3.2.78 \(\int \frac {\text {csch}(c+d x)}{a+b \sinh ^3(c+d x)} \, dx\) [178]

Optimal. Leaf size=286 \[ \frac {2 \sqrt [3]{b} \text {ArcTan}\left (\frac {(-1)^{5/6} \left (\sqrt [6]{-1} \sqrt [3]{b}+i \sqrt [3]{a} \tanh \left (\frac {1}{2} (c+d x)\right )\right )}{\sqrt {-(-1)^{2/3} a^{2/3}-b^{2/3}}}\right )}{3 a \sqrt {-(-1)^{2/3} a^{2/3}-b^{2/3}} d}+\frac {2 \sqrt [3]{b} \text {ArcTan}\left (\frac {\sqrt [6]{-1} \left ((-1)^{5/6} \sqrt [3]{b}+i \sqrt [3]{a} \tanh \left (\frac {1}{2} (c+d x)\right )\right )}{\sqrt {\sqrt [3]{-1} a^{2/3}-b^{2/3}}}\right )}{3 a \sqrt {\sqrt [3]{-1} a^{2/3}-b^{2/3}} d}-\frac {\tanh ^{-1}(\cosh (c+d x))}{a d}+\frac {2 \sqrt [3]{b} \tanh ^{-1}\left (\frac {\sqrt [3]{b}-\sqrt [3]{a} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^{2/3}+b^{2/3}}}\right )}{3 a \sqrt {a^{2/3}+b^{2/3}} d} \]

[Out]

-arctanh(cosh(d*x+c))/a/d+2/3*b^(1/3)*arctan((-1)^(1/6)*((-1)^(5/6)*b^(1/3)+I*a^(1/3)*tanh(1/2*d*x+1/2*c))/((-
1)^(1/3)*a^(2/3)-b^(2/3))^(1/2))/a/d/((-1)^(1/3)*a^(2/3)-b^(2/3))^(1/2)+2/3*b^(1/3)*arctan((-1)^(5/6)*((-1)^(1
/6)*b^(1/3)+I*a^(1/3)*tanh(1/2*d*x+1/2*c))/(-(-1)^(2/3)*a^(2/3)-b^(2/3))^(1/2))/a/d/(-(-1)^(2/3)*a^(2/3)-b^(2/
3))^(1/2)+2/3*b^(1/3)*arctanh((b^(1/3)-a^(1/3)*tanh(1/2*d*x+1/2*c))/(a^(2/3)+b^(2/3))^(1/2))/a/d/(a^(2/3)+b^(2
/3))^(1/2)

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Rubi [A]
time = 0.36, antiderivative size = 286, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3299, 3855, 2739, 632, 212, 210} \begin {gather*} \frac {2 \sqrt [3]{b} \text {ArcTan}\left (\frac {(-1)^{5/6} \left (\sqrt [6]{-1} \sqrt [3]{b}+i \sqrt [3]{a} \tanh \left (\frac {1}{2} (c+d x)\right )\right )}{\sqrt {-(-1)^{2/3} a^{2/3}-b^{2/3}}}\right )}{3 a d \sqrt {-(-1)^{2/3} a^{2/3}-b^{2/3}}}+\frac {2 \sqrt [3]{b} \text {ArcTan}\left (\frac {\sqrt [6]{-1} \left ((-1)^{5/6} \sqrt [3]{b}+i \sqrt [3]{a} \tanh \left (\frac {1}{2} (c+d x)\right )\right )}{\sqrt {\sqrt [3]{-1} a^{2/3}-b^{2/3}}}\right )}{3 a d \sqrt {\sqrt [3]{-1} a^{2/3}-b^{2/3}}}+\frac {2 \sqrt [3]{b} \tanh ^{-1}\left (\frac {\sqrt [3]{b}-\sqrt [3]{a} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^{2/3}+b^{2/3}}}\right )}{3 a d \sqrt {a^{2/3}+b^{2/3}}}-\frac {\tanh ^{-1}(\cosh (c+d x))}{a d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*b^(1/3)*ArcTan[((-1)^(5/6)*((-1)^(1/6)*b^(1/3) + I*a^(1/3)*Tanh[(c + d*x)/2]))/Sqrt[-((-1)^(2/3)*a^(2/3)) -
b^(2/3)]])/(3*a*Sqrt[-((-1)^(2/3)*a^(2/3)) - b^(2/3)]*d) + (2*b^(1/3)*ArcTan[((-1)^(1/6)*((-1)^(5/6)*b^(1/3)
+ I*a^(1/3)*Tanh[(c + d*x)/2]))/Sqrt[(-1)^(1/3)*a^(2/3) - b^(2/3)]])/(3*a*Sqrt[(-1)^(1/3)*a^(2/3) - b^(2/3)]*d
) - ArcTanh[Cosh[c + d*x]]/(a*d) + (2*b^(1/3)*ArcTanh[(b^(1/3) - a^(1/3)*Tanh[(c + d*x)/2])/Sqrt[a^(2/3) + b^(
2/3)]])/(3*a*Sqrt[a^(2/3) + b^(2/3)]*d)

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 212

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

Rule 632

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

Rule 2739

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x]}, Dis
t[2*(e/d), Subst[Int[1/(a + 2*b*e*x + a*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] &&
NeQ[a^2 - b^2, 0]

Rule 3299

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^(n_))^(p_.), x_Symbol] :> Int[ExpandTr
ig[sin[e + f*x]^m*(a + b*sin[e + f*x]^n)^p, x], x] /; FreeQ[{a, b, e, f}, x] && IntegersQ[m, p] && (EqQ[n, 4]
|| GtQ[p, 0] || (EqQ[p, -1] && IntegerQ[n]))

Rule 3855

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin {align*} \int \frac {\text {csch}(c+d x)}{a+b \sinh ^3(c+d x)} \, dx &=i \int \left (-\frac {i \text {csch}(c+d x)}{a}+\frac {i b \sinh ^2(c+d x)}{a \left (a+b \sinh ^3(c+d x)\right )}\right ) \, dx\\ &=\frac {\int \text {csch}(c+d x) \, dx}{a}-\frac {b \int \frac {\sinh ^2(c+d x)}{a+b \sinh ^3(c+d x)} \, dx}{a}\\ &=-\frac {\tanh ^{-1}(\cosh (c+d x))}{a d}+\frac {b \int \left (\frac {i}{3 b^{2/3} \left (-i \sqrt [3]{a}-i \sqrt [3]{b} \sinh (c+d x)\right )}+\frac {i}{3 b^{2/3} \left (\sqrt [6]{-1} \sqrt [3]{a}-i \sqrt [3]{b} \sinh (c+d x)\right )}+\frac {i}{3 b^{2/3} \left ((-1)^{5/6} \sqrt [3]{a}-i \sqrt [3]{b} \sinh (c+d x)\right )}\right ) \, dx}{a}\\ &=-\frac {\tanh ^{-1}(\cosh (c+d x))}{a d}+\frac {\left (i \sqrt [3]{b}\right ) \int \frac {1}{-i \sqrt [3]{a}-i \sqrt [3]{b} \sinh (c+d x)} \, dx}{3 a}+\frac {\left (i \sqrt [3]{b}\right ) \int \frac {1}{\sqrt [6]{-1} \sqrt [3]{a}-i \sqrt [3]{b} \sinh (c+d x)} \, dx}{3 a}+\frac {\left (i \sqrt [3]{b}\right ) \int \frac {1}{(-1)^{5/6} \sqrt [3]{a}-i \sqrt [3]{b} \sinh (c+d x)} \, dx}{3 a}\\ &=-\frac {\tanh ^{-1}(\cosh (c+d x))}{a d}+\frac {\left (2 \sqrt [3]{b}\right ) \text {Subst}\left (\int \frac {1}{-i \sqrt [3]{a}-2 \sqrt [3]{b} x-i \sqrt [3]{a} x^2} \, dx,x,\tan \left (\frac {1}{2} (i c+i d x)\right )\right )}{3 a d}+\frac {\left (2 \sqrt [3]{b}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [6]{-1} \sqrt [3]{a}-2 \sqrt [3]{b} x+\sqrt [6]{-1} \sqrt [3]{a} x^2} \, dx,x,\tan \left (\frac {1}{2} (i c+i d x)\right )\right )}{3 a d}+\frac {\left (2 \sqrt [3]{b}\right ) \text {Subst}\left (\int \frac {1}{(-1)^{5/6} \sqrt [3]{a}-2 \sqrt [3]{b} x+(-1)^{5/6} \sqrt [3]{a} x^2} \, dx,x,\tan \left (\frac {1}{2} (i c+i d x)\right )\right )}{3 a d}\\ &=-\frac {\tanh ^{-1}(\cosh (c+d x))}{a d}-\frac {\left (4 \sqrt [3]{b}\right ) \text {Subst}\left (\int \frac {1}{-4 \left (\sqrt [3]{-1} a^{2/3}-b^{2/3}\right )-x^2} \, dx,x,-2 \sqrt [3]{b}+2 \sqrt [6]{-1} \sqrt [3]{a} \tan \left (\frac {1}{2} (i c+i d x)\right )\right )}{3 a d}-\frac {\left (4 \sqrt [3]{b}\right ) \text {Subst}\left (\int \frac {1}{4 \left (a^{2/3}+b^{2/3}\right )-x^2} \, dx,x,-2 \sqrt [3]{b}-2 i \sqrt [3]{a} \tan \left (\frac {1}{2} (i c+i d x)\right )\right )}{3 a d}-\frac {\left (4 \sqrt [3]{b}\right ) \text {Subst}\left (\int \frac {1}{4 \left ((-1)^{2/3} a^{2/3}+b^{2/3}\right )-x^2} \, dx,x,-2 \sqrt [3]{b}+2 (-1)^{5/6} \sqrt [3]{a} \tan \left (\frac {1}{2} (i c+i d x)\right )\right )}{3 a d}\\ &=-\frac {2 \sqrt [3]{b} \tan ^{-1}\left (\frac {\sqrt [3]{b}+\sqrt [3]{-1} \sqrt [3]{a} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-(-1)^{2/3} a^{2/3}-b^{2/3}}}\right )}{3 a \sqrt {-(-1)^{2/3} a^{2/3}-b^{2/3}} d}-\frac {2 \sqrt [3]{b} \tan ^{-1}\left (\frac {\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{a} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {\sqrt [3]{-1} a^{2/3}-b^{2/3}}}\right )}{3 a \sqrt {\sqrt [3]{-1} a^{2/3}-b^{2/3}} d}-\frac {\tanh ^{-1}(\cosh (c+d x))}{a d}+\frac {2 \sqrt [3]{b} \tanh ^{-1}\left (\frac {\sqrt [3]{b}-\sqrt [3]{a} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^{2/3}+b^{2/3}}}\right )}{3 a \sqrt {a^{2/3}+b^{2/3}} d}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 0.16, size = 295, normalized size = 1.03 \begin {gather*} \frac {6 \log \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )-b \text {RootSum}\left [-b+3 b \text {$\#$1}^2+8 a \text {$\#$1}^3-3 b \text {$\#$1}^4+b \text {$\#$1}^6\&,\frac {c+d x+2 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right )-2 c \text {$\#$1}^2-2 d x \text {$\#$1}^2-4 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^2+c \text {$\#$1}^4+d x \text {$\#$1}^4+2 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^4}{b \text {$\#$1}+4 a \text {$\#$1}^2-2 b \text {$\#$1}^3+b \text {$\#$1}^5}\&\right ]}{6 a d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(6*Log[Tanh[(c + d*x)/2]] - b*RootSum[-b + 3*b*#1^2 + 8*a*#1^3 - 3*b*#1^4 + b*#1^6 & , (c + d*x + 2*Log[-Cosh[
(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1] - 2*c*#1^2 - 2*d*x*#1^2 - 4*Lo
g[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^2 + c*#1^4 + d*x*#1
^4 + 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)/(b*#1 +
4*a*#1^2 - 2*b*#1^3 + b*#1^5) & ])/(6*a*d)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 2.59, size = 98, normalized size = 0.34

method	result	size
derivativedivides	\(\frac {\frac {4 b \left (\munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{6}-3 a \,\textit {\_Z}^{4}-8 b \,\textit {\_Z}^{3}+3 a \,\textit {\_Z}^{2}-a \right )}{\sum }\frac {\textit {\_R}^{2} \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5} a -2 \textit {\_R}^{3} a -4 \textit {\_R}^{2} b +\textit {\_R} a}\right )}{3 a}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}}{d}\)	\(98\)
default	\(\frac {\frac {4 b \left (\munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{6}-3 a \,\textit {\_Z}^{4}-8 b \,\textit {\_Z}^{3}+3 a \,\textit {\_Z}^{2}-a \right )}{\sum }\frac {\textit {\_R}^{2} \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5} a -2 \textit {\_R}^{3} a -4 \textit {\_R}^{2} b +\textit {\_R} a}\right )}{3 a}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}}{d}\)	\(98\)
risch	\(\frac {\ln \left ({\mathrm e}^{d x +c}-1\right )}{d a}+2 \left (\munderset {\textit {\_R} =\RootOf \left (\left (46656 a^{8} d^{6}+46656 a^{6} b^{2} d^{6}\right ) \textit {\_Z}^{6}-3888 a^{4} b^{2} d^{4} \textit {\_Z}^{4}+108 a^{2} b^{2} d^{2} \textit {\_Z}^{2}-b^{2}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{d x +c}+\left (\frac {7776 a^{7} d^{5}}{b^{2}}+7776 a^{5} d^{5}\right ) \textit {\_R}^{5}+\left (\frac {1296 d^{4} a^{5}}{b}+1296 b \,d^{4} a^{3}\right ) \textit {\_R}^{4}-648 a^{3} d^{3} \textit {\_R}^{3}+\left (\frac {36 a^{3} d^{2}}{b}-72 a \,d^{2} b \right ) \textit {\_R}^{2}+18 a d \textit {\_R} +\frac {b}{a}\right )\right )-\frac {\ln \left ({\mathrm e}^{d x +c}+1\right )}{d a}\)	\(202\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(4/3/a*b*sum(_R^2/(_R^5*a-2*_R^3*a-4*_R^2*b+_R*a)*ln(tanh(1/2*d*x+1/2*c)-_R),_R=RootOf(_Z^6*a-3*_Z^4*a-8*_
Z^3*b+3*_Z^2*a-a))+1/a*ln(tanh(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)^3),x, algorithm="maxima")

[Out]

-log((e^(d*x + c) + 1)*e^(-c))/(a*d) + log((e^(d*x + c) - 1)*e^(-c))/(a*d) - 2*integrate((b*e^(5*d*x + 5*c) -
2*b*e^(3*d*x + 3*c) + b*e^(d*x + c))/(a*b*e^(6*d*x + 6*c) - 3*a*b*e^(4*d*x + 4*c) + 8*a^2*e^(3*d*x + 3*c) + 3*
a*b*e^(2*d*x + 2*c) - a*b), x)

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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 28005 vs. \(2 (205) = 410\).
time = 2.62, size = 28005, normalized size = 97.92 \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)^3),x, algorithm="fricas")

[Out]

-1/2*(sqrt(2/3)*sqrt(1/6)*a*d*sqrt(((a^4 + a^2*b^2)*(2*(1/2)^(2/3)*(b^4/(a^4*d^2 + a^2*b^2*d^2)^2 - b^2/(a^6*d
^4 + a^4*b^2*d^4))*(-I*sqrt(3) + 1)/(2*b^6/(a^4*d^2 + a^2*b^2*d^2)^3 - 3*b^4/((a^6*d^4 + a^4*b^2*d^4)*(a^4*d^2
+ a^2*b^2*d^2)) + b^2/(a^8*d^6 + a^6*b^2*d^6) + b^2/((a^2 + b^2)^2*a^4*d^6))^(1/3) + (1/2)^(1/3)*(2*b^6/(a^4*
d^2 + a^2*b^2*d^2)^3 - 3*b^4/((a^6*d^4 + a^4*b^2*d^4)*(a^4*d^2 + a^2*b^2*d^2)) + b^2/(a^8*d^6 + a^6*b^2*d^6) +
b^2/((a^2 + b^2)^2*a^4*d^6))^(1/3)*(I*sqrt(3) + 1) - 2*b^2/(a^4*d^2 + a^2*b^2*d^2))*d^2 + 3*sqrt(1/3)*(a^4 +
a^2*b^2)*d^2*sqrt(-((a^8 + 2*a^6*b^2 + a^4*b^4)*(2*(1/2)^(2/3)*(b^4/(a^4*d^2 + a^2*b^2*d^2)^2 - b^2/(a^6*d^4 +
a^4*b^2*d^4))*(-I*sqrt(3) + 1)/(2*b^6/(a^4*d^2 + a^2*b^2*d^2)^3 - 3*b^4/((a^6*d^4 + a^4*b^2*d^4)*(a^4*d^2 + a
^2*b^2*d^2)) + b^2/(a^8*d^6 + a^6*b^2*d^6) + b^2/((a^2 + b^2)^2*a^4*d^6))^(1/3) + (1/2)^(1/3)*(2*b^6/(a^4*d^2
+ a^2*b^2*d^2)^3 - 3*b^4/((a^6*d^4 + a^4*b^2*d^4)*(a^4*d^2 + a^2*b^2*d^2)) + b^2/(a^8*d^6 + a^6*b^2*d^6) + b^2
/((a^2 + b ...

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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)**3),x)

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not 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)^3),x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 55.94, size = 2500, normalized size = 8.74 \begin {gather*} \text {Too large to display} \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)^3)),x)

[Out]

symsum(log(-(2147483648*a*b*exp(root(729*a^6*b^2*d^6*z^6 + 729*a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*
d^2*z^2 - b^2, z, k) + d*x) - 1073741824*b^2 - 86973087744*root(729*a^6*b^2*d^6*z^6 + 729*a^8*d^6*z^6 - 243*a^
4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z, k)^4*a^6*d^4 + 86973087744*root(729*a^6*b^2*d^6*z^6 + 729*a^8*d^6
*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z, k)^6*a^8*d^6 + 134217728*root(729*a^6*b^2*d^6*z^6 +
729*a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z, k)*b^3*d + 3221225472*root(729*a^6*b^2*d^
6*z^6 + 729*a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z, k)*a^2*b*d + 18589155328*root(729
*a^6*b^2*d^6*z^6 + 729*a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z, k)^2*a^2*b^2*d^2 - 281
8572288*root(729*a^6*b^2*d^6*z^6 + 729*a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z, k)^3*a
^2*b^3*d^3 - 88181047296*root(729*a^6*b^2*d^6*z^6 + 729*a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2
- b^2, z, k)^4*a^4*b^2*d^4 + 18119393280*root(729*a^6*b^2*d^6*z^6 + 729*a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 + 2
7*a^2*b^2*d^2*z^2 - b^2, z, k)^5*a^4*b^3*d^5 + 70665633792*root(729*a^6*b^2*d^6*z^6 + 729*a^8*d^6*z^6 - 243*a^
4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z, k)^6*a^6*b^2*d^6 - 32614907904*root(729*a^6*b^2*d^6*z^6 + 729*a^8
*d^6*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z, k)^7*a^6*b^3*d^7 - 57982058496*root(729*a^6*b^2*
d^6*z^6 + 729*a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z, k)^3*a^5*d^3*exp(root(729*a^6*b
^2*d^6*z^6 + 729*a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z, k) + d*x) - 333396836352*roo
t(729*a^6*b^2*d^6*z^6 + 729*a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z, k)^5*a^7*d^5*exp(
root(729*a^6*b^2*d^6*z^6 + 729*a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z, k) + d*x) + 39
1378894848*root(729*a^6*b^2*d^6*z^6 + 729*a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z, k)^
7*a^9*d^7*exp(root(729*a^6*b^2*d^6*z^6 + 729*a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z,
k) + d*x) - 17716740096*root(729*a^6*b^2*d^6*z^6 + 729*a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2
- b^2, z, k)^3*a^4*b*d^3 + 30802968576*root(729*a^6*b^2*d^6*z^6 + 729*a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a
^2*b^2*d^2*z^2 - b^2, z, k)^5*a^6*b*d^5 - 40768634880*root(729*a^6*b^2*d^6*z^6 + 729*a^8*d^6*z^6 - 243*a^4*b^2
*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z, k)^7*a^8*b*d^7 + 268435456*root(729*a^6*b^2*d^6*z^6 + 729*a^8*d^6*z^6
- 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z, k)^2*a*b^3*d^2*exp(root(729*a^6*b^2*d^6*z^6 + 729*a^8*d^6
*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z, k) + d*x) - 16642998272*root(729*a^6*b^2*d^6*z^6 + 7
29*a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z, k)^2*a^3*b*d^2*exp(root(729*a^6*b^2*d^6*z^
6 + 729*a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z, k) + d*x) + 36238786560*root(729*a^6*
b^2*d^6*z^6 + 729*a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z, k)^4*a^5*b*d^4*exp(root(729
*a^6*b^2*d^6*z^6 + 729*a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z, k) + d*x) + 2717908992
*root(729*a^6*b^2*d^6*z^6 + 729*a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z, k)^6*a^7*b*d^
6*exp(root(729*a^6*b^2*d^6*z^6 + 729*a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z, k) + d*x
) - 5637144576*root(729*a^6*b^2*d^6*z^6 + 729*a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z,
k)*a*b^2*d*exp(root(729*a^6*b^2*d^6*z^6 + 729*a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z
, k) + d*x) + 100763959296*root(729*a^6*b^2*d^6*z^6 + 729*a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z
^2 - b^2, z, k)^3*a^3*b^2*d^3*exp(root(729*a^6*b^2*d^6*z^6 + 729*a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^
2*d^2*z^2 - b^2, z, k) + d*x) - 4831838208*root(729*a^6*b^2*d^6*z^6 + 729*a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 +
27*a^2*b^2*d^2*z^2 - b^2, z, k)^4*a^3*b^3*d^4*exp(root(729*a^6*b^2*d^6*z^6 + 729*a^8*d^6*z^6 - 243*a^4*b^2*d^4
*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z, k) + d*x) - 494659436544*root(729*a^6*b^2*d^6*z^6 + 729*a^8*d^6*z^6 - 243*
a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z, k)^5*a^5*b^2*d^5*exp(root(729*a^6*b^2*d^6*z^6 + 729*a^8*d^6*z^6
- 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z, k) + d*x) + 21743271936*root(729*a^6*b^2*d^6*z^6 + 729*a
^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z, k)^6*a^5*b^3*d^6*exp(root(729*a^6*b^2*d^6*z^6
+ 729*a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z, k) + d*x) + 399532621824*root(729*a^6*b
^2*d^6*z^6 + 729*a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z, k)^7*a^7*b^2*d^7*exp(root(72
9*a^6*b^2*d^6*z^6 + 729*a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z, k) + d*x))/b^9)*root(
729*a^6*b^2*d^6*z^6 + 729*a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z, k), k, 1, 6) + log(
exp(d*x + 1/(a*d)) - 1)/(a*d) - log(exp(d*x - 1...

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