Integrand size = 24, antiderivative size = 229 \[ \int \frac {\csc ^5(c+d x)}{a-b \sin ^4(c+d x)} \, dx=-\frac {b^{5/4} \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 a^2 \sqrt {\sqrt {a}-\sqrt {b}} d}-\frac {(3 a+8 b) \text {arctanh}(\cos (c+d x))}{8 a^2 d}+\frac {b^{5/4} \text {arctanh}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 a^2 \sqrt {\sqrt {a}+\sqrt {b}} d}-\frac {1}{16 a d (1-\cos (c+d x))^2}-\frac {3}{16 a d (1-\cos (c+d x))}+\frac {1}{16 a d (1+\cos (c+d x))^2}+\frac {3}{16 a d (1+\cos (c+d x))} \]
-1/8*(3*a+8*b)*arctanh(cos(d*x+c))/a^2/d-1/16/a/d/(1-cos(d*x+c))^2-3/16/a/ d/(1-cos(d*x+c))+1/16/a/d/(1+cos(d*x+c))^2+3/16/a/d/(1+cos(d*x+c))-1/2*b^( 5/4)*arctan(b^(1/4)*cos(d*x+c)/(a^(1/2)-b^(1/2))^(1/2))/a^2/d/(a^(1/2)-b^( 1/2))^(1/2)+1/2*b^(5/4)*arctanh(b^(1/4)*cos(d*x+c)/(a^(1/2)+b^(1/2))^(1/2) )/a^2/d/(a^(1/2)+b^(1/2))^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 1.33 (sec) , antiderivative size = 409, normalized size of antiderivative = 1.79 \[ \int \frac {\csc ^5(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\frac {-6 a \csc ^2\left (\frac {1}{2} (c+d x)\right )-a \csc ^4\left (\frac {1}{2} (c+d x)\right )-24 a \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-64 b \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+24 a \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+64 b \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-8 i b^2 \text {RootSum}\left [b-4 b \text {$\#$1}^2-16 a \text {$\#$1}^4+6 b \text {$\#$1}^4-4 b \text {$\#$1}^6+b \text {$\#$1}^8\&,\frac {-2 \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )+i \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right )+6 \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^2-3 i \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^2-6 \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^4+3 i \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^4+2 \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^6-i \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^6}{-b \text {$\#$1}-8 a \text {$\#$1}^3+3 b \text {$\#$1}^3-3 b \text {$\#$1}^5+b \text {$\#$1}^7}\&\right ]+6 a \sec ^2\left (\frac {1}{2} (c+d x)\right )+a \sec ^4\left (\frac {1}{2} (c+d x)\right )}{64 a^2 d} \]
(-6*a*Csc[(c + d*x)/2]^2 - a*Csc[(c + d*x)/2]^4 - 24*a*Log[Cos[(c + d*x)/2 ]] - 64*b*Log[Cos[(c + d*x)/2]] + 24*a*Log[Sin[(c + d*x)/2]] + 64*b*Log[Si n[(c + d*x)/2]] - (8*I)*b^2*RootSum[b - 4*b*#1^2 - 16*a*#1^4 + 6*b*#1^4 - 4*b*#1^6 + b*#1^8 & , (-2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)] + I*Log [1 - 2*Cos[c + d*x]*#1 + #1^2] + 6*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1) ]*#1^2 - (3*I)*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 - 6*ArcTan[Sin[c + d *x]/(Cos[c + d*x] - #1)]*#1^4 + (3*I)*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1 ^4 + 2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^6 - I*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^6)/(-(b*#1) - 8*a*#1^3 + 3*b*#1^3 - 3*b*#1^5 + b*#1^7) & ] + 6*a*Sec[(c + d*x)/2]^2 + a*Sec[(c + d*x)/2]^4)/(64*a^2*d)
Time = 0.41 (sec) , antiderivative size = 213, normalized size of antiderivative = 0.93, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3042, 3694, 1484, 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.
| \(\displaystyle \int \frac {\csc ^5(c+d x)}{a-b \sin ^4(c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\sin (c+d x)^5 \left (a-b \sin (c+d x)^4\right )}dx\) |
| \(\Big \downarrow \) 3694 |
| \(\displaystyle -\frac {\int \frac {1}{\left (1-\cos ^2(c+d x)\right )^3 \left (-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)+a-b\right )}d\cos (c+d x)}{d}\) |
| \(\Big \downarrow \) 1484 |
| \(\displaystyle -\frac {\int \left (-\frac {\left (\cos ^2(c+d x)-1\right ) b^2}{a^2 \left (-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)+a-b\right )}+\frac {-3 a-8 b}{8 a^2 \left (\cos ^2(c+d x)-1\right )}+\frac {3}{16 a (\cos (c+d x)-1)^2}+\frac {3}{16 a (\cos (c+d x)+1)^2}-\frac {1}{8 a (\cos (c+d x)-1)^3}+\frac {1}{8 a (\cos (c+d x)+1)^3}\right )d\cos (c+d x)}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\frac {b^{5/4} \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 a^2 \sqrt {\sqrt {a}-\sqrt {b}}}-\frac {b^{5/4} \text {arctanh}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 a^2 \sqrt {\sqrt {a}+\sqrt {b}}}+\frac {(3 a+8 b) \text {arctanh}(\cos (c+d x))}{8 a^2}+\frac {3}{16 a (1-\cos (c+d x))}-\frac {3}{16 a (\cos (c+d x)+1)}+\frac {1}{16 a (1-\cos (c+d x))^2}-\frac {1}{16 a (\cos (c+d x)+1)^2}}{d}\) |
-(((b^(5/4)*ArcTan[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(2*a^2 *Sqrt[Sqrt[a] - Sqrt[b]]) + ((3*a + 8*b)*ArcTanh[Cos[c + d*x]])/(8*a^2) - (b^(5/4)*ArcTanh[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]])/(2*a^2*S qrt[Sqrt[a] + Sqrt[b]]) + 1/(16*a*(1 - Cos[c + d*x])^2) + 3/(16*a*(1 - Cos [c + d*x])) - 1/(16*a*(1 + Cos[c + d*x])^2) - 3/(16*a*(1 + Cos[c + d*x]))) /d)
3.3.2.3.1 Defintions of rubi rules used
Int[((d_) + (e_.)*(x_)^2)^(q_)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symb ol] :> Int[ExpandIntegrand[(d + e*x^2)^q/(a + b*x^2 + c*x^4), x], x] /; Fre eQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[q]
Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^4)^ (p_.), x_Symbol] :> With[{ff = FreeFactors[Cos[e + f*x], x]}, Simp[-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]
Time = 1.93 (sec) , antiderivative size = 193, normalized size of antiderivative = 0.84
| method | result | size |
| derivativedivides | \(\frac {\frac {1}{16 a \left (1+\cos \left (d x +c \right )\right )^{2}}+\frac {3}{16 a \left (1+\cos \left (d x +c \right )\right )}+\frac {\left (-3 a -8 b \right ) \ln \left (1+\cos \left (d x +c \right )\right )}{16 a^{2}}-\frac {b^{3} \left (\frac {\arctan \left (\frac {\cos \left (d x +c \right ) b}{\sqrt {\left (\sqrt {a b}-b \right ) b}}\right )}{2 b \sqrt {\left (\sqrt {a b}-b \right ) b}}-\frac {\operatorname {arctanh}\left (\frac {\cos \left (d x +c \right ) b}{\sqrt {\left (\sqrt {a b}+b \right ) b}}\right )}{2 b \sqrt {\left (\sqrt {a b}+b \right ) b}}\right )}{a^{2}}-\frac {1}{16 a \left (\cos \left (d x +c \right )-1\right )^{2}}+\frac {3}{16 a \left (\cos \left (d x +c \right )-1\right )}+\frac {\left (3 a +8 b \right ) \ln \left (\cos \left (d x +c \right )-1\right )}{16 a^{2}}}{d}\) | \(193\) |
| default | \(\frac {\frac {1}{16 a \left (1+\cos \left (d x +c \right )\right )^{2}}+\frac {3}{16 a \left (1+\cos \left (d x +c \right )\right )}+\frac {\left (-3 a -8 b \right ) \ln \left (1+\cos \left (d x +c \right )\right )}{16 a^{2}}-\frac {b^{3} \left (\frac {\arctan \left (\frac {\cos \left (d x +c \right ) b}{\sqrt {\left (\sqrt {a b}-b \right ) b}}\right )}{2 b \sqrt {\left (\sqrt {a b}-b \right ) b}}-\frac {\operatorname {arctanh}\left (\frac {\cos \left (d x +c \right ) b}{\sqrt {\left (\sqrt {a b}+b \right ) b}}\right )}{2 b \sqrt {\left (\sqrt {a b}+b \right ) b}}\right )}{a^{2}}-\frac {1}{16 a \left (\cos \left (d x +c \right )-1\right )^{2}}+\frac {3}{16 a \left (\cos \left (d x +c \right )-1\right )}+\frac {\left (3 a +8 b \right ) \ln \left (\cos \left (d x +c \right )-1\right )}{16 a^{2}}}{d}\) | \(193\) |
| risch | \(\frac {3 \,{\mathrm e}^{7 i \left (d x +c \right )}-11 \,{\mathrm e}^{5 i \left (d x +c \right )}-11 \,{\mathrm e}^{3 i \left (d x +c \right )}+3 \,{\mathrm e}^{i \left (d x +c \right )}}{4 d a \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{4}}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{8 d a}+\frac {b \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{a^{2} d}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{8 d a}-\frac {b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{a^{2} d}+32 i \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (268435456 a^{9} d^{4}-268435456 a^{8} b \,d^{4}\right ) \textit {\_Z}^{4}-32768 a^{4} b^{3} d^{2} \textit {\_Z}^{2}-b^{5}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\left (\left (-\frac {4194304 i d^{3} a^{7}}{b^{4}}+\frac {4194304 i a^{6} d^{3}}{b^{3}}\right ) \textit {\_R}^{3}+\frac {512 i a^{2} d \textit {\_R}}{b}\right ) {\mathrm e}^{i \left (d x +c \right )}+1\right )\right )\) | \(263\) |
1/d*(1/16/a/(1+cos(d*x+c))^2+3/16/a/(1+cos(d*x+c))+1/16/a^2*(-3*a-8*b)*ln( 1+cos(d*x+c))-b^3/a^2*(1/2/b/(((a*b)^(1/2)-b)*b)^(1/2)*arctan(cos(d*x+c)*b /(((a*b)^(1/2)-b)*b)^(1/2))-1/2/b/(((a*b)^(1/2)+b)*b)^(1/2)*arctanh(cos(d* x+c)*b/(((a*b)^(1/2)+b)*b)^(1/2)))-1/16/a/(cos(d*x+c)-1)^2+3/16/a/(cos(d*x +c)-1)+1/16*(3*a+8*b)/a^2*ln(cos(d*x+c)-1))
Leaf count of result is larger than twice the leaf count of optimal. 1089 vs. \(2 (179) = 358\).
Time = 0.43 (sec) , antiderivative size = 1089, normalized size of antiderivative = 4.76 \[ \int \frac {\csc ^5(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\text {Too large to display} \]
1/16*(6*a*cos(d*x + c)^3 + 4*(a^2*d*cos(d*x + c)^4 - 2*a^2*d*cos(d*x + c)^ 2 + a^2*d)*sqrt(-((a^5 - a^4*b)*d^2*sqrt(b^5/((a^9 - 2*a^8*b + a^7*b^2)*d^ 4)) + b^3)/((a^5 - a^4*b)*d^2))*log(b^4*cos(d*x + c) + (a^2*b^3*d - (a^7 - a^6*b)*d^3*sqrt(b^5/((a^9 - 2*a^8*b + a^7*b^2)*d^4)))*sqrt(-((a^5 - a^4*b )*d^2*sqrt(b^5/((a^9 - 2*a^8*b + a^7*b^2)*d^4)) + b^3)/((a^5 - a^4*b)*d^2) )) - 4*(a^2*d*cos(d*x + c)^4 - 2*a^2*d*cos(d*x + c)^2 + a^2*d)*sqrt(((a^5 - a^4*b)*d^2*sqrt(b^5/((a^9 - 2*a^8*b + a^7*b^2)*d^4)) - b^3)/((a^5 - a^4* b)*d^2))*log(b^4*cos(d*x + c) - (a^2*b^3*d + (a^7 - a^6*b)*d^3*sqrt(b^5/(( a^9 - 2*a^8*b + a^7*b^2)*d^4)))*sqrt(((a^5 - a^4*b)*d^2*sqrt(b^5/((a^9 - 2 *a^8*b + a^7*b^2)*d^4)) - b^3)/((a^5 - a^4*b)*d^2))) - 4*(a^2*d*cos(d*x + c)^4 - 2*a^2*d*cos(d*x + c)^2 + a^2*d)*sqrt(-((a^5 - a^4*b)*d^2*sqrt(b^5/( (a^9 - 2*a^8*b + a^7*b^2)*d^4)) + b^3)/((a^5 - a^4*b)*d^2))*log(-b^4*cos(d *x + c) + (a^2*b^3*d - (a^7 - a^6*b)*d^3*sqrt(b^5/((a^9 - 2*a^8*b + a^7*b^ 2)*d^4)))*sqrt(-((a^5 - a^4*b)*d^2*sqrt(b^5/((a^9 - 2*a^8*b + a^7*b^2)*d^4 )) + b^3)/((a^5 - a^4*b)*d^2))) + 4*(a^2*d*cos(d*x + c)^4 - 2*a^2*d*cos(d* x + c)^2 + a^2*d)*sqrt(((a^5 - a^4*b)*d^2*sqrt(b^5/((a^9 - 2*a^8*b + a^7*b ^2)*d^4)) - b^3)/((a^5 - a^4*b)*d^2))*log(-b^4*cos(d*x + c) - (a^2*b^3*d + (a^7 - a^6*b)*d^3*sqrt(b^5/((a^9 - 2*a^8*b + a^7*b^2)*d^4)))*sqrt(((a^5 - a^4*b)*d^2*sqrt(b^5/((a^9 - 2*a^8*b + a^7*b^2)*d^4)) - b^3)/((a^5 - a^4*b )*d^2))) - 10*a*cos(d*x + c) - ((3*a + 8*b)*cos(d*x + c)^4 - 2*(3*a + 8...
\[ \int \frac {\csc ^5(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\int \frac {\csc ^{5}{\left (c + d x \right )}}{a - b \sin ^{4}{\left (c + d x \right )}}\, dx \]
\[ \int \frac {\csc ^5(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\int { -\frac {\csc \left (d x + c\right )^{5}}{b \sin \left (d x + c\right )^{4} - a} \,d x } \]
-1/16*(48*a*cos(2*d*x + 2*c)*cos(d*x + c) - 176*a*sin(3*d*x + 3*c)*sin(2*d *x + 2*c) + 48*a*sin(2*d*x + 2*c)*sin(d*x + c) - 4*(3*a*cos(7*d*x + 7*c) - 11*a*cos(5*d*x + 5*c) - 11*a*cos(3*d*x + 3*c) + 3*a*cos(d*x + c))*cos(8*d *x + 8*c) + 12*(4*a*cos(6*d*x + 6*c) - 6*a*cos(4*d*x + 4*c) + 4*a*cos(2*d* x + 2*c) - a)*cos(7*d*x + 7*c) - 16*(11*a*cos(5*d*x + 5*c) + 11*a*cos(3*d* x + 3*c) - 3*a*cos(d*x + c))*cos(6*d*x + 6*c) + 44*(6*a*cos(4*d*x + 4*c) - 4*a*cos(2*d*x + 2*c) + a)*cos(5*d*x + 5*c) + 24*(11*a*cos(3*d*x + 3*c) - 3*a*cos(d*x + c))*cos(4*d*x + 4*c) - 44*(4*a*cos(2*d*x + 2*c) - a)*cos(3*d *x + 3*c) - 12*a*cos(d*x + c) + 16*(a^2*d*cos(8*d*x + 8*c)^2 + 16*a^2*d*co s(6*d*x + 6*c)^2 + 36*a^2*d*cos(4*d*x + 4*c)^2 + 16*a^2*d*cos(2*d*x + 2*c) ^2 + a^2*d*sin(8*d*x + 8*c)^2 + 16*a^2*d*sin(6*d*x + 6*c)^2 + 36*a^2*d*sin (4*d*x + 4*c)^2 - 48*a^2*d*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 16*a^2*d*si n(2*d*x + 2*c)^2 - 8*a^2*d*cos(2*d*x + 2*c) + a^2*d - 2*(4*a^2*d*cos(6*d*x + 6*c) - 6*a^2*d*cos(4*d*x + 4*c) + 4*a^2*d*cos(2*d*x + 2*c) - a^2*d)*cos (8*d*x + 8*c) - 8*(6*a^2*d*cos(4*d*x + 4*c) - 4*a^2*d*cos(2*d*x + 2*c) + a ^2*d)*cos(6*d*x + 6*c) - 12*(4*a^2*d*cos(2*d*x + 2*c) - a^2*d)*cos(4*d*x + 4*c) - 4*(2*a^2*d*sin(6*d*x + 6*c) - 3*a^2*d*sin(4*d*x + 4*c) + 2*a^2*d*s in(2*d*x + 2*c))*sin(8*d*x + 8*c) - 16*(3*a^2*d*sin(4*d*x + 4*c) - 2*a^2*d *sin(2*d*x + 2*c))*sin(6*d*x + 6*c))*integrate(-2*(12*b^3*cos(3*d*x + 3*c) *sin(2*d*x + 2*c) - 4*b^3*cos(d*x + c)*sin(2*d*x + 2*c) + 4*b^3*cos(2*d...
\[ \int \frac {\csc ^5(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\int { -\frac {\csc \left (d x + c\right )^{5}}{b \sin \left (d x + c\right )^{4} - a} \,d x } \]
Time = 15.24 (sec) , antiderivative size = 3692, normalized size of antiderivative = 16.12 \[ \int \frac {\csc ^5(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\text {Too large to display} \]
(atan(((((768*a^3*b^8 - 144*a^5*b^6)/(64*a^5) + (((10240*a^8*b^5 - 12288*a ^7*b^6 + 6144*a^9*b^4)/(64*a^5) - (cos(c + d*x)*(12288*a^8*b^5 - 8192*a^9* b^4)*(((a^9*b^5)^(1/2) + a^4*b^3)/(16*(a^8*b - a^9)))^(1/2))/(16*a^4))*((( a^9*b^5)^(1/2) + a^4*b^3)/(16*(a^8*b - a^9)))^(1/2) + (cos(c + d*x)*(2304* a^4*b^7 + 768*a^5*b^6 + 144*a^6*b^5))/(16*a^4))*(((a^9*b^5)^(1/2) + a^4*b^ 3)/(16*(a^8*b - a^9)))^(1/2))*(((a^9*b^5)^(1/2) + a^4*b^3)/(16*(a^8*b - a^ 9)))^(1/2) - (cos(c + d*x)*(48*a*b^8 + 96*b^9 + 9*a^2*b^7))/(16*a^4))*(((a ^9*b^5)^(1/2) + a^4*b^3)/(16*(a^8*b - a^9)))^(1/2)*1i - (((768*a^3*b^8 - 1 44*a^5*b^6)/(64*a^5) + (((10240*a^8*b^5 - 12288*a^7*b^6 + 6144*a^9*b^4)/(6 4*a^5) + (cos(c + d*x)*(12288*a^8*b^5 - 8192*a^9*b^4)*(((a^9*b^5)^(1/2) + a^4*b^3)/(16*(a^8*b - a^9)))^(1/2))/(16*a^4))*(((a^9*b^5)^(1/2) + a^4*b^3) /(16*(a^8*b - a^9)))^(1/2) - (cos(c + d*x)*(2304*a^4*b^7 + 768*a^5*b^6 + 1 44*a^6*b^5))/(16*a^4))*(((a^9*b^5)^(1/2) + a^4*b^3)/(16*(a^8*b - a^9)))^(1 /2))*(((a^9*b^5)^(1/2) + a^4*b^3)/(16*(a^8*b - a^9)))^(1/2) + (cos(c + d*x )*(48*a*b^8 + 96*b^9 + 9*a^2*b^7))/(16*a^4))*(((a^9*b^5)^(1/2) + a^4*b^3)/ (16*(a^8*b - a^9)))^(1/2)*1i)/((((768*a^3*b^8 - 144*a^5*b^6)/(64*a^5) + (( (10240*a^8*b^5 - 12288*a^7*b^6 + 6144*a^9*b^4)/(64*a^5) - (cos(c + d*x)*(1 2288*a^8*b^5 - 8192*a^9*b^4)*(((a^9*b^5)^(1/2) + a^4*b^3)/(16*(a^8*b - a^9 )))^(1/2))/(16*a^4))*(((a^9*b^5)^(1/2) + a^4*b^3)/(16*(a^8*b - a^9)))^(1/2 ) + (cos(c + d*x)*(2304*a^4*b^7 + 768*a^5*b^6 + 144*a^6*b^5))/(16*a^4))...