2 \left (1-e^{2 b x}\right )^2 \left (e^{2 b x}+1\right )^2}\right )+\frac {e^{5 b x}}{12 \left (1-e^{2 b x}\right )^3 \left (e^{2 b x}+1\right )^2}\right )}{b}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {128 e^{2 a} \left (\frac {1}{12} \left (\frac {1}{2} \left (-\frac {3}{8} \left (\frac {7 e^{b x}}{2 \left (e^{2 b x}+1\right )}-\frac {1}{2} \int \frac {5+7 e^{2 b x}}{\left (1-e^{2 b x}\right ) \left (1+e^{2 b x}\right )}de^{b x}\right )-\frac {13 e^{b x}}{8 \left (e^{2 b x}+1\right )^2}+\frac {2 e^{b x}}{\left (1-e^{2 b x}\right ) \left (e^{2 b x}+1\right )^2}\right )-\frac {3 e^{3 b x}}{2 \left (1-e^{2 b x}\right )^2 \left (e^{2 b x}+1\right )^2}\right )+\frac {e^{5 b x}}{12 \left (1-e^{2 b x}\right )^3 \left (e^{2 b x}+1\right )^2}\right )}{b}\)

\(\Big \downarrow \) 397

\(\displaystyle \frac {128 e^{2 a} \left (\frac {1}{12} \left (\frac {1}{2} \left (-\frac {3}{8} \left (\frac {1}{2} \left (\int \frac {1}{1+e^{2 b x}}de^{b x}-6 \int \frac {1}{1-e^{2 b x}}de^{b x}\right )+\frac {7 e^{b x}}{2 \left (e^{2 b x}+1\right )}\right )-\frac {13 e^{b x}}{8 \left (e^{2 b x}+1\right )^2}+\frac {2 e^{b x}}{\left (1-e^{2 b x}\right ) \left (e^{2 b x}+1\right )^2}\right )-\frac {3 e^{3 b x}}{2 \left (1-e^{2 b x}\right )^2 \left (e^{2 b x}+1\right )^2}\right )+\frac {e^{5 b x}}{12 \left (1-e^{2 b x}\right )^3 \left (e^{2 b x}+1\right )^2}\right )}{b}\)

\(\Big \downarrow \) 216

\(\displaystyle \frac {128 e^{2 a} \left (\frac {1}{12} \left (\frac {1}{2} \left (-\frac {3}{8} \left (\frac {1}{2} \left (\arctan \left (e^{b x}\right )-6 \int \frac {1}{1-e^{2 b x}}de^{b x}\right )+\frac {7 e^{b x}}{2 \left (e^{2 b x}+1\right )}\right )-\frac {13 e^{b x}}{8 \left (e^{2 b x}+1\right )^2}+\frac {2 e^{b x}}{\left (1-e^{2 b x}\right ) \left (e^{2 b x}+1\right )^2}\right )-\frac {3 e^{3 b x}}{2 \left (1-e^{2 b x}\right )^2 \left (e^{2 b x}+1\right )^2}\right )+\frac {e^{5 b x}}{12 \left (1-e^{2 b x}\right )^3 \left (e^{2 b x}+1\right )^2}\right )}{b}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {128 e^{2 a} \left (\frac {1}{12} \left (\frac {1}{2} \left (-\frac {3}{8} \left (\frac {1}{2} \left (\arctan \left (e^{b x}\right )-6 \text {arctanh}\left (e^{b x}\right )\right )+\frac {7 e^{b x}}{2 \left (e^{2 b x}+1\right )}\right )-\frac {13 e^{b x}}{8 \left (e^{2 b x}+1\right )^2}+\frac {2 e^{b x}}{\left (1-e^{2 b x}\right ) \left (e^{2 b x}+1\right )^2}\right )-\frac {3 e^{3 b x}}{2 \left (1-e^{2 b x}\right )^2 \left (e^{2 b x}+1\right )^2}\right )+\frac {e^{5 b x}}{12 \left (1-e^{2 b x}\right )^3 \left (e^{2 b x}+1\right )^2}\right )}{b}\)

Input:

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

Output:

(128*E^(2*a)*(E^(5*b*x)/(12*(1 - E^(2*b*x))^3*(1 + E^(2*b*x))^2) + ((-3*E^ 
(3*b*x))/(2*(1 - E^(2*b*x))^2*(1 + E^(2*b*x))^2) + ((-13*E^(b*x))/(8*(1 + 
E^(2*b*x))^2) + (2*E^(b*x))/((1 - E^(2*b*x))*(1 + E^(2*b*x))^2) - (3*((7*E 
^(b*x))/(2*(1 + E^(2*b*x))) + (ArcTan[E^(b*x)] - 6*ArcTanh[E^(b*x)])/2))/8 
)/2)/12))/b
 

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 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 = 0.27 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.24

\[-\frac {\left (-21 \,{\mathrm e}^{8 b x +8 a +8 d}+16 \,{\mathrm e}^{6 b x +8 a +6 d}-34 \,{\mathrm e}^{4 b x +8 a +4 d}-8 \,{\mathrm e}^{2 b x +8 a +2 d}+15 \,{\mathrm e}^{8 a}\right ) {\mathrm e}^{b x +4 a -d}}{3 \left (-{\mathrm e}^{2 b x +2 a +2 d}+{\mathrm e}^{2 a}\right )^{3} \left ({\mathrm e}^{2 b x +2 a +2 d}+{\mathrm e}^{2 a}\right )^{2} b}-\frac {3 \ln \left ({\mathrm e}^{b x +a}-{\mathrm e}^{a -d}\right ) {\mathrm e}^{2 a -2 d}}{b}+\frac {i \ln \left ({\mathrm e}^{b x +a}-i {\mathrm e}^{a -d}\right ) {\mathrm e}^{2 a -2 d}}{2 b}-\frac {i \ln \left ({\mathrm e}^{b x +a}+i {\mathrm e}^{a -d}\right ) {\mathrm e}^{2 a -2 d}}{2 b}+\frac {3 \ln \left ({\mathrm e}^{b x +a}+{\mathrm e}^{a -d}\right ) {\mathrm e}^{2 a -2 d}}{b}\]

Input:

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

Output:

-1/3/(-exp(2*b*x+2*a+2*d)+exp(2*a))^3/(exp(2*b*x+2*a+2*d)+exp(2*a))^2/b*(- 
21*exp(8*b*x+8*a+8*d)+16*exp(6*b*x+8*a+6*d)-34*exp(4*b*x+8*a+4*d)-8*exp(2* 
b*x+8*a+2*d)+15*exp(8*a))*exp(b*x+4*a-d)-3*ln(exp(b*x+a)-exp(a-d))/b*exp(2 
*a-2*d)+1/2*I*ln(exp(b*x+a)-I*exp(a-d))/b*exp(2*a-2*d)-1/2*I*ln(exp(b*x+a) 
+I*exp(a-d))/b*exp(2*a-2*d)+3*ln(exp(b*x+a)+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. 4685 vs. \(2 (168) = 336\).

Time = 0.14 (sec) , antiderivative size = 4685, normalized size of antiderivative = 24.27 \[ \int e^{2 (a+b x)} \text {csch}^4(d+b x) \text {sech}^3(d+b x) \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F]

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

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

Output:

exp(2*a)*Integral(exp(2*b*x)*csch(b*x + d)**4*sech(b*x + d)**3, x)
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.02 \[ \int e^{2 (a+b x)} \text {csch}^4(d+b x) \text {sech}^3(d+b x) \, dx=\frac {\arctan \left (e^{\left (-b x - d\right )}\right ) e^{\left (2 \, a - 2 \, d\right )}}{b} + \frac {3 \, e^{\left (2 \, a - 2 \, d\right )} \log \left (e^{\left (-b x - d\right )} + 1\right )}{b} - \frac {3 \, e^{\left (2 \, a - 2 \, d\right )} \log \left (e^{\left (-b x - d\right )} - 1\right )}{b} + \frac {{\left (21 \, e^{\left (-b x - d\right )} - 16 \, e^{\left (-3 \, b x - 3 \, d\right )} + 34 \, e^{\left (-5 \, b x - 5 \, d\right )} + 8 \, e^{\left (-7 \, b x - 7 \, d\right )} - 15 \, e^{\left (-9 \, b x - 9 \, d\right )}\right )} e^{\left (2 \, a - 2 \, d\right )}}{3 \, b {\left (e^{\left (-2 \, b x - 2 \, d\right )} + 2 \, e^{\left (-4 \, b x - 4 \, d\right )} - 2 \, e^{\left (-6 \, b x - 6 \, d\right )} - e^{\left (-8 \, b x - 8 \, d\right )} + e^{\left (-10 \, b x - 10 \, d\right )} - 1\right )}} \] Input:

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

Output:

arctan(e^(-b*x - d))*e^(2*a - 2*d)/b + 3*e^(2*a - 2*d)*log(e^(-b*x - d) + 
1)/b - 3*e^(2*a - 2*d)*log(e^(-b*x - d) - 1)/b + 1/3*(21*e^(-b*x - d) - 16 
*e^(-3*b*x - 3*d) + 34*e^(-5*b*x - 5*d) + 8*e^(-7*b*x - 7*d) - 15*e^(-9*b* 
x - 9*d))*e^(2*a - 2*d)/(b*(e^(-2*b*x - 2*d) + 2*e^(-4*b*x - 4*d) - 2*e^(- 
6*b*x - 6*d) - e^(-8*b*x - 8*d) + e^(-10*b*x - 10*d) - 1))
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.74 \[ \int e^{2 (a+b x)} \text {csch}^4(d+b x) \text {sech}^3(d+b x) \, dx=-\frac {{\left (3 \, \arctan \left (e^{\left (b x + d\right )}\right ) e^{\left (-2 \, d\right )} - 9 \, e^{\left (-2 \, d\right )} \log \left (e^{\left (b x + d\right )} + 1\right ) + 9 \, e^{\left (-2 \, d\right )} \log \left ({\left | e^{\left (b x + d\right )} - 1 \right |}\right ) + \frac {3 \, {\left (5 \, e^{\left (3 \, b x + 3 \, d\right )} + 3 \, e^{\left (b x + d\right )}\right )} e^{\left (-2 \, d\right )}}{{\left (e^{\left (2 \, b x + 2 \, d\right )} + 1\right )}^{2}} + \frac {2 \, {\left (3 \, e^{\left (5 \, b x + 5 \, d\right )} + 4 \, e^{\left (3 \, b x + 3 \, d\right )} - 3 \, e^{\left (b x + d\right )}\right )} e^{\left (-2 \, d\right )}}{{\left (e^{\left (2 \, b x + 2 \, d\right )} - 1\right )}^{3}}\right )} e^{\left (2 \, a\right )}}{3 \, b} \] Input:

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

Output:

-1/3*(3*arctan(e^(b*x + d))*e^(-2*d) - 9*e^(-2*d)*log(e^(b*x + d) + 1) + 9 
*e^(-2*d)*log(abs(e^(b*x + d) - 1)) + 3*(5*e^(3*b*x + 3*d) + 3*e^(b*x + d) 
)*e^(-2*d)/(e^(2*b*x + 2*d) + 1)^2 + 2*(3*e^(5*b*x + 5*d) + 4*e^(3*b*x + 3 
*d) - 3*e^(b*x + d))*e^(-2*d)/(e^(2*b*x + 2*d) - 1)^3)*e^(2*a)/b
 

Mupad [B] (verification not implemented)

Time = 4.42 (sec) , antiderivative size = 347, normalized size of antiderivative = 1.80 \[ \int e^{2 (a+b x)} \text {csch}^4(d+b x) \text {sech}^3(d+b x) \, dx=\frac {2\,{\mathrm {e}}^{2\,a+b\,x}}{2\,b\,{\mathrm {e}}^{3\,d+2\,b\,x}+b\,{\mathrm {e}}^{5\,d+4\,b\,x}+b\,{\mathrm {e}}^d}-\frac {20\,{\mathrm {e}}^{2\,a+b\,x}}{3\,\left (b\,{\mathrm {e}}^{5\,d+4\,b\,x}-2\,b\,{\mathrm {e}}^{3\,d+2\,b\,x}+b\,{\mathrm {e}}^d\right )}+\frac {8\,{\mathrm {e}}^{2\,a-d+b\,x}}{3\,\left (b-3\,b\,{\mathrm {e}}^{2\,d+2\,b\,x}+3\,b\,{\mathrm {e}}^{4\,d+4\,b\,x}-b\,{\mathrm {e}}^{6\,d+6\,b\,x}\right )}-\frac {5\,{\mathrm {e}}^{2\,a+b\,x}}{b\,{\mathrm {e}}^{3\,d+2\,b\,x}+b\,{\mathrm {e}}^d}-\frac {2\,{\mathrm {e}}^{2\,a+b\,x}}{b\,{\mathrm {e}}^{3\,d+2\,b\,x}-b\,{\mathrm {e}}^d}-\frac {{\mathrm {e}}^{2\,a-2\,d}\,\left (\ln \left (-148\,{\mathrm {e}}^{6\,a}\,{\mathrm {e}}^{-5\,d}\,{\mathrm {e}}^{b\,x}-{\mathrm {e}}^{6\,a}\,{\mathrm {e}}^{-6\,d}\,148{}\mathrm {i}\right )\,1{}\mathrm {i}-\ln \left (-148\,{\mathrm {e}}^{6\,a}\,{\mathrm {e}}^{-5\,d}\,{\mathrm {e}}^{b\,x}+{\mathrm {e}}^{6\,a}\,{\mathrm {e}}^{-6\,d}\,148{}\mathrm {i}\right )\,1{}\mathrm {i}\right )}{2\,b}-\frac {3\,{\mathrm {e}}^{2\,a-2\,d}\,\ln \left (888\,{\mathrm {e}}^{6\,a}\,{\mathrm {e}}^{-5\,d}\,{\mathrm {e}}^{b\,x}-888\,{\mathrm {e}}^{6\,a}\,{\mathrm {e}}^{-6\,d}\right )}{b}+\frac {3\,{\mathrm {e}}^{2\,a-2\,d}\,\ln \left (888\,{\mathrm {e}}^{6\,a}\,{\mathrm {e}}^{-6\,d}+888\,{\mathrm {e}}^{6\,a}\,{\mathrm {e}}^{-5\,d}\,{\mathrm {e}}^{b\,x}\right )}{b} \] Input:

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

Output:

(2*exp(2*a + b*x))/(2*b*exp(3*d + 2*b*x) + b*exp(5*d + 4*b*x) + b*exp(d)) 
- (20*exp(2*a + b*x))/(3*(b*exp(5*d + 4*b*x) - 2*b*exp(3*d + 2*b*x) + b*ex 
p(d))) + (8*exp(2*a - d + b*x))/(3*(b - 3*b*exp(2*d + 2*b*x) + 3*b*exp(4*d 
 + 4*b*x) - b*exp(6*d + 6*b*x))) - (5*exp(2*a + b*x))/(b*exp(3*d + 2*b*x) 
+ b*exp(d)) - (2*exp(2*a + b*x))/(b*exp(3*d + 2*b*x) - b*exp(d)) - (exp(2* 
a - 2*d)*(log(- exp(6*a)*exp(-6*d)*148i - 148*exp(6*a)*exp(-5*d)*exp(b*x)) 
*1i - log(exp(6*a)*exp(-6*d)*148i - 148*exp(6*a)*exp(-5*d)*exp(b*x))*1i))/ 
(2*b) - (3*exp(2*a - 2*d)*log(888*exp(6*a)*exp(-5*d)*exp(b*x) - 888*exp(6* 
a)*exp(-6*d)))/b + (3*exp(2*a - 2*d)*log(888*exp(6*a)*exp(-6*d) + 888*exp( 
6*a)*exp(-5*d)*exp(b*x)))/b
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 489, normalized size of antiderivative = 2.53 \[ \int e^{2 (a+b x)} \text {csch}^4(d+b x) \text {sech}^3(d+b x) \, dx=\frac {e^{2 a} \left (-3 e^{10 b x +10 d} \mathit {atan} \left (e^{b x +d}\right )+3 e^{8 b x +8 d} \mathit {atan} \left (e^{b x +d}\right )+6 e^{6 b x +6 d} \mathit {atan} \left (e^{b x +d}\right )-6 e^{4 b x +4 d} \mathit {atan} \left (e^{b x +d}\right )-3 e^{2 b x +2 d} \mathit {atan} \left (e^{b x +d}\right )+3 \mathit {atan} \left (e^{b x +d}\right )-9 e^{10 b x +10 d} \mathrm {log}\left (e^{b x +d}-1\right )+9 e^{10 b x +10 d} \mathrm {log}\left (e^{b x +d}+1\right )-21 e^{9 b x +9 d}+9 e^{8 b x +8 d} \mathrm {log}\left (e^{b x +d}-1\right )-9 e^{8 b x +8 d} \mathrm {log}\left (e^{b x +d}+1\right )+16 e^{7 b x +7 d}+18 e^{6 b x +6 d} \mathrm {log}\left (e^{b x +d}-1\right )-18 e^{6 b x +6 d} \mathrm {log}\left (e^{b x +d}+1\right )-34 e^{5 b x +5 d}-18 e^{4 b x +4 d} \mathrm {log}\left (e^{b x +d}-1\right )+18 e^{4 b x +4 d} \mathrm {log}\left (e^{b x +d}+1\right )-8 e^{3 b x +3 d}-9 e^{2 b x +2 d} \mathrm {log}\left (e^{b x +d}-1\right )+9 e^{2 b x +2 d} \mathrm {log}\left (e^{b x +d}+1\right )+15 e^{b x +d}+9 \,\mathrm {log}\left (e^{b x +d}-1\right )-9 \,\mathrm {log}\left (e^{b x +d}+1\right )\right )}{3 e^{2 d} b \left (e^{10 b x +10 d}-e^{8 b x +8 d}-2 e^{6 b x +6 d}+2 e^{4 b x +4 d}+e^{2 b x +2 d}-1\right )} \] Input:

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

Output:

(e**(2*a)*( - 3*e**(10*b*x + 10*d)*atan(e**(b*x + d)) + 3*e**(8*b*x + 8*d) 
*atan(e**(b*x + d)) + 6*e**(6*b*x + 6*d)*atan(e**(b*x + d)) - 6*e**(4*b*x 
+ 4*d)*atan(e**(b*x + d)) - 3*e**(2*b*x + 2*d)*atan(e**(b*x + d)) + 3*atan 
(e**(b*x + d)) - 9*e**(10*b*x + 10*d)*log(e**(b*x + d) - 1) + 9*e**(10*b*x 
 + 10*d)*log(e**(b*x + d) + 1) - 21*e**(9*b*x + 9*d) + 9*e**(8*b*x + 8*d)* 
log(e**(b*x + d) - 1) - 9*e**(8*b*x + 8*d)*log(e**(b*x + d) + 1) + 16*e**( 
7*b*x + 7*d) + 18*e**(6*b*x + 6*d)*log(e**(b*x + d) - 1) - 18*e**(6*b*x + 
6*d)*log(e**(b*x + d) + 1) - 34*e**(5*b*x + 5*d) - 18*e**(4*b*x + 4*d)*log 
(e**(b*x + d) - 1) + 18*e**(4*b*x + 4*d)*log(e**(b*x + d) + 1) - 8*e**(3*b 
*x + 3*d) - 9*e**(2*b*x + 2*d)*log(e**(b*x + d) - 1) + 9*e**(2*b*x + 2*d)* 
log(e**(b*x + d) + 1) + 15*e**(b*x + d) + 9*log(e**(b*x + d) - 1) - 9*log( 
e**(b*x + d) + 1)))/(3*e**(2*d)*b*(e**(10*b*x + 10*d) - e**(8*b*x + 8*d) - 
 2*e**(6*b*x + 6*d) + 2*e**(4*b*x + 4*d) + e**(2*b*x + 2*d) - 1))