Integrand size = 24, antiderivative size = 138 \[ \int \frac {\sin ^5(c+d x)}{a-b \sin ^4(c+d x)} \, dx=-\frac {\sqrt {a} \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 \sqrt {\sqrt {a}-\sqrt {b}} b^{5/4} d}-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 \sqrt {\sqrt {a}+\sqrt {b}} b^{5/4} d}+\frac {\cos (c+d x)}{b d} \]
cos(d*x+c)/b/d-1/2*arctan(b^(1/4)*cos(d*x+c)/(a^(1/2)-b^(1/2))^(1/2))*a^(1 /2)/b^(5/4)/d/(a^(1/2)-b^(1/2))^(1/2)-1/2*arctanh(b^(1/4)*cos(d*x+c)/(a^(1 /2)+b^(1/2))^(1/2))*a^(1/2)/b^(5/4)/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 = 0.82 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.43 \[ \int \frac {\sin ^5(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\frac {2 \cos (c+d x)+i a \text {RootSum}\left [b-4 b \text {$\#$1}^2-16 a \text {$\#$1}^4+6 b \text {$\#$1}^4-4 b \text {$\#$1}^6+b \text {$\#$1}^8\&,\frac {-2 \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}+i \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}+2 \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^3-i \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^3}{-b-8 a \text {$\#$1}^2+3 b \text {$\#$1}^2-3 b \text {$\#$1}^4+b \text {$\#$1}^6}\&\right ]}{2 b d} \]
(2*Cos[c + d*x] + I*a*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)]*#1 + I*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1 + 2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1) ]*#1^3 - I*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^3)/(-b - 8*a*#1^2 + 3*b*#1 ^2 - 3*b*#1^4 + b*#1^6) & ])/(2*b*d)
Time = 0.34 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.98, 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.
| \(\displaystyle \int \frac {\sin ^5(c+d x)}{a-b \sin ^4(c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (c+d x)^5}{a-b \sin (c+d x)^4}dx\) |
| \(\Big \downarrow \) 3694 |
| \(\displaystyle -\frac {\int \frac {\left (1-\cos ^2(c+d x)\right )^2}{-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)+a-b}d\cos (c+d x)}{d}\) |
| \(\Big \downarrow \) 1484 |
| \(\displaystyle -\frac {\int \left (\frac {a}{b \left (-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)+a-b\right )}-\frac {1}{b}\right )d\cos (c+d x)}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\frac {\sqrt {a} \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 b^{5/4} \sqrt {\sqrt {a}-\sqrt {b}}}+\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 b^{5/4} \sqrt {\sqrt {a}+\sqrt {b}}}-\frac {\cos (c+d x)}{b}}{d}\) |
-(((Sqrt[a]*ArcTan[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(2*Sqr t[Sqrt[a] - Sqrt[b]]*b^(5/4)) + (Sqrt[a]*ArcTanh[(b^(1/4)*Cos[c + d*x])/Sq rt[Sqrt[a] + Sqrt[b]]])/(2*Sqrt[Sqrt[a] + Sqrt[b]]*b^(5/4)) - Cos[c + d*x] /b)/d)
3.2.97.3.1 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 = 1.13 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.72
| method | result | size |
| derivativedivides | \(\frac {\frac {\cos \left (d x +c \right )}{b}+a \left (-\frac {\operatorname {arctanh}\left (\frac {\cos \left (d x +c \right ) b}{\sqrt {\left (\sqrt {a b}+b \right ) b}}\right )}{2 \sqrt {a b}\, \sqrt {\left (\sqrt {a b}+b \right ) b}}-\frac {\arctan \left (\frac {\cos \left (d x +c \right ) b}{\sqrt {\left (\sqrt {a b}-b \right ) b}}\right )}{2 \sqrt {a b}\, \sqrt {\left (\sqrt {a b}-b \right ) b}}\right )}{d}\) | \(99\) |
| default | \(\frac {\frac {\cos \left (d x +c \right )}{b}+a \left (-\frac {\operatorname {arctanh}\left (\frac {\cos \left (d x +c \right ) b}{\sqrt {\left (\sqrt {a b}+b \right ) b}}\right )}{2 \sqrt {a b}\, \sqrt {\left (\sqrt {a b}+b \right ) b}}-\frac {\arctan \left (\frac {\cos \left (d x +c \right ) b}{\sqrt {\left (\sqrt {a b}-b \right ) b}}\right )}{2 \sqrt {a b}\, \sqrt {\left (\sqrt {a b}-b \right ) b}}\right )}{d}\) | \(99\) |
| risch | \(\frac {{\mathrm e}^{i \left (d x +c \right )}}{2 b d}+\frac {{\mathrm e}^{-i \left (d x +c \right )}}{2 b d}-\frac {i \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (a \,b^{5} d^{4}-b^{6} d^{4}\right ) \textit {\_Z}^{4}-128 a \,d^{2} \textit {\_Z}^{2} b^{3}-4096 a^{2}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\left (\left (-\frac {i b^{4} d^{3}}{256 a}+\frac {i b^{5} d^{3}}{256 a^{2}}\right ) \textit {\_R}^{3}+\left (\frac {i d b}{4}+\frac {i b^{2} d}{4 a}\right ) \textit {\_R} \right ) {\mathrm e}^{i \left (d x +c \right )}+1\right )\right )}{32}\) | \(154\) |
1/d*(cos(d*x+c)/b+a*(-1/2/(a*b)^(1/2)/(((a*b)^(1/2)+b)*b)^(1/2)*arctanh(co s(d*x+c)*b/(((a*b)^(1/2)+b)*b)^(1/2))-1/2/(a*b)^(1/2)/(((a*b)^(1/2)-b)*b)^ (1/2)*arctan(cos(d*x+c)*b/(((a*b)^(1/2)-b)*b)^(1/2))))
Leaf count of result is larger than twice the leaf count of optimal. 815 vs. \(2 (98) = 196\).
Time = 0.36 (sec) , antiderivative size = 815, normalized size of antiderivative = 5.91 \[ \int \frac {\sin ^5(c+d x)}{a-b \sin ^4(c+d x)} \, dx=-\frac {b d \sqrt {-\frac {{\left (a b^{2} - b^{3}\right )} d^{2} \sqrt {\frac {a^{3}}{{\left (a^{2} b^{5} - 2 \, a b^{6} + b^{7}\right )} d^{4}}} + a}{{\left (a b^{2} - b^{3}\right )} d^{2}}} \log \left (a^{2} \cos \left (d x + c\right ) - {\left ({\left (a b^{4} - b^{5}\right )} d^{3} \sqrt {\frac {a^{3}}{{\left (a^{2} b^{5} - 2 \, a b^{6} + b^{7}\right )} d^{4}}} - a^{2} b d\right )} \sqrt {-\frac {{\left (a b^{2} - b^{3}\right )} d^{2} \sqrt {\frac {a^{3}}{{\left (a^{2} b^{5} - 2 \, a b^{6} + b^{7}\right )} d^{4}}} + a}{{\left (a b^{2} - b^{3}\right )} d^{2}}}\right ) - b d \sqrt {\frac {{\left (a b^{2} - b^{3}\right )} d^{2} \sqrt {\frac {a^{3}}{{\left (a^{2} b^{5} - 2 \, a b^{6} + b^{7}\right )} d^{4}}} - a}{{\left (a b^{2} - b^{3}\right )} d^{2}}} \log \left (a^{2} \cos \left (d x + c\right ) - {\left ({\left (a b^{4} - b^{5}\right )} d^{3} \sqrt {\frac {a^{3}}{{\left (a^{2} b^{5} - 2 \, a b^{6} + b^{7}\right )} d^{4}}} + a^{2} b d\right )} \sqrt {\frac {{\left (a b^{2} - b^{3}\right )} d^{2} \sqrt {\frac {a^{3}}{{\left (a^{2} b^{5} - 2 \, a b^{6} + b^{7}\right )} d^{4}}} - a}{{\left (a b^{2} - b^{3}\right )} d^{2}}}\right ) - b d \sqrt {-\frac {{\left (a b^{2} - b^{3}\right )} d^{2} \sqrt {\frac {a^{3}}{{\left (a^{2} b^{5} - 2 \, a b^{6} + b^{7}\right )} d^{4}}} + a}{{\left (a b^{2} - b^{3}\right )} d^{2}}} \log \left (-a^{2} \cos \left (d x + c\right ) - {\left ({\left (a b^{4} - b^{5}\right )} d^{3} \sqrt {\frac {a^{3}}{{\left (a^{2} b^{5} - 2 \, a b^{6} + b^{7}\right )} d^{4}}} - a^{2} b d\right )} \sqrt {-\frac {{\left (a b^{2} - b^{3}\right )} d^{2} \sqrt {\frac {a^{3}}{{\left (a^{2} b^{5} - 2 \, a b^{6} + b^{7}\right )} d^{4}}} + a}{{\left (a b^{2} - b^{3}\right )} d^{2}}}\right ) + b d \sqrt {\frac {{\left (a b^{2} - b^{3}\right )} d^{2} \sqrt {\frac {a^{3}}{{\left (a^{2} b^{5} - 2 \, a b^{6} + b^{7}\right )} d^{4}}} - a}{{\left (a b^{2} - b^{3}\right )} d^{2}}} \log \left (-a^{2} \cos \left (d x + c\right ) - {\left ({\left (a b^{4} - b^{5}\right )} d^{3} \sqrt {\frac {a^{3}}{{\left (a^{2} b^{5} - 2 \, a b^{6} + b^{7}\right )} d^{4}}} + a^{2} b d\right )} \sqrt {\frac {{\left (a b^{2} - b^{3}\right )} d^{2} \sqrt {\frac {a^{3}}{{\left (a^{2} b^{5} - 2 \, a b^{6} + b^{7}\right )} d^{4}}} - a}{{\left (a b^{2} - b^{3}\right )} d^{2}}}\right ) - 4 \, \cos \left (d x + c\right )}{4 \, b d} \]
-1/4*(b*d*sqrt(-((a*b^2 - b^3)*d^2*sqrt(a^3/((a^2*b^5 - 2*a*b^6 + b^7)*d^4 )) + a)/((a*b^2 - b^3)*d^2))*log(a^2*cos(d*x + c) - ((a*b^4 - b^5)*d^3*sqr t(a^3/((a^2*b^5 - 2*a*b^6 + b^7)*d^4)) - a^2*b*d)*sqrt(-((a*b^2 - b^3)*d^2 *sqrt(a^3/((a^2*b^5 - 2*a*b^6 + b^7)*d^4)) + a)/((a*b^2 - b^3)*d^2))) - b* d*sqrt(((a*b^2 - b^3)*d^2*sqrt(a^3/((a^2*b^5 - 2*a*b^6 + b^7)*d^4)) - a)/( (a*b^2 - b^3)*d^2))*log(a^2*cos(d*x + c) - ((a*b^4 - b^5)*d^3*sqrt(a^3/((a ^2*b^5 - 2*a*b^6 + b^7)*d^4)) + a^2*b*d)*sqrt(((a*b^2 - b^3)*d^2*sqrt(a^3/ ((a^2*b^5 - 2*a*b^6 + b^7)*d^4)) - a)/((a*b^2 - b^3)*d^2))) - b*d*sqrt(-(( a*b^2 - b^3)*d^2*sqrt(a^3/((a^2*b^5 - 2*a*b^6 + b^7)*d^4)) + a)/((a*b^2 - b^3)*d^2))*log(-a^2*cos(d*x + c) - ((a*b^4 - b^5)*d^3*sqrt(a^3/((a^2*b^5 - 2*a*b^6 + b^7)*d^4)) - a^2*b*d)*sqrt(-((a*b^2 - b^3)*d^2*sqrt(a^3/((a^2*b ^5 - 2*a*b^6 + b^7)*d^4)) + a)/((a*b^2 - b^3)*d^2))) + b*d*sqrt(((a*b^2 - b^3)*d^2*sqrt(a^3/((a^2*b^5 - 2*a*b^6 + b^7)*d^4)) - a)/((a*b^2 - b^3)*d^2 ))*log(-a^2*cos(d*x + c) - ((a*b^4 - b^5)*d^3*sqrt(a^3/((a^2*b^5 - 2*a*b^6 + b^7)*d^4)) + a^2*b*d)*sqrt(((a*b^2 - b^3)*d^2*sqrt(a^3/((a^2*b^5 - 2*a* b^6 + b^7)*d^4)) - a)/((a*b^2 - b^3)*d^2))) - 4*cos(d*x + c))/(b*d)
Timed out. \[ \int \frac {\sin ^5(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\text {Timed out} \]
\[ \int \frac {\sin ^5(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\int { -\frac {\sin \left (d x + c\right )^{5}}{b \sin \left (d x + c\right )^{4} - a} \,d x } \]
(b*d*integrate(8*(4*a*b*cos(3*d*x + 3*c)*sin(2*d*x + 2*c) + 2*(8*a^2 - 3*a *b)*cos(3*d*x + 3*c)*sin(4*d*x + 4*c) - 2*(8*a^2 - 3*a*b)*cos(4*d*x + 4*c) *sin(3*d*x + 3*c) - (a*b*sin(5*d*x + 5*c) - a*b*sin(3*d*x + 3*c))*cos(8*d* x + 8*c) + 4*(a*b*sin(5*d*x + 5*c) - a*b*sin(3*d*x + 3*c))*cos(6*d*x + 6*c ) - 2*(2*a*b*sin(2*d*x + 2*c) + (8*a^2 - 3*a*b)*sin(4*d*x + 4*c))*cos(5*d* x + 5*c) + (a*b*cos(5*d*x + 5*c) - a*b*cos(3*d*x + 3*c))*sin(8*d*x + 8*c) - 4*(a*b*cos(5*d*x + 5*c) - a*b*cos(3*d*x + 3*c))*sin(6*d*x + 6*c) + (4*a* b*cos(2*d*x + 2*c) - a*b + 2*(8*a^2 - 3*a*b)*cos(4*d*x + 4*c))*sin(5*d*x + 5*c) - (4*a*b*cos(2*d*x + 2*c) - a*b)*sin(3*d*x + 3*c))/(b^3*cos(8*d*x + 8*c)^2 + 16*b^3*cos(6*d*x + 6*c)^2 + 16*b^3*cos(2*d*x + 2*c)^2 + b^3*sin(8 *d*x + 8*c)^2 + 16*b^3*sin(6*d*x + 6*c)^2 + 16*b^3*sin(2*d*x + 2*c)^2 - 8* b^3*cos(2*d*x + 2*c) + b^3 + 4*(64*a^2*b - 48*a*b^2 + 9*b^3)*cos(4*d*x + 4 *c)^2 + 4*(64*a^2*b - 48*a*b^2 + 9*b^3)*sin(4*d*x + 4*c)^2 + 16*(8*a*b^2 - 3*b^3)*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) - 2*(4*b^3*cos(6*d*x + 6*c) + 4* b^3*cos(2*d*x + 2*c) - b^3 + 2*(8*a*b^2 - 3*b^3)*cos(4*d*x + 4*c))*cos(8*d *x + 8*c) + 8*(4*b^3*cos(2*d*x + 2*c) - b^3 + 2*(8*a*b^2 - 3*b^3)*cos(4*d* x + 4*c))*cos(6*d*x + 6*c) - 4*(8*a*b^2 - 3*b^3 - 4*(8*a*b^2 - 3*b^3)*cos( 2*d*x + 2*c))*cos(4*d*x + 4*c) - 4*(2*b^3*sin(6*d*x + 6*c) + 2*b^3*sin(2*d *x + 2*c) + (8*a*b^2 - 3*b^3)*sin(4*d*x + 4*c))*sin(8*d*x + 8*c) + 16*(2*b ^3*sin(2*d*x + 2*c) + (8*a*b^2 - 3*b^3)*sin(4*d*x + 4*c))*sin(6*d*x + 6...
\[ \int \frac {\sin ^5(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\int { -\frac {\sin \left (d x + c\right )^{5}}{b \sin \left (d x + c\right )^{4} - a} \,d x } \]
Time = 13.50 (sec) , antiderivative size = 1001, normalized size of antiderivative = 7.25 \[ \int \frac {\sin ^5(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\frac {\cos \left (c+d\,x\right )}{b\,d}-\frac {2\,\mathrm {atanh}\left (\frac {8\,a^2\,b^7\,\cos \left (c+d\,x\right )\,\sqrt {-\frac {\sqrt {a^3\,b^5}}{16\,\left (a\,b^5-b^6\right )}-\frac {a\,b^3}{16\,\left (a\,b^5-b^6\right )}}}{\frac {2\,a^3\,b^{11}}{a\,b^5-b^6}-\frac {2\,a^4\,b^{10}}{a\,b^5-b^6}+\frac {2\,a^2\,b^8\,\sqrt {a^3\,b^5}}{a\,b^5-b^6}-\frac {2\,a^3\,b^7\,\sqrt {a^3\,b^5}}{a\,b^5-b^6}}-\frac {8\,a^2\,b\,\cos \left (c+d\,x\right )\,\sqrt {-\frac {\sqrt {a^3\,b^5}}{16\,\left (a\,b^5-b^6\right )}-\frac {a\,b^3}{16\,\left (a\,b^5-b^6\right )}}}{\frac {2\,a^3\,b^5}{a\,b^5-b^6}+\frac {2\,a^2\,b^2\,\sqrt {a^3\,b^5}}{a\,b^5-b^6}}+\frac {8\,a\,b^4\,\cos \left (c+d\,x\right )\,\sqrt {-\frac {\sqrt {a^3\,b^5}}{16\,\left (a\,b^5-b^6\right )}-\frac {a\,b^3}{16\,\left (a\,b^5-b^6\right )}}\,\sqrt {a^3\,b^5}}{\frac {2\,a^3\,b^{11}}{a\,b^5-b^6}-\frac {2\,a^4\,b^{10}}{a\,b^5-b^6}+\frac {2\,a^2\,b^8\,\sqrt {a^3\,b^5}}{a\,b^5-b^6}-\frac {2\,a^3\,b^7\,\sqrt {a^3\,b^5}}{a\,b^5-b^6}}\right )\,\sqrt {-\frac {\sqrt {a^3\,b^5}+a\,b^3}{16\,\left (a\,b^5-b^6\right )}}}{d}+\frac {2\,\mathrm {atanh}\left (\frac {8\,a^2\,b\,\cos \left (c+d\,x\right )\,\sqrt {\frac {\sqrt {a^3\,b^5}}{16\,\left (a\,b^5-b^6\right )}-\frac {a\,b^3}{16\,\left (a\,b^5-b^6\right )}}}{\frac {2\,a^3\,b^5}{a\,b^5-b^6}-\frac {2\,a^2\,b^2\,\sqrt {a^3\,b^5}}{a\,b^5-b^6}}-\frac {8\,a^2\,b^7\,\cos \left (c+d\,x\right )\,\sqrt {\frac {\sqrt {a^3\,b^5}}{16\,\left (a\,b^5-b^6\right )}-\frac {a\,b^3}{16\,\left (a\,b^5-b^6\right )}}}{\frac {2\,a^3\,b^{11}}{a\,b^5-b^6}-\frac {2\,a^4\,b^{10}}{a\,b^5-b^6}-\frac {2\,a^2\,b^8\,\sqrt {a^3\,b^5}}{a\,b^5-b^6}+\frac {2\,a^3\,b^7\,\sqrt {a^3\,b^5}}{a\,b^5-b^6}}+\frac {8\,a\,b^4\,\cos \left (c+d\,x\right )\,\sqrt {\frac {\sqrt {a^3\,b^5}}{16\,\left (a\,b^5-b^6\right )}-\frac {a\,b^3}{16\,\left (a\,b^5-b^6\right )}}\,\sqrt {a^3\,b^5}}{\frac {2\,a^3\,b^{11}}{a\,b^5-b^6}-\frac {2\,a^4\,b^{10}}{a\,b^5-b^6}-\frac {2\,a^2\,b^8\,\sqrt {a^3\,b^5}}{a\,b^5-b^6}+\frac {2\,a^3\,b^7\,\sqrt {a^3\,b^5}}{a\,b^5-b^6}}\right )\,\sqrt {\frac {\sqrt {a^3\,b^5}-a\,b^3}{16\,\left (a\,b^5-b^6\right )}}}{d} \]
cos(c + d*x)/(b*d) - (2*atanh((8*a^2*b^7*cos(c + d*x)*(- (a^3*b^5)^(1/2)/( 16*(a*b^5 - b^6)) - (a*b^3)/(16*(a*b^5 - b^6)))^(1/2))/((2*a^3*b^11)/(a*b^ 5 - b^6) - (2*a^4*b^10)/(a*b^5 - b^6) + (2*a^2*b^8*(a^3*b^5)^(1/2))/(a*b^5 - b^6) - (2*a^3*b^7*(a^3*b^5)^(1/2))/(a*b^5 - b^6)) - (8*a^2*b*cos(c + d* x)*(- (a^3*b^5)^(1/2)/(16*(a*b^5 - b^6)) - (a*b^3)/(16*(a*b^5 - b^6)))^(1/ 2))/((2*a^3*b^5)/(a*b^5 - b^6) + (2*a^2*b^2*(a^3*b^5)^(1/2))/(a*b^5 - b^6) ) + (8*a*b^4*cos(c + d*x)*(- (a^3*b^5)^(1/2)/(16*(a*b^5 - b^6)) - (a*b^3)/ (16*(a*b^5 - b^6)))^(1/2)*(a^3*b^5)^(1/2))/((2*a^3*b^11)/(a*b^5 - b^6) - ( 2*a^4*b^10)/(a*b^5 - b^6) + (2*a^2*b^8*(a^3*b^5)^(1/2))/(a*b^5 - b^6) - (2 *a^3*b^7*(a^3*b^5)^(1/2))/(a*b^5 - b^6)))*(-((a^3*b^5)^(1/2) + a*b^3)/(16* (a*b^5 - b^6)))^(1/2))/d + (2*atanh((8*a^2*b*cos(c + d*x)*((a^3*b^5)^(1/2) /(16*(a*b^5 - b^6)) - (a*b^3)/(16*(a*b^5 - b^6)))^(1/2))/((2*a^3*b^5)/(a*b ^5 - b^6) - (2*a^2*b^2*(a^3*b^5)^(1/2))/(a*b^5 - b^6)) - (8*a^2*b^7*cos(c + d*x)*((a^3*b^5)^(1/2)/(16*(a*b^5 - b^6)) - (a*b^3)/(16*(a*b^5 - b^6)))^( 1/2))/((2*a^3*b^11)/(a*b^5 - b^6) - (2*a^4*b^10)/(a*b^5 - b^6) - (2*a^2*b^ 8*(a^3*b^5)^(1/2))/(a*b^5 - b^6) + (2*a^3*b^7*(a^3*b^5)^(1/2))/(a*b^5 - b^ 6)) + (8*a*b^4*cos(c + d*x)*((a^3*b^5)^(1/2)/(16*(a*b^5 - b^6)) - (a*b^3)/ (16*(a*b^5 - b^6)))^(1/2)*(a^3*b^5)^(1/2))/((2*a^3*b^11)/(a*b^5 - b^6) - ( 2*a^4*b^10)/(a*b^5 - b^6) - (2*a^2*b^8*(a^3*b^5)^(1/2))/(a*b^5 - b^6) + (2 *a^3*b^7*(a^3*b^5)^(1/2))/(a*b^5 - b^6)))*(((a^3*b^5)^(1/2) - a*b^3)/(1...