Integrand size = 27, antiderivative size = 160 \[ \int \frac {\text {csch}(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {b^3 \arctan (\sinh (c+d x))}{\left (a^2+b^2\right )^2 d}-\frac {b \arctan (\sinh (c+d x))}{2 \left (a^2+b^2\right ) d}-\frac {a \left (a^2+2 b^2\right ) \log (\cosh (c+d x))}{\left (a^2+b^2\right )^2 d}+\frac {\log (\sinh (c+d x))}{a d}-\frac {b^4 \log (a+b \sinh (c+d x))}{a \left (a^2+b^2\right )^2 d}+\frac {\text {sech}^2(c+d x) (a-b \sinh (c+d x))}{2 \left (a^2+b^2\right ) d} \]
-b^3*arctan(sinh(d*x+c))/(a^2+b^2)^2/d-1/2*b*arctan(sinh(d*x+c))/(a^2+b^2) /d-a*(a^2+2*b^2)*ln(cosh(d*x+c))/(a^2+b^2)^2/d+ln(sinh(d*x+c))/a/d-b^4*ln( a+b*sinh(d*x+c))/a/(a^2+b^2)^2/d+1/2*sech(d*x+c)^2*(a-b*sinh(d*x+c))/(a^2+ b^2)/d
Time = 0.54 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.22 \[ \int \frac {\text {csch}(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {a b \left (a^2+b^2\right ) \arctan (\sinh (c+d x))-2 \left (a^2+b^2\right )^2 \log (\sinh (c+d x))+a \left (a^3+2 a b^2+\left (-b^2\right )^{3/2}\right ) \log \left (\sqrt {-b^2}-b \sinh (c+d x)\right )+2 b^4 \log (a+b \sinh (c+d x))+a \left (a^3+2 a b^2-\left (-b^2\right )^{3/2}\right ) \log \left (\sqrt {-b^2}+b \sinh (c+d x)\right )-a^2 \left (a^2+b^2\right ) \text {sech}^2(c+d x)+a b \left (a^2+b^2\right ) \text {sech}(c+d x) \tanh (c+d x)}{2 a \left (a^2+b^2\right )^2 d} \]
-1/2*(a*b*(a^2 + b^2)*ArcTan[Sinh[c + d*x]] - 2*(a^2 + b^2)^2*Log[Sinh[c + d*x]] + a*(a^3 + 2*a*b^2 + (-b^2)^(3/2))*Log[Sqrt[-b^2] - b*Sinh[c + d*x] ] + 2*b^4*Log[a + b*Sinh[c + d*x]] + a*(a^3 + 2*a*b^2 - (-b^2)^(3/2))*Log[ Sqrt[-b^2] + b*Sinh[c + d*x]] - a^2*(a^2 + b^2)*Sech[c + d*x]^2 + a*b*(a^2 + b^2)*Sech[c + d*x]*Tanh[c + d*x])/(a*(a^2 + b^2)^2*d)
Time = 0.44 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.13, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {3042, 26, 3316, 26, 27, 615, 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.
| \(\displaystyle \int \frac {\text {csch}(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {i}{\sin (i c+i d x) \cos (i c+i d x)^3 (a-i b \sin (i c+i d x))}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \int \frac {1}{\cos (i c+i d x)^3 \sin (i c+i d x) (a-i b \sin (i c+i d x))}dx\) |
| \(\Big \downarrow \) 3316 |
| \(\displaystyle \frac {i b^3 \int -\frac {i \text {csch}(c+d x)}{(a+b \sinh (c+d x)) \left (\sinh ^2(c+d x) b^2+b^2\right )^2}d(b \sinh (c+d x))}{d}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {b^3 \int \frac {\text {csch}(c+d x)}{(a+b \sinh (c+d x)) \left (\sinh ^2(c+d x) b^2+b^2\right )^2}d(b \sinh (c+d x))}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b^4 \int \frac {\text {csch}(c+d x)}{b (a+b \sinh (c+d x)) \left (\sinh ^2(c+d x) b^2+b^2\right )^2}d(b \sinh (c+d x))}{d}\) |
| \(\Big \downarrow \) 615 |
| \(\displaystyle \frac {b^4 \int \left (\frac {\text {csch}(c+d x)}{a b^5}-\frac {1}{a \left (a^2+b^2\right )^2 (a+b \sinh (c+d x))}+\frac {-b^4-a \left (a^2+2 b^2\right ) \sinh (c+d x) b}{b^4 \left (a^2+b^2\right )^2 \left (\sinh ^2(c+d x) b^2+b^2\right )}+\frac {-b^2-a \sinh (c+d x) b}{b^2 \left (a^2+b^2\right ) \left (\sinh ^2(c+d x) b^2+b^2\right )^2}\right )d(b \sinh (c+d x))}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {b^4 \left (-\frac {\arctan (\sinh (c+d x))}{b \left (a^2+b^2\right )^2}-\frac {\arctan (\sinh (c+d x))}{2 b^3 \left (a^2+b^2\right )}+\frac {a-b \sinh (c+d x)}{2 b^2 \left (a^2+b^2\right ) \left (b^2 \sinh ^2(c+d x)+b^2\right )}-\frac {\log (a+b \sinh (c+d x))}{a \left (a^2+b^2\right )^2}-\frac {a \left (a^2+2 b^2\right ) \log \left (b^2 \sinh ^2(c+d x)+b^2\right )}{2 b^4 \left (a^2+b^2\right )^2}+\frac {\log (b \sinh (c+d x))}{a b^4}\right )}{d}\) |
(b^4*(-(ArcTan[Sinh[c + d*x]]/(b*(a^2 + b^2)^2)) - ArcTan[Sinh[c + d*x]]/( 2*b^3*(a^2 + b^2)) + Log[b*Sinh[c + d*x]]/(a*b^4) - Log[a + b*Sinh[c + d*x ]]/(a*(a^2 + b^2)^2) - (a*(a^2 + 2*b^2)*Log[b^2 + b^2*Sinh[c + d*x]^2])/(2 *b^4*(a^2 + b^2)^2) + (a - b*Sinh[c + d*x])/(2*b^2*(a^2 + b^2)*(b^2 + b^2* Sinh[c + d*x]^2))))/d
3.5.47.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Int[ExpandIntegrand[(e*x)^m*(c + d*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && ILtQ[p, 0]
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]
Time = 15.01 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.44
| method | result | size |
| derivativedivides | \(\frac {-\frac {2 \left (\frac {\left (-\frac {1}{2} a^{2} b -\frac {1}{2} b^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (a^{3}+a \,b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\left (\frac {1}{2} a^{2} b +\frac {1}{2} b^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2}}+\frac {\left (2 a^{3}+4 a \,b^{2}\right ) \ln \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}{4}+\frac {\left (a^{2} b +3 b^{3}\right ) \arctan \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}\right )}{a^{4}+2 a^{2} b^{2}+b^{4}}-\frac {b^{4} \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-a \right )}{a \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}}{d}\) | \(230\) |
| default | \(\frac {-\frac {2 \left (\frac {\left (-\frac {1}{2} a^{2} b -\frac {1}{2} b^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (a^{3}+a \,b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\left (\frac {1}{2} a^{2} b +\frac {1}{2} b^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2}}+\frac {\left (2 a^{3}+4 a \,b^{2}\right ) \ln \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}{4}+\frac {\left (a^{2} b +3 b^{3}\right ) \arctan \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}\right )}{a^{4}+2 a^{2} b^{2}+b^{4}}-\frac {b^{4} \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-a \right )}{a \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}}{d}\) | \(230\) |
| risch | \(\frac {2 d^{2} a^{3} x}{a^{4} d^{2}+2 a^{2} b^{2} d^{2}+b^{4} d^{2}}+\frac {2 d \,a^{3} c}{a^{4} d^{2}+2 a^{2} b^{2} d^{2}+b^{4} d^{2}}+\frac {4 a \,b^{2} d^{2} x}{a^{4} d^{2}+2 a^{2} b^{2} d^{2}+b^{4} d^{2}}+\frac {4 a \,b^{2} d c}{a^{4} d^{2}+2 a^{2} b^{2} d^{2}+b^{4} d^{2}}-\frac {2 x}{a}-\frac {2 c}{d a}+\frac {2 b^{4} x}{a \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {2 b^{4} c}{a d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {{\mathrm e}^{d x +c} \left (-b \,{\mathrm e}^{2 d x +2 c}+2 a \,{\mathrm e}^{d x +c}+b \right )}{d \left (a^{2}+b^{2}\right ) \left (1+{\mathrm e}^{2 d x +2 c}\right )^{2}}+\frac {3 i \ln \left ({\mathrm e}^{d x +c}-i\right ) b^{3}}{2 \left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) d}+\frac {i \ln \left ({\mathrm e}^{d x +c}-i\right ) a^{2} b}{2 \left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) d}-\frac {\ln \left ({\mathrm e}^{d x +c}-i\right ) a^{3}}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) d}-\frac {2 \ln \left ({\mathrm e}^{d x +c}-i\right ) a \,b^{2}}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) d}-\frac {i \ln \left ({\mathrm e}^{d x +c}+i\right ) a^{2} b}{2 \left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) d}-\frac {3 i \ln \left ({\mathrm e}^{d x +c}+i\right ) b^{3}}{2 \left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) d}-\frac {\ln \left ({\mathrm e}^{d x +c}+i\right ) a^{3}}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) d}-\frac {2 \ln \left ({\mathrm e}^{d x +c}+i\right ) a \,b^{2}}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) d}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}-1\right )}{d a}-\frac {b^{4} \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \,{\mathrm e}^{d x +c}}{b}-1\right )}{a d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}\) | \(632\) |
1/d*(-2/(a^4+2*a^2*b^2+b^4)*(((-1/2*a^2*b-1/2*b^3)*tanh(1/2*d*x+1/2*c)^3+( a^3+a*b^2)*tanh(1/2*d*x+1/2*c)^2+(1/2*a^2*b+1/2*b^3)*tanh(1/2*d*x+1/2*c))/ (1+tanh(1/2*d*x+1/2*c)^2)^2+1/4*(2*a^3+4*a*b^2)*ln(1+tanh(1/2*d*x+1/2*c)^2 )+1/2*(a^2*b+3*b^3)*arctan(tanh(1/2*d*x+1/2*c)))-b^4/a/(a^4+2*a^2*b^2+b^4) *ln(tanh(1/2*d*x+1/2*c)^2*a-2*b*tanh(1/2*d*x+1/2*c)-a)+1/a*ln(tanh(1/2*d*x +1/2*c)))
Leaf count of result is larger than twice the leaf count of optimal. 1279 vs. \(2 (157) = 314\).
Time = 0.44 (sec) , antiderivative size = 1279, normalized size of antiderivative = 7.99 \[ \int \frac {\text {csch}(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \]
-((a^3*b + a*b^3)*cosh(d*x + c)^3 + (a^3*b + a*b^3)*sinh(d*x + c)^3 - 2*(a ^4 + a^2*b^2)*cosh(d*x + c)^2 - (2*a^4 + 2*a^2*b^2 - 3*(a^3*b + a*b^3)*cos h(d*x + c))*sinh(d*x + c)^2 + ((a^3*b + 3*a*b^3)*cosh(d*x + c)^4 + 4*(a^3* b + 3*a*b^3)*cosh(d*x + c)*sinh(d*x + c)^3 + (a^3*b + 3*a*b^3)*sinh(d*x + c)^4 + a^3*b + 3*a*b^3 + 2*(a^3*b + 3*a*b^3)*cosh(d*x + c)^2 + 2*(a^3*b + 3*a*b^3 + 3*(a^3*b + 3*a*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*((a^3*b + 3*a*b^3)*cosh(d*x + c)^3 + (a^3*b + 3*a*b^3)*cosh(d*x + c))*sinh(d*x + c))*arctan(cosh(d*x + c) + sinh(d*x + c)) - (a^3*b + a*b^3)*cosh(d*x + c) + (b^4*cosh(d*x + c)^4 + 4*b^4*cosh(d*x + c)*sinh(d*x + c)^3 + b^4*sinh(d* x + c)^4 + 2*b^4*cosh(d*x + c)^2 + b^4 + 2*(3*b^4*cosh(d*x + c)^2 + b^4)*s inh(d*x + c)^2 + 4*(b^4*cosh(d*x + c)^3 + b^4*cosh(d*x + c))*sinh(d*x + c) )*log(2*(b*sinh(d*x + c) + a)/(cosh(d*x + c) - sinh(d*x + c))) + ((a^4 + 2 *a^2*b^2)*cosh(d*x + c)^4 + 4*(a^4 + 2*a^2*b^2)*cosh(d*x + c)*sinh(d*x + c )^3 + (a^4 + 2*a^2*b^2)*sinh(d*x + c)^4 + a^4 + 2*a^2*b^2 + 2*(a^4 + 2*a^2 *b^2)*cosh(d*x + c)^2 + 2*(a^4 + 2*a^2*b^2 + 3*(a^4 + 2*a^2*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*((a^4 + 2*a^2*b^2)*cosh(d*x + c)^3 + (a^4 + 2* a^2*b^2)*cosh(d*x + c))*sinh(d*x + c))*log(2*cosh(d*x + c)/(cosh(d*x + c) - sinh(d*x + c))) - ((a^4 + 2*a^2*b^2 + b^4)*cosh(d*x + c)^4 + 4*(a^4 + 2* a^2*b^2 + b^4)*cosh(d*x + c)*sinh(d*x + c)^3 + (a^4 + 2*a^2*b^2 + b^4)*sin h(d*x + c)^4 + a^4 + 2*a^2*b^2 + b^4 + 2*(a^4 + 2*a^2*b^2 + b^4)*cosh(d...
Timed out. \[ \int \frac {\text {csch}(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \]
Time = 0.33 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.66 \[ \int \frac {\text {csch}(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {b^{4} \log \left (-2 \, a e^{\left (-d x - c\right )} + b e^{\left (-2 \, d x - 2 \, c\right )} - b\right )}{{\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} d} + \frac {{\left (a^{2} b + 3 \, b^{3}\right )} \arctan \left (e^{\left (-d x - c\right )}\right )}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d} - \frac {{\left (a^{3} + 2 \, a b^{2}\right )} \log \left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d} - \frac {b e^{\left (-d x - c\right )} - 2 \, a e^{\left (-2 \, d x - 2 \, c\right )} - b e^{\left (-3 \, d x - 3 \, c\right )}}{{\left (a^{2} + b^{2} + 2 \, {\left (a^{2} + b^{2}\right )} e^{\left (-2 \, d x - 2 \, c\right )} + {\left (a^{2} + b^{2}\right )} e^{\left (-4 \, d x - 4 \, c\right )}\right )} d} + \frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{a d} + \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{a d} \]
-b^4*log(-2*a*e^(-d*x - c) + b*e^(-2*d*x - 2*c) - b)/((a^5 + 2*a^3*b^2 + a *b^4)*d) + (a^2*b + 3*b^3)*arctan(e^(-d*x - c))/((a^4 + 2*a^2*b^2 + b^4)*d ) - (a^3 + 2*a*b^2)*log(e^(-2*d*x - 2*c) + 1)/((a^4 + 2*a^2*b^2 + b^4)*d) - (b*e^(-d*x - c) - 2*a*e^(-2*d*x - 2*c) - b*e^(-3*d*x - 3*c))/((a^2 + b^2 + 2*(a^2 + b^2)*e^(-2*d*x - 2*c) + (a^2 + b^2)*e^(-4*d*x - 4*c))*d) + log (e^(-d*x - c) + 1)/(a*d) + log(e^(-d*x - c) - 1)/(a*d)
Leaf count of result is larger than twice the leaf count of optimal. 343 vs. \(2 (157) = 314\).
Time = 0.29 (sec) , antiderivative size = 343, normalized size of antiderivative = 2.14 \[ \int \frac {\text {csch}(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {\frac {4 \, b^{5} \log \left ({\left | b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 2 \, a \right |}\right )}{a^{5} b + 2 \, a^{3} b^{3} + a b^{5}} + \frac {{\left (\pi + 2 \, \arctan \left (\frac {1}{2} \, {\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )} e^{\left (-d x - c\right )}\right )\right )} {\left (a^{2} b + 3 \, b^{3}\right )}}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {2 \, {\left (a^{3} + 2 \, a b^{2}\right )} \log \left ({\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} + 4\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} - \frac {4 \, \log \left ({\left | e^{\left (d x + c\right )} - e^{\left (-d x - c\right )} \right |}\right )}{a} - \frac {2 \, {\left (a^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} + 2 \, a b^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} - 2 \, a^{2} b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} - 2 \, b^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 8 \, a^{3} + 12 \, a b^{2}\right )}}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} {\left ({\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} + 4\right )}}}{4 \, d} \]
-1/4*(4*b^5*log(abs(b*(e^(d*x + c) - e^(-d*x - c)) + 2*a))/(a^5*b + 2*a^3* b^3 + a*b^5) + (pi + 2*arctan(1/2*(e^(2*d*x + 2*c) - 1)*e^(-d*x - c)))*(a^ 2*b + 3*b^3)/(a^4 + 2*a^2*b^2 + b^4) + 2*(a^3 + 2*a*b^2)*log((e^(d*x + c) - e^(-d*x - c))^2 + 4)/(a^4 + 2*a^2*b^2 + b^4) - 4*log(abs(e^(d*x + c) - e ^(-d*x - c)))/a - 2*(a^3*(e^(d*x + c) - e^(-d*x - c))^2 + 2*a*b^2*(e^(d*x + c) - e^(-d*x - c))^2 - 2*a^2*b*(e^(d*x + c) - e^(-d*x - c)) - 2*b^3*(e^( d*x + c) - e^(-d*x - c)) + 8*a^3 + 12*a*b^2)/((a^4 + 2*a^2*b^2 + b^4)*((e^ (d*x + c) - e^(-d*x - c))^2 + 4)))/d
Timed out. \[ \int \frac {\text {csch}(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {1}{{\mathrm {cosh}\left (c+d\,x\right )}^3\,\mathrm {sinh}\left (c+d\,x\right )\,\left (a+b\,\mathrm {sinh}\left (c+d\,x\right )\right )} \,d x \]