Integrand size = 22, antiderivative size = 325 \[ \int \frac {\text {csch}(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^2} \, dx=-\frac {\sqrt [4]{b} \arctan \left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{8 a^{3/2} \left (\sqrt {a}-\sqrt {b}\right )^{3/2} d}-\frac {\sqrt [4]{b} \arctan \left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 a^2 \sqrt {\sqrt {a}-\sqrt {b}} d}-\frac {\text {arctanh}(\cosh (c+d x))}{a^2 d}+\frac {\sqrt [4]{b} \text {arctanh}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{8 a^{3/2} \left (\sqrt {a}+\sqrt {b}\right )^{3/2} d}+\frac {\sqrt [4]{b} \text {arctanh}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 a^2 \sqrt {\sqrt {a}+\sqrt {b}} d}-\frac {b \cosh (c+d x) \left (2-\cosh ^2(c+d x)\right )}{4 a (a-b) d \left (a-b+2 b \cosh ^2(c+d x)-b \cosh ^4(c+d x)\right )} \]
-arctanh(cosh(d*x+c))/a^2/d-1/4*b*cosh(d*x+c)*(2-cosh(d*x+c)^2)/a/(a-b)/d/ (a-b+2*b*cosh(d*x+c)^2-b*cosh(d*x+c)^4)-1/8*b^(1/4)*arctan(b^(1/4)*cosh(d* x+c)/(a^(1/2)-b^(1/2))^(1/2))/a^(3/2)/d/(a^(1/2)-b^(1/2))^(3/2)+1/8*b^(1/4 )*arctanh(b^(1/4)*cosh(d*x+c)/(a^(1/2)+b^(1/2))^(1/2))/a^(3/2)/d/(a^(1/2)+ b^(1/2))^(3/2)-1/2*b^(1/4)*arctan(b^(1/4)*cosh(d*x+c)/(a^(1/2)-b^(1/2))^(1 /2))/a^2/d/(a^(1/2)-b^(1/2))^(1/2)+1/2*b^(1/4)*arctanh(b^(1/4)*cosh(d*x+c) /(a^(1/2)+b^(1/2))^(1/2))/a^2/d/(a^(1/2)+b^(1/2))^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 2.39 (sec) , antiderivative size = 774, normalized size of antiderivative = 2.38 \[ \int \frac {\text {csch}(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^2} \, dx=\frac {\frac {16 a b (-5 \cosh (c+d x)+\cosh (3 (c+d x)))}{(a-b) (8 a-3 b+4 b \cosh (2 (c+d x))-b \cosh (4 (c+d x)))}-32 \log \left (\cosh \left (\frac {1}{2} (c+d x)\right )\right )+32 \log \left (\sinh \left (\frac {1}{2} (c+d x)\right )\right )-\frac {b \text {RootSum}\left [b-4 b \text {$\#$1}^2-16 a \text {$\#$1}^4+6 b \text {$\#$1}^4-4 b \text {$\#$1}^6+b \text {$\#$1}^8\&,\frac {-5 a c+4 b c-5 a d x+4 b d x-10 a \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right )+8 b \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right )+19 a c \text {$\#$1}^2-12 b c \text {$\#$1}^2+19 a d x \text {$\#$1}^2-12 b d x \text {$\#$1}^2+38 a \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^2-24 b \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^2-19 a c \text {$\#$1}^4+12 b c \text {$\#$1}^4-19 a d x \text {$\#$1}^4+12 b d x \text {$\#$1}^4-38 a \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^4+24 b \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^4+5 a c \text {$\#$1}^6-4 b c \text {$\#$1}^6+5 a d x \text {$\#$1}^6-4 b d x \text {$\#$1}^6+10 a \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^6-8 b \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^6}{-b \text {$\#$1}-8 a \text {$\#$1}^3+3 b \text {$\#$1}^3-3 b \text {$\#$1}^5+b \text {$\#$1}^7}\&\right ]}{a-b}}{32 a^2 d} \]
((16*a*b*(-5*Cosh[c + d*x] + Cosh[3*(c + d*x)]))/((a - b)*(8*a - 3*b + 4*b *Cosh[2*(c + d*x)] - b*Cosh[4*(c + d*x)])) - 32*Log[Cosh[(c + d*x)/2]] + 3 2*Log[Sinh[(c + d*x)/2]] - (b*RootSum[b - 4*b*#1^2 - 16*a*#1^4 + 6*b*#1^4 - 4*b*#1^6 + b*#1^8 & , (-5*a*c + 4*b*c - 5*a*d*x + 4*b*d*x - 10*a*Log[-Co sh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x )/2]*#1] + 8*b*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x) /2]*#1 - Sinh[(c + d*x)/2]*#1] + 19*a*c*#1^2 - 12*b*c*#1^2 + 19*a*d*x*#1^2 - 12*b*d*x*#1^2 + 38*a*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[ (c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^2 - 24*b*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^2 - 19*a*c*#1^4 + 12*b*c*#1^4 - 19*a*d*x*#1^4 + 12*b*d*x*#1^4 - 38*a*Log[-Cos h[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x) /2]*#1]*#1^4 + 24*b*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^4 + 5*a*c*#1^6 - 4*b*c*#1^6 + 5*a*d *x*#1^6 - 4*b*d*x*#1^6 + 10*a*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^6 - 8*b*Log[-Cosh[(c + d* x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*# 1^6)/(-(b*#1) - 8*a*#1^3 + 3*b*#1^3 - 3*b*#1^5 + b*#1^7) & ])/(a - b))/(32 *a^2*d)
Time = 0.57 (sec) , antiderivative size = 311, normalized size of antiderivative = 0.96, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {3042, 26, 3694, 1567, 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.
| \(\displaystyle \int \frac {\text {csch}(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {i}{\sin (i c+i d x) \left (a-b \sin (i c+i d x)^4\right )^2}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \int \frac {1}{\sin (i c+i d x) \left (a-b \sin (i c+i d x)^4\right )^2}dx\) |
| \(\Big \downarrow \) 3694 |
| \(\displaystyle -\frac {\int \frac {1}{\left (1-\cosh ^2(c+d x)\right ) \left (-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)+a-b\right )^2}d\cosh (c+d x)}{d}\) |
| \(\Big \downarrow \) 1567 |
| \(\displaystyle -\frac {\int \left (\frac {b-b \cosh ^2(c+d x)}{a^2 \left (-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)+a-b\right )}+\frac {b-b \cosh ^2(c+d x)}{a \left (-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)+a-b\right )^2}-\frac {1}{a^2 \left (\cosh ^2(c+d x)-1\right )}\right )d\cosh (c+d x)}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\frac {\sqrt [4]{b} \arctan \left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{8 a^{3/2} \left (\sqrt {a}-\sqrt {b}\right )^{3/2}}-\frac {\sqrt [4]{b} \text {arctanh}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{8 a^{3/2} \left (\sqrt {a}+\sqrt {b}\right )^{3/2}}+\frac {\sqrt [4]{b} \arctan \left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 a^2 \sqrt {\sqrt {a}-\sqrt {b}}}-\frac {\sqrt [4]{b} \text {arctanh}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 a^2 \sqrt {\sqrt {a}+\sqrt {b}}}+\frac {\text {arctanh}(\cosh (c+d x))}{a^2}+\frac {b \cosh (c+d x) \left (2-\cosh ^2(c+d x)\right )}{4 a (a-b) \left (a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right )}}{d}\) |
-(((b^(1/4)*ArcTan[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(8*a^ (3/2)*(Sqrt[a] - Sqrt[b])^(3/2)) + (b^(1/4)*ArcTan[(b^(1/4)*Cosh[c + d*x]) /Sqrt[Sqrt[a] - Sqrt[b]]])/(2*a^2*Sqrt[Sqrt[a] - Sqrt[b]]) + ArcTanh[Cosh[ c + d*x]]/a^2 - (b^(1/4)*ArcTanh[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] + Sq rt[b]]])/(8*a^(3/2)*(Sqrt[a] + Sqrt[b])^(3/2)) - (b^(1/4)*ArcTanh[(b^(1/4) *Cosh[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]])/(2*a^2*Sqrt[Sqrt[a] + Sqrt[b]]) + (b*Cosh[c + d*x]*(2 - Cosh[c + d*x]^2))/(4*a*(a - b)*(a - b + 2*b*Cosh[c + d*x]^2 - b*Cosh[c + d*x]^4)))/d)
3.3.46.3.1 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[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x _Symbol] :> Int[ExpandIntegrand[(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b^2 - 4*a*c, 0] && ((IntegerQ[p] && IntegerQ[q]) || IGtQ[p, 0] || IGtQ[q, 0])
Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^4)^ (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 - 2*b*ff^2*x^2 + b*ff^4*x^4)^p, x], x, Cos[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]
Time = 2.26 (sec) , antiderivative size = 364, normalized size of antiderivative = 1.12
| method | result | size |
| derivativedivides | \(\frac {\frac {16 b \left (\frac {-\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{32 \left (a -b \right )}-\frac {\left (3 a -8 b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{32 \left (a -b \right )}+\frac {5 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{32 \left (a -b \right )}-\frac {a}{32 \left (a -b \right )}}{\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{8} a -4 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} a +6 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a -16 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-4 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +a}+\frac {a \left (\frac {\left (-5 \sqrt {a b}\, a +4 \sqrt {a b}\, b -a b \right ) \arctan \left (\frac {2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +4 \sqrt {a b}-2 a}{4 \sqrt {-a b +\sqrt {a b}\, a}}\right )}{4 a b \sqrt {-a b +\sqrt {a b}\, a}}-\frac {\left (5 \sqrt {a b}\, a -4 \sqrt {a b}\, b -a b \right ) \arctan \left (\frac {-2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +4 \sqrt {a b}+2 a}{4 \sqrt {-a b -\sqrt {a b}\, a}}\right )}{4 a b \sqrt {-a b -\sqrt {a b}\, a}}\right )}{32 a -32 b}\right )}{a^{2}}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{2}}}{d}\) | \(364\) |
| default | \(\frac {\frac {16 b \left (\frac {-\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{32 \left (a -b \right )}-\frac {\left (3 a -8 b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{32 \left (a -b \right )}+\frac {5 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{32 \left (a -b \right )}-\frac {a}{32 \left (a -b \right )}}{\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{8} a -4 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} a +6 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a -16 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-4 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +a}+\frac {a \left (\frac {\left (-5 \sqrt {a b}\, a +4 \sqrt {a b}\, b -a b \right ) \arctan \left (\frac {2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +4 \sqrt {a b}-2 a}{4 \sqrt {-a b +\sqrt {a b}\, a}}\right )}{4 a b \sqrt {-a b +\sqrt {a b}\, a}}-\frac {\left (5 \sqrt {a b}\, a -4 \sqrt {a b}\, b -a b \right ) \arctan \left (\frac {-2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +4 \sqrt {a b}+2 a}{4 \sqrt {-a b -\sqrt {a b}\, a}}\right )}{4 a b \sqrt {-a b -\sqrt {a b}\, a}}\right )}{32 a -32 b}\right )}{a^{2}}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{2}}}{d}\) | \(364\) |
| risch | \(\frac {b \,{\mathrm e}^{d x +c} \left ({\mathrm e}^{6 d x +6 c}-5 \,{\mathrm e}^{4 d x +4 c}-5 \,{\mathrm e}^{2 d x +2 c}+1\right )}{2 a \left (a -b \right ) d \left (-b \,{\mathrm e}^{8 d x +8 c}+4 b \,{\mathrm e}^{6 d x +6 c}+16 \,{\mathrm e}^{4 d x +4 c} a -6 b \,{\mathrm e}^{4 d x +4 c}+4 b \,{\mathrm e}^{2 d x +2 c}-b \right )}+\frac {\ln \left ({\mathrm e}^{d x +c}-1\right )}{a^{2} d}-\frac {\ln \left ({\mathrm e}^{d x +c}+1\right )}{a^{2} d}+2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (1048576 a^{11} d^{4}-3145728 a^{10} b \,d^{4}+3145728 a^{9} b^{2} d^{4}-1048576 a^{8} b^{3} d^{4}\right ) \textit {\_Z}^{4}+\left (71680 a^{6} b \,d^{2}-96256 a^{5} b^{2} d^{2}+32768 a^{4} b^{3} d^{2}\right ) \textit {\_Z}^{2}-625 a^{2} b +800 a \,b^{2}-256 b^{3}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 d x +2 c}+\left (\left (\frac {327680 a^{10} d^{3}}{625 a^{3} b -1125 a^{2} b^{2}+664 a \,b^{3}-128 b^{4}}-\frac {1179648 a^{9} d^{3} b}{625 a^{3} b -1125 a^{2} b^{2}+664 a \,b^{3}-128 b^{4}}+\frac {1572864 a^{8} d^{3} b^{2}}{625 a^{3} b -1125 a^{2} b^{2}+664 a \,b^{3}-128 b^{4}}-\frac {917504 a^{7} d^{3} b^{3}}{625 a^{3} b -1125 a^{2} b^{2}+664 a \,b^{3}-128 b^{4}}+\frac {196608 a^{6} b^{4} d^{3}}{625 a^{3} b -1125 a^{2} b^{2}+664 a \,b^{3}-128 b^{4}}\right ) \textit {\_R}^{3}+\left (\frac {20800 a^{5} d b}{625 a^{3} b -1125 a^{2} b^{2}+664 a \,b^{3}-128 b^{4}}-\frac {39296 a^{4} d \,b^{2}}{625 a^{3} b -1125 a^{2} b^{2}+664 a \,b^{3}-128 b^{4}}+\frac {24640 a^{3} b^{3} d}{625 a^{3} b -1125 a^{2} b^{2}+664 a \,b^{3}-128 b^{4}}-\frac {5120 a^{2} b^{4} d}{625 a^{3} b -1125 a^{2} b^{2}+664 a \,b^{3}-128 b^{4}}\right ) \textit {\_R} \right ) {\mathrm e}^{d x +c}+1\right )\right )\) | \(626\) |
1/d*(16*b/a^2*((-1/32*a/(a-b)*tanh(1/2*d*x+1/2*c)^6-1/32*(3*a-8*b)/(a-b)*t anh(1/2*d*x+1/2*c)^4+5/32*a/(a-b)*tanh(1/2*d*x+1/2*c)^2-1/32*a/(a-b))/(tan h(1/2*d*x+1/2*c)^8*a-4*tanh(1/2*d*x+1/2*c)^6*a+6*tanh(1/2*d*x+1/2*c)^4*a-1 6*b*tanh(1/2*d*x+1/2*c)^4-4*tanh(1/2*d*x+1/2*c)^2*a+a)+1/32/(a-b)*a*(1/4*( -5*(a*b)^(1/2)*a+4*(a*b)^(1/2)*b-a*b)/a/b/(-a*b+(a*b)^(1/2)*a)^(1/2)*arcta n(1/4*(2*tanh(1/2*d*x+1/2*c)^2*a+4*(a*b)^(1/2)-2*a)/(-a*b+(a*b)^(1/2)*a)^( 1/2))-1/4*(5*(a*b)^(1/2)*a-4*(a*b)^(1/2)*b-a*b)/a/b/(-a*b-(a*b)^(1/2)*a)^( 1/2)*arctan(1/4*(-2*tanh(1/2*d*x+1/2*c)^2*a+4*(a*b)^(1/2)+2*a)/(-a*b-(a*b) ^(1/2)*a)^(1/2))))+1/a^2*ln(tanh(1/2*d*x+1/2*c)))
Leaf count of result is larger than twice the leaf count of optimal. 7793 vs. \(2 (244) = 488\).
Time = 0.60 (sec) , antiderivative size = 7793, normalized size of antiderivative = 23.98 \[ \int \frac {\text {csch}(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^2} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {\text {csch}(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^2} \, dx=\text {Timed out} \]
\[ \int \frac {\text {csch}(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^2} \, dx=\int { \frac {\operatorname {csch}\left (d x + c\right )}{{\left (b \sinh \left (d x + c\right )^{4} - a\right )}^{2}} \,d x } \]
-1/2*(b*e^(7*d*x + 7*c) - 5*b*e^(5*d*x + 5*c) - 5*b*e^(3*d*x + 3*c) + b*e^ (d*x + c))/(a^2*b*d - a*b^2*d + (a^2*b*d*e^(8*c) - a*b^2*d*e^(8*c))*e^(8*d *x) - 4*(a^2*b*d*e^(6*c) - a*b^2*d*e^(6*c))*e^(6*d*x) - 2*(8*a^3*d*e^(4*c) - 11*a^2*b*d*e^(4*c) + 3*a*b^2*d*e^(4*c))*e^(4*d*x) - 4*(a^2*b*d*e^(2*c) - a*b^2*d*e^(2*c))*e^(2*d*x)) - log((e^(d*x + c) + 1)*e^(-c))/(a^2*d) + lo g((e^(d*x + c) - 1)*e^(-c))/(a^2*d) - 2*integrate(1/4*((5*a*b*e^(7*c) - 4* b^2*e^(7*c))*e^(7*d*x) - (19*a*b*e^(5*c) - 12*b^2*e^(5*c))*e^(5*d*x) + (19 *a*b*e^(3*c) - 12*b^2*e^(3*c))*e^(3*d*x) - (5*a*b*e^c - 4*b^2*e^c)*e^(d*x) )/(a^3*b - a^2*b^2 + (a^3*b*e^(8*c) - a^2*b^2*e^(8*c))*e^(8*d*x) - 4*(a^3* b*e^(6*c) - a^2*b^2*e^(6*c))*e^(6*d*x) - 2*(8*a^4*e^(4*c) - 11*a^3*b*e^(4* c) + 3*a^2*b^2*e^(4*c))*e^(4*d*x) - 4*(a^3*b*e^(2*c) - a^2*b^2*e^(2*c))*e^ (2*d*x)), x)
\[ \int \frac {\text {csch}(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^2} \, dx=\int { \frac {\operatorname {csch}\left (d x + c\right )}{{\left (b \sinh \left (d x + c\right )^{4} - a\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {\text {csch}(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^2} \, dx=\int \frac {1}{\mathrm {sinh}\left (c+d\,x\right )\,{\left (a-b\,{\mathrm {sinh}\left (c+d\,x\right )}^4\right )}^2} \,d x \]