\(\int \frac {\text {csch}^3(x)}{a+b \text {sech}(x)} \, dx\) [66]

Optimal result

Integrand size = 13, antiderivative size = 85 \[ \int \frac {\text {csch}^3(x)}{a+b \text {sech}(x)} \, dx=\frac {(b-a \cosh (x)) \text {csch}^2(x)}{2 \left (a^2-b^2\right )}-\frac {a \log (1-\cosh (x))}{4 (a+b)^2}+\frac {a \log (1+\cosh (x))}{4 (a-b)^2}-\frac {a^2 b \log (b+a \cosh (x))}{\left (a^2-b^2\right )^2} \] Output:

(b-a*cosh(x))*csch(x)^2/(2*a^2-2*b^2)-1/4*a*ln(1-cosh(x))/(a+b)^2+1/4*a*ln 
(1+cosh(x))/(a-b)^2-a^2*b*ln(b+a*cosh(x))/(a^2-b^2)^2
 

Mathematica [A] (verified)

Time = 0.31 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.09 \[ \int \frac {\text {csch}^3(x)}{a+b \text {sech}(x)} \, dx=\frac {1}{8} \left (-\frac {\text {csch}^2\left (\frac {x}{2}\right )}{a+b}+\frac {4 a \left ((a+b)^2 \log \left (\cosh \left (\frac {x}{2}\right )\right )-2 a b \log (b+a \cosh (x))-(a-b)^2 \log \left (\sinh \left (\frac {x}{2}\right )\right )\right )}{(a-b)^2 (a+b)^2}-\frac {\text {sech}^2\left (\frac {x}{2}\right )}{a-b}\right ) \] Input:

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

Output:

(-(Csch[x/2]^2/(a + b)) + (4*a*((a + b)^2*Log[Cosh[x/2]] - 2*a*b*Log[b + a 
*Cosh[x]] - (a - b)^2*Log[Sinh[x/2]]))/((a - b)^2*(a + b)^2) - Sech[x/2]^2 
/(a - b))/8
 

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.51, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.077, Rules used = {3042, 26, 4360, 26, 25, 3042, 26, 3316, 25, 27, 593, 25, 657, 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.

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

\(\Big \downarrow \) 3042

\(\displaystyle \int -\frac {i}{\cos \left (-\frac {\pi }{2}+i x\right )^3 \left (a-b \csc \left (-\frac {\pi }{2}+i x\right )\right )}dx\)

\(\Big \downarrow \) 26

\(\displaystyle -i \int \frac {1}{\cos \left (i x-\frac {\pi }{2}\right )^3 \left (a-b \csc \left (i x-\frac {\pi }{2}\right )\right )}dx\)

\(\Big \downarrow \) 4360

\(\displaystyle -i \int -\frac {i \coth (x) \text {csch}^2(x)}{-b-a \cosh (x)}dx\)

\(\Big \downarrow \) 26

\(\displaystyle -\int -\frac {\coth (x) \text {csch}^2(x)}{b+a \cosh (x)}dx\)

\(\Big \downarrow \) 25

\(\displaystyle \int \frac {\coth (x) \text {csch}^2(x)}{a \cosh (x)+b}dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {i \sin \left (-\frac {\pi }{2}+i x\right )}{\cos \left (-\frac {\pi }{2}+i x\right )^3 \left (b-a \sin \left (-\frac {\pi }{2}+i x\right )\right )}dx\)

\(\Big \downarrow \) 26

\(\displaystyle i \int \frac {\sin \left (i x-\frac {\pi }{2}\right )}{\cos \left (i x-\frac {\pi }{2}\right )^3 \left (b-a \sin \left (i x-\frac {\pi }{2}\right )\right )}dx\)

\(\Big \downarrow \) 3316

\(\displaystyle -a^3 \int -\frac {\cosh (x)}{(b+a \cosh (x)) \left (a^2-a^2 \cosh ^2(x)\right )^2}d(a \cosh (x))\)

\(\Big \downarrow \) 25

\(\displaystyle a^3 \int \frac {\cosh (x)}{(b+a \cosh (x)) \left (a^2-a^2 \cosh ^2(x)\right )^2}d(a \cosh (x))\)

\(\Big \downarrow \) 27

\(\displaystyle a^2 \int \frac {a \cosh (x)}{(b+a \cosh (x)) \left (a^2-a^2 \cosh ^2(x)\right )^2}d(a \cosh (x))\)

\(\Big \downarrow \) 593

\(\displaystyle a^2 \left (\frac {\int -\frac {b-a \cosh (x)}{(b+a \cosh (x)) \left (a^2-a^2 \cosh ^2(x)\right )}d(a \cosh (x))}{2 \left (a^2-b^2\right )}-\frac {b-a \cosh (x)}{2 \left (a^2-b^2\right ) \left (a^2-a^2 \cosh ^2(x)\right )}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle a^2 \left (-\frac {\int \frac {b-a \cosh (x)}{(b+a \cosh (x)) \left (a^2-a^2 \cosh ^2(x)\right )}d(a \cosh (x))}{2 \left (a^2-b^2\right )}-\frac {b-a \cosh (x)}{2 \left (a^2-b^2\right ) \left (a^2-a^2 \cosh ^2(x)\right )}\right )\)

\(\Big \downarrow \) 657

\(\displaystyle a^2 \left (-\frac {\int \left (\frac {-a-b}{2 a (a-b) (\cosh (x) a+a)}+\frac {b-a}{2 a (a+b) (a-a \cosh (x))}+\frac {2 b}{(a-b) (a+b) (b+a \cosh (x))}\right )d(a \cosh (x))}{2 \left (a^2-b^2\right )}-\frac {b-a \cosh (x)}{2 \left (a^2-b^2\right ) \left (a^2-a^2 \cosh ^2(x)\right )}\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle a^2 \left (-\frac {b-a \cosh (x)}{2 \left (a^2-b^2\right ) \left (a^2-a^2 \cosh ^2(x)\right )}-\frac {\frac {2 b \log (a \cosh (x)+b)}{a^2-b^2}+\frac {(a-b) \log (a-a \cosh (x))}{2 a (a+b)}-\frac {(a+b) \log (a \cosh (x)+a)}{2 a (a-b)}}{2 \left (a^2-b^2\right )}\right )\)

Input:

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

Output:

a^2*(-1/2*(b - a*Cosh[x])/((a^2 - b^2)*(a^2 - a^2*Cosh[x]^2)) - (((a - b)* 
Log[a - a*Cosh[x]])/(2*a*(a + b)) - ((a + b)*Log[a + a*Cosh[x]])/(2*a*(a - 
 b)) + (2*b*Log[b + a*Cosh[x]])/(a^2 - b^2))/(2*(a^2 - b^2)))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

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 593

Int[(x_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> 
 Simp[(c + d*x)^(n + 1)*(c - d*x)*((a + b*x^2)^(p + 1)/(2*(p + 1)*(b*c^2 + 
a*d^2))), x] - Simp[d/(2*(p + 1)*(b*c^2 + a*d^2))   Int[(c + d*x)^n*(a + b* 
x^2)^(p + 1)*(c*n - d*(n + 2*p + 4)*x), x], x] /; FreeQ[{a, b, c, d, n}, x] 
 && LtQ[p, -1] && NeQ[b*c^2 + a*d^2, 0]
 

rule 657

Int[(((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.))/((a_) + (c_.)*( 
x_)^2), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*((f + g*x)^n/(a + c*x^ 
2)), x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && IntegersQ[n]
 

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 3316

Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_ 
.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Simp[1/(b^p* 
f)   Subst[Int[(a + x)^m*(c + (d/b)*x)^n*(b^2 - x^2)^((p - 1)/2), x], x, b* 
Sin[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IntegerQ[(p - 1) 
/2] && NeQ[a^2 - b^2, 0]
 

rule 4360

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + 
(a_))^(m_.), x_Symbol] :> Int[(g*Cos[e + f*x])^p*((b + a*Sin[e + f*x])^m/Si 
n[e + f*x]^m), x] /; FreeQ[{a, b, e, f, g, p}, x] && IntegerQ[m]
 

Maple [A] (verified)

Time = 0.42 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.96

method	result	size
default	\(\frac {\tanh \left (\frac {x}{2}\right )^{2}}{8 a -8 b}-\frac {a^{2} b \ln \left (a \tanh \left (\frac {x}{2}\right )^{2}-b \tanh \left (\frac {x}{2}\right )^{2}+a +b \right )}{\left (a +b \right )^{2} \left (a -b \right )^{2}}-\frac {1}{8 \left (a +b \right ) \tanh \left (\frac {x}{2}\right )^{2}}-\frac {a \ln \left (\tanh \left (\frac {x}{2}\right )\right )}{2 \left (a +b \right )^{2}}\)	\(82\)
risch	\(\frac {x a}{2 a^{2}+4 a b +2 b^{2}}-\frac {x a}{2 \left (a^{2}-2 a b +b^{2}\right )}+\frac {2 a^{2} b x}{a^{4}-2 a^{2} b^{2}+b^{4}}-\frac {{\mathrm e}^{x} \left ({\mathrm e}^{2 x} a -2 b \,{\mathrm e}^{x}+a \right )}{\left ({\mathrm e}^{2 x}-1\right )^{2} \left (a^{2}-b^{2}\right )}-\frac {a \ln \left ({\mathrm e}^{x}-1\right )}{2 \left (a^{2}+2 a b +b^{2}\right )}+\frac {a \ln \left (1+{\mathrm e}^{x}\right )}{2 a^{2}-4 a b +2 b^{2}}-\frac {a^{2} b \ln \left ({\mathrm e}^{2 x}+\frac {2 b \,{\mathrm e}^{x}}{a}+1\right )}{a^{4}-2 a^{2} b^{2}+b^{4}}\)	\(176\)

Input:

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

Output:

1/8*tanh(1/2*x)^2/(a-b)-a^2*b/(a+b)^2/(a-b)^2*ln(a*tanh(1/2*x)^2-b*tanh(1/ 
2*x)^2+a+b)-1/8/(a+b)/tanh(1/2*x)^2-1/2/(a+b)^2*a*ln(tanh(1/2*x))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 828 vs. \(2 (80) = 160\).

Time = 0.10 (sec) , antiderivative size = 828, normalized size of antiderivative = 9.74 \[ \int \frac {\text {csch}^3(x)}{a+b \text {sech}(x)} \, dx=\text {Too large to display} \] Input:

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

Output:

-1/2*(2*(a^3 - a*b^2)*cosh(x)^3 + 2*(a^3 - a*b^2)*sinh(x)^3 - 4*(a^2*b - b 
^3)*cosh(x)^2 - 2*(2*a^2*b - 2*b^3 - 3*(a^3 - a*b^2)*cosh(x))*sinh(x)^2 + 
2*(a^3 - a*b^2)*cosh(x) + 2*(a^2*b*cosh(x)^4 + 4*a^2*b*cosh(x)*sinh(x)^3 + 
 a^2*b*sinh(x)^4 - 2*a^2*b*cosh(x)^2 + a^2*b + 2*(3*a^2*b*cosh(x)^2 - a^2* 
b)*sinh(x)^2 + 4*(a^2*b*cosh(x)^3 - a^2*b*cosh(x))*sinh(x))*log(2*(a*cosh( 
x) + b)/(cosh(x) - sinh(x))) - ((a^3 + 2*a^2*b + a*b^2)*cosh(x)^4 + 4*(a^3 
 + 2*a^2*b + a*b^2)*cosh(x)*sinh(x)^3 + (a^3 + 2*a^2*b + a*b^2)*sinh(x)^4 
+ a^3 + 2*a^2*b + a*b^2 - 2*(a^3 + 2*a^2*b + a*b^2)*cosh(x)^2 - 2*(a^3 + 2 
*a^2*b + a*b^2 - 3*(a^3 + 2*a^2*b + a*b^2)*cosh(x)^2)*sinh(x)^2 + 4*((a^3 
+ 2*a^2*b + a*b^2)*cosh(x)^3 - (a^3 + 2*a^2*b + a*b^2)*cosh(x))*sinh(x))*l 
og(cosh(x) + sinh(x) + 1) + ((a^3 - 2*a^2*b + a*b^2)*cosh(x)^4 + 4*(a^3 - 
2*a^2*b + a*b^2)*cosh(x)*sinh(x)^3 + (a^3 - 2*a^2*b + a*b^2)*sinh(x)^4 + a 
^3 - 2*a^2*b + a*b^2 - 2*(a^3 - 2*a^2*b + a*b^2)*cosh(x)^2 - 2*(a^3 - 2*a^ 
2*b + a*b^2 - 3*(a^3 - 2*a^2*b + a*b^2)*cosh(x)^2)*sinh(x)^2 + 4*((a^3 - 2 
*a^2*b + a*b^2)*cosh(x)^3 - (a^3 - 2*a^2*b + a*b^2)*cosh(x))*sinh(x))*log( 
cosh(x) + sinh(x) - 1) + 2*(a^3 - a*b^2 + 3*(a^3 - a*b^2)*cosh(x)^2 - 4*(a 
^2*b - b^3)*cosh(x))*sinh(x))/((a^4 - 2*a^2*b^2 + b^4)*cosh(x)^4 + 4*(a^4 
- 2*a^2*b^2 + b^4)*cosh(x)*sinh(x)^3 + (a^4 - 2*a^2*b^2 + b^4)*sinh(x)^4 + 
 a^4 - 2*a^2*b^2 + b^4 - 2*(a^4 - 2*a^2*b^2 + b^4)*cosh(x)^2 - 2*(a^4 - 2* 
a^2*b^2 + b^4 - 3*(a^4 - 2*a^2*b^2 + b^4)*cosh(x)^2)*sinh(x)^2 + 4*((a^...
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.74 \[ \int \frac {\text {csch}^3(x)}{a+b \text {sech}(x)} \, dx=-\frac {a^{2} b \log \left (2 \, b e^{\left (-x\right )} + a e^{\left (-2 \, x\right )} + a\right )}{a^{4} - 2 \, a^{2} b^{2} + b^{4}} + \frac {a \log \left (e^{\left (-x\right )} + 1\right )}{2 \, {\left (a^{2} - 2 \, a b + b^{2}\right )}} - \frac {a \log \left (e^{\left (-x\right )} - 1\right )}{2 \, {\left (a^{2} + 2 \, a b + b^{2}\right )}} - \frac {a e^{\left (-x\right )} - 2 \, b e^{\left (-2 \, x\right )} + a e^{\left (-3 \, x\right )}}{a^{2} - b^{2} - 2 \, {\left (a^{2} - b^{2}\right )} e^{\left (-2 \, x\right )} + {\left (a^{2} - b^{2}\right )} e^{\left (-4 \, x\right )}} \] Input:

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

Output:

-a^2*b*log(2*b*e^(-x) + a*e^(-2*x) + a)/(a^4 - 2*a^2*b^2 + b^4) + 1/2*a*lo 
g(e^(-x) + 1)/(a^2 - 2*a*b + b^2) - 1/2*a*log(e^(-x) - 1)/(a^2 + 2*a*b + b 
^2) - (a*e^(-x) - 2*b*e^(-2*x) + a*e^(-3*x))/(a^2 - b^2 - 2*(a^2 - b^2)*e^ 
(-2*x) + (a^2 - b^2)*e^(-4*x))
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 174 vs. \(2 (80) = 160\).

Time = 0.11 (sec) , antiderivative size = 174, normalized size of antiderivative = 2.05 \[ \int \frac {\text {csch}^3(x)}{a+b \text {sech}(x)} \, dx=-\frac {a^{3} b \log \left ({\left | a {\left (e^{\left (-x\right )} + e^{x}\right )} + 2 \, b \right |}\right )}{a^{5} - 2 \, a^{3} b^{2} + a b^{4}} + \frac {a \log \left (e^{\left (-x\right )} + e^{x} + 2\right )}{4 \, {\left (a^{2} - 2 \, a b + b^{2}\right )}} - \frac {a \log \left (e^{\left (-x\right )} + e^{x} - 2\right )}{4 \, {\left (a^{2} + 2 \, a b + b^{2}\right )}} - \frac {a^{2} b {\left (e^{\left (-x\right )} + e^{x}\right )}^{2} + 2 \, a^{3} {\left (e^{\left (-x\right )} + e^{x}\right )} - 2 \, a b^{2} {\left (e^{\left (-x\right )} + e^{x}\right )} - 8 \, a^{2} b + 4 \, b^{3}}{2 \, {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} {\left ({\left (e^{\left (-x\right )} + e^{x}\right )}^{2} - 4\right )}} \] Input:

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

Output:

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

Mupad [B] (verification not implemented)

Time = 2.82 (sec) , antiderivative size = 255, normalized size of antiderivative = 3.00 \[ \int \frac {\text {csch}^3(x)}{a+b \text {sech}(x)} \, dx=\frac {\frac {2\,\left (a^2\,b-b^3\right )}{{\left (a^2-b^2\right )}^2}+\frac {{\mathrm {e}}^x\,\left (a\,b^2-a^3\right )}{{\left (a^2-b^2\right )}^2}}{{\mathrm {e}}^{2\,x}-1}+\frac {\frac {2\,b}{a^2-b^2}-\frac {2\,a\,{\mathrm {e}}^x}{a^2-b^2}}{{\mathrm {e}}^{4\,x}-2\,{\mathrm {e}}^{2\,x}+1}-\frac {a\,\ln \left ({\mathrm {e}}^x-1\right )}{2\,a^2+4\,a\,b+2\,b^2}+\frac {a\,\ln \left ({\mathrm {e}}^x+1\right )}{2\,a^2-4\,a\,b+2\,b^2}-\frac {a^2\,b\,\ln \left (a^6\,{\mathrm {e}}^{2\,x}+a^6+a^2\,b^4-14\,a^4\,b^2+a^2\,b^4\,{\mathrm {e}}^{2\,x}-14\,a^4\,b^2\,{\mathrm {e}}^{2\,x}+2\,a\,b^5\,{\mathrm {e}}^x+2\,a^5\,b\,{\mathrm {e}}^x-28\,a^3\,b^3\,{\mathrm {e}}^x\right )}{a^4-2\,a^2\,b^2+b^4} \] Input:

int(1/(sinh(x)^3*(a + b/cosh(x))),x)
 

Output:

((2*(a^2*b - b^3))/(a^2 - b^2)^2 + (exp(x)*(a*b^2 - a^3))/(a^2 - b^2)^2)/( 
exp(2*x) - 1) + ((2*b)/(a^2 - b^2) - (2*a*exp(x))/(a^2 - b^2))/(exp(4*x) - 
 2*exp(2*x) + 1) - (a*log(exp(x) - 1))/(4*a*b + 2*a^2 + 2*b^2) + (a*log(ex 
p(x) + 1))/(2*a^2 - 4*a*b + 2*b^2) - (a^2*b*log(a^6*exp(2*x) + a^6 + a^2*b 
^4 - 14*a^4*b^2 + a^2*b^4*exp(2*x) - 14*a^4*b^2*exp(2*x) + 2*a*b^5*exp(x) 
+ 2*a^5*b*exp(x) - 28*a^3*b^3*exp(x)))/(a^4 + b^4 - 2*a^2*b^2)
 

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 501, normalized size of antiderivative = 5.89 \[ \int \frac {\text {csch}^3(x)}{a+b \text {sech}(x)} \, dx=\frac {-2 b^{3}-\mathrm {log}\left (e^{x}-1\right ) a^{3}+\mathrm {log}\left (e^{x}+1\right ) a^{3}-2 e^{3 x} a^{3}-2 e^{x} a^{3}+2 e^{3 x} a \,b^{2}+2 e^{2 x} \mathrm {log}\left (e^{x}-1\right ) a^{3}-2 e^{2 x} \mathrm {log}\left (e^{x}+1\right ) a^{3}+2 e^{x} a \,b^{2}+2 \,\mathrm {log}\left (e^{x}-1\right ) a^{2} b +2 \,\mathrm {log}\left (e^{x}+1\right ) a^{2} b -2 \,\mathrm {log}\left (e^{2 x} a +2 e^{x} b +a \right ) a^{2} b -e^{4 x} \mathrm {log}\left (e^{x}-1\right ) a^{3}-e^{4 x} \mathrm {log}\left (e^{x}-1\right ) a \,b^{2}+e^{4 x} \mathrm {log}\left (e^{x}+1\right ) a \,b^{2}+e^{4 x} \mathrm {log}\left (e^{x}+1\right ) a^{3}-\mathrm {log}\left (e^{x}-1\right ) a \,b^{2}-2 e^{4 x} b^{3}+2 e^{4 x} a^{2} b +2 a^{2} b +\mathrm {log}\left (e^{x}+1\right ) a \,b^{2}+2 e^{4 x} \mathrm {log}\left (e^{x}-1\right ) a^{2} b +2 e^{4 x} \mathrm {log}\left (e^{x}+1\right ) a^{2} b -2 e^{4 x} \mathrm {log}\left (e^{2 x} a +2 e^{x} b +a \right ) a^{2} b -4 e^{2 x} \mathrm {log}\left (e^{x}-1\right ) a^{2} b +2 e^{2 x} \mathrm {log}\left (e^{x}-1\right ) a \,b^{2}-4 e^{2 x} \mathrm {log}\left (e^{x}+1\right ) a^{2} b -2 e^{2 x} \mathrm {log}\left (e^{x}+1\right ) a \,b^{2}+4 e^{2 x} \mathrm {log}\left (e^{2 x} a +2 e^{x} b +a \right ) a^{2} b}{2 e^{4 x} a^{4}-4 e^{4 x} a^{2} b^{2}+2 e^{4 x} b^{4}-4 e^{2 x} a^{4}+8 e^{2 x} a^{2} b^{2}-4 e^{2 x} b^{4}+2 a^{4}-4 a^{2} b^{2}+2 b^{4}} \] Input:

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

Output:

( - e**(4*x)*log(e**x - 1)*a**3 + 2*e**(4*x)*log(e**x - 1)*a**2*b - e**(4* 
x)*log(e**x - 1)*a*b**2 + e**(4*x)*log(e**x + 1)*a**3 + 2*e**(4*x)*log(e** 
x + 1)*a**2*b + e**(4*x)*log(e**x + 1)*a*b**2 - 2*e**(4*x)*log(e**(2*x)*a 
+ 2*e**x*b + a)*a**2*b + 2*e**(4*x)*a**2*b - 2*e**(4*x)*b**3 - 2*e**(3*x)* 
a**3 + 2*e**(3*x)*a*b**2 + 2*e**(2*x)*log(e**x - 1)*a**3 - 4*e**(2*x)*log( 
e**x - 1)*a**2*b + 2*e**(2*x)*log(e**x - 1)*a*b**2 - 2*e**(2*x)*log(e**x + 
 1)*a**3 - 4*e**(2*x)*log(e**x + 1)*a**2*b - 2*e**(2*x)*log(e**x + 1)*a*b* 
*2 + 4*e**(2*x)*log(e**(2*x)*a + 2*e**x*b + a)*a**2*b - 2*e**x*a**3 + 2*e* 
*x*a*b**2 - log(e**x - 1)*a**3 + 2*log(e**x - 1)*a**2*b - log(e**x - 1)*a* 
b**2 + log(e**x + 1)*a**3 + 2*log(e**x + 1)*a**2*b + log(e**x + 1)*a*b**2 
- 2*log(e**(2*x)*a + 2*e**x*b + a)*a**2*b + 2*a**2*b - 2*b**3)/(2*(e**(4*x 
)*a**4 - 2*e**(4*x)*a**2*b**2 + e**(4*x)*b**4 - 2*e**(2*x)*a**4 + 4*e**(2* 
x)*a**2*b**2 - 2*e**(2*x)*b**4 + a**4 - 2*a**2*b**2 + b**4))