Integrand size = 21, antiderivative size = 166 \[ \int \frac {\text {csch}(c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^3} \, dx=-\frac {\sqrt {b} \left (15 a^2-20 a b+8 b^2\right ) \arctan \left (\frac {\sqrt {b} \cosh (c+d x)}{\sqrt {a-b}}\right )}{8 a^3 (a-b)^{5/2} d}-\frac {\text {arctanh}(\cosh (c+d x))}{a^3 d}-\frac {b \cosh (c+d x)}{4 a (a-b) d \left (a-b+b \cosh ^2(c+d x)\right )^2}-\frac {(7 a-4 b) b \cosh (c+d x)}{8 a^2 (a-b)^2 d \left (a-b+b \cosh ^2(c+d x)\right )} \]
-arctanh(cosh(d*x+c))/a^3/d-1/4*b*cosh(d*x+c)/a/(a-b)/d/(a-b+b*cosh(d*x+c) ^2)^2-1/8*(7*a-4*b)*b*cosh(d*x+c)/a^2/(a-b)^2/d/(a-b+b*cosh(d*x+c)^2)-1/8* (15*a^2-20*a*b+8*b^2)*arctan(cosh(d*x+c)*b^(1/2)/(a-b)^(1/2))*b^(1/2)/a^3/ (a-b)^(5/2)/d
Result contains complex when optimal does not.
Time = 7.24 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.51 \[ \int \frac {\text {csch}(c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^3} \, dx=-\frac {\frac {\sqrt {b} \left (15 a^2-20 a b+8 b^2\right ) \arctan \left (\frac {\sqrt {b}-i \sqrt {a} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a-b}}\right )}{(a-b)^{5/2}}+\frac {\sqrt {b} \left (15 a^2-20 a b+8 b^2\right ) \arctan \left (\frac {\sqrt {b}+i \sqrt {a} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a-b}}\right )}{(a-b)^{5/2}}+\frac {8 a^2 b \cosh (c+d x)}{(a-b) (2 a-b+b \cosh (2 (c+d x)))^2}+\frac {2 a (7 a-4 b) b \cosh (c+d x)}{(a-b)^2 (2 a-b+b \cosh (2 (c+d x)))}+8 \log \left (\cosh \left (\frac {1}{2} (c+d x)\right )\right )-8 \log \left (\sinh \left (\frac {1}{2} (c+d x)\right )\right )}{8 a^3 d} \]
-1/8*((Sqrt[b]*(15*a^2 - 20*a*b + 8*b^2)*ArcTan[(Sqrt[b] - I*Sqrt[a]*Tanh[ (c + d*x)/2])/Sqrt[a - b]])/(a - b)^(5/2) + (Sqrt[b]*(15*a^2 - 20*a*b + 8* b^2)*ArcTan[(Sqrt[b] + I*Sqrt[a]*Tanh[(c + d*x)/2])/Sqrt[a - b]])/(a - b)^ (5/2) + (8*a^2*b*Cosh[c + d*x])/((a - b)*(2*a - b + b*Cosh[2*(c + d*x)])^2 ) + (2*a*(7*a - 4*b)*b*Cosh[c + d*x])/((a - b)^2*(2*a - b + b*Cosh[2*(c + d*x)])) + 8*Log[Cosh[(c + d*x)/2]] - 8*Log[Sinh[(c + d*x)/2]])/(a^3*d)
Time = 0.41 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.16, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {3042, 26, 3665, 316, 25, 402, 25, 397, 218, 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.
| \(\displaystyle \int \frac {\text {csch}(c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {i}{\sin (i c+i d x) \left (a-b \sin (i c+i d x)^2\right )^3}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \int \frac {1}{\sin (i c+i d x) \left (a-b \sin (i c+i d x)^2\right )^3}dx\) |
| \(\Big \downarrow \) 3665 |
| \(\displaystyle -\frac {\int \frac {1}{\left (1-\cosh ^2(c+d x)\right ) \left (b \cosh ^2(c+d x)+a-b\right )^3}d\cosh (c+d x)}{d}\) |
| \(\Big \downarrow \) 316 |
| \(\displaystyle -\frac {\frac {b \cosh (c+d x)}{4 a (a-b) \left (a+b \cosh ^2(c+d x)-b\right )^2}-\frac {\int -\frac {-3 b \cosh ^2(c+d x)+4 a-b}{\left (1-\cosh ^2(c+d x)\right ) \left (b \cosh ^2(c+d x)+a-b\right )^2}d\cosh (c+d x)}{4 a (a-b)}}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {\int \frac {-3 b \cosh ^2(c+d x)+4 a-b}{\left (1-\cosh ^2(c+d x)\right ) \left (b \cosh ^2(c+d x)+a-b\right )^2}d\cosh (c+d x)}{4 a (a-b)}+\frac {b \cosh (c+d x)}{4 a (a-b) \left (a+b \cosh ^2(c+d x)-b\right )^2}}{d}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle -\frac {\frac {\frac {b (7 a-4 b) \cosh (c+d x)}{2 a (a-b) \left (a+b \cosh ^2(c+d x)-b\right )}-\frac {\int -\frac {8 a^2-9 b a+4 b^2-(7 a-4 b) b \cosh ^2(c+d x)}{\left (1-\cosh ^2(c+d x)\right ) \left (b \cosh ^2(c+d x)+a-b\right )}d\cosh (c+d x)}{2 a (a-b)}}{4 a (a-b)}+\frac {b \cosh (c+d x)}{4 a (a-b) \left (a+b \cosh ^2(c+d x)-b\right )^2}}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {\frac {\int \frac {8 a^2-9 b a+4 b^2-(7 a-4 b) b \cosh ^2(c+d x)}{\left (1-\cosh ^2(c+d x)\right ) \left (b \cosh ^2(c+d x)+a-b\right )}d\cosh (c+d x)}{2 a (a-b)}+\frac {b (7 a-4 b) \cosh (c+d x)}{2 a (a-b) \left (a+b \cosh ^2(c+d x)-b\right )}}{4 a (a-b)}+\frac {b \cosh (c+d x)}{4 a (a-b) \left (a+b \cosh ^2(c+d x)-b\right )^2}}{d}\) |
| \(\Big \downarrow \) 397 |
| \(\displaystyle -\frac {\frac {\frac {\frac {b \left (15 a^2-20 a b+8 b^2\right ) \int \frac {1}{b \cosh ^2(c+d x)+a-b}d\cosh (c+d x)}{a}+\frac {8 (a-b)^2 \int \frac {1}{1-\cosh ^2(c+d x)}d\cosh (c+d x)}{a}}{2 a (a-b)}+\frac {b (7 a-4 b) \cosh (c+d x)}{2 a (a-b) \left (a+b \cosh ^2(c+d x)-b\right )}}{4 a (a-b)}+\frac {b \cosh (c+d x)}{4 a (a-b) \left (a+b \cosh ^2(c+d x)-b\right )^2}}{d}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle -\frac {\frac {\frac {\frac {8 (a-b)^2 \int \frac {1}{1-\cosh ^2(c+d x)}d\cosh (c+d x)}{a}+\frac {\sqrt {b} \left (15 a^2-20 a b+8 b^2\right ) \arctan \left (\frac {\sqrt {b} \cosh (c+d x)}{\sqrt {a-b}}\right )}{a \sqrt {a-b}}}{2 a (a-b)}+\frac {b (7 a-4 b) \cosh (c+d x)}{2 a (a-b) \left (a+b \cosh ^2(c+d x)-b\right )}}{4 a (a-b)}+\frac {b \cosh (c+d x)}{4 a (a-b) \left (a+b \cosh ^2(c+d x)-b\right )^2}}{d}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {\frac {\frac {\frac {\sqrt {b} \left (15 a^2-20 a b+8 b^2\right ) \arctan \left (\frac {\sqrt {b} \cosh (c+d x)}{\sqrt {a-b}}\right )}{a \sqrt {a-b}}+\frac {8 (a-b)^2 \text {arctanh}(\cosh (c+d x))}{a}}{2 a (a-b)}+\frac {b (7 a-4 b) \cosh (c+d x)}{2 a (a-b) \left (a+b \cosh ^2(c+d x)-b\right )}}{4 a (a-b)}+\frac {b \cosh (c+d x)}{4 a (a-b) \left (a+b \cosh ^2(c+d x)-b\right )^2}}{d}\) |
-(((b*Cosh[c + d*x])/(4*a*(a - b)*(a - b + b*Cosh[c + d*x]^2)^2) + (((Sqrt [b]*(15*a^2 - 20*a*b + 8*b^2)*ArcTan[(Sqrt[b]*Cosh[c + d*x])/Sqrt[a - b]]) /(a*Sqrt[a - b]) + (8*(a - b)^2*ArcTanh[Cosh[c + d*x]])/a)/(2*a*(a - b)) + ((7*a - 4*b)*b*Cosh[c + d*x])/(2*a*(a - b)*(a - b + b*Cosh[c + d*x]^2)))/ (4*a*(a - b)))/d)
3.1.56.3.1 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_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
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])
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim p[(-b)*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(2*a*(p + 1)*(b*c - a*d)) ), x] + Simp[1/(2*a*(p + 1)*(b*c - a*d)) Int[(a + b*x^2)^(p + 1)*(c + d*x ^2)^q*Simp[b*c + 2*(p + 1)*(b*c - a*d) + d*b*(2*(p + q + 2) + 1)*x^2, x], x ], x] /; FreeQ[{a, b, c, d, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && ! ( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomialQ[a, b, c, d, 2, p, q, x]
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]
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]
Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^ (p_.), x_Symbol] :> With[{ff = FreeFactors[Cos[e + f*x], x]}, Simp[-ff/f Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b - b*ff^2*x^2)^p, x], x, Cos[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]
Leaf count of result is larger than twice the leaf count of optimal. \(307\) vs. \(2(152)=304\).
Time = 0.79 (sec) , antiderivative size = 308, normalized size of antiderivative = 1.86
| method | result | size |
| derivativedivides | \(\frac {\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{3}}-\frac {2 b \left (\frac {-\frac {\left (9 a^{2}-28 a b +16 b^{2}\right ) a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{8 \left (a^{2}-2 a b +b^{2}\right )}+\frac {3 \left (9 a^{3}-30 a^{2} b +40 a \,b^{2}-16 b^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{8 \left (a^{2}-2 a b +b^{2}\right )}-\frac {a \left (27 a^{2}-68 a b +32 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{8 \left (a^{2}-2 a b +b^{2}\right )}+\frac {3 a^{2} \left (3 a -2 b \right )}{8 \left (a^{2}-2 a b +b^{2}\right )}}{\left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a -2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +4 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+a \right )^{2}}+\frac {\left (15 a^{2}-20 a b +8 b^{2}\right ) \arctan \left (\frac {2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 a +4 b}{4 \sqrt {a b -b^{2}}}\right )}{16 \left (a^{2}-2 a b +b^{2}\right ) \sqrt {a b -b^{2}}}\right )}{a^{3}}}{d}\) | \(308\) |
| default | \(\frac {\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{3}}-\frac {2 b \left (\frac {-\frac {\left (9 a^{2}-28 a b +16 b^{2}\right ) a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{8 \left (a^{2}-2 a b +b^{2}\right )}+\frac {3 \left (9 a^{3}-30 a^{2} b +40 a \,b^{2}-16 b^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{8 \left (a^{2}-2 a b +b^{2}\right )}-\frac {a \left (27 a^{2}-68 a b +32 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{8 \left (a^{2}-2 a b +b^{2}\right )}+\frac {3 a^{2} \left (3 a -2 b \right )}{8 \left (a^{2}-2 a b +b^{2}\right )}}{\left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a -2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +4 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+a \right )^{2}}+\frac {\left (15 a^{2}-20 a b +8 b^{2}\right ) \arctan \left (\frac {2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 a +4 b}{4 \sqrt {a b -b^{2}}}\right )}{16 \left (a^{2}-2 a b +b^{2}\right ) \sqrt {a b -b^{2}}}\right )}{a^{3}}}{d}\) | \(308\) |
| risch | \(-\frac {{\mathrm e}^{d x +c} b \left (7 \,{\mathrm e}^{6 d x +6 c} a b -4 b^{2} {\mathrm e}^{6 d x +6 c}+36 \,{\mathrm e}^{4 d x +4 c} a^{2}-31 \,{\mathrm e}^{4 d x +4 c} a b +4 b^{2} {\mathrm e}^{4 d x +4 c}+36 \,{\mathrm e}^{2 d x +2 c} a^{2}-31 \,{\mathrm e}^{2 d x +2 c} b a +4 b^{2} {\mathrm e}^{2 d x +2 c}+7 a b -4 b^{2}\right )}{4 d \,a^{2} \left (a -b \right )^{2} \left (b \,{\mathrm e}^{4 d x +4 c}+4 a \,{\mathrm e}^{2 d x +2 c}-2 b \,{\mathrm e}^{2 d x +2 c}+b \right )^{2}}-\frac {\ln \left ({\mathrm e}^{d x +c}+1\right )}{a^{3} d}+\frac {\ln \left ({\mathrm e}^{d x +c}-1\right )}{a^{3} d}+\frac {15 \sqrt {-b \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {-b \left (a -b \right )}\, {\mathrm e}^{d x +c}}{b}+1\right )}{16 \left (a -b \right )^{3} d a}-\frac {5 \sqrt {-b \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {-b \left (a -b \right )}\, {\mathrm e}^{d x +c}}{b}+1\right ) b}{4 \left (a -b \right )^{3} d \,a^{2}}+\frac {\sqrt {-b \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {-b \left (a -b \right )}\, {\mathrm e}^{d x +c}}{b}+1\right ) b^{2}}{2 \left (a -b \right )^{3} d \,a^{3}}-\frac {15 \sqrt {-b \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {-b \left (a -b \right )}\, {\mathrm e}^{d x +c}}{b}+1\right )}{16 \left (a -b \right )^{3} d a}+\frac {5 \sqrt {-b \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {-b \left (a -b \right )}\, {\mathrm e}^{d x +c}}{b}+1\right ) b}{4 \left (a -b \right )^{3} d \,a^{2}}-\frac {\sqrt {-b \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {-b \left (a -b \right )}\, {\mathrm e}^{d x +c}}{b}+1\right ) b^{2}}{2 \left (a -b \right )^{3} d \,a^{3}}\) | \(571\) |
1/d*(1/a^3*ln(tanh(1/2*d*x+1/2*c))-2*b/a^3*((-1/8*(9*a^2-28*a*b+16*b^2)*a/ (a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^6+3/8*(9*a^3-30*a^2*b+40*a*b^2-16*b^3) /(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^4-1/8*a*(27*a^2-68*a*b+32*b^2)/(a^2-2 *a*b+b^2)*tanh(1/2*d*x+1/2*c)^2+3/8*a^2*(3*a-2*b)/(a^2-2*a*b+b^2))/(tanh(1 /2*d*x+1/2*c)^4*a-2*tanh(1/2*d*x+1/2*c)^2*a+4*b*tanh(1/2*d*x+1/2*c)^2+a)^2 +1/16*(15*a^2-20*a*b+8*b^2)/(a^2-2*a*b+b^2)/(a*b-b^2)^(1/2)*arctan(1/4*(2* tanh(1/2*d*x+1/2*c)^2*a-2*a+4*b)/(a*b-b^2)^(1/2))))
Leaf count of result is larger than twice the leaf count of optimal. 5242 vs. \(2 (152) = 304\).
Time = 0.42 (sec) , antiderivative size = 9815, normalized size of antiderivative = 59.13 \[ \int \frac {\text {csch}(c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^3} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {\text {csch}(c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^3} \, dx=\text {Timed out} \]
\[ \int \frac {\text {csch}(c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^3} \, dx=\int { \frac {\operatorname {csch}\left (d x + c\right )}{{\left (b \sinh \left (d x + c\right )^{2} + a\right )}^{3}} \,d x } \]
-1/4*((7*a*b^2*e^(7*c) - 4*b^3*e^(7*c))*e^(7*d*x) + (36*a^2*b*e^(5*c) - 31 *a*b^2*e^(5*c) + 4*b^3*e^(5*c))*e^(5*d*x) + (36*a^2*b*e^(3*c) - 31*a*b^2*e ^(3*c) + 4*b^3*e^(3*c))*e^(3*d*x) + (7*a*b^2*e^c - 4*b^3*e^c)*e^(d*x))/(a^ 4*b^2*d - 2*a^3*b^3*d + a^2*b^4*d + (a^4*b^2*d*e^(8*c) - 2*a^3*b^3*d*e^(8* c) + a^2*b^4*d*e^(8*c))*e^(8*d*x) + 4*(2*a^5*b*d*e^(6*c) - 5*a^4*b^2*d*e^( 6*c) + 4*a^3*b^3*d*e^(6*c) - a^2*b^4*d*e^(6*c))*e^(6*d*x) + 2*(8*a^6*d*e^( 4*c) - 24*a^5*b*d*e^(4*c) + 27*a^4*b^2*d*e^(4*c) - 14*a^3*b^3*d*e^(4*c) + 3*a^2*b^4*d*e^(4*c))*e^(4*d*x) + 4*(2*a^5*b*d*e^(2*c) - 5*a^4*b^2*d*e^(2*c ) + 4*a^3*b^3*d*e^(2*c) - a^2*b^4*d*e^(2*c))*e^(2*d*x)) - log((e^(d*x + c) + 1)*e^(-c))/(a^3*d) + log((e^(d*x + c) - 1)*e^(-c))/(a^3*d) - 2*integrat e(1/8*((15*a^2*b*e^(3*c) - 20*a*b^2*e^(3*c) + 8*b^3*e^(3*c))*e^(3*d*x) - ( 15*a^2*b*e^c - 20*a*b^2*e^c + 8*b^3*e^c)*e^(d*x))/(a^5*b - 2*a^4*b^2 + a^3 *b^3 + (a^5*b*e^(4*c) - 2*a^4*b^2*e^(4*c) + a^3*b^3*e^(4*c))*e^(4*d*x) + 2 *(2*a^6*e^(2*c) - 5*a^5*b*e^(2*c) + 4*a^4*b^2*e^(2*c) - a^3*b^3*e^(2*c))*e ^(2*d*x)), x)
\[ \int \frac {\text {csch}(c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^3} \, dx=\int { \frac {\operatorname {csch}\left (d x + c\right )}{{\left (b \sinh \left (d x + c\right )^{2} + a\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {\text {csch}(c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^3} \, dx=\int \frac {1}{\mathrm {sinh}\left (c+d\,x\right )\,{\left (b\,{\mathrm {sinh}\left (c+d\,x\right )}^2+a\right )}^3} \,d x \]