3.3.58 \(\int \frac {\text {csch}(c+d x)}{(a-b \sinh ^4(c+d x))^3} \, dx\) [258]

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

[Out]

-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*cosh(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)
*arctan(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)

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Rubi [A]
time = 0.66, antiderivative size = 617, normalized size of antiderivative = 1.00, 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 = {3294, 1252, 213, 1192, 1180, 211, 214} \begin {gather*} -\frac {\sqrt [4]{b} \text {ArcTan}\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 ) \text {ArcTan}\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 )}{8 a^{5/2} d \left (\sqrt {a}+\sqrt {b}\right )^{3/2}}-\frac {\sqrt [4]{b} \text {ArcTan}\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 )}{2 a^3 d \sqrt {\sqrt {a}+\sqrt {b}}}-\frac {\tanh ^{-1}(\cosh (c+d x))}{a^3 d}-\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 (-\left ((5 a+b) \cosh ^2(c+d x)\right )+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 {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} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/64*((5*Sqrt[a] - 2*Sqrt[b])*b^(1/4)*ArcTan[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(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
[Sqrt[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 211

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

Rule 213

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

Rule 214

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

Rule 1180

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 1192

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 1252

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 3294

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]

Rubi steps

\begin {align*} \int \frac {\text {csch}(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^3} \, dx &=-\frac {\text {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 {\text {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 {\text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\cosh (c+d x)\right )}{a^3 d}-\frac {\text {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 {\text {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 {\text {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 {\text {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 {\text {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 \text {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 \text {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 {\text {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 \text {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 \text {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 ) \text {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 ) \text {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*}

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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 3.85, size = 1189, normalized size = 1.93 \begin {gather*} \frac {\frac {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)))}+\frac {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 \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )-\frac {b \text {RootSum}\left [b-4 b \text {$\#$1}^2-16 a \text {$\#$1}^4+6 b \text {$\#$1}^4-4 b \text {$\#$1}^6+b \text {$\#$1}^8\&,\frac {-45 a^2 c+71 a b c-32 b^2 c-45 a^2 d x+71 a b d x-32 b^2 d x-90 a^2 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right )+142 a b \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right )-64 b^2 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right )+199 a^2 c \text {$\#$1}^2-253 a b c \text {$\#$1}^2+96 b^2 c \text {$\#$1}^2+199 a^2 d x \text {$\#$1}^2-253 a b d x \text {$\#$1}^2+96 b^2 d x \text {$\#$1}^2+398 a^2 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^2-506 a b \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^2+192 b^2 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^2-199 a^2 c \text {$\#$1}^4+253 a b c \text {$\#$1}^4-96 b^2 c \text {$\#$1}^4-199 a^2 d x \text {$\#$1}^4+253 a b d x \text {$\#$1}^4-96 b^2 d x \text {$\#$1}^4-398 a^2 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^4+506 a b \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^4-192 b^2 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^4+45 a^2 c \text {$\#$1}^6-71 a b c \text {$\#$1}^6+32 b^2 c \text {$\#$1}^6+45 a^2 d x \text {$\#$1}^6-71 a b d x \text {$\#$1}^6+32 b^2 d x \text {$\#$1}^6+90 a^2 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^6-142 a b \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^6+64 b^2 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^6}{-b \text {$\#$1}-8 a \text {$\#$1}^3+3 b \text {$\#$1}^3-3 b \text {$\#$1}^5+b \text {$\#$1}^7}\&\right ]}{(a-b)^2}}{256 a^3 d} \end {gather*}

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 [A]
time = 4.05, size = 647, normalized size = 1.05

method	result	size
derivativedivides	\(\frac {\frac {8 b \left (\frac {-\frac {a^{2} \left (8 a -5 b \right ) \left (\tanh ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 \left (a^{2}-2 a b +b^{2}\right )}+\frac {a \left (5 a^{2}+86 a b -64 b^{2}\right ) \left (\tanh ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 a^{2}-128 a b +64 b^{2}}+\frac {a \left (104 a^{2}-327 a b +208 b^{2}\right ) \left (\tanh ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{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 ) \left (\tanh ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 \left (a^{2}-2 a b +b^{2}\right )}+\frac {a \left (400 a^{2}-1161 a b +560 b^{2}\right ) \left (\tanh ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 a^{2}-128 a b +64 b^{2}}-\frac {a \left (257 a^{2}-370 a b +128 b^{2}\right ) \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 \left (a^{2}-2 a b +b^{2}\right )}+\frac {\left (80 a -53 b \right ) a^{2} \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{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 (a \left (\tanh ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-4 a \left (\tanh ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+6 a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-16 b \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-4 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+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 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 \sqrt {a b}-2 a}{4 \sqrt {\sqrt {a b}\, a -a b}}\right )}{8 a b \sqrt {\sqrt {a b}\, a -a b}}-\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 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 \sqrt {a b}+2 a}{4 \sqrt {-\sqrt {a b}\, a -a b}}\right )}{8 a b \sqrt {-\sqrt {a b}\, a -a b}}\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 ) \left (\tanh ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 \left (a^{2}-2 a b +b^{2}\right )}+\frac {a \left (5 a^{2}+86 a b -64 b^{2}\right ) \left (\tanh ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 a^{2}-128 a b +64 b^{2}}+\frac {a \left (104 a^{2}-327 a b +208 b^{2}\right ) \left (\tanh ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{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 ) \left (\tanh ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 \left (a^{2}-2 a b +b^{2}\right )}+\frac {a \left (400 a^{2}-1161 a b +560 b^{2}\right ) \left (\tanh ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 a^{2}-128 a b +64 b^{2}}-\frac {a \left (257 a^{2}-370 a b +128 b^{2}\right ) \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 \left (a^{2}-2 a b +b^{2}\right )}+\frac {\left (80 a -53 b \right ) a^{2} \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{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 (a \left (\tanh ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-4 a \left (\tanh ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+6 a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-16 b \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-4 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+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 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 \sqrt {a b}-2 a}{4 \sqrt {\sqrt {a b}\, a -a b}}\right )}{8 a b \sqrt {\sqrt {a b}\, a -a b}}-\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 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 \sqrt {a b}+2 a}{4 \sqrt {-\sqrt {a b}\, a -a b}}\right )}{8 a b \sqrt {-\sqrt {a b}\, a -a b}}\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	\(\text {Expression too large to display}\)	\(1496\)

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,method=_RETURNVERBOSE)

[Out]

1/d*(8/a^3*b*((-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*(
105*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)^6-1/64*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))/(a*tanh(1/2*d*x+1
/2*c)^8-4*a*tanh(1/2*d*x+1/2*c)^6+6*a*tanh(1/2*d*x+1/2*c)^4-16*b*tanh(1/2*d*x+1/2*c)^4-4*a*tanh(1/2*d*x+1/2*c)
^2+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)^(1/2)*a-a*b)^(1/2)*arctan(1/4*(2*a*tanh(1/2*d*x+1/2*c)^2+4*(a*b)^(1/2)-2*a)/((a*b)^(1/2)*a-a*b)
^(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+10*a*b^2)/a/b/(-(a*b)^(1/2)*a-a
*b)^(1/2)*arctan(1/4*(-2*a*tanh(1/2*d*x+1/2*c)^2+4*(a*b)^(1/2)+2*a)/(-(a*b)^(1/2)*a-a*b)^(1/2))))+1/a^3*ln(tan
h(1/2*d*x+1/2*c)))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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
)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 28586 vs. \(2 (481) = 962\).
time = 1.43, size = 28586, normalized size = 46.33 \begin {gather*} \text {too large to display} \end {gather*}

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]

-1/128*(8*(13*a^2*b^2 - 7*a*b^3)*cosh(d*x + c)^15 + 120*(13*a^2*b^2 - 7*a*b^3)*cosh(d*x + c)*sinh(d*x + c)^14
+ 8*(13*a^2*b^2 - 7*a*b^3)*sinh(d*x + c)^15 - 8*(121*a^2*b^2 - 67*a*b^3)*cosh(d*x + c)^13 - 8*(121*a^2*b^2 - 6
7*a*b^3 - 105*(13*a^2*b^2 - 7*a*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^13 + 104*(35*(13*a^2*b^2 - 7*a*b^3)*cosh(d
*x + c)^3 - (121*a^2*b^2 - 67*a*b^3)*cosh(d*x + c))*sinh(d*x + c)^12 - 8*(272*a^3*b - 461*a^2*b^2 + 159*a*b^3)
*cosh(d*x + c)^11 + 8*(1365*(13*a^2*b^2 - 7*a*b^3)*cosh(d*x + c)^4 - 272*a^3*b + 461*a^2*b^2 - 159*a*b^3 - 78*
(121*a^2*b^2 - 67*a*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^11 + 88*(273*(13*a^2*b^2 - 7*a*b^3)*cosh(d*x + c)^5 -
26*(121*a^2*b^2 - 67*a*b^3)*cosh(d*x + c)^3 - (272*a^3*b - 461*a^2*b^2 + 159*a*b^3)*cosh(d*x + c))*sinh(d*x +
c)^10 + 8*(1424*a^3*b - 1121*a^2*b^2 + 99*a*b^3)*cosh(d*x + c)^9 + 8*(5005*(13*a^2*b^2 - 7*a*b^3)*cosh(d*x + c
)^6 - 715*(121*a^2*b^2 - 67*a*b^3)*cosh(d*x + c)^4 + 1424*a^3*b - 1121*a^2*b^2 + 99*a*b^3 - 55*(272*a^3*b - 46
1*a^2*b^2  ...

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1783 vs. \(2 (481) = 962\).
time = 0.64, size = 1783, normalized size = 2.89 \begin {gather*} \frac {\frac {{\left ({\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )}^{2} {\left (180 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} a^{3} - 59 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} a^{2} b - 227 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} a b^{2} + 160 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} b^{3}\right )} {\left | b \right |} - {\left (244 \, \sqrt {-b^{2} - \sqrt {a b} b} a^{8} b - 507 \, \sqrt {-b^{2} - \sqrt {a b} b} a^{7} b^{2} + 5 \, \sqrt {-b^{2} - \sqrt {a b} b} a^{6} b^{3} + 695 \, \sqrt {-b^{2} - \sqrt {a b} b} a^{5} b^{4} - 597 \, \sqrt {-b^{2} - \sqrt {a b} b} a^{4} b^{5} + 160 \, \sqrt {-b^{2} - \sqrt {a b} b} a^{3} b^{6}\right )} {\left | a^{5} - 2 \, a^{4} b + a^{3} b^{2} \right |} {\left | b \right |} + 2 \, {\left (32 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} a^{12} b - 108 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} a^{11} b^{2} + 87 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} a^{10} b^{3} + 92 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} a^{9} b^{4} - 198 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} a^{8} b^{5} + 120 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} a^{7} b^{6} - 25 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} a^{6} b^{7}\right )} {\left | b \right |}\right )} \arctan \left (\frac {e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}}{2 \, \sqrt {-\frac {a^{5} b - 2 \, a^{4} b^{2} + a^{3} b^{3} + \sqrt {{\left (a^{6} - 3 \, a^{5} b + 3 \, a^{4} b^{2} - a^{3} b^{3}\right )} {\left (a^{5} b - 2 \, a^{4} b^{2} + a^{3} b^{3}\right )} + {\left (a^{5} b - 2 \, a^{4} b^{2} + a^{3} b^{3}\right )}^{2}}}{a^{5} b - 2 \, a^{4} b^{2} + a^{3} b^{3}}}}\right )}{{\left (4 \, a^{12} b^{2} - 15 \, a^{11} b^{3} + 15 \, a^{10} b^{4} + 10 \, a^{9} b^{5} - 30 \, a^{8} b^{6} + 21 \, a^{7} b^{7} - 5 \, a^{6} b^{8}\right )} {\left | a^{5} - 2 \, a^{4} b + a^{3} b^{2} \right |}} - \frac {{\left ({\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )}^{2} {\left (180 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a^{3} - 59 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a^{2} b - 227 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a b^{2} + 160 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} b^{3}\right )} {\left | b \right |} + {\left (244 \, \sqrt {-b^{2} + \sqrt {a b} b} a^{8} b - 507 \, \sqrt {-b^{2} + \sqrt {a b} b} a^{7} b^{2} + 5 \, \sqrt {-b^{2} + \sqrt {a b} b} a^{6} b^{3} + 695 \, \sqrt {-b^{2} + \sqrt {a b} b} a^{5} b^{4} - 597 \, \sqrt {-b^{2} + \sqrt {a b} b} a^{4} b^{5} + 160 \, \sqrt {-b^{2} + \sqrt {a b} b} a^{3} b^{6}\right )} {\left | a^{5} - 2 \, a^{4} b + a^{3} b^{2} \right |} {\left | b \right |} + 2 \, {\left (32 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a^{12} b - 108 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a^{11} b^{2} + 87 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a^{10} b^{3} + 92 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a^{9} b^{4} - 198 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a^{8} b^{5} + 120 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a^{7} b^{6} - 25 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a^{6} b^{7}\right )} {\left | b \right |}\right )} \arctan \left (\frac {e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}}{2 \, \sqrt {-\frac {a^{5} b - 2 \, a^{4} b^{2} + a^{3} b^{3} - \sqrt {{\left (a^{6} - 3 \, a^{5} b + 3 \, a^{4} b^{2} - a^{3} b^{3}\right )} {\left (a^{5} b - 2 \, a^{4} b^{2} + a^{3} b^{3}\right )} + {\left (a^{5} b - 2 \, a^{4} b^{2} + a^{3} b^{3}\right )}^{2}}}{a^{5} b - 2 \, a^{4} b^{2} + a^{3} b^{3}}}}\right )}{{\left (4 \, a^{12} b^{2} - 15 \, a^{11} b^{3} + 15 \, a^{10} b^{4} + 10 \, a^{9} b^{5} - 30 \, a^{8} b^{6} + 21 \, a^{7} b^{7} - 5 \, a^{6} b^{8}\right )} {\left | a^{5} - 2 \, a^{4} b + a^{3} b^{2} \right |}} - \frac {4 \, {\left (13 \, a b^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{7} - 7 \, b^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{7} - 212 \, a b^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{5} + 116 \, b^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{5} - 272 \, a^{2} b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3} + 1248 \, a b^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3} - 592 \, b^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3} + 2240 \, a^{2} b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )} - 3200 \, a b^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )} + 960 \, b^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}\right )}}{{\left (b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{4} - 8 \, b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{2} - 16 \, a + 16 \, b\right )}^{2} {\left (a^{4} - 2 \, a^{3} b + a^{2} b^{2}\right )}} - \frac {32 \, \log \left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )} + 2\right )}{a^{3}} + \frac {32 \, \log \left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )} - 2\right )}{a^{3}}}{64 \, d} \end {gather*}

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]

1/64*(((a^5 - 2*a^4*b + a^3*b^2)^2*(180*sqrt(a*b)*sqrt(-b^2 - sqrt(a*b)*b)*a^3 - 59*sqrt(a*b)*sqrt(-b^2 - sqrt
(a*b)*b)*a^2*b - 227*sqrt(a*b)*sqrt(-b^2 - sqrt(a*b)*b)*a*b^2 + 160*sqrt(a*b)*sqrt(-b^2 - sqrt(a*b)*b)*b^3)*ab
s(b) - (244*sqrt(-b^2 - sqrt(a*b)*b)*a^8*b - 507*sqrt(-b^2 - sqrt(a*b)*b)*a^7*b^2 + 5*sqrt(-b^2 - sqrt(a*b)*b)
*a^6*b^3 + 695*sqrt(-b^2 - sqrt(a*b)*b)*a^5*b^4 - 597*sqrt(-b^2 - sqrt(a*b)*b)*a^4*b^5 + 160*sqrt(-b^2 - sqrt(
a*b)*b)*a^3*b^6)*abs(a^5 - 2*a^4*b + a^3*b^2)*abs(b) + 2*(32*sqrt(a*b)*sqrt(-b^2 - sqrt(a*b)*b)*a^12*b - 108*s
qrt(a*b)*sqrt(-b^2 - sqrt(a*b)*b)*a^11*b^2 + 87*sqrt(a*b)*sqrt(-b^2 - sqrt(a*b)*b)*a^10*b^3 + 92*sqrt(a*b)*sqr
t(-b^2 - sqrt(a*b)*b)*a^9*b^4 - 198*sqrt(a*b)*sqrt(-b^2 - sqrt(a*b)*b)*a^8*b^5 + 120*sqrt(a*b)*sqrt(-b^2 - sqr
t(a*b)*b)*a^7*b^6 - 25*sqrt(a*b)*sqrt(-b^2 - sqrt(a*b)*b)*a^6*b^7)*abs(b))*arctan(1/2*(e^(d*x + c) + e^(-d*x -
c))/sqrt(-(a^5*b - 2*a^4*b^2 + a^3*b^3 + sqrt((a^6 - 3*a^5*b + 3*a^4*b^2 - a^3*b^3)*(a^5*b - 2*a^4*b^2 + a^3*
b^3) + (a^5*b - 2*a^4*b^2 + a^3*b^3)^2))/(a^5*b - 2*a^4*b^2 + a^3*b^3)))/((4*a^12*b^2 - 15*a^11*b^3 + 15*a^10*
b^4 + 10*a^9*b^5 - 30*a^8*b^6 + 21*a^7*b^7 - 5*a^6*b^8)*abs(a^5 - 2*a^4*b + a^3*b^2)) - ((a^5 - 2*a^4*b + a^3*
b^2)^2*(180*sqrt(a*b)*sqrt(-b^2 + sqrt(a*b)*b)*a^3 - 59*sqrt(a*b)*sqrt(-b^2 + sqrt(a*b)*b)*a^2*b - 227*sqrt(a*
b)*sqrt(-b^2 + sqrt(a*b)*b)*a*b^2 + 160*sqrt(a*b)*sqrt(-b^2 + sqrt(a*b)*b)*b^3)*abs(b) + (244*sqrt(-b^2 + sqrt
(a*b)*b)*a^8*b - 507*sqrt(-b^2 + sqrt(a*b)*b)*a^7*b^2 + 5*sqrt(-b^2 + sqrt(a*b)*b)*a^6*b^3 + 695*sqrt(-b^2 + s
qrt(a*b)*b)*a^5*b^4 - 597*sqrt(-b^2 + sqrt(a*b)*b)*a^4*b^5 + 160*sqrt(-b^2 + sqrt(a*b)*b)*a^3*b^6)*abs(a^5 - 2
*a^4*b + a^3*b^2)*abs(b) + 2*(32*sqrt(a*b)*sqrt(-b^2 + sqrt(a*b)*b)*a^12*b - 108*sqrt(a*b)*sqrt(-b^2 + sqrt(a*
b)*b)*a^11*b^2 + 87*sqrt(a*b)*sqrt(-b^2 + sqrt(a*b)*b)*a^10*b^3 + 92*sqrt(a*b)*sqrt(-b^2 + sqrt(a*b)*b)*a^9*b^
4 - 198*sqrt(a*b)*sqrt(-b^2 + sqrt(a*b)*b)*a^8*b^5 + 120*sqrt(a*b)*sqrt(-b^2 + sqrt(a*b)*b)*a^7*b^6 - 25*sqrt(
a*b)*sqrt(-b^2 + sqrt(a*b)*b)*a^6*b^7)*abs(b))*arctan(1/2*(e^(d*x + c) + e^(-d*x - c))/sqrt(-(a^5*b - 2*a^4*b^
2 + a^3*b^3 - sqrt((a^6 - 3*a^5*b + 3*a^4*b^2 - a^3*b^3)*(a^5*b - 2*a^4*b^2 + a^3*b^3) + (a^5*b - 2*a^4*b^2 +
a^3*b^3)^2))/(a^5*b - 2*a^4*b^2 + a^3*b^3)))/((4*a^12*b^2 - 15*a^11*b^3 + 15*a^10*b^4 + 10*a^9*b^5 - 30*a^8*b^
6 + 21*a^7*b^7 - 5*a^6*b^8)*abs(a^5 - 2*a^4*b + a^3*b^2)) - 4*(13*a*b^2*(e^(d*x + c) + e^(-d*x - c))^7 - 7*b^3
*(e^(d*x + c) + e^(-d*x - c))^7 - 212*a*b^2*(e^(d*x + c) + e^(-d*x - c))^5 + 116*b^3*(e^(d*x + c) + e^(-d*x -
c))^5 - 272*a^2*b*(e^(d*x + c) + e^(-d*x - c))^3 + 1248*a*b^2*(e^(d*x + c) + e^(-d*x - c))^3 - 592*b^3*(e^(d*x
+ c) + e^(-d*x - c))^3 + 2240*a^2*b*(e^(d*x + c) + e^(-d*x - c)) - 3200*a*b^2*(e^(d*x + c) + e^(-d*x - c)) +
960*b^3*(e^(d*x + c) + e^(-d*x - c)))/((b*(e^(d*x + c) + e^(-d*x - c))^4 - 8*b*(e^(d*x + c) + e^(-d*x - c))^2
- 16*a + 16*b)^2*(a^4 - 2*a^3*b + a^2*b^2)) - 32*log(e^(d*x + c) + e^(-d*x - c) + 2)/a^3 + 32*log(e^(d*x + c)
+ e^(-d*x - c) - 2)/a^3)/d

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(sinh(c + d*x)*(a - b*sinh(c + d*x)^4)^3),x)

[Out]

int(1/(sinh(c + d*x)*(a - b*sinh(c + d*x)^4)^3), x)

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