\(\int \frac {\text {csch}^4(c+d x)}{a+b \tanh ^3(c+d x)} \, dx\) [80]

Optimal result

Integrand size = 23, antiderivative size = 215 \[ \int \frac {\text {csch}^4(c+d x)}{a+b \tanh ^3(c+d x)} \, dx=-\frac {\sqrt [3]{b} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \tanh (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{4/3} d}+\frac {\coth (c+d x)}{a d}-\frac {\coth ^3(c+d x)}{3 a d}-\frac {b \log (\tanh (c+d x))}{a^2 d}-\frac {\sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \tanh (c+d x)\right )}{3 a^{4/3} d}+\frac {\sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \tanh (c+d x)+b^{2/3} \tanh ^2(c+d x)\right )}{6 a^{4/3} d}+\frac {b \log \left (a+b \tanh ^3(c+d x)\right )}{3 a^2 d} \]

[Out]

coth(d*x+c)/a/d-1/3*coth(d*x+c)^3/a/d-b*ln(tanh(d*x+c))/a^2/d-1/3*b^(1/3)*ln(a^(1/3)+b^(1/3)*tanh(d*x+c))/a^(4
/3)/d+1/6*b^(1/3)*ln(a^(2/3)-a^(1/3)*b^(1/3)*tanh(d*x+c)+b^(2/3)*tanh(d*x+c)^2)/a^(4/3)/d+1/3*b*ln(a+b*tanh(d*
x+c)^3)/a^2/d-1/3*b^(1/3)*arctan(1/3*(a^(1/3)-2*b^(1/3)*tanh(d*x+c))/a^(1/3)*3^(1/2))/a^(4/3)/d*3^(1/2)

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {3744, 1848, 1885, 12, 298, 31, 648, 631, 210, 642, 266} \[ \int \frac {\text {csch}^4(c+d x)}{a+b \tanh ^3(c+d x)} \, dx=-\frac {\sqrt [3]{b} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \tanh (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{4/3} d}+\frac {\sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \tanh (c+d x)+b^{2/3} \tanh ^2(c+d x)\right )}{6 a^{4/3} d}-\frac {\sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \tanh (c+d x)\right )}{3 a^{4/3} d}+\frac {b \log \left (a+b \tanh ^3(c+d x)\right )}{3 a^2 d}-\frac {b \log (\tanh (c+d x))}{a^2 d}-\frac {\coth ^3(c+d x)}{3 a d}+\frac {\coth (c+d x)}{a d} \]

[In]

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

[Out]

-((b^(1/3)*ArcTan[(a^(1/3) - 2*b^(1/3)*Tanh[c + d*x])/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*a^(4/3)*d)) + Coth[c + d*x]
/(a*d) - Coth[c + d*x]^3/(3*a*d) - (b*Log[Tanh[c + d*x]])/(a^2*d) - (b^(1/3)*Log[a^(1/3) + b^(1/3)*Tanh[c + d*
x]])/(3*a^(4/3)*d) + (b^(1/3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*Tanh[c + d*x] + b^(2/3)*Tanh[c + d*x]^2])/(6*a^(4/
3)*d) + (b*Log[a + b*Tanh[c + d*x]^3])/(3*a^2*d)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 210

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

Rule 266

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 298

Int[(x_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> Dist[-(3*Rt[a, 3]*Rt[b, 3])^(-1), Int[1/(Rt[a, 3] + Rt[b, 3]*x),
x], x] + Dist[1/(3*Rt[a, 3]*Rt[b, 3]), Int[(Rt[a, 3] + Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3
]^2*x^2), x], x] /; FreeQ[{a, b}, x]

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 648

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

Rule 1848

Int[((Pq_)*((c_.)*(x_))^(m_.))/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(Pq/(a + b*x
^n)), x], x] /; FreeQ[{a, b, c, m}, x] && PolyQ[Pq, x] && IntegerQ[n] &&  !IGtQ[m, 0]

Rule 1885

Int[(P2_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> With[{A = Coeff[P2, x, 0], B = Coeff[P2, x, 1], C = Coeff[P2, x,
2]}, Int[(A + B*x)/(a + b*x^3), x] + Dist[C, Int[x^2/(a + b*x^3), x], x] /; EqQ[a*B^3 - b*A^3, 0] ||  !Ration
alQ[a/b]] /; FreeQ[{a, b}, x] && PolyQ[P2, x, 2]

Rule 3744

Int[sin[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_.), x_Symbol] :> With[
{ff = FreeFactors[Tan[e + f*x], x]}, Dist[c*(ff^(m + 1)/f), Subst[Int[x^m*((a + b*(ff*x)^n)^p/(c^2 + ff^2*x^2)
^(m/2 + 1)), x], x, c*(Tan[e + f*x]/ff)], x]] /; FreeQ[{a, b, c, e, f, n, p}, x] && IntegerQ[m/2]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1-x^2}{x^4 \left (a+b x^3\right )} \, dx,x,\tanh (c+d x)\right )}{d} \\ & = \frac {\text {Subst}\left (\int \left (\frac {1}{a x^4}-\frac {1}{a x^2}-\frac {b}{a^2 x}+\frac {b x (a+b x)}{a^2 \left (a+b x^3\right )}\right ) \, dx,x,\tanh (c+d x)\right )}{d} \\ & = \frac {\coth (c+d x)}{a d}-\frac {\coth ^3(c+d x)}{3 a d}-\frac {b \log (\tanh (c+d x))}{a^2 d}+\frac {b \text {Subst}\left (\int \frac {x (a+b x)}{a+b x^3} \, dx,x,\tanh (c+d x)\right )}{a^2 d} \\ & = \frac {\coth (c+d x)}{a d}-\frac {\coth ^3(c+d x)}{3 a d}-\frac {b \log (\tanh (c+d x))}{a^2 d}+\frac {b \text {Subst}\left (\int \frac {a x}{a+b x^3} \, dx,x,\tanh (c+d x)\right )}{a^2 d}+\frac {b^2 \text {Subst}\left (\int \frac {x^2}{a+b x^3} \, dx,x,\tanh (c+d x)\right )}{a^2 d} \\ & = \frac {\coth (c+d x)}{a d}-\frac {\coth ^3(c+d x)}{3 a d}-\frac {b \log (\tanh (c+d x))}{a^2 d}+\frac {b \log \left (a+b \tanh ^3(c+d x)\right )}{3 a^2 d}+\frac {b \text {Subst}\left (\int \frac {x}{a+b x^3} \, dx,x,\tanh (c+d x)\right )}{a d} \\ & = \frac {\coth (c+d x)}{a d}-\frac {\coth ^3(c+d x)}{3 a d}-\frac {b \log (\tanh (c+d x))}{a^2 d}+\frac {b \log \left (a+b \tanh ^3(c+d x)\right )}{3 a^2 d}-\frac {b^{2/3} \text {Subst}\left (\int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx,x,\tanh (c+d x)\right )}{3 a^{4/3} d}+\frac {b^{2/3} \text {Subst}\left (\int \frac {\sqrt [3]{a}+\sqrt [3]{b} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\tanh (c+d x)\right )}{3 a^{4/3} d} \\ & = \frac {\coth (c+d x)}{a d}-\frac {\coth ^3(c+d x)}{3 a d}-\frac {b \log (\tanh (c+d x))}{a^2 d}-\frac {\sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \tanh (c+d x)\right )}{3 a^{4/3} d}+\frac {b \log \left (a+b \tanh ^3(c+d x)\right )}{3 a^2 d}+\frac {\sqrt [3]{b} \text {Subst}\left (\int \frac {-\sqrt [3]{a} \sqrt [3]{b}+2 b^{2/3} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\tanh (c+d x)\right )}{6 a^{4/3} d}+\frac {b^{2/3} \text {Subst}\left (\int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\tanh (c+d x)\right )}{2 a d} \\ & = \frac {\coth (c+d x)}{a d}-\frac {\coth ^3(c+d x)}{3 a d}-\frac {b \log (\tanh (c+d x))}{a^2 d}-\frac {\sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \tanh (c+d x)\right )}{3 a^{4/3} d}+\frac {\sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \tanh (c+d x)+b^{2/3} \tanh ^2(c+d x)\right )}{6 a^{4/3} d}+\frac {b \log \left (a+b \tanh ^3(c+d x)\right )}{3 a^2 d}+\frac {\sqrt [3]{b} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} \tanh (c+d x)}{\sqrt [3]{a}}\right )}{a^{4/3} d} \\ & = -\frac {\sqrt [3]{b} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} \tanh (c+d x)}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt {3} a^{4/3} d}+\frac {\coth (c+d x)}{a d}-\frac {\coth ^3(c+d x)}{3 a d}-\frac {b \log (\tanh (c+d x))}{a^2 d}-\frac {\sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \tanh (c+d x)\right )}{3 a^{4/3} d}+\frac {\sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \tanh (c+d x)+b^{2/3} \tanh ^2(c+d x)\right )}{6 a^{4/3} d}+\frac {b \log \left (a+b \tanh ^3(c+d x)\right )}{3 a^2 d} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.

Time = 1.12 (sec) , antiderivative size = 322, normalized size of antiderivative = 1.50 \[ \int \frac {\text {csch}^4(c+d x)}{a+b \tanh ^3(c+d x)} \, dx=\frac {-a \coth (c+d x) \left (-2+\text {csch}^2(c+d x)\right )+3 b (c+d x-\log (\sinh (c+d x)))+b \text {RootSum}\left [a-b+3 a \text {$\#$1}+3 b \text {$\#$1}+3 a \text {$\#$1}^2-3 b \text {$\#$1}^2+a \text {$\#$1}^3+b \text {$\#$1}^3\&,\frac {-2 a c+2 b c-2 a d x+2 b d x+a \log \left (e^{2 (c+d x)}-\text {$\#$1}\right )-b \log \left (e^{2 (c+d x)}-\text {$\#$1}\right )-8 a c \text {$\#$1}-4 b c \text {$\#$1}-8 a d x \text {$\#$1}-4 b d x \text {$\#$1}+4 a \log \left (e^{2 (c+d x)}-\text {$\#$1}\right ) \text {$\#$1}+2 b \log \left (e^{2 (c+d x)}-\text {$\#$1}\right ) \text {$\#$1}+2 a c \text {$\#$1}^2+2 b c \text {$\#$1}^2+2 a d x \text {$\#$1}^2+2 b d x \text {$\#$1}^2-a \log \left (e^{2 (c+d x)}-\text {$\#$1}\right ) \text {$\#$1}^2-b \log \left (e^{2 (c+d x)}-\text {$\#$1}\right ) \text {$\#$1}^2}{a-b+2 a \text {$\#$1}+2 b \text {$\#$1}+a \text {$\#$1}^2-b \text {$\#$1}^2}\&\right ]}{3 a^2 d} \]

[In]

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

[Out]

(-(a*Coth[c + d*x]*(-2 + Csch[c + d*x]^2)) + 3*b*(c + d*x - Log[Sinh[c + d*x]]) + b*RootSum[a - b + 3*a*#1 + 3
*b*#1 + 3*a*#1^2 - 3*b*#1^2 + a*#1^3 + b*#1^3 & , (-2*a*c + 2*b*c - 2*a*d*x + 2*b*d*x + a*Log[E^(2*(c + d*x))
- #1] - b*Log[E^(2*(c + d*x)) - #1] - 8*a*c*#1 - 4*b*c*#1 - 8*a*d*x*#1 - 4*b*d*x*#1 + 4*a*Log[E^(2*(c + d*x))
- #1]*#1 + 2*b*Log[E^(2*(c + d*x)) - #1]*#1 + 2*a*c*#1^2 + 2*b*c*#1^2 + 2*a*d*x*#1^2 + 2*b*d*x*#1^2 - a*Log[E^
(2*(c + d*x)) - #1]*#1^2 - b*Log[E^(2*(c + d*x)) - #1]*#1^2)/(a - b + 2*a*#1 + 2*b*#1 + a*#1^2 - b*#1^2) & ])/
(3*a^2*d)

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 1.60 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.80

method	result	size
derivativedivides	\(\frac {\frac {b \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a \,\textit {\_Z}^{6}+3 a \,\textit {\_Z}^{4}+8 b \,\textit {\_Z}^{3}+3 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (\textit {\_R}^{5} a +4 \textit {\_R}^{2} b +3 \textit {\_R} a \right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5} a +2 \textit {\_R}^{3} a +4 \textit {\_R}^{2} b +\textit {\_R} a}\right )}{3 a^{2}}-\frac {\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3}-3 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a}-\frac {1}{24 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}+\frac {3}{8 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {b \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{2}}}{d}\)	\(173\)
default	\(\frac {\frac {b \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a \,\textit {\_Z}^{6}+3 a \,\textit {\_Z}^{4}+8 b \,\textit {\_Z}^{3}+3 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (\textit {\_R}^{5} a +4 \textit {\_R}^{2} b +3 \textit {\_R} a \right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5} a +2 \textit {\_R}^{3} a +4 \textit {\_R}^{2} b +\textit {\_R} a}\right )}{3 a^{2}}-\frac {\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3}-3 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a}-\frac {1}{24 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}+\frac {3}{8 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {b \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{2}}}{d}\)	\(173\)
risch	\(-\frac {4 \left (3 \,{\mathrm e}^{2 d x +2 c}-1\right )}{3 d a \left ({\mathrm e}^{2 d x +2 c}-1\right )^{3}}+16 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (110592 a^{6} d^{3} \textit {\_Z}^{3}-6912 a^{4} b \,d^{2} \textit {\_Z}^{2}+144 a^{2} b^{2} d \textit {\_Z} +a^{2} b -b^{3}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {4608 a^{4} d^{2} \textit {\_R}^{2}}{a^{2}+a b}+\left (\frac {96 d \,a^{3}}{a^{2}+a b}-\frac {192 a^{2} b d}{a^{2}+a b}\right ) \textit {\_R} +\frac {a^{2}}{a^{2}+a b}-\frac {3 a b}{a^{2}+a b}+\frac {2 b^{2}}{a^{2}+a b}\right )\right )-\frac {b \ln \left ({\mathrm e}^{2 d x +2 c}-1\right )}{a^{2} d}\)	\(214\)

[In]

int(csch(d*x+c)^4/(a+b*tanh(d*x+c)^3),x,method=_RETURNVERBOSE)

[Out]

1/d*(1/3/a^2*b*sum((_R^5*a+4*_R^2*b+3*_R*a)/(_R^5*a+2*_R^3*a+4*_R^2*b+_R*a)*ln(tanh(1/2*d*x+1/2*c)-_R),_R=Root
Of(_Z^6*a+3*_Z^4*a+8*_Z^3*b+3*_Z^2*a+a))-1/8/a*(1/3*tanh(1/2*d*x+1/2*c)^3-3*tanh(1/2*d*x+1/2*c))-1/24/a/tanh(1
/2*d*x+1/2*c)^3+3/8/a/tanh(1/2*d*x+1/2*c)-1/a^2*b*ln(tanh(1/2*d*x+1/2*c)))

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.25 (sec) , antiderivative size = 3726, normalized size of antiderivative = 17.33 \[ \int \frac {\text {csch}^4(c+d x)}{a+b \tanh ^3(c+d x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/12*(48*a*cosh(d*x + c)^2 + 2*(a^2*d*cosh(d*x + c)^6 + 6*a^2*d*cosh(d*x + c)*sinh(d*x + c)^5 + a^2*d*sinh(d*
x + c)^6 - 3*a^2*d*cosh(d*x + c)^4 + 3*a^2*d*cosh(d*x + c)^2 + 3*(5*a^2*d*cosh(d*x + c)^2 - a^2*d)*sinh(d*x +
c)^4 + 4*(5*a^2*d*cosh(d*x + c)^3 - 3*a^2*d*cosh(d*x + c))*sinh(d*x + c)^3 - a^2*d + 3*(5*a^2*d*cosh(d*x + c)^
4 - 6*a^2*d*cosh(d*x + c)^2 + a^2*d)*sinh(d*x + c)^2 + 6*(a^2*d*cosh(d*x + c)^5 - 2*a^2*d*cosh(d*x + c)^3 + a^
2*d*cosh(d*x + c))*sinh(d*x + c))*((1/2)^(1/3)*(I*sqrt(3) + 1)*(b/(a^4*d^3) - b^3/(a^6*d^3) - (a^2*b - b^3)/(a
^6*d^3))^(1/3) - 2*b/(a^2*d))*log(1/2*((1/2)^(1/3)*(I*sqrt(3) + 1)*(b/(a^4*d^3) - b^3/(a^6*d^3) - (a^2*b - b^3
)/(a^6*d^3))^(1/3) - 2*b/(a^2*d))^2*a^4*d^2 - (a^3 - 2*a^2*b)*((1/2)^(1/3)*(I*sqrt(3) + 1)*(b/(a^4*d^3) - b^3/
(a^6*d^3) - (a^2*b - b^3)/(a^6*d^3))^(1/3) - 2*b/(a^2*d))*d + (a^2 + a*b)*cosh(d*x + c)^2 + 2*(a^2 + a*b)*cosh
(d*x + c)*sinh(d*x + c) + (a^2 + a*b)*sinh(d*x + c)^2 + a^2 - 3*a*b + 2*b^2) + 96*a*cosh(d*x + c)*sinh(d*x + c
) + 48*a*sinh(d*x + c)^2 - (6*b*cosh(d*x + c)^6 + 36*b*cosh(d*x + c)*sinh(d*x + c)^5 + 6*b*sinh(d*x + c)^6 - 1
8*b*cosh(d*x + c)^4 + 18*(5*b*cosh(d*x + c)^2 - b)*sinh(d*x + c)^4 + 24*(5*b*cosh(d*x + c)^3 - 3*b*cosh(d*x +
c))*sinh(d*x + c)^3 + 18*b*cosh(d*x + c)^2 + 18*(5*b*cosh(d*x + c)^4 - 6*b*cosh(d*x + c)^2 + b)*sinh(d*x + c)^
2 + (a^2*d*cosh(d*x + c)^6 + 6*a^2*d*cosh(d*x + c)*sinh(d*x + c)^5 + a^2*d*sinh(d*x + c)^6 - 3*a^2*d*cosh(d*x
+ c)^4 + 3*a^2*d*cosh(d*x + c)^2 + 3*(5*a^2*d*cosh(d*x + c)^2 - a^2*d)*sinh(d*x + c)^4 + 4*(5*a^2*d*cosh(d*x +
c)^3 - 3*a^2*d*cosh(d*x + c))*sinh(d*x + c)^3 - a^2*d + 3*(5*a^2*d*cosh(d*x + c)^4 - 6*a^2*d*cosh(d*x + c)^2
+ a^2*d)*sinh(d*x + c)^2 + 6*(a^2*d*cosh(d*x + c)^5 - 2*a^2*d*cosh(d*x + c)^3 + a^2*d*cosh(d*x + c))*sinh(d*x
+ c))*((1/2)^(1/3)*(I*sqrt(3) + 1)*(b/(a^4*d^3) - b^3/(a^6*d^3) - (a^2*b - b^3)/(a^6*d^3))^(1/3) - 2*b/(a^2*d)
) + 3*sqrt(1/3)*(a^2*d*cosh(d*x + c)^6 + 6*a^2*d*cosh(d*x + c)*sinh(d*x + c)^5 + a^2*d*sinh(d*x + c)^6 - 3*a^2
*d*cosh(d*x + c)^4 + 3*a^2*d*cosh(d*x + c)^2 + 3*(5*a^2*d*cosh(d*x + c)^2 - a^2*d)*sinh(d*x + c)^4 + 4*(5*a^2*
d*cosh(d*x + c)^3 - 3*a^2*d*cosh(d*x + c))*sinh(d*x + c)^3 - a^2*d + 3*(5*a^2*d*cosh(d*x + c)^4 - 6*a^2*d*cosh
(d*x + c)^2 + a^2*d)*sinh(d*x + c)^2 + 6*(a^2*d*cosh(d*x + c)^5 - 2*a^2*d*cosh(d*x + c)^3 + a^2*d*cosh(d*x + c
))*sinh(d*x + c))*sqrt(-(((1/2)^(1/3)*(I*sqrt(3) + 1)*(b/(a^4*d^3) - b^3/(a^6*d^3) - (a^2*b - b^3)/(a^6*d^3))^
(1/3) - 2*b/(a^2*d))^2*a^4*d^2 + 4*((1/2)^(1/3)*(I*sqrt(3) + 1)*(b/(a^4*d^3) - b^3/(a^6*d^3) - (a^2*b - b^3)/(
a^6*d^3))^(1/3) - 2*b/(a^2*d))*a^2*b*d + 4*b^2)/(a^4*d^2)) + 36*(b*cosh(d*x + c)^5 - 2*b*cosh(d*x + c)^3 + b*c
osh(d*x + c))*sinh(d*x + c) - 6*b)*log(-1/4*((1/2)^(1/3)*(I*sqrt(3) + 1)*(b/(a^4*d^3) - b^3/(a^6*d^3) - (a^2*b
- b^3)/(a^6*d^3))^(1/3) - 2*b/(a^2*d))^2*a^4*d^2 + 1/2*(a^3 - 2*a^2*b)*((1/2)^(1/3)*(I*sqrt(3) + 1)*(b/(a^4*d
^3) - b^3/(a^6*d^3) - (a^2*b - b^3)/(a^6*d^3))^(1/3) - 2*b/(a^2*d))*d + (a^2 + a*b)*cosh(d*x + c)^2 + 2*(a^2 +
a*b)*cosh(d*x + c)*sinh(d*x + c) + (a^2 + a*b)*sinh(d*x + c)^2 + a^2 - b^2 + 3/4*sqrt(1/3)*(((1/2)^(1/3)*(I*s
qrt(3) + 1)*(b/(a^4*d^3) - b^3/(a^6*d^3) - (a^2*b - b^3)/(a^6*d^3))^(1/3) - 2*b/(a^2*d))*a^4*d^2 + 2*(a^3 + a^
2*b)*d)*sqrt(-(((1/2)^(1/3)*(I*sqrt(3) + 1)*(b/(a^4*d^3) - b^3/(a^6*d^3) - (a^2*b - b^3)/(a^6*d^3))^(1/3) - 2*
b/(a^2*d))^2*a^4*d^2 + 4*((1/2)^(1/3)*(I*sqrt(3) + 1)*(b/(a^4*d^3) - b^3/(a^6*d^3) - (a^2*b - b^3)/(a^6*d^3))^
(1/3) - 2*b/(a^2*d))*a^2*b*d + 4*b^2)/(a^4*d^2))) - (6*b*cosh(d*x + c)^6 + 36*b*cosh(d*x + c)*sinh(d*x + c)^5
+ 6*b*sinh(d*x + c)^6 - 18*b*cosh(d*x + c)^4 + 18*(5*b*cosh(d*x + c)^2 - b)*sinh(d*x + c)^4 + 24*(5*b*cosh(d*x
+ c)^3 - 3*b*cosh(d*x + c))*sinh(d*x + c)^3 + 18*b*cosh(d*x + c)^2 + 18*(5*b*cosh(d*x + c)^4 - 6*b*cosh(d*x +
c)^2 + b)*sinh(d*x + c)^2 + (a^2*d*cosh(d*x + c)^6 + 6*a^2*d*cosh(d*x + c)*sinh(d*x + c)^5 + a^2*d*sinh(d*x +
c)^6 - 3*a^2*d*cosh(d*x + c)^4 + 3*a^2*d*cosh(d*x + c)^2 + 3*(5*a^2*d*cosh(d*x + c)^2 - a^2*d)*sinh(d*x + c)^
4 + 4*(5*a^2*d*cosh(d*x + c)^3 - 3*a^2*d*cosh(d*x + c))*sinh(d*x + c)^3 - a^2*d + 3*(5*a^2*d*cosh(d*x + c)^4 -
6*a^2*d*cosh(d*x + c)^2 + a^2*d)*sinh(d*x + c)^2 + 6*(a^2*d*cosh(d*x + c)^5 - 2*a^2*d*cosh(d*x + c)^3 + a^2*d
*cosh(d*x + c))*sinh(d*x + c))*((1/2)^(1/3)*(I*sqrt(3) + 1)*(b/(a^4*d^3) - b^3/(a^6*d^3) - (a^2*b - b^3)/(a^6*
d^3))^(1/3) - 2*b/(a^2*d)) - 3*sqrt(1/3)*(a^2*d*cosh(d*x + c)^6 + 6*a^2*d*cosh(d*x + c)*sinh(d*x + c)^5 + a^2*
d*sinh(d*x + c)^6 - 3*a^2*d*cosh(d*x + c)^4 + 3*a^2*d*cosh(d*x + c)^2 + 3*(5*a^2*d*cosh(d*x + c)^2 - a^2*d)*si
nh(d*x + c)^4 + 4*(5*a^2*d*cosh(d*x + c)^3 - 3*a^2*d*cosh(d*x + c))*sinh(d*x + c)^3 - a^2*d + 3*(5*a^2*d*cosh(
d*x + c)^4 - 6*a^2*d*cosh(d*x + c)^2 + a^2*d)*sinh(d*x + c)^2 + 6*(a^2*d*cosh(d*x + c)^5 - 2*a^2*d*cosh(d*x +
c)^3 + a^2*d*cosh(d*x + c))*sinh(d*x + c))*sqrt(-(((1/2)^(1/3)*(I*sqrt(3) + 1)*(b/(a^4*d^3) - b^3/(a^6*d^3) -
(a^2*b - b^3)/(a^6*d^3))^(1/3) - 2*b/(a^2*d))^2*a^4*d^2 + 4*((1/2)^(1/3)*(I*sqrt(3) + 1)*(b/(a^4*d^3) - b^3/(a
^6*d^3) - (a^2*b - b^3)/(a^6*d^3))^(1/3) - 2*b/(a^2*d))*a^2*b*d + 4*b^2)/(a^4*d^2)) + 36*(b*cosh(d*x + c)^5 -
2*b*cosh(d*x + c)^3 + b*cosh(d*x + c))*sinh(d*x + c) - 6*b)*log(-1/4*((1/2)^(1/3)*(I*sqrt(3) + 1)*(b/(a^4*d^3)
- b^3/(a^6*d^3) - (a^2*b - b^3)/(a^6*d^3))^(1/3) - 2*b/(a^2*d))^2*a^4*d^2 + 1/2*(a^3 - 2*a^2*b)*((1/2)^(1/3)*
(I*sqrt(3) + 1)*(b/(a^4*d^3) - b^3/(a^6*d^3) - (a^2*b - b^3)/(a^6*d^3))^(1/3) - 2*b/(a^2*d))*d + (a^2 + a*b)*c
osh(d*x + c)^2 + 2*(a^2 + a*b)*cosh(d*x + c)*sinh(d*x + c) + (a^2 + a*b)*sinh(d*x + c)^2 + a^2 - b^2 - 3/4*sqr
t(1/3)*(((1/2)^(1/3)*(I*sqrt(3) + 1)*(b/(a^4*d^3) - b^3/(a^6*d^3) - (a^2*b - b^3)/(a^6*d^3))^(1/3) - 2*b/(a^2*
d))*a^4*d^2 + 2*(a^3 + a^2*b)*d)*sqrt(-(((1/2)^(1/3)*(I*sqrt(3) + 1)*(b/(a^4*d^3) - b^3/(a^6*d^3) - (a^2*b - b
^3)/(a^6*d^3))^(1/3) - 2*b/(a^2*d))^2*a^4*d^2 + 4*((1/2)^(1/3)*(I*sqrt(3) + 1)*(b/(a^4*d^3) - b^3/(a^6*d^3) -
(a^2*b - b^3)/(a^6*d^3))^(1/3) - 2*b/(a^2*d))*a^2*b*d + 4*b^2)/(a^4*d^2))) + 12*(b*cosh(d*x + c)^6 + 6*b*cosh(
d*x + c)*sinh(d*x + c)^5 + b*sinh(d*x + c)^6 - 3*b*cosh(d*x + c)^4 + 3*(5*b*cosh(d*x + c)^2 - b)*sinh(d*x + c)
^4 + 4*(5*b*cosh(d*x + c)^3 - 3*b*cosh(d*x + c))*sinh(d*x + c)^3 + 3*b*cosh(d*x + c)^2 + 3*(5*b*cosh(d*x + c)^
4 - 6*b*cosh(d*x + c)^2 + b)*sinh(d*x + c)^2 + 6*(b*cosh(d*x + c)^5 - 2*b*cosh(d*x + c)^3 + b*cosh(d*x + c))*s
inh(d*x + c) - b)*log(2*sinh(d*x + c)/(cosh(d*x + c) - sinh(d*x + c))) - 16*a)/(a^2*d*cosh(d*x + c)^6 + 6*a^2*
d*cosh(d*x + c)*sinh(d*x + c)^5 + a^2*d*sinh(d*x + c)^6 - 3*a^2*d*cosh(d*x + c)^4 + 3*a^2*d*cosh(d*x + c)^2 +
3*(5*a^2*d*cosh(d*x + c)^2 - a^2*d)*sinh(d*x + c)^4 + 4*(5*a^2*d*cosh(d*x + c)^3 - 3*a^2*d*cosh(d*x + c))*sinh
(d*x + c)^3 - a^2*d + 3*(5*a^2*d*cosh(d*x + c)^4 - 6*a^2*d*cosh(d*x + c)^2 + a^2*d)*sinh(d*x + c)^2 + 6*(a^2*d
*cosh(d*x + c)^5 - 2*a^2*d*cosh(d*x + c)^3 + a^2*d*cosh(d*x + c))*sinh(d*x + c))

Sympy [F]

\[ \int \frac {\text {csch}^4(c+d x)}{a+b \tanh ^3(c+d x)} \, dx=\int \frac {\operatorname {csch}^{4}{\left (c + d x \right )}}{a + b \tanh ^{3}{\left (c + d x \right )}}\, dx \]

[In]

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

[Out]

Integral(csch(c + d*x)**4/(a + b*tanh(c + d*x)**3), x)

Maxima [F]

\[ \int \frac {\text {csch}^4(c+d x)}{a+b \tanh ^3(c+d x)} \, dx=\int { \frac {\operatorname {csch}\left (d x + c\right )^{4}}{b \tanh \left (d x + c\right )^{3} + a} \,d x } \]

[In]

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

[Out]

2*a*b*(integrate(((a + b)*e^(4*d*x + 4*c) + 3*(a - b)*e^(2*d*x + 2*c) + 3*a + 3*b)*e^(2*d*x + 2*c)/((a + b)*e^
(6*d*x + 6*c) + 3*(a - b)*e^(4*d*x + 4*c) + 3*(a + b)*e^(2*d*x + 2*c) + a - b), x)/(a^3 - a^2*b) - (d*x + c)/(
(a^3 - a^2*b)*d)) - 2*b^2*(integrate(((a + b)*e^(4*d*x + 4*c) + 3*(a - b)*e^(2*d*x + 2*c) + 3*a + 3*b)*e^(2*d*
x + 2*c)/((a + b)*e^(6*d*x + 6*c) + 3*(a - b)*e^(4*d*x + 4*c) + 3*(a + b)*e^(2*d*x + 2*c) + a - b), x)/(a^3 -
a^2*b) - (d*x + c)/((a^3 - a^2*b)*d)) + 2*b*integrate(e^(4*d*x + 4*c)/((a + b)*e^(6*d*x + 6*c) + 3*(a - b)*e^(
4*d*x + 4*c) + 3*(a + b)*e^(2*d*x + 2*c) + a - b), x)/a + 2*b^2*integrate(e^(4*d*x + 4*c)/((a + b)*e^(6*d*x +
6*c) + 3*(a - b)*e^(4*d*x + 4*c) + 3*(a + b)*e^(2*d*x + 2*c) + a - b), x)/a^2 - 8*b*integrate(e^(2*d*x + 2*c)/
((a + b)*e^(6*d*x + 6*c) + 3*(a - b)*e^(4*d*x + 4*c) + 3*(a + b)*e^(2*d*x + 2*c) + a - b), x)/a - 4*b^2*integr
ate(e^(2*d*x + 2*c)/((a + b)*e^(6*d*x + 6*c) + 3*(a - b)*e^(4*d*x + 4*c) + 3*(a + b)*e^(2*d*x + 2*c) + a - b),
x)/a^2 + 2/3*(3*b*d*x*e^(6*d*x + 6*c) - 9*b*d*x*e^(4*d*x + 4*c) - 3*b*d*x + 3*(3*b*d*x*e^(2*c) - 2*a*e^(2*c))
*e^(2*d*x) + 2*a)/(a^2*d*e^(6*d*x + 6*c) - 3*a^2*d*e^(4*d*x + 4*c) + 3*a^2*d*e^(2*d*x + 2*c) - a^2*d) - b*log(
(e^(d*x + c) + 1)*e^(-c))/(a^2*d) - b*log((e^(d*x + c) - 1)*e^(-c))/(a^2*d)

Giac [A] (verification not implemented)

none

Time = 0.37 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.84 \[ \int \frac {\text {csch}^4(c+d x)}{a+b \tanh ^3(c+d x)} \, dx=\frac {\frac {2 \, b \log \left ({\left | a e^{\left (6 \, d x + 6 \, c\right )} + b e^{\left (6 \, d x + 6 \, c\right )} + 3 \, a e^{\left (4 \, d x + 4 \, c\right )} - 3 \, b e^{\left (4 \, d x + 4 \, c\right )} + 3 \, a e^{\left (2 \, d x + 2 \, c\right )} + 3 \, b e^{\left (2 \, d x + 2 \, c\right )} + a - b \right |}\right )}{a^{2}} - \frac {6 \, b \log \left ({\left | e^{\left (2 \, d x + 2 \, c\right )} - 1 \right |}\right )}{a^{2}} + \frac {11 \, b e^{\left (6 \, d x + 6 \, c\right )} - 33 \, b e^{\left (4 \, d x + 4 \, c\right )} - 24 \, a e^{\left (2 \, d x + 2 \, c\right )} + 33 \, b e^{\left (2 \, d x + 2 \, c\right )} + 8 \, a - 11 \, b}{a^{2} {\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{3}}}{6 \, d} \]

[In]

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

[Out]

1/6*(2*b*log(abs(a*e^(6*d*x + 6*c) + b*e^(6*d*x + 6*c) + 3*a*e^(4*d*x + 4*c) - 3*b*e^(4*d*x + 4*c) + 3*a*e^(2*
d*x + 2*c) + 3*b*e^(2*d*x + 2*c) + a - b))/a^2 - 6*b*log(abs(e^(2*d*x + 2*c) - 1))/a^2 + (11*b*e^(6*d*x + 6*c)
- 33*b*e^(4*d*x + 4*c) - 24*a*e^(2*d*x + 2*c) + 33*b*e^(2*d*x + 2*c) + 8*a - 11*b)/(a^2*(e^(2*d*x + 2*c) - 1)
^3))/d

Mupad [B] (verification not implemented)

Time = 3.73 (sec) , antiderivative size = 4563, normalized size of antiderivative = 21.22 \[ \int \frac {\text {csch}^4(c+d x)}{a+b \tanh ^3(c+d x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

8/(3*(a*d - 3*a*d*exp(2*c + 2*d*x) + 3*a*d*exp(4*c + 4*d*x) - a*d*exp(6*c + 6*d*x))) - 4/(a*d - 2*a*d*exp(2*c
+ 2*d*x) + a*d*exp(4*c + 4*d*x)) + symsum(log((1507328*a*b^9 + 1572864*b^10 - 5242880*a^2*b^8 - 7479296*a^3*b^
7 + 3948544*a^4*b^6 + 5963776*a^5*b^5 - 278528*a^6*b^4 + 8192*a^7*b^3 - 1572864*b^10*exp(2*root(27*a^6*d^3*z^3
- 27*a^4*b*d^2*z^2 + 9*a^2*b^2*d*z + a^2*b - b^3, z, k))*exp(2*d*x) - 1769472*a*b^9*exp(2*root(27*a^6*d^3*z^3
- 27*a^4*b*d^2*z^2 + 9*a^2*b^2*d*z + a^2*b - b^3, z, k))*exp(2*d*x) + 42467328*root(27*a^6*d^3*z^3 - 27*a^4*b
*d^2*z^2 + 9*a^2*b^2*d*z + a^2*b - b^3, z, k)^2*a^4*b^8*d^2 + 21626880*root(27*a^6*d^3*z^3 - 27*a^4*b*d^2*z^2
+ 9*a^2*b^2*d*z + a^2*b - b^3, z, k)^2*a^5*b^7*d^2 - 70189056*root(27*a^6*d^3*z^3 - 27*a^4*b*d^2*z^2 + 9*a^2*b
^2*d*z + a^2*b - b^3, z, k)^2*a^6*b^6*d^2 + 18038784*root(27*a^6*d^3*z^3 - 27*a^4*b*d^2*z^2 + 9*a^2*b^2*d*z +
a^2*b - b^3, z, k)^2*a^7*b^5*d^2 - 11993088*root(27*a^6*d^3*z^3 - 27*a^4*b*d^2*z^2 + 9*a^2*b^2*d*z + a^2*b - b
^3, z, k)^2*a^8*b^4*d^2 + 147456*root(27*a^6*d^3*z^3 - 27*a^4*b*d^2*z^2 + 9*a^2*b^2*d*z + a^2*b - b^3, z, k)^2
*a^9*b^3*d^2 - 98304*root(27*a^6*d^3*z^3 - 27*a^4*b*d^2*z^2 + 9*a^2*b^2*d*z + a^2*b - b^3, z, k)^2*a^10*b^2*d^
2 - 42467328*root(27*a^6*d^3*z^3 - 27*a^4*b*d^2*z^2 + 9*a^2*b^2*d*z + a^2*b - b^3, z, k)^3*a^6*b^7*d^3 - 12091
392*root(27*a^6*d^3*z^3 - 27*a^4*b*d^2*z^2 + 9*a^2*b^2*d*z + a^2*b - b^3, z, k)^3*a^7*b^6*d^3 + 22708224*root(
27*a^6*d^3*z^3 - 27*a^4*b*d^2*z^2 + 9*a^2*b^2*d*z + a^2*b - b^3, z, k)^3*a^8*b^5*d^3 + 12386304*root(27*a^6*d^
3*z^3 - 27*a^4*b*d^2*z^2 + 9*a^2*b^2*d*z + a^2*b - b^3, z, k)^3*a^9*b^4*d^3 + 19759104*root(27*a^6*d^3*z^3 - 2
7*a^4*b*d^2*z^2 + 9*a^2*b^2*d*z + a^2*b - b^3, z, k)^3*a^10*b^3*d^3 - 294912*root(27*a^6*d^3*z^3 - 27*a^4*b*d^
2*z^2 + 9*a^2*b^2*d*z + a^2*b - b^3, z, k)^3*a^11*b^2*d^3 - 14155776*root(27*a^6*d^3*z^3 - 27*a^4*b*d^2*z^2 +
9*a^2*b^2*d*z + a^2*b - b^3, z, k)*a^2*b^9*d - 10387456*root(27*a^6*d^3*z^3 - 27*a^4*b*d^2*z^2 + 9*a^2*b^2*d*z
+ a^2*b - b^3, z, k)*a^3*b^8*d + 32407552*root(27*a^6*d^3*z^3 - 27*a^4*b*d^2*z^2 + 9*a^2*b^2*d*z + a^2*b - b^
3, z, k)*a^4*b^7*d + 16187392*root(27*a^6*d^3*z^3 - 27*a^4*b*d^2*z^2 + 9*a^2*b^2*d*z + a^2*b - b^3, z, k)*a^5*
b^6*d - 29818880*root(27*a^6*d^3*z^3 - 27*a^4*b*d^2*z^2 + 9*a^2*b^2*d*z + a^2*b - b^3, z, k)*a^6*b^5*d + 61358
08*root(27*a^6*d^3*z^3 - 27*a^4*b*d^2*z^2 + 9*a^2*b^2*d*z + a^2*b - b^3, z, k)*a^7*b^4*d - 376832*root(27*a^6*
d^3*z^3 - 27*a^4*b*d^2*z^2 + 9*a^2*b^2*d*z + a^2*b - b^3, z, k)*a^8*b^3*d + 8192*root(27*a^6*d^3*z^3 - 27*a^4*
b*d^2*z^2 + 9*a^2*b^2*d*z + a^2*b - b^3, z, k)*a^9*b^2*d - 3571712*a^2*b^8*exp(2*root(27*a^6*d^3*z^3 - 27*a^4*
b*d^2*z^2 + 9*a^2*b^2*d*z + a^2*b - b^3, z, k))*exp(2*d*x) + 30990336*a^3*b^7*exp(2*root(27*a^6*d^3*z^3 - 27*a
^4*b*d^2*z^2 + 9*a^2*b^2*d*z + a^2*b - b^3, z, k))*exp(2*d*x) + 43139072*a^4*b^6*exp(2*root(27*a^6*d^3*z^3 - 2
7*a^4*b*d^2*z^2 + 9*a^2*b^2*d*z + a^2*b - b^3, z, k))*exp(2*d*x) + 8519680*a^5*b^5*exp(2*root(27*a^6*d^3*z^3 -
27*a^4*b*d^2*z^2 + 9*a^2*b^2*d*z + a^2*b - b^3, z, k))*exp(2*d*x) - 245760*a^6*b^4*exp(2*root(27*a^6*d^3*z^3
- 27*a^4*b*d^2*z^2 + 9*a^2*b^2*d*z + a^2*b - b^3, z, k))*exp(2*d*x) + 8192*a^7*b^3*exp(2*root(27*a^6*d^3*z^3 -
27*a^4*b*d^2*z^2 + 9*a^2*b^2*d*z + a^2*b - b^3, z, k))*exp(2*d*x) - 42467328*root(27*a^6*d^3*z^3 - 27*a^4*b*d
^2*z^2 + 9*a^2*b^2*d*z + a^2*b - b^3, z, k)^2*a^4*b^8*d^2*exp(2*root(27*a^6*d^3*z^3 - 27*a^4*b*d^2*z^2 + 9*a^2
*b^2*d*z + a^2*b - b^3, z, k))*exp(2*d*x) - 22413312*root(27*a^6*d^3*z^3 - 27*a^4*b*d^2*z^2 + 9*a^2*b^2*d*z +
a^2*b - b^3, z, k)^2*a^5*b^7*d^2*exp(2*root(27*a^6*d^3*z^3 - 27*a^4*b*d^2*z^2 + 9*a^2*b^2*d*z + a^2*b - b^3, z
, k))*exp(2*d*x) + 54853632*root(27*a^6*d^3*z^3 - 27*a^4*b*d^2*z^2 + 9*a^2*b^2*d*z + a^2*b - b^3, z, k)^2*a^6*
b^6*d^2*exp(2*root(27*a^6*d^3*z^3 - 27*a^4*b*d^2*z^2 + 9*a^2*b^2*d*z + a^2*b - b^3, z, k))*exp(2*d*x) + 679772
16*root(27*a^6*d^3*z^3 - 27*a^4*b*d^2*z^2 + 9*a^2*b^2*d*z + a^2*b - b^3, z, k)^2*a^7*b^5*d^2*exp(2*root(27*a^6
*d^3*z^3 - 27*a^4*b*d^2*z^2 + 9*a^2*b^2*d*z + a^2*b - b^3, z, k))*exp(2*d*x) - 60014592*root(27*a^6*d^3*z^3 -
27*a^4*b*d^2*z^2 + 9*a^2*b^2*d*z + a^2*b - b^3, z, k)^2*a^8*b^4*d^2*exp(2*root(27*a^6*d^3*z^3 - 27*a^4*b*d^2*z
^2 + 9*a^2*b^2*d*z + a^2*b - b^3, z, k))*exp(2*d*x) + 2211840*root(27*a^6*d^3*z^3 - 27*a^4*b*d^2*z^2 + 9*a^2*b
^2*d*z + a^2*b - b^3, z, k)^2*a^9*b^3*d^2*exp(2*root(27*a^6*d^3*z^3 - 27*a^4*b*d^2*z^2 + 9*a^2*b^2*d*z + a^2*b
- b^3, z, k))*exp(2*d*x) - 147456*root(27*a^6*d^3*z^3 - 27*a^4*b*d^2*z^2 + 9*a^2*b^2*d*z + a^2*b - b^3, z, k)
^2*a^10*b^2*d^2*exp(2*root(27*a^6*d^3*z^3 - 27*a^4*b*d^2*z^2 + 9*a^2*b^2*d*z + a^2*b - b^3, z, k))*exp(2*d*x)
+ 42467328*root(27*a^6*d^3*z^3 - 27*a^4*b*d^2*z^2 + 9*a^2*b^2*d*z + a^2*b - b^3, z, k)^3*a^6*b^7*d^3*exp(2*roo
t(27*a^6*d^3*z^3 - 27*a^4*b*d^2*z^2 + 9*a^2*b^2*d*z + a^2*b - b^3, z, k))*exp(2*d*x) + 9732096*root(27*a^6*d^3
*z^3 - 27*a^4*b*d^2*z^2 + 9*a^2*b^2*d*z + a^2*b - b^3, z, k)^3*a^7*b^6*d^3*exp(2*root(27*a^6*d^3*z^3 - 27*a^4*
b*d^2*z^2 + 9*a^2*b^2*d*z + a^2*b - b^3, z, k))*exp(2*d*x) - 85377024*root(27*a^6*d^3*z^3 - 27*a^4*b*d^2*z^2 +
9*a^2*b^2*d*z + a^2*b - b^3, z, k)^3*a^8*b^5*d^3*exp(2*root(27*a^6*d^3*z^3 - 27*a^4*b*d^2*z^2 + 9*a^2*b^2*d*z
+ a^2*b - b^3, z, k))*exp(2*d*x) + 246398976*root(27*a^6*d^3*z^3 - 27*a^4*b*d^2*z^2 + 9*a^2*b^2*d*z + a^2*b -
b^3, z, k)^3*a^9*b^4*d^3*exp(2*root(27*a^6*d^3*z^3 - 27*a^4*b*d^2*z^2 + 9*a^2*b^2*d*z + a^2*b - b^3, z, k))*e
xp(2*d*x) + 12828672*root(27*a^6*d^3*z^3 - 27*a^4*b*d^2*z^2 + 9*a^2*b^2*d*z + a^2*b - b^3, z, k)^3*a^10*b^3*d^
3*exp(2*root(27*a^6*d^3*z^3 - 27*a^4*b*d^2*z^2 + 9*a^2*b^2*d*z + a^2*b - b^3, z, k))*exp(2*d*x) + 442368*root(
27*a^6*d^3*z^3 - 27*a^4*b*d^2*z^2 + 9*a^2*b^2*d*z + a^2*b - b^3, z, k)^3*a^11*b^2*d^3*exp(2*root(27*a^6*d^3*z^
3 - 27*a^4*b*d^2*z^2 + 9*a^2*b^2*d*z + a^2*b - b^3, z, k))*exp(2*d*x) + 14155776*root(27*a^6*d^3*z^3 - 27*a^4*
b*d^2*z^2 + 9*a^2*b^2*d*z + a^2*b - b^3, z, k)*a^2*b^9*d*exp(2*root(27*a^6*d^3*z^3 - 27*a^4*b*d^2*z^2 + 9*a^2*
b^2*d*z + a^2*b - b^3, z, k))*exp(2*d*x) + 11698176*root(27*a^6*d^3*z^3 - 27*a^4*b*d^2*z^2 + 9*a^2*b^2*d*z + a
^2*b - b^3, z, k)*a^3*b^8*d*exp(2*root(27*a^6*d^3*z^3 - 27*a^4*b*d^2*z^2 + 9*a^2*b^2*d*z + a^2*b - b^3, z, k))
*exp(2*d*x) + 6111232*root(27*a^6*d^3*z^3 - 27*a^4*b*d^2*z^2 + 9*a^2*b^2*d*z + a^2*b - b^3, z, k)*a^4*b^7*d*ex
p(2*root(27*a^6*d^3*z^3 - 27*a^4*b*d^2*z^2 + 9*a^2*b^2*d*z + a^2*b - b^3, z, k))*exp(2*d*x) - 165445632*root(2
7*a^6*d^3*z^3 - 27*a^4*b*d^2*z^2 + 9*a^2*b^2*d*z + a^2*b - b^3, z, k)*a^5*b^6*d*exp(2*root(27*a^6*d^3*z^3 - 27
*a^4*b*d^2*z^2 + 9*a^2*b^2*d*z + a^2*b - b^3, z, k))*exp(2*d*x) - 27688960*root(27*a^6*d^3*z^3 - 27*a^4*b*d^2*
z^2 + 9*a^2*b^2*d*z + a^2*b - b^3, z, k)*a^6*b^5*d*exp(2*root(27*a^6*d^3*z^3 - 27*a^4*b*d^2*z^2 + 9*a^2*b^2*d*
z + a^2*b - b^3, z, k))*exp(2*d*x) + 10559488*root(27*a^6*d^3*z^3 - 27*a^4*b*d^2*z^2 + 9*a^2*b^2*d*z + a^2*b -
b^3, z, k)*a^7*b^4*d*exp(2*root(27*a^6*d^3*z^3 - 27*a^4*b*d^2*z^2 + 9*a^2*b^2*d*z + a^2*b - b^3, z, k))*exp(2
*d*x) - 393216*root(27*a^6*d^3*z^3 - 27*a^4*b*d^2*z^2 + 9*a^2*b^2*d*z + a^2*b - b^3, z, k)*a^8*b^3*d*exp(2*roo
t(27*a^6*d^3*z^3 - 27*a^4*b*d^2*z^2 + 9*a^2*b^2*d*z + a^2*b - b^3, z, k))*exp(2*d*x) + 8192*root(27*a^6*d^3*z^
3 - 27*a^4*b*d^2*z^2 + 9*a^2*b^2*d*z + a^2*b - b^3, z, k)*a^9*b^2*d*exp(2*root(27*a^6*d^3*z^3 - 27*a^4*b*d^2*z
^2 + 9*a^2*b^2*d*z + a^2*b - b^3, z, k))*exp(2*d*x))/(24*a^14*b + 3*a^15 + 3*a^7*b^8 + 24*a^8*b^7 + 84*a^9*b^6
+ 168*a^10*b^5 + 210*a^11*b^4 + 168*a^12*b^3 + 84*a^13*b^2))*root(27*a^6*d^3*z^3 - 27*a^4*b*d^2*z^2 + 9*a^2*b
^2*d*z + a^2*b - b^3, z, k), k, 1, 3) - (b*log(45613056*a*b^9 + 100663296*b^10 - 130547712*a^2*b^8 - 18014208*
a^3*b^7 + 2015232*a^4*b^6 + 270336*a^5*b^5 - 100663296*b^10*exp(2*d*x)*exp(-(2*b)/(a^2*d)) + 130547712*a^2*b^8
*exp(2*d*x)*exp(-(2*b)/(a^2*d)) + 18014208*a^3*b^7*exp(2*d*x)*exp(-(2*b)/(a^2*d)) - 2015232*a^4*b^6*exp(2*d*x)
*exp(-(2*b)/(a^2*d)) - 270336*a^5*b^5*exp(2*d*x)*exp(-(2*b)/(a^2*d)) - 45613056*a*b^9*exp(2*d*x)*exp(-(2*b)/(a
^2*d))))/(a^2*d)