Integrand size = 23, antiderivative size = 295 \[ \int \frac {\sinh ^5(c+d x)}{a+b \sinh ^3(c+d x)} \, dx=-\frac {x}{2 b}+\frac {2 a \arctan \left (\frac {(-1)^{5/6} \left (\sqrt [6]{-1} \sqrt [3]{b}+i \sqrt [3]{a} \tanh \left (\frac {1}{2} (c+d x)\right )\right )}{\sqrt {-(-1)^{2/3} a^{2/3}-b^{2/3}}}\right )}{3 \sqrt {-(-1)^{2/3} a^{2/3}-b^{2/3}} b^{5/3} d}+\frac {2 a \arctan \left (\frac {\sqrt [6]{-1} \left ((-1)^{5/6} \sqrt [3]{b}+i \sqrt [3]{a} \tanh \left (\frac {1}{2} (c+d x)\right )\right )}{\sqrt {\sqrt [3]{-1} a^{2/3}-b^{2/3}}}\right )}{3 \sqrt {\sqrt [3]{-1} a^{2/3}-b^{2/3}} b^{5/3} d}+\frac {2 a \text {arctanh}\left (\frac {\sqrt [3]{b}-\sqrt [3]{a} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^{2/3}+b^{2/3}}}\right )}{3 \sqrt {a^{2/3}+b^{2/3}} b^{5/3} d}+\frac {\cosh (c+d x) \sinh (c+d x)}{2 b d} \]
-1/2*x/b+1/2*cosh(d*x+c)*sinh(d*x+c)/b/d+2/3*a*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))/b^(5/3)/d/((-1)^(1/3)*a^(2/3)-b^(2/3))^(1/2)+2/3*a*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))/b^(5/3)/d/(-(-1)^(2/3)*a^(2/3)-b^(2/3))^(1/2)+2/3*a*arctanh(( b^(1/3)-a^(1/3)*tanh(1/2*d*x+1/2*c))/(a^(2/3)+b^(2/3))^(1/2))/b^(5/3)/d/(a ^(2/3)+b^(2/3))^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.84 (sec) , antiderivative size = 299, normalized size of antiderivative = 1.01 \[ \int \frac {\sinh ^5(c+d x)}{a+b \sinh ^3(c+d x)} \, dx=\frac {-6 (c+d x)-2 a \text {RootSum}\left [-b+3 b \text {$\#$1}^2+8 a \text {$\#$1}^3-3 b \text {$\#$1}^4+b \text {$\#$1}^6\&,\frac {c+d x+2 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right )-2 c \text {$\#$1}^2-2 d x \text {$\#$1}^2-4 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^2+c \text {$\#$1}^4+d x \text {$\#$1}^4+2 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^4}{b \text {$\#$1}+4 a \text {$\#$1}^2-2 b \text {$\#$1}^3+b \text {$\#$1}^5}\&\right ]+3 \sinh (2 (c+d x))}{12 b d} \]
(-6*(c + d*x) - 2*a*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*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 + 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) & ] + 3*Sinh[2*(c + d*x)])/(12*b*d)
Time = 0.83 (sec) , antiderivative size = 309, normalized size of antiderivative = 1.05, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3042, 26, 3699, 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.
| \(\displaystyle \int \frac {\sinh ^5(c+d x)}{a+b \sinh ^3(c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {i \sin (i c+i d x)^5}{a+i b \sin (i c+i d x)^3}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i \int \frac {\sin (i c+i d x)^5}{i b \sin (i c+i d x)^3+a}dx\) |
| \(\Big \downarrow \) 3699 |
| \(\displaystyle -i \int \left (\frac {i \sinh ^2(c+d x)}{b}-\frac {i a \sinh ^2(c+d x)}{b \left (b \sinh ^3(c+d x)+a\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -i \left (\frac {2 i a \arctan \left (\frac {(-1)^{5/6} \left (\sqrt [6]{-1} \sqrt [3]{b}+i \sqrt [3]{a} \tanh \left (\frac {1}{2} (c+d x)\right )\right )}{\sqrt {-(-1)^{2/3} a^{2/3}-b^{2/3}}}\right )}{3 b^{5/3} d \sqrt {-(-1)^{2/3} a^{2/3}-b^{2/3}}}+\frac {2 i a \arctan \left (\frac {\sqrt [6]{-1} \left ((-1)^{5/6} \sqrt [3]{b}+i \sqrt [3]{a} \tanh \left (\frac {1}{2} (c+d x)\right )\right )}{\sqrt {\sqrt [3]{-1} a^{2/3}-b^{2/3}}}\right )}{3 b^{5/3} d \sqrt {\sqrt [3]{-1} a^{2/3}-b^{2/3}}}+\frac {2 i a \text {arctanh}\left (\frac {\sqrt [3]{b}-\sqrt [3]{a} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^{2/3}+b^{2/3}}}\right )}{3 b^{5/3} d \sqrt {a^{2/3}+b^{2/3}}}+\frac {i \sinh (c+d x) \cosh (c+d x)}{2 b d}-\frac {i x}{2 b}\right )\) |
(-I)*(((-1/2*I)*x)/b + (((2*I)/3)*a*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)]])/( Sqrt[-((-1)^(2/3)*a^(2/3)) - b^(2/3)]*b^(5/3)*d) + (((2*I)/3)*a*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)]])/(Sqrt[(-1)^(1/3)*a^(2/3) - b^(2/3)]*b^(5/3)*d) + ( ((2*I)/3)*a*ArcTanh[(b^(1/3) - a^(1/3)*Tanh[(c + d*x)/2])/Sqrt[a^(2/3) + b ^(2/3)]])/(Sqrt[a^(2/3) + b^(2/3)]*b^(5/3)*d) + ((I/2)*Cosh[c + d*x]*Sinh[ c + d*x])/(b*d))
3.2.72.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^(n_ ))^(p_.), x_Symbol] :> Int[ExpandTrig[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] || Gt Q[p, 0] || (EqQ[p, -1] && IntegerQ[n]))
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.72 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.64
| method | result | size |
| derivativedivides | \(\frac {\frac {4 a \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a \,\textit {\_Z}^{6}-3 a \,\textit {\_Z}^{4}-8 b \,\textit {\_Z}^{3}+3 a \,\textit {\_Z}^{2}-a \right )}{\sum }\frac {\textit {\_R}^{2} \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5} a -2 \textit {\_R}^{3} a -4 \textit {\_R}^{2} b +\textit {\_R} a}\right )}{3 b}+\frac {1}{2 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {1}{2 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 b}-\frac {1}{2 b \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {1}{2 b \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}-\frac {\ln \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 b}}{d}\) | \(190\) |
| default | \(\frac {\frac {4 a \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a \,\textit {\_Z}^{6}-3 a \,\textit {\_Z}^{4}-8 b \,\textit {\_Z}^{3}+3 a \,\textit {\_Z}^{2}-a \right )}{\sum }\frac {\textit {\_R}^{2} \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5} a -2 \textit {\_R}^{3} a -4 \textit {\_R}^{2} b +\textit {\_R} a}\right )}{3 b}+\frac {1}{2 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {1}{2 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 b}-\frac {1}{2 b \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {1}{2 b \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}-\frac {\ln \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 b}}{d}\) | \(190\) |
| risch | \(-\frac {x}{2 b}+\frac {{\mathrm e}^{2 d x +2 c}}{8 b d}-\frac {{\mathrm e}^{-2 d x -2 c}}{8 b d}+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (729 a^{2} b^{10} d^{6}+729 b^{12} d^{6}\right ) \textit {\_Z}^{6}-243 a^{2} b^{8} d^{4} \textit {\_Z}^{4}+27 a^{4} b^{4} d^{2} \textit {\_Z}^{2}-a^{6}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{d x +c}+\left (\frac {243 d^{5} b^{8}}{a^{3}}+\frac {243 d^{5} b^{10}}{a^{5}}\right ) \textit {\_R}^{5}+\left (\frac {81 b^{7} d^{4}}{a^{3}}+\frac {81 b^{9} d^{4}}{a^{5}}\right ) \textit {\_R}^{4}-\frac {81 d^{3} b^{6} \textit {\_R}^{3}}{a^{3}}+\left (\frac {9 d^{2} b^{3}}{a}-\frac {18 d^{2} b^{5}}{a^{3}}\right ) \textit {\_R}^{2}+\frac {9 d \,b^{2} \textit {\_R}}{a}+\frac {b}{a}\right )\right )\) | \(224\) |
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))+1/2/b/(tanh(1/2*d*x +1/2*c)-1)^2+1/2/b/(tanh(1/2*d*x+1/2*c)-1)+1/2/b*ln(tanh(1/2*d*x+1/2*c)-1) -1/2/b/(1+tanh(1/2*d*x+1/2*c))^2+1/2/b/(1+tanh(1/2*d*x+1/2*c))-1/2/b*ln(1+ tanh(1/2*d*x+1/2*c)))
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 28427 vs. \(2 (210) = 420\).
Time = 2.11 (sec) , antiderivative size = 28427, normalized size of antiderivative = 96.36 \[ \int \frac {\sinh ^5(c+d x)}{a+b \sinh ^3(c+d x)} \, dx=\text {Too large to display} \]
\[ \int \frac {\sinh ^5(c+d x)}{a+b \sinh ^3(c+d x)} \, dx=\int \frac {\sinh ^{5}{\left (c + d x \right )}}{a + b \sinh ^{3}{\left (c + d x \right )}}\, dx \]
\[ \int \frac {\sinh ^5(c+d x)}{a+b \sinh ^3(c+d x)} \, dx=\int { \frac {\sinh \left (d x + c\right )^{5}}{b \sinh \left (d x + c\right )^{3} + a} \,d x } \]
-1/8*(4*d*x*e^(2*d*x + 2*c) - e^(4*d*x + 4*c) + 1)*e^(-2*d*x - 2*c)/(b*d) - 1/32*integrate(64*(a*e^(5*d*x + 5*c) - 2*a*e^(3*d*x + 3*c) + a*e^(d*x + c))/(b^2*e^(6*d*x + 6*c) - 3*b^2*e^(4*d*x + 4*c) + 8*a*b*e^(3*d*x + 3*c) + 3*b^2*e^(2*d*x + 2*c) - b^2), x)
\[ \int \frac {\sinh ^5(c+d x)}{a+b \sinh ^3(c+d x)} \, dx=\int { \frac {\sinh \left (d x + c\right )^{5}}{b \sinh \left (d x + c\right )^{3} + a} \,d x } \]
Time = 12.30 (sec) , antiderivative size = 1114, normalized size of antiderivative = 3.78 \[ \int \frac {\sinh ^5(c+d x)}{a+b \sinh ^3(c+d x)} \, dx=\left (\sum _{k=1}^6\ln \left (-\mathrm {root}\left (729\,a^2\,b^{10}\,d^6\,z^6+729\,b^{12}\,d^6\,z^6-243\,a^2\,b^8\,d^4\,z^4+27\,a^4\,b^4\,d^2\,z^2-a^6,z,k\right )\,\left (\mathrm {root}\left (729\,a^2\,b^{10}\,d^6\,z^6+729\,b^{12}\,d^6\,z^6-243\,a^2\,b^8\,d^4\,z^4+27\,a^4\,b^4\,d^2\,z^2-a^6,z,k\right )\,\left (\mathrm {root}\left (729\,a^2\,b^{10}\,d^6\,z^6+729\,b^{12}\,d^6\,z^6-243\,a^2\,b^8\,d^4\,z^4+27\,a^4\,b^4\,d^2\,z^2-a^6,z,k\right )\,\left (\mathrm {root}\left (729\,a^2\,b^{10}\,d^6\,z^6+729\,b^{12}\,d^6\,z^6-243\,a^2\,b^8\,d^4\,z^4+27\,a^4\,b^4\,d^2\,z^2-a^6,z,k\right )\,\left (\frac {\left (8\,a^6\,d^4+4\,a^4\,b^2\,d^4-a^5\,b\,d^4\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^{\mathrm {root}\left (729\,a^2\,b^{10}\,d^6\,z^6+729\,b^{12}\,d^6\,z^6-243\,a^2\,b^8\,d^4\,z^4+27\,a^4\,b^4\,d^2\,z^2-a^6,z,k\right )}\,5\right )\,663552}{b^7}+\frac {\mathrm {root}\left (729\,a^2\,b^{10}\,d^6\,z^6+729\,b^{12}\,d^6\,z^6-243\,a^2\,b^8\,d^4\,z^4+27\,a^4\,b^4\,d^2\,z^2-a^6,z,k\right )\,\left (-a^4\,b\,d^5+a^5\,d^5\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^{\mathrm {root}\left (729\,a^2\,b^{10}\,d^6\,z^6+729\,b^{12}\,d^6\,z^6-243\,a^2\,b^8\,d^4\,z^4+27\,a^4\,b^4\,d^2\,z^2-a^6,z,k\right )}\,4+a^3\,b^2\,d^5\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^{\mathrm {root}\left (729\,a^2\,b^{10}\,d^6\,z^6+729\,b^{12}\,d^6\,z^6-243\,a^2\,b^8\,d^4\,z^4+27\,a^4\,b^4\,d^2\,z^2-a^6,z,k\right )}\,5\right )\,1990656}{b^5}\right )+\frac {\left (4\,a^6\,b\,d^3+a^7\,d^3\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^{\mathrm {root}\left (729\,a^2\,b^{10}\,d^6\,z^6+729\,b^{12}\,d^6\,z^6-243\,a^2\,b^8\,d^4\,z^4+27\,a^4\,b^4\,d^2\,z^2-a^6,z,k\right )}\,8-a^5\,b^2\,d^3\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^{\mathrm {root}\left (729\,a^2\,b^{10}\,d^6\,z^6+729\,b^{12}\,d^6\,z^6-243\,a^2\,b^8\,d^4\,z^4+27\,a^4\,b^4\,d^2\,z^2-a^6,z,k\right )}\,5\right )\,442368}{b^9}\right )-\frac {a^6\,d^2\,\left (2\,b-a\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^{\mathrm {root}\left (729\,a^2\,b^{10}\,d^6\,z^6+729\,b^{12}\,d^6\,z^6-243\,a^2\,b^8\,d^4\,z^4+27\,a^4\,b^4\,d^2\,z^2-a^6,z,k\right )}\,5\right )\,294912}{b^{10}}\right )-\frac {a^7\,d\,\left (8\,a-b\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^{\mathrm {root}\left (729\,a^2\,b^{10}\,d^6\,z^6+729\,b^{12}\,d^6\,z^6-243\,a^2\,b^8\,d^4\,z^4+27\,a^4\,b^4\,d^2\,z^2-a^6,z,k\right )}\,5\right )\,24576}{b^{12}}\right )-\frac {a^8\,\left (b-a\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^{\mathrm {root}\left (729\,a^2\,b^{10}\,d^6\,z^6+729\,b^{12}\,d^6\,z^6-243\,a^2\,b^8\,d^4\,z^4+27\,a^4\,b^4\,d^2\,z^2-a^6,z,k\right )}\,4\right )\,32768}{b^{14}}\right )\,\mathrm {root}\left (729\,a^2\,b^{10}\,d^6\,z^6+729\,b^{12}\,d^6\,z^6-243\,a^2\,b^8\,d^4\,z^4+27\,a^4\,b^4\,d^2\,z^2-a^6,z,k\right )\right )-\frac {x}{2\,b}-\frac {{\mathrm {e}}^{-2\,c-2\,d\,x}}{8\,b\,d}+\frac {{\mathrm {e}}^{2\,c+2\,d\,x}}{8\,b\,d} \]
symsum(log(- root(729*a^2*b^10*d^6*z^6 + 729*b^12*d^6*z^6 - 243*a^2*b^8*d^ 4*z^4 + 27*a^4*b^4*d^2*z^2 - a^6, z, k)*(root(729*a^2*b^10*d^6*z^6 + 729*b ^12*d^6*z^6 - 243*a^2*b^8*d^4*z^4 + 27*a^4*b^4*d^2*z^2 - a^6, z, k)*(root( 729*a^2*b^10*d^6*z^6 + 729*b^12*d^6*z^6 - 243*a^2*b^8*d^4*z^4 + 27*a^4*b^4 *d^2*z^2 - a^6, z, k)*(root(729*a^2*b^10*d^6*z^6 + 729*b^12*d^6*z^6 - 243* a^2*b^8*d^4*z^4 + 27*a^4*b^4*d^2*z^2 - a^6, z, k)*((663552*(8*a^6*d^4 + 4* a^4*b^2*d^4 - 5*a^5*b*d^4*exp(d*x)*exp(root(729*a^2*b^10*d^6*z^6 + 729*b^1 2*d^6*z^6 - 243*a^2*b^8*d^4*z^4 + 27*a^4*b^4*d^2*z^2 - a^6, z, k))))/b^7 + (1990656*root(729*a^2*b^10*d^6*z^6 + 729*b^12*d^6*z^6 - 243*a^2*b^8*d^4*z ^4 + 27*a^4*b^4*d^2*z^2 - a^6, z, k)*(4*a^5*d^5*exp(d*x)*exp(root(729*a^2* b^10*d^6*z^6 + 729*b^12*d^6*z^6 - 243*a^2*b^8*d^4*z^4 + 27*a^4*b^4*d^2*z^2 - a^6, z, k)) - a^4*b*d^5 + 5*a^3*b^2*d^5*exp(d*x)*exp(root(729*a^2*b^10* d^6*z^6 + 729*b^12*d^6*z^6 - 243*a^2*b^8*d^4*z^4 + 27*a^4*b^4*d^2*z^2 - a^ 6, z, k))))/b^5) + (442368*(4*a^6*b*d^3 + 8*a^7*d^3*exp(d*x)*exp(root(729* a^2*b^10*d^6*z^6 + 729*b^12*d^6*z^6 - 243*a^2*b^8*d^4*z^4 + 27*a^4*b^4*d^2 *z^2 - a^6, z, k)) - 5*a^5*b^2*d^3*exp(d*x)*exp(root(729*a^2*b^10*d^6*z^6 + 729*b^12*d^6*z^6 - 243*a^2*b^8*d^4*z^4 + 27*a^4*b^4*d^2*z^2 - a^6, z, k) )))/b^9) - (294912*a^6*d^2*(2*b - 5*a*exp(d*x)*exp(root(729*a^2*b^10*d^6*z ^6 + 729*b^12*d^6*z^6 - 243*a^2*b^8*d^4*z^4 + 27*a^4*b^4*d^2*z^2 - a^6, z, k))))/b^10) - (24576*a^7*d*(8*a - 5*b*exp(d*x)*exp(root(729*a^2*b^10*d...