Integrand size = 23, antiderivative size = 215 \[ \int \frac {\text {csch}^4(c+d x)}{a+b \tanh ^3(c+d x)} \, dx=-\frac {\sqrt [3]{b} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \tanh (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{4/3} d}+\frac {\coth (c+d x)}{a d}-\frac {\coth ^3(c+d x)}{3 a d}-\frac {b \log (\tanh (c+d x))}{a^2 d}-\frac {\sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \tanh (c+d x)\right )}{3 a^{4/3} d}+\frac {\sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \tanh (c+d x)+b^{2/3} \tanh ^2(c+d x)\right )}{6 a^{4/3} d}+\frac {b \log \left (a+b \tanh ^3(c+d x)\right )}{3 a^2 d} \]
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)
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 \[ \int \frac {\text {csch}^4(c+d x)}{a+b \tanh ^3(c+d x)} \, dx=\frac {-a \coth (c+d x) \left (-2+\text {csch}^2(c+d x)\right )+3 b (c+d x-\log (\sinh (c+d x)))+b \text {RootSum}\left [a-b+3 a \text {$\#$1}+3 b \text {$\#$1}+3 a \text {$\#$1}^2-3 b \text {$\#$1}^2+a \text {$\#$1}^3+b \text {$\#$1}^3\&,\frac {-2 a c+2 b c-2 a d x+2 b d x+a \log \left (e^{2 (c+d x)}-\text {$\#$1}\right )-b \log \left (e^{2 (c+d x)}-\text {$\#$1}\right )-8 a c \text {$\#$1}-4 b c \text {$\#$1}-8 a d x \text {$\#$1}-4 b d x \text {$\#$1}+4 a \log \left (e^{2 (c+d x)}-\text {$\#$1}\right ) \text {$\#$1}+2 b \log \left (e^{2 (c+d x)}-\text {$\#$1}\right ) \text {$\#$1}+2 a c \text {$\#$1}^2+2 b c \text {$\#$1}^2+2 a d x \text {$\#$1}^2+2 b d x \text {$\#$1}^2-a \log \left (e^{2 (c+d x)}-\text {$\#$1}\right ) \text {$\#$1}^2-b \log \left (e^{2 (c+d x)}-\text {$\#$1}\right ) \text {$\#$1}^2}{a-b+2 a \text {$\#$1}+2 b \text {$\#$1}+a \text {$\#$1}^2-b \text {$\#$1}^2}\&\right ]}{3 a^2 d} \]
(-(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)
Time = 0.48 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.92, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3042, 4146, 2373, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\text {csch}^4(c+d x)}{a+b \tanh ^3(c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\sin (i c+i d x)^4 \left (a+i b \tan (i c+i d x)^3\right )}dx\) |
| \(\Big \downarrow \) 4146 |
| \(\displaystyle \frac {\int \frac {\coth ^4(c+d x) \left (1-\tanh ^2(c+d x)\right )}{b \tanh ^3(c+d x)+a}d\tanh (c+d x)}{d}\) |
| \(\Big \downarrow \) 2373 |
| \(\displaystyle \frac {\int \left (\frac {\coth ^4(c+d x)}{a}-\frac {\coth ^2(c+d x)}{a}-\frac {b \coth (c+d x)}{a^2}+\frac {b \tanh (c+d x) (a+b \tanh (c+d x))}{a^2 \left (b \tanh ^3(c+d x)+a\right )}\right )d\tanh (c+d x)}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {\sqrt [3]{b} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \tanh (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{4/3}}+\frac {\sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \tanh (c+d x)+b^{2/3} \tanh ^2(c+d x)\right )}{6 a^{4/3}}-\frac {\sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \tanh (c+d x)\right )}{3 a^{4/3}}+\frac {b \log \left (a+b \tanh ^3(c+d x)\right )}{3 a^2}-\frac {b \log (\tanh (c+d x))}{a^2}-\frac {\coth ^3(c+d x)}{3 a}+\frac {\coth (c+d x)}{a}}{d}\) |
(-((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))) + Coth[c + d*x]/a - Coth[c + d*x]^3/(3*a) - (b*Log[Tan h[c + d*x]])/a^2 - (b^(1/3)*Log[a^(1/3) + b^(1/3)*Tanh[c + d*x]])/(3*a^(4/ 3)) + (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)) + (b*Log[a + b*Tanh[c + d*x]^3])/(3*a^2))/d
3.1.80.3.1 Defintions of rubi rules used
Int[((Pq_)*((c_.)*(x_))^(m_.))/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[E xpandIntegrand[(c*x)^m*(Pq/(a + b*x^n)), x], x] /; FreeQ[{a, b, c, m}, x] & & PolyQ[Pq, x] && IntegerQ[n] && !IGtQ[m, 0]
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]}, Sim p[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]
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 | \(-\frac {4 \left (3 \,{\mathrm e}^{2 d x +2 c}-1\right )}{3 d a \left ({\mathrm e}^{2 d x +2 c}-1\right )^{3}}+16 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (110592 a^{6} d^{3} \textit {\_Z}^{3}-6912 a^{4} b \,d^{2} \textit {\_Z}^{2}+144 a^{2} b^{2} d \textit {\_Z} +a^{2} b -b^{3}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {4608 a^{4} d^{2} \textit {\_R}^{2}}{a^{2}+a b}+\left (\frac {96 d \,a^{3}}{a^{2}+a b}-\frac {192 a^{2} b d}{a^{2}+a b}\right ) \textit {\_R} +\frac {a^{2}}{a^{2}+a b}-\frac {3 a b}{a^{2}+a b}+\frac {2 b^{2}}{a^{2}+a b}\right )\right )-\frac {b \ln \left ({\mathrm e}^{2 d x +2 c}-1\right )}{a^{2} d}\) | \(214\) |
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=RootOf(_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)))
Result contains complex when optimal does not.
Time = 1.25 (sec) , antiderivative size = 3726, normalized size of antiderivative = 17.33 \[ \int \frac {\text {csch}^4(c+d x)}{a+b \tanh ^3(c+d x)} \, dx=\text {Too large to display} \]
-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)*sin h(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*sq rt(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 - 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*c osh(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*c...
\[ \int \frac {\text {csch}^4(c+d x)}{a+b \tanh ^3(c+d x)} \, dx=\int \frac {\operatorname {csch}^{4}{\left (c + d x \right )}}{a + b \tanh ^{3}{\left (c + d x \right )}}\, dx \]
\[ \int \frac {\text {csch}^4(c+d x)}{a+b \tanh ^3(c+d x)} \, dx=\int { \frac {\operatorname {csch}\left (d x + c\right )^{4}}{b \tanh \left (d x + c\right )^{3} + a} \,d x } \]
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*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^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)
Time = 0.37 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.84 \[ \int \frac {\text {csch}^4(c+d x)}{a+b \tanh ^3(c+d x)} \, dx=\frac {\frac {2 \, b \log \left ({\left | a e^{\left (6 \, d x + 6 \, c\right )} + b e^{\left (6 \, d x + 6 \, c\right )} + 3 \, a e^{\left (4 \, d x + 4 \, c\right )} - 3 \, b e^{\left (4 \, d x + 4 \, c\right )} + 3 \, a e^{\left (2 \, d x + 2 \, c\right )} + 3 \, b e^{\left (2 \, d x + 2 \, c\right )} + a - b \right |}\right )}{a^{2}} - \frac {6 \, b \log \left ({\left | e^{\left (2 \, d x + 2 \, c\right )} - 1 \right |}\right )}{a^{2}} + \frac {11 \, b e^{\left (6 \, d x + 6 \, c\right )} - 33 \, b e^{\left (4 \, d x + 4 \, c\right )} - 24 \, a e^{\left (2 \, d x + 2 \, c\right )} + 33 \, b e^{\left (2 \, d x + 2 \, c\right )} + 8 \, a - 11 \, b}{a^{2} {\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{3}}}{6 \, d} \]
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
Time = 3.73 (sec) , antiderivative size = 4563, normalized size of antiderivative = 21.22 \[ \int \frac {\text {csch}^4(c+d x)}{a+b \tanh ^3(c+d x)} \, dx=\text {Too large to display} \]
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)) + sym sum(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 - 1572 864*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 - 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) + 4246732 8*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 - 12091392*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...