\(\int e^{a+b x} \text {csch}^3(d+b x) \text {sech}(d+b x) \, dx\) [28]

Optimal result

Integrand size = 22, antiderivative size = 96 \[ \int e^{a+b x} \text {csch}^3(d+b x) \text {sech}(d+b x) \, dx=-\frac {2 e^{a+b x}}{b \left (1-e^{2 d+2 b x}\right )^2}+\frac {3 e^{a+b x}}{b \left (1-e^{2 d+2 b x}\right )}-\frac {2 e^{a-d} \arctan \left (e^{d+b x}\right )}{b}+\frac {e^{a-d} \text {arctanh}\left (e^{d+b x}\right )}{b} \] Output:

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

Mathematica [A] (verified)

Time = 0.32 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.05 \[ \int e^{a+b x} \text {csch}^3(d+b x) \text {sech}(d+b x) \, dx=\frac {e^a \left (-\frac {4 e^{b x}}{\left (-1+e^{2 (d+b x)}\right )^2}-\frac {6 e^{b x}}{-1+e^{2 (d+b x)}}-4 e^{-d} \arctan \left (e^{d+b x}\right )-e^{-d} \log \left (1-e^{d+b x}\right )+e^{-d} \log \left (1+e^{d+b x}\right )\right )}{2 b} \] Input:

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

Output:

(E^a*((-4*E^(b*x))/(-1 + E^(2*(d + b*x)))^2 - (6*E^(b*x))/(-1 + E^(2*(d + 
b*x))) - (4*ArcTan[E^(d + b*x)])/E^d - Log[1 - E^(d + b*x)]/E^d + Log[1 + 
E^(d + b*x)]/E^d))/(2*b)
 

Rubi [A] (warning: unable to verify)

Time = 0.41 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.80, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {2720, 27, 372, 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.

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

\(\Big \downarrow \) 2720

\(\displaystyle \frac {\int -\frac {16 e^{a+4 b x}}{\left (1-e^{2 b x}\right )^3 \left (1+e^{2 b x}\right )}de^{b x}}{b}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {16 e^a \int \frac {e^{4 b x}}{\left (1-e^{2 b x}\right )^3 \left (1+e^{2 b x}\right )}de^{b x}}{b}\)

\(\Big \downarrow \) 372

\(\displaystyle -\frac {16 e^a \left (\frac {e^{b x}}{8 \left (1-e^{2 b x}\right )^2}-\frac {1}{8} \int \frac {1+5 e^{2 b x}}{\left (1-e^{2 b x}\right )^2 \left (1+e^{2 b x}\right )}de^{b x}\right )}{b}\)

\(\Big \downarrow \) 402

\(\displaystyle -\frac {16 e^a \left (\frac {1}{8} \left (-\frac {1}{4} \int -\frac {2 \left (1-3 e^{2 b x}\right )}{\left (1-e^{2 b x}\right ) \left (1+e^{2 b x}\right )}de^{b x}-\frac {3 e^{b x}}{2 \left (1-e^{2 b x}\right )}\right )+\frac {e^{b x}}{8 \left (1-e^{2 b x}\right )^2}\right )}{b}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {16 e^a \left (\frac {1}{8} \left (\frac {1}{2} \int \frac {1-3 e^{2 b x}}{\left (1-e^{2 b x}\right ) \left (1+e^{2 b x}\right )}de^{b x}-\frac {3 e^{b x}}{2 \left (1-e^{2 b x}\right )}\right )+\frac {e^{b x}}{8 \left (1-e^{2 b x}\right )^2}\right )}{b}\)

\(\Big \downarrow \) 397

\(\displaystyle -\frac {16 e^a \left (\frac {1}{8} \left (\frac {1}{2} \left (2 \int \frac {1}{1+e^{2 b x}}de^{b x}-\int \frac {1}{1-e^{2 b x}}de^{b x}\right )-\frac {3 e^{b x}}{2 \left (1-e^{2 b x}\right )}\right )+\frac {e^{b x}}{8 \left (1-e^{2 b x}\right )^2}\right )}{b}\)

\(\Big \downarrow \) 216

\(\displaystyle -\frac {16 e^a \left (\frac {1}{8} \left (\frac {1}{2} \left (2 \arctan \left (e^{b x}\right )-\int \frac {1}{1-e^{2 b x}}de^{b x}\right )-\frac {3 e^{b x}}{2 \left (1-e^{2 b x}\right )}\right )+\frac {e^{b x}}{8 \left (1-e^{2 b x}\right )^2}\right )}{b}\)

\(\Big \downarrow \) 219

\(\displaystyle -\frac {16 e^a \left (\frac {1}{8} \left (\frac {1}{2} \left (2 \arctan \left (e^{b x}\right )-\text {arctanh}\left (e^{b x}\right )\right )-\frac {3 e^{b x}}{2 \left (1-e^{2 b x}\right )}\right )+\frac {e^{b x}}{8 \left (1-e^{2 b x}\right )^2}\right )}{b}\)

Input:

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

Output:

(-16*E^a*(E^(b*x)/(8*(1 - E^(2*b*x))^2) + ((-3*E^(b*x))/(2*(1 - E^(2*b*x)) 
) + (2*ArcTan[E^(b*x)] - ArcTanh[E^(b*x)])/2)/8))/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 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 = 36.08 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.71

method	result	size
risch	\(\frac {\left (-3 \,{\mathrm e}^{2 b x +2 a +2 d}+{\mathrm e}^{2 a}\right ) {\mathrm e}^{b x +3 a}}{\left (-{\mathrm e}^{2 b x +2 a +2 d}+{\mathrm e}^{2 a}\right )^{2} b}+\frac {i \ln \left ({\mathrm e}^{b x +a}-i {\mathrm e}^{a -d}\right ) {\mathrm e}^{a -d}}{b}-\frac {i \ln \left ({\mathrm e}^{b x +a}+i {\mathrm e}^{a -d}\right ) {\mathrm e}^{a -d}}{b}+\frac {\ln \left ({\mathrm e}^{b x +a}+{\mathrm e}^{a -d}\right ) {\mathrm e}^{a -d}}{2 b}-\frac {\ln \left ({\mathrm e}^{b x +a}-{\mathrm e}^{a -d}\right ) {\mathrm e}^{a -d}}{2 b}\)	\(164\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1068 vs. \(2 (84) = 168\).

Time = 0.10 (sec) , antiderivative size = 1068, normalized size of antiderivative = 11.12 \[ \int e^{a+b x} \text {csch}^3(d+b x) \text {sech}(d+b x) \, dx=\text {Too large to display} \] Input:

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

Output:

-1/2*(6*cosh(b*x + d)^3*cosh(-a + d) + 6*(cosh(-a + d) - sinh(-a + d))*sin 
h(b*x + d)^3 + 18*(cosh(b*x + d)*cosh(-a + d) - cosh(b*x + d)*sinh(-a + d) 
)*sinh(b*x + d)^2 + 4*(cosh(b*x + d)^4*cosh(-a + d) + (cosh(-a + d) - sinh 
(-a + d))*sinh(b*x + d)^4 + 4*(cosh(b*x + d)*cosh(-a + d) - cosh(b*x + d)* 
sinh(-a + d))*sinh(b*x + d)^3 - 2*cosh(b*x + d)^2*cosh(-a + d) + 2*(3*cosh 
(b*x + d)^2*cosh(-a + d) - (3*cosh(b*x + d)^2 - 1)*sinh(-a + d) - cosh(-a 
+ d))*sinh(b*x + d)^2 + 4*(cosh(b*x + d)^3*cosh(-a + d) - cosh(b*x + d)*co 
sh(-a + d) - (cosh(b*x + d)^3 - cosh(b*x + d))*sinh(-a + d))*sinh(b*x + d) 
 - (cosh(b*x + d)^4 - 2*cosh(b*x + d)^2 + 1)*sinh(-a + d) + cosh(-a + d))* 
arctan(cosh(b*x + d) + sinh(b*x + d)) - 2*cosh(b*x + d)*cosh(-a + d) - (co 
sh(b*x + d)^4*cosh(-a + d) + (cosh(-a + d) - sinh(-a + d))*sinh(b*x + d)^4 
 + 4*(cosh(b*x + d)*cosh(-a + d) - cosh(b*x + d)*sinh(-a + d))*sinh(b*x + 
d)^3 - 2*cosh(b*x + d)^2*cosh(-a + d) + 2*(3*cosh(b*x + d)^2*cosh(-a + d) 
- (3*cosh(b*x + d)^2 - 1)*sinh(-a + d) - cosh(-a + d))*sinh(b*x + d)^2 + 4 
*(cosh(b*x + d)^3*cosh(-a + d) - cosh(b*x + d)*cosh(-a + d) - (cosh(b*x + 
d)^3 - cosh(b*x + d))*sinh(-a + d))*sinh(b*x + d) - (cosh(b*x + d)^4 - 2*c 
osh(b*x + d)^2 + 1)*sinh(-a + d) + cosh(-a + d))*log(cosh(b*x + d) + sinh( 
b*x + d) + 1) + (cosh(b*x + d)^4*cosh(-a + d) + (cosh(-a + d) - sinh(-a + 
d))*sinh(b*x + d)^4 + 4*(cosh(b*x + d)*cosh(-a + d) - cosh(b*x + d)*sinh(- 
a + d))*sinh(b*x + d)^3 - 2*cosh(b*x + d)^2*cosh(-a + d) + 2*(3*cosh(b*...
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.33 \[ \int e^{a+b x} \text {csch}^3(d+b x) \text {sech}(d+b x) \, dx=-\frac {2 \, \arctan \left (e^{\left (b x + d\right )}\right ) e^{\left (a - d\right )}}{b} + \frac {e^{\left (a - d\right )} \log \left (e^{\left (b x + a + d\right )} + e^{a}\right )}{2 \, b} - \frac {e^{\left (a - d\right )} \log \left (e^{\left (b x + a + d\right )} - e^{a}\right )}{2 \, b} - \frac {3 \, e^{\left (3 \, b x + 5 \, a + 2 \, d\right )} - e^{\left (b x + 5 \, a\right )}}{b {\left (e^{\left (4 \, b x + 4 \, a + 4 \, d\right )} - 2 \, e^{\left (2 \, b x + 4 \, a + 2 \, d\right )} + e^{\left (4 \, a\right )}\right )}} \] Input:

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.05 \[ \int e^{a+b x} \text {csch}^3(d+b x) \text {sech}(d+b x) \, dx=-\frac {1}{2} \, {\left (\frac {4 \, \arctan \left (e^{\left (b x + d\right )}\right ) e^{\left (-5 \, d\right )}}{b} - \frac {e^{\left (-5 \, d\right )} \log \left (e^{\left (b x + d\right )} + 1\right )}{b} + \frac {e^{\left (-5 \, d\right )} \log \left ({\left | e^{\left (b x + d\right )} - 1 \right |}\right )}{b} + \frac {2 \, {\left (3 \, e^{\left (3 \, b x + 2 \, d\right )} - e^{\left (b x\right )}\right )} e^{\left (-4 \, d\right )}}{b {\left (e^{\left (2 \, b x + 2 \, d\right )} - 1\right )}^{2}}\right )} e^{\left (a + 4 \, d\right )} \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int e^{a+b x} \text {csch}^3(d+b x) \text {sech}(d+b x) \, dx=\int \frac {{\mathrm {e}}^{a+b\,x}}{\mathrm {cosh}\left (d+b\,x\right )\,{\mathrm {sinh}\left (d+b\,x\right )}^3} \,d x \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 220, normalized size of antiderivative = 2.29 \[ \int e^{a+b x} \text {csch}^3(d+b x) \text {sech}(d+b x) \, dx=\frac {e^{a} \left (-4 e^{4 b x +4 d} \mathit {atan} \left (e^{b x +d}\right )+8 e^{2 b x +2 d} \mathit {atan} \left (e^{b x +d}\right )-4 \mathit {atan} \left (e^{b x +d}\right )-e^{4 b x +4 d} \mathrm {log}\left (e^{b x +d}-1\right )+e^{4 b x +4 d} \mathrm {log}\left (e^{b x +d}+1\right )-6 e^{3 b x +3 d}+2 e^{2 b x +2 d} \mathrm {log}\left (e^{b x +d}-1\right )-2 e^{2 b x +2 d} \mathrm {log}\left (e^{b x +d}+1\right )+2 e^{b x +d}-\mathrm {log}\left (e^{b x +d}-1\right )+\mathrm {log}\left (e^{b x +d}+1\right )\right )}{2 e^{d} b \left (e^{4 b x +4 d}-2 e^{2 b x +2 d}+1\right )} \] Input:

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

Output:

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