Integrand size = 22, antiderivative size = 617 \[ \int \frac {\text {csch}(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^3} \, dx=-\frac {\left (5 \sqrt {a}-2 \sqrt {b}\right ) \sqrt [4]{b} \arctan \left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{64 a^{5/2} \left (\sqrt {a}-\sqrt {b}\right )^{5/2} d}-\frac {\sqrt [4]{b} \arctan \left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{8 a^{5/2} \left (\sqrt {a}-\sqrt {b}\right )^{3/2} d}-\frac {\sqrt [4]{b} \arctan \left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 a^3 \sqrt {\sqrt {a}-\sqrt {b}} d}-\frac {\text {arctanh}(\cosh (c+d x))}{a^3 d}+\frac {\sqrt [4]{b} \text {arctanh}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{8 a^{5/2} \left (\sqrt {a}+\sqrt {b}\right )^{3/2} d}+\frac {\sqrt [4]{b} \text {arctanh}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 a^3 \sqrt {\sqrt {a}+\sqrt {b}} d}+\frac {\left (5 \sqrt {a}+2 \sqrt {b}\right ) \sqrt [4]{b} \text {arctanh}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{64 a^{5/2} \left (\sqrt {a}+\sqrt {b}\right )^{5/2} d}-\frac {b \cosh (c+d x) \left (2-\cosh ^2(c+d x)\right )}{8 a (a-b) d \left (a-b+2 b \cosh ^2(c+d x)-b \cosh ^4(c+d x)\right )^2}-\frac {b \cosh (c+d x) \left (2-\cosh ^2(c+d x)\right )}{4 a^2 (a-b) d \left (a-b+2 b \cosh ^2(c+d x)-b \cosh ^4(c+d x)\right )}-\frac {b \cosh (c+d x) \left (11 a+b-(5 a+b) \cosh ^2(c+d x)\right )}{32 a^2 (a-b)^2 d \left (a-b+2 b \cosh ^2(c+d x)-b \cosh ^4(c+d x)\right )} \]
-arctanh(cosh(d*x+c))/a^3/d-1/8*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)^2-1/4*b*cosh(d*x+c)*(2-cosh(d*x+c) ^2)/a^2/(a-b)/d/(a-b+2*b*cosh(d*x+c)^2-b*cosh(d*x+c)^4)-1/32*b*cosh(d*x+c) *(11*a+b-(5*a+b)*cosh(d*x+c)^2)/a^2/(a-b)^2/d/(a-b+2*b*cosh(d*x+c)^2-b*cos h(d*x+c)^4)-1/64*b^(1/4)*arctan(b^(1/4)*cosh(d*x+c)/(a^(1/2)-b^(1/2))^(1/2 ))*(5*a^(1/2)-2*b^(1/2))/a^(5/2)/d/(a^(1/2)-b^(1/2))^(5/2)-1/8*b^(1/4)*arc tan(b^(1/4)*cosh(d*x+c)/(a^(1/2)-b^(1/2))^(1/2))/a^(5/2)/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^(5/2)/d/(a^(1/2)+b^(1/2))^(3/2)+1/64*b^(1/4)*arctanh(b^(1/4)*cosh(d*x+c) /(a^(1/2)+b^(1/2))^(1/2))*(5*a^(1/2)+2*b^(1/2))/a^(5/2)/d/(a^(1/2)+b^(1/2) )^(5/2)-1/2*b^(1/4)*arctan(b^(1/4)*cosh(d*x+c)/(a^(1/2)-b^(1/2))^(1/2))/a^ 3/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^3/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 = 11.78 (sec) , antiderivative size = 1202, normalized size of antiderivative = 1.95 \[ \int \frac {\text {csch}(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^3} \, dx =\text {Too large to display} \]
((32*a*b*Cosh[c + d*x]*(-41*a + 23*b + (13*a - 7*b)*Cosh[2*(c + d*x)]))/(( a - b)^2*(8*a - 3*b + 4*b*Cosh[2*(c + d*x)] - b*Cosh[4*(c + d*x)])) + (512 *a^2*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)])^2) - 256*Log[Cosh[(c + d*x)/2]] + 256*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 & , (-45*a^2*c + 71*a*b*c - 32*b^2*c - 45*a^2*d*x + 71*a*b*d*x - 32*b^2*d*x - 90*a^2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/ 2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1] + 142*a*b*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1 ] - 64*b^2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]* #1 - Sinh[(c + d*x)/2]*#1] + 199*a^2*c*#1^2 - 253*a*b*c*#1^2 + 96*b^2*c*#1 ^2 + 199*a^2*d*x*#1^2 - 253*a*b*d*x*#1^2 + 96*b^2*d*x*#1^2 + 398*a^2*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 - 506*a*b*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cos h[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^2 + 192*b^2*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 - 199*a^2*c*#1^4 + 253*a*b*c*#1^4 - 96*b^2*c*#1^4 - 199*a^2*d*x*#1^4 + 253*a*b*d*x*#1^4 - 96*b^2*d*x*#1^4 - 398*a^2*Log[-Cosh[(c + d*x)/2] - Sin h[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^4 + 506*a *b*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - ...
Time = 0.99 (sec) , antiderivative size = 591, 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.
| \(\displaystyle \int \frac {\text {csch}(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {i}{\sin (i c+i d x) \left (a-b \sin (i c+i d x)^4\right )^3}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \int \frac {1}{\sin (i c+i d x) \left (a-b \sin (i c+i d x)^4\right )^3}dx\) |
| \(\Big \downarrow \) 3694 |
| \(\displaystyle -\frac {\int \frac {1}{\left (1-\cosh ^2(c+d x)\right ) \left (-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)+a-b\right )^3}d\cosh (c+d x)}{d}\) |
| \(\Big \downarrow \) 1567 |
| \(\displaystyle -\frac {\int \left (\frac {b-b \cosh ^2(c+d x)}{a^3 \left (-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)+a-b\right )}+\frac {b-b \cosh ^2(c+d x)}{a^2 \left (-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)+a-b\right )^2}+\frac {b-b \cosh ^2(c+d x)}{a \left (-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)+a-b\right )^3}-\frac {1}{a^3 \left (\cosh ^2(c+d x)-1\right )}\right )d\cosh (c+d x)}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\frac {\sqrt [4]{b} \arctan \left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{8 a^{5/2} \left (\sqrt {a}-\sqrt {b}\right )^{3/2}}+\frac {\sqrt [4]{b} \left (5 \sqrt {a}-2 \sqrt {b}\right ) \arctan \left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{64 a^{5/2} \left (\sqrt {a}-\sqrt {b}\right )^{5/2}}-\frac {\sqrt [4]{b} \left (5 \sqrt {a}+2 \sqrt {b}\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{64 a^{5/2} \left (\sqrt {a}+\sqrt {b}\right )^{5/2}}-\frac {\sqrt [4]{b} \text {arctanh}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{8 a^{5/2} \left (\sqrt {a}+\sqrt {b}\right )^{3/2}}+\frac {\sqrt [4]{b} \arctan \left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 a^3 \sqrt {\sqrt {a}-\sqrt {b}}}-\frac {\sqrt [4]{b} \text {arctanh}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 a^3 \sqrt {\sqrt {a}+\sqrt {b}}}+\frac {\text {arctanh}(\cosh (c+d x))}{a^3}+\frac {b \cosh (c+d x) \left (2-\cosh ^2(c+d x)\right )}{4 a^2 (a-b) \left (a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right )}+\frac {b \cosh (c+d x) \left (-\left ((5 a+b) \cosh ^2(c+d x)\right )+11 a+b\right )}{32 a^2 (a-b)^2 \left (a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right )}+\frac {b \cosh (c+d x) \left (2-\cosh ^2(c+d x)\right )}{8 a (a-b) \left (a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right )^2}}{d}\) |
-((((5*Sqrt[a] - 2*Sqrt[b])*b^(1/4)*ArcTan[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sq rt[a] - Sqrt[b]]])/(64*a^(5/2)*(Sqrt[a] - Sqrt[b])^(5/2)) + (b^(1/4)*ArcTa n[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(8*a^(5/2)*(Sqrt[a] - Sqrt[b])^(3/2)) + (b^(1/4)*ArcTan[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] - S qrt[b]]])/(2*a^3*Sqrt[Sqrt[a] - Sqrt[b]]) + ArcTanh[Cosh[c + d*x]]/a^3 - ( b^(1/4)*ArcTanh[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]])/(8*a^(5/ 2)*(Sqrt[a] + Sqrt[b])^(3/2)) - (b^(1/4)*ArcTanh[(b^(1/4)*Cosh[c + d*x])/S qrt[Sqrt[a] + Sqrt[b]]])/(2*a^3*Sqrt[Sqrt[a] + Sqrt[b]]) - ((5*Sqrt[a] + 2 *Sqrt[b])*b^(1/4)*ArcTanh[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]] )/(64*a^(5/2)*(Sqrt[a] + Sqrt[b])^(5/2)) + (b*Cosh[c + d*x]*(2 - Cosh[c + d*x]^2))/(8*a*(a - b)*(a - b + 2*b*Cosh[c + d*x]^2 - b*Cosh[c + d*x]^4)^2) + (b*Cosh[c + d*x]*(2 - Cosh[c + d*x]^2))/(4*a^2*(a - b)*(a - b + 2*b*Cos h[c + d*x]^2 - b*Cosh[c + d*x]^4)) + (b*Cosh[c + d*x]*(11*a + b - (5*a + b )*Cosh[c + d*x]^2))/(32*a^2*(a - b)^2*(a - b + 2*b*Cosh[c + d*x]^2 - b*Cos h[c + d*x]^4)))/d)
3.3.58.3.1 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 = 7.80 (sec) , antiderivative size = 647, normalized size of antiderivative = 1.05
| method | result | size |
| derivativedivides | \(\frac {\frac {8 b \left (\frac {-\frac {a^{2} \left (8 a -5 b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}}{64 \left (a^{2}-2 a b +b^{2}\right )}+\frac {a \left (5 a^{2}+86 a b -64 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{64 a^{2}-128 a b +64 b^{2}}+\frac {a \left (104 a^{2}-327 a b +208 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{64 a^{2}-128 a b +64 b^{2}}-\frac {3 \left (105 a^{3}-358 a^{2} b +576 a \,b^{2}-256 b^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{64 \left (a^{2}-2 a b +b^{2}\right )}+\frac {a \left (400 a^{2}-1161 a b +560 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{64 a^{2}-128 a b +64 b^{2}}-\frac {a \left (257 a^{2}-370 a b +128 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{64 \left (a^{2}-2 a b +b^{2}\right )}+\frac {\left (80 a -53 b \right ) a^{2} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{64 a^{2}-128 a b +64 b^{2}}-\frac {3 a^{2} \left (3 a -2 b \right )}{64 \left (a^{2}-2 a b +b^{2}\right )}}{\left (\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 \right )^{2}}+\frac {a \left (\frac {\left (-45 \sqrt {a b}\, a^{2}+71 a b \sqrt {a b}-32 \sqrt {a b}\, b^{2}-16 a^{2} b +10 a \,b^{2}\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 )}{8 a b \sqrt {-a b +\sqrt {a b}\, a}}-\frac {\left (45 \sqrt {a b}\, a^{2}-71 a b \sqrt {a b}+32 \sqrt {a b}\, b^{2}-16 a^{2} b +10 a \,b^{2}\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 )}{8 a b \sqrt {-a b -\sqrt {a b}\, a}}\right )}{64 a^{2}-128 a b +64 b^{2}}\right )}{a^{3}}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{3}}}{d}\) | \(647\) |
| default | \(\frac {\frac {8 b \left (\frac {-\frac {a^{2} \left (8 a -5 b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}}{64 \left (a^{2}-2 a b +b^{2}\right )}+\frac {a \left (5 a^{2}+86 a b -64 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{64 a^{2}-128 a b +64 b^{2}}+\frac {a \left (104 a^{2}-327 a b +208 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{64 a^{2}-128 a b +64 b^{2}}-\frac {3 \left (105 a^{3}-358 a^{2} b +576 a \,b^{2}-256 b^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{64 \left (a^{2}-2 a b +b^{2}\right )}+\frac {a \left (400 a^{2}-1161 a b +560 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{64 a^{2}-128 a b +64 b^{2}}-\frac {a \left (257 a^{2}-370 a b +128 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{64 \left (a^{2}-2 a b +b^{2}\right )}+\frac {\left (80 a -53 b \right ) a^{2} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{64 a^{2}-128 a b +64 b^{2}}-\frac {3 a^{2} \left (3 a -2 b \right )}{64 \left (a^{2}-2 a b +b^{2}\right )}}{\left (\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 \right )^{2}}+\frac {a \left (\frac {\left (-45 \sqrt {a b}\, a^{2}+71 a b \sqrt {a b}-32 \sqrt {a b}\, b^{2}-16 a^{2} b +10 a \,b^{2}\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 )}{8 a b \sqrt {-a b +\sqrt {a b}\, a}}-\frac {\left (45 \sqrt {a b}\, a^{2}-71 a b \sqrt {a b}+32 \sqrt {a b}\, b^{2}-16 a^{2} b +10 a \,b^{2}\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 )}{8 a b \sqrt {-a b -\sqrt {a b}\, a}}\right )}{64 a^{2}-128 a b +64 b^{2}}\right )}{a^{3}}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{3}}}{d}\) | \(647\) |
| risch | \(\text {Expression too large to display}\) | \(1496\) |
1/d*(8*b/a^3*((-1/64*a^2*(8*a-5*b)/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^14+ 1/64*a*(5*a^2+86*a*b-64*b^2)/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^12+1/64*a *(104*a^2-327*a*b+208*b^2)/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^10-3/64*(10 5*a^3-358*a^2*b+576*a*b^2-256*b^3)/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^8+1 /64*a*(400*a^2-1161*a*b+560*b^2)/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^6-1/6 4*a*(257*a^2-370*a*b+128*b^2)/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^4+1/64*( 80*a-53*b)*a^2/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^2-3/64*a^2*(3*a-2*b)/(a ^2-2*a*b+b^2))/(tanh(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-16*b*tanh(1/2*d*x+1/2*c)^4-4*tanh(1/2*d*x+1/2*c)^2*a+a)^ 2+1/64/(a^2-2*a*b+b^2)*a*(1/8*(-45*(a*b)^(1/2)*a^2+71*a*b*(a*b)^(1/2)-32*( a*b)^(1/2)*b^2-16*a^2*b+10*a*b^2)/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) )-1/8*(45*(a*b)^(1/2)*a^2-71*a*b*(a*b)^(1/2)+32*(a*b)^(1/2)*b^2-16*a^2*b+1 0*a*b^2)/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^3*ln(tanh(1/2*d* x+1/2*c)))
Leaf count of result is larger than twice the leaf count of optimal. 28586 vs. \(2 (481) = 962\).
Time = 1.33 (sec) , antiderivative size = 28586, normalized size of antiderivative = 46.33 \[ \int \frac {\text {csch}(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^3} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {\text {csch}(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^3} \, dx=\text {Timed out} \]
\[ \int \frac {\text {csch}(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^3} \, dx=\int { -\frac {\operatorname {csch}\left (d x + c\right )}{{\left (b \sinh \left (d x + c\right )^{4} - a\right )}^{3}} \,d x } \]
-1/16*((13*a*b^2*e^(15*c) - 7*b^3*e^(15*c))*e^(15*d*x) - (121*a*b^2*e^(13* c) - 67*b^3*e^(13*c))*e^(13*d*x) - (272*a^2*b*e^(11*c) - 461*a*b^2*e^(11*c ) + 159*b^3*e^(11*c))*e^(11*d*x) + (1424*a^2*b*e^(9*c) - 1121*a*b^2*e^(9*c ) + 99*b^3*e^(9*c))*e^(9*d*x) + (1424*a^2*b*e^(7*c) - 1121*a*b^2*e^(7*c) + 99*b^3*e^(7*c))*e^(7*d*x) - (272*a^2*b*e^(5*c) - 461*a*b^2*e^(5*c) + 159* b^3*e^(5*c))*e^(5*d*x) - (121*a*b^2*e^(3*c) - 67*b^3*e^(3*c))*e^(3*d*x) + (13*a*b^2*e^c - 7*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^(16*c) - 2*a^3*b^3*d*e^(16*c) + a^2*b^4*d*e^(16*c))*e^(16*d* x) - 8*(a^4*b^2*d*e^(14*c) - 2*a^3*b^3*d*e^(14*c) + a^2*b^4*d*e^(14*c))*e^ (14*d*x) - 4*(8*a^5*b*d*e^(12*c) - 23*a^4*b^2*d*e^(12*c) + 22*a^3*b^3*d*e^ (12*c) - 7*a^2*b^4*d*e^(12*c))*e^(12*d*x) + 8*(16*a^5*b*d*e^(10*c) - 39*a^ 4*b^2*d*e^(10*c) + 30*a^3*b^3*d*e^(10*c) - 7*a^2*b^4*d*e^(10*c))*e^(10*d*x ) + 2*(128*a^6*d*e^(8*c) - 352*a^5*b*d*e^(8*c) + 355*a^4*b^2*d*e^(8*c) - 1 66*a^3*b^3*d*e^(8*c) + 35*a^2*b^4*d*e^(8*c))*e^(8*d*x) + 8*(16*a^5*b*d*e^( 6*c) - 39*a^4*b^2*d*e^(6*c) + 30*a^3*b^3*d*e^(6*c) - 7*a^2*b^4*d*e^(6*c))* e^(6*d*x) - 4*(8*a^5*b*d*e^(4*c) - 23*a^4*b^2*d*e^(4*c) + 22*a^3*b^3*d*e^( 4*c) - 7*a^2*b^4*d*e^(4*c))*e^(4*d*x) - 8*(a^4*b^2*d*e^(2*c) - 2*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*integrate(1/32*((45*a^2 *b*e^(7*c) - 71*a*b^2*e^(7*c) + 32*b^3*e^(7*c))*e^(7*d*x) - (199*a^2*b*...
\[ \int \frac {\text {csch}(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^3} \, dx=\int { -\frac {\operatorname {csch}\left (d x + c\right )}{{\left (b \sinh \left (d x + c\right )^{4} - a\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {\text {csch}(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^3} \, dx=\int \frac {1}{\mathrm {sinh}\left (c+d\,x\right )\,{\left (a-b\,{\mathrm {sinh}\left (c+d\,x\right )}^4\right )}^3} \,d x \]