3.258 \(\int \frac{\text{csch}(c+d x)}{(a-b \sinh ^4(c+d x))^3} \, dx\)

Optimal. Leaf size=617 \[ -\frac{b \cosh (c+d x) \left (2-\cosh ^2(c+d x)\right )}{4 a^2 d (a-b) \left (a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right )}-\frac{b \cosh (c+d x) \left (-(5 a+b) \cosh ^2(c+d x)+11 a+b\right )}{32 a^2 d (a-b)^2 \left (a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right )}-\frac{\sqrt [4]{b} \tan ^{-1}\left (\frac{\sqrt [4]{b} \cosh (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right )}{2 a^3 d \sqrt{\sqrt{a}-\sqrt{b}}}-\frac{\sqrt [4]{b} \tan ^{-1}\left (\frac{\sqrt [4]{b} \cosh (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right )}{8 a^{5/2} d \left (\sqrt{a}-\sqrt{b}\right )^{3/2}}-\frac{\sqrt [4]{b} \left (5 \sqrt{a}-2 \sqrt{b}\right ) \tan ^{-1}\left (\frac{\sqrt [4]{b} \cosh (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right )}{64 a^{5/2} d \left (\sqrt{a}-\sqrt{b}\right )^{5/2}}+\frac{\sqrt [4]{b} \left (5 \sqrt{a}+2 \sqrt{b}\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{b} \cosh (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right )}{64 a^{5/2} d \left (\sqrt{a}+\sqrt{b}\right )^{5/2}}+\frac{\sqrt [4]{b} \tanh ^{-1}\left (\frac{\sqrt [4]{b} \cosh (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right )}{2 a^3 d \sqrt{\sqrt{a}+\sqrt{b}}}+\frac{\sqrt [4]{b} \tanh ^{-1}\left (\frac{\sqrt [4]{b} \cosh (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right )}{8 a^{5/2} d \left (\sqrt{a}+\sqrt{b}\right )^{3/2}}-\frac{\tanh ^{-1}(\cosh (c+d x))}{a^3 d}-\frac{b \cosh (c+d x) \left (2-\cosh ^2(c+d x)\right )}{8 a d (a-b) \left (a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right )^2} \]

[Out]

-((5*Sqrt[a] - 2*Sqrt[b])*b^(1/4)*ArcTan[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(64*a^(5/2)*(Sqrt[a
] - Sqrt[b])^(5/2)*d) - (b^(1/4)*ArcTan[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(8*a^(5/2)*(Sqrt[a]
- Sqrt[b])^(3/2)*d) - (b^(1/4)*ArcTan[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(2*a^3*Sqrt[Sqrt[a] -
Sqrt[b]]*d) - ArcTanh[Cosh[c + d*x]]/(a^3*d) + (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)*d) + (b^(1/4)*ArcTanh[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]
])/(2*a^3*Sqrt[Sqrt[a] + Sqrt[b]]*d) + ((5*Sqrt[a] + 2*Sqrt[b])*b^(1/4)*ArcTanh[(b^(1/4)*Cosh[c + d*x])/Sqrt[S
qrt[a] + Sqrt[b]]])/(64*a^(5/2)*(Sqrt[a] + Sqrt[b])^(5/2)*d) - (b*Cosh[c + d*x]*(2 - Cosh[c + d*x]^2))/(8*a*(a
- b)*d*(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)*d*(a - b + 2*b*Cosh[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*d*(a - b + 2*b*Cosh[c + d*x]^2 - b*Cosh[c + d*x]^4))

________________________________________________________________________________________

Rubi [A]  time = 0.81855, antiderivative size = 617, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318, Rules used = {3215, 1238, 207, 1178, 1166, 205, 208} \[ -\frac{b \cosh (c+d x) \left (2-\cosh ^2(c+d x)\right )}{4 a^2 d (a-b) \left (a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right )}-\frac{b \cosh (c+d x) \left (-(5 a+b) \cosh ^2(c+d x)+11 a+b\right )}{32 a^2 d (a-b)^2 \left (a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right )}-\frac{\sqrt [4]{b} \tan ^{-1}\left (\frac{\sqrt [4]{b} \cosh (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right )}{2 a^3 d \sqrt{\sqrt{a}-\sqrt{b}}}-\frac{\sqrt [4]{b} \tan ^{-1}\left (\frac{\sqrt [4]{b} \cosh (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right )}{8 a^{5/2} d \left (\sqrt{a}-\sqrt{b}\right )^{3/2}}-\frac{\sqrt [4]{b} \left (5 \sqrt{a}-2 \sqrt{b}\right ) \tan ^{-1}\left (\frac{\sqrt [4]{b} \cosh (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right )}{64 a^{5/2} d \left (\sqrt{a}-\sqrt{b}\right )^{5/2}}+\frac{\sqrt [4]{b} \left (5 \sqrt{a}+2 \sqrt{b}\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{b} \cosh (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right )}{64 a^{5/2} d \left (\sqrt{a}+\sqrt{b}\right )^{5/2}}+\frac{\sqrt [4]{b} \tanh ^{-1}\left (\frac{\sqrt [4]{b} \cosh (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right )}{2 a^3 d \sqrt{\sqrt{a}+\sqrt{b}}}+\frac{\sqrt [4]{b} \tanh ^{-1}\left (\frac{\sqrt [4]{b} \cosh (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right )}{8 a^{5/2} d \left (\sqrt{a}+\sqrt{b}\right )^{3/2}}-\frac{\tanh ^{-1}(\cosh (c+d x))}{a^3 d}-\frac{b \cosh (c+d x) \left (2-\cosh ^2(c+d x)\right )}{8 a d (a-b) \left (a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[Csch[c + d*x]/(a - b*Sinh[c + d*x]^4)^3,x]

[Out]

-((5*Sqrt[a] - 2*Sqrt[b])*b^(1/4)*ArcTan[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(64*a^(5/2)*(Sqrt[a
] - Sqrt[b])^(5/2)*d) - (b^(1/4)*ArcTan[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(8*a^(5/2)*(Sqrt[a]
- Sqrt[b])^(3/2)*d) - (b^(1/4)*ArcTan[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(2*a^3*Sqrt[Sqrt[a] -
Sqrt[b]]*d) - ArcTanh[Cosh[c + d*x]]/(a^3*d) + (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)*d) + (b^(1/4)*ArcTanh[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]
])/(2*a^3*Sqrt[Sqrt[a] + Sqrt[b]]*d) + ((5*Sqrt[a] + 2*Sqrt[b])*b^(1/4)*ArcTanh[(b^(1/4)*Cosh[c + d*x])/Sqrt[S
qrt[a] + Sqrt[b]]])/(64*a^(5/2)*(Sqrt[a] + Sqrt[b])^(5/2)*d) - (b*Cosh[c + d*x]*(2 - Cosh[c + d*x]^2))/(8*a*(a
- b)*d*(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)*d*(a - b + 2*b*Cosh[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*d*(a - b + 2*b*Cosh[c + d*x]^2 - b*Cosh[c + d*x]^4))

Rule 3215

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^4)^(p_.), x_Symbol] :> With[{ff = Free
Factors[Cos[e + f*x], x]}, -Dist[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]

Rule 1238

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] && ((Intege
rQ[p] && IntegerQ[q]) || IGtQ[p, 0] || IGtQ[q, 0])

Rule 207

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTanh[(Rt[b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rule 1178

Int[((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[(x*(a*b*e - d*(b^2 - 2*
a*c) - c*(b*d - 2*a*e)*x^2)*(a + b*x^2 + c*x^4)^(p + 1))/(2*a*(p + 1)*(b^2 - 4*a*c)), x] + Dist[1/(2*a*(p + 1)
*(b^2 - 4*a*c)), Int[Simp[(2*p + 3)*d*b^2 - a*b*e - 2*a*c*d*(4*p + 5) + (4*p + 7)*(d*b - 2*a*e)*c*x^2, x]*(a +
b*x^2 + c*x^4)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e
^2, 0] && LtQ[p, -1] && IntegerQ[2*p]

Rule 1166

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rubi steps

\begin{align*} \int \frac{\text{csch}(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^3} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{1}{\left (1-x^2\right ) \left (a-b+2 b x^2-b x^4\right )^3} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac{\operatorname{Subst}\left (\int \left (-\frac{1}{a^3 \left (-1+x^2\right )}+\frac{b-b x^2}{a \left (a-b+2 b x^2-b x^4\right )^3}+\frac{b-b x^2}{a^2 \left (a-b+2 b x^2-b x^4\right )^2}+\frac{b-b x^2}{a^3 \left (a-b+2 b x^2-b x^4\right )}\right ) \, dx,x,\cosh (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,\cosh (c+d x)\right )}{a^3 d}-\frac{\operatorname{Subst}\left (\int \frac{b-b x^2}{a-b+2 b x^2-b x^4} \, dx,x,\cosh (c+d x)\right )}{a^3 d}-\frac{\operatorname{Subst}\left (\int \frac{b-b x^2}{\left (a-b+2 b x^2-b x^4\right )^2} \, dx,x,\cosh (c+d x)\right )}{a^2 d}-\frac{\operatorname{Subst}\left (\int \frac{b-b x^2}{\left (a-b+2 b x^2-b x^4\right )^3} \, dx,x,\cosh (c+d x)\right )}{a d}\\ &=-\frac{\tanh ^{-1}(\cosh (c+d x))}{a^3 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{\operatorname{Subst}\left (\int \frac{-4 a b^2+2 a b^2 x^2}{a-b+2 b x^2-b x^4} \, dx,x,\cosh (c+d x)\right )}{8 a^3 (a-b) b d}+\frac{\operatorname{Subst}\left (\int \frac{-12 a b^2+10 a b^2 x^2}{\left (a-b+2 b x^2-b x^4\right )^2} \, dx,x,\cosh (c+d x)\right )}{16 a^2 (a-b) b d}+\frac{b \operatorname{Subst}\left (\int \frac{1}{-\sqrt{a} \sqrt{b}+b-b x^2} \, dx,x,\cosh (c+d x)\right )}{2 a^3 d}+\frac{b \operatorname{Subst}\left (\int \frac{1}{\sqrt{a} \sqrt{b}+b-b x^2} \, dx,x,\cosh (c+d x)\right )}{2 a^3 d}\\ &=-\frac{\sqrt [4]{b} \tan ^{-1}\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{\tanh ^{-1}(\cosh (c+d x))}{a^3 d}+\frac{\sqrt [4]{b} \tanh ^{-1}\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{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 )}-\frac{\operatorname{Subst}\left (\int \frac{4 a (13 a-b) b^3-4 a b^3 (5 a+b) x^2}{a-b+2 b x^2-b x^4} \, dx,x,\cosh (c+d x)\right )}{128 a^3 (a-b)^2 b^2 d}+\frac{b \operatorname{Subst}\left (\int \frac{1}{-\sqrt{a} \sqrt{b}+b-b x^2} \, dx,x,\cosh (c+d x)\right )}{8 a^{5/2} \left (\sqrt{a}-\sqrt{b}\right ) d}+\frac{b \operatorname{Subst}\left (\int \frac{1}{\sqrt{a} \sqrt{b}+b-b x^2} \, dx,x,\cosh (c+d x)\right )}{8 a^{5/2} \left (\sqrt{a}+\sqrt{b}\right ) d}\\ &=-\frac{\sqrt [4]{b} \tan ^{-1}\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} \tan ^{-1}\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{\tanh ^{-1}(\cosh (c+d x))}{a^3 d}+\frac{\sqrt [4]{b} \tanh ^{-1}\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} \tanh ^{-1}\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{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 )}+\frac{\left (\left (5 \sqrt{a}-2 \sqrt{b}\right ) b\right ) \operatorname{Subst}\left (\int \frac{1}{-\sqrt{a} \sqrt{b}+b-b x^2} \, dx,x,\cosh (c+d x)\right )}{64 a^{5/2} \left (\sqrt{a}-\sqrt{b}\right )^2 d}+\frac{\left (\left (5 \sqrt{a}+2 \sqrt{b}\right ) b\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a} \sqrt{b}+b-b x^2} \, dx,x,\cosh (c+d x)\right )}{64 a^{5/2} \left (\sqrt{a}+\sqrt{b}\right )^2 d}\\ &=-\frac{\left (5 \sqrt{a}-2 \sqrt{b}\right ) \sqrt [4]{b} \tan ^{-1}\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} \tan ^{-1}\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} \tan ^{-1}\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{\tanh ^{-1}(\cosh (c+d x))}{a^3 d}+\frac{\sqrt [4]{b} \tanh ^{-1}\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} \tanh ^{-1}\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} \tanh ^{-1}\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 )}\\ \end{align*}

Mathematica [C]  time = 5.75688, size = 1189, normalized size = 1.93 \[ \text{result too large to display} \]

Antiderivative was successfully verified.

[In]

Integrate[Csch[c + d*x]/(a - b*Sinh[c + d*x]^4)^3,x]

[Out]

((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[Tanh[(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*L
og[-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] - S
inh[(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] + Cosh[(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] - Sinh[(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 - Sinh[(c + d*x)/2]*#1]*#1^4 - 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]*#1^4 + 45*a^2*c*#1^6 - 71*a*b*c*#1^6 + 32*b^2*c*#1^6 + 45*a^2*d*x*#1^6 - 71
*a*b*d*x*#1^6 + 32*b^2*d*x*#1^6 + 90*a^2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - S
inh[(c + d*x)/2]*#1]*#1^6 - 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]*#1^6 + 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]*#1^6)/(-(b*#1) - 8*a*#1^3 + 3*b*#1^3 - 3*b*#1^5 + b*#1^7) & ])/(a - b)^2)/(256*a^3*d)

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Maple [B]  time = 0.138, size = 3159, normalized size = 5.1 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(d*x+c)/(a-b*sinh(d*x+c)^4)^3,x)

[Out]

-1/2/d*b^2/a^3/(a^2-2*a*b+b^2)/(-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))*(a*b)^(1/2)-1/2/d*b^2/a^3/(a^2-2*a*b+b^2)/(-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))*(a*b)^(1/2)+26/d/(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/a^2/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^10*b^3+70/d/(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/a^2/(a^2-2*a*b+b^2)*tan
h(1/2*d*x+1/2*c)^6*b^3-53/8/d/(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/a/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^2*b^2+5/8/d/(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/a/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^14*b^2-5/32/d/a^2/(a^2-2*a*b+b^2)/(-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))*b^2+5/32/d/a^2/
(a^2-2*a*b+b^2)/(-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))*b^2-45/64/d/(a^2-2*a*b+b^2)/a/(-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))*(a*b)^(1/2)-1161/8/d/(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/a*b^2/(a^2-2*
a*b+b^2)*tanh(1/2*d*x+1/2*c)^6+1/4/d*b/(a^2-2*a*b+b^2)/a/(-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))-1/4/d*b/(a^2-2*a*b+b^2)/a/(-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))-45/64/d/(a^2-2*a*b+b^2
)/a/(-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))*(a*b)^(1/2)-327/8/d/(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*ta
nh(1/2*d*x+1/2*c)^4-4*tanh(1/2*d*x+1/2*c)^2*a+a)^2*b^2/(a^2-2*a*b+b^2)/a*tanh(1/2*d*x+1/2*c)^10+185/4/d/(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/a/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^4*b^2+43/4/d/(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/(a^2-2*a*b+b^2)
/a*tanh(1/2*d*x+1/2*c)^12*b^2+537/4/d/(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/a*b^2/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^8+71/
64/d/a^2/(a^2-2*a*b+b^2)/(-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))*(a*b)^(1/2)*b+71/64/d/a^2/(a^2-2*a*b+b^2)/(-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))*(a*b)^(1/2)*b-216/d/(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/a^2/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^8*b^3-315/8/d/(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*b/(a^2-2*a*b+b^2)*tanh(1
/2*d*x+1/2*c)^8+50/d/(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*b/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^6+10/d/(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*b/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^2-1/d/(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*b/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*
c)^14+5/8/d/(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*b/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^12+3/4/d*b^2/a/(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/(a^2-2*a*b+b^2)+13/d/(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*ta
nh(1/2*d*x+1/2*c)^4-4*tanh(1/2*d*x+1/2*c)^2*a+a)^2*b/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^10-16/d*b^3/a^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/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^4+96/d*b^4/a^3/(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/(a^2-2*a*b+b^2
)*tanh(1/2*d*x+1/2*c)^8-8/d*b^3/a^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/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^12-9/8/d/(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-16*b*tanh(1/2*d*x+1/2*c)^4-4*tanh(1/2
*d*x+1/2*c)^2*a+a)^2/(a^2-2*a*b+b^2)*b+1/d/a^3*ln(tanh(1/2*d*x+1/2*c))-257/8/d/(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*b/(a^2
-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^4

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)/(a-b*sinh(d*x+c)^4)^3,x, algorithm="maxima")

[Out]

-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) - 166*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*e^(5*c) - 253*a*b^2*e^(5*c) + 96*b^3*e^(5*c))*e^(5*d*x) + (199*
a^2*b*e^(3*c) - 253*a*b^2*e^(3*c) + 96*b^3*e^(3*c))*e^(3*d*x) - (45*a^2*b*e^c - 71*a*b^2*e^c + 32*b^3*e^c)*e^(
d*x))/(a^5*b - 2*a^4*b^2 + a^3*b^3 + (a^5*b*e^(8*c) - 2*a^4*b^2*e^(8*c) + a^3*b^3*e^(8*c))*e^(8*d*x) - 4*(a^5*
b*e^(6*c) - 2*a^4*b^2*e^(6*c) + a^3*b^3*e^(6*c))*e^(6*d*x) - 2*(8*a^6*e^(4*c) - 19*a^5*b*e^(4*c) + 14*a^4*b^2*
e^(4*c) - 3*a^3*b^3*e^(4*c))*e^(4*d*x) - 4*(a^5*b*e^(2*c) - 2*a^4*b^2*e^(2*c) + a^3*b^3*e^(2*c))*e^(2*d*x)), x
)

________________________________________________________________________________________

Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)/(a-b*sinh(d*x+c)^4)^3,x, algorithm="fricas")

[Out]

Timed out

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)/(a-b*sinh(d*x+c)**4)**3,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)/(a-b*sinh(d*x+c)^4)^3,x, algorithm="giac")

[Out]

Exception raised: NotImplementedError