Integrand size = 24, antiderivative size = 184 \[ \int \frac {\csc ^3(c+d x)}{a-b \sin ^4(c+d x)} \, dx=-\frac {b^{3/4} \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 a^{3/2} \sqrt {\sqrt {a}-\sqrt {b}} d}-\frac {\text {arctanh}(\cos (c+d x))}{2 a d}-\frac {b^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 a^{3/2} \sqrt {\sqrt {a}+\sqrt {b}} d}-\frac {1}{4 a d (1-\cos (c+d x))}+\frac {1}{4 a d (1+\cos (c+d x))} \]
-1/2*arctanh(cos(d*x+c))/a/d-1/4/a/d/(1-cos(d*x+c))+1/4/a/d/(1+cos(d*x+c)) -1/2*b^(3/4)*arctan(b^(1/4)*cos(d*x+c)/(a^(1/2)-b^(1/2))^(1/2))/a^(3/2)/d/ (a^(1/2)-b^(1/2))^(1/2)-1/2*b^(3/4)*arctanh(b^(1/4)*cos(d*x+c)/(a^(1/2)+b^ (1/2))^(1/2))/a^(3/2)/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 = 0.86 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.32 \[ \int \frac {\csc ^3(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\frac {-\csc ^2\left (\frac {1}{2} (c+d x)\right )-4 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+4 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+4 i b \text {RootSum}\left [b-4 b \text {$\#$1}^2-16 a \text {$\#$1}^4+6 b \text {$\#$1}^4-4 b \text {$\#$1}^6+b \text {$\#$1}^8\&,\frac {-2 \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}+i \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}+2 \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^3-i \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^3}{-b-8 a \text {$\#$1}^2+3 b \text {$\#$1}^2-3 b \text {$\#$1}^4+b \text {$\#$1}^6}\&\right ]+\sec ^2\left (\frac {1}{2} (c+d x)\right )}{8 a d} \]
(-Csc[(c + d*x)/2]^2 - 4*Log[Cos[(c + d*x)/2]] + 4*Log[Sin[(c + d*x)/2]] + (4*I)*b*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*Lo g[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) & ] + Sec[(c + d*x)/2]^2)/(8*a*d)
Time = 0.37 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.95, 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.
| \(\displaystyle \int \frac {\csc ^3(c+d x)}{a-b \sin ^4(c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\sin (c+d x)^3 \left (a-b \sin (c+d x)^4\right )}dx\) |
| \(\Big \downarrow \) 3694 |
| \(\displaystyle -\frac {\int \frac {1}{\left (1-\cos ^2(c+d x)\right )^2 \left (-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)+a-b\right )}d\cos (c+d x)}{d}\) |
| \(\Big \downarrow \) 1484 |
| \(\displaystyle -\frac {\int \left (\frac {b}{a \left (-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)+a-b\right )}-\frac {1}{2 a \left (\cos ^2(c+d x)-1\right )}+\frac {1}{4 a (\cos (c+d x)-1)^2}+\frac {1}{4 a (\cos (c+d x)+1)^2}\right )d\cos (c+d x)}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\frac {b^{3/4} \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 a^{3/2} \sqrt {\sqrt {a}-\sqrt {b}}}+\frac {b^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 a^{3/2} \sqrt {\sqrt {a}+\sqrt {b}}}+\frac {\text {arctanh}(\cos (c+d x))}{2 a}+\frac {1}{4 a (1-\cos (c+d x))}-\frac {1}{4 a (\cos (c+d x)+1)}}{d}\) |
-(((b^(3/4)*ArcTan[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(2*a^( 3/2)*Sqrt[Sqrt[a] - Sqrt[b]]) + ArcTanh[Cos[c + d*x]]/(2*a) + (b^(3/4)*Arc Tanh[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]])/(2*a^(3/2)*Sqrt[Sqrt [a] + Sqrt[b]]) + 1/(4*a*(1 - Cos[c + d*x])) - 1/(4*a*(1 + Cos[c + d*x]))) /d)
3.3.1.3.1 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 = 1.28 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.83
| method | result | size |
| derivativedivides | \(\frac {\frac {1}{4 a \left (\cos \left (d x +c \right )-1\right )}+\frac {\ln \left (\cos \left (d x +c \right )-1\right )}{4 a}+\frac {1}{4 a \left (1+\cos \left (d x +c \right )\right )}-\frac {\ln \left (1+\cos \left (d x +c \right )\right )}{4 a}+\frac {b^{2} \left (-\frac {\operatorname {arctanh}\left (\frac {\cos \left (d x +c \right ) b}{\sqrt {\left (\sqrt {a b}+b \right ) b}}\right )}{2 \sqrt {a b}\, \sqrt {\left (\sqrt {a b}+b \right ) b}}-\frac {\arctan \left (\frac {\cos \left (d x +c \right ) b}{\sqrt {\left (\sqrt {a b}-b \right ) b}}\right )}{2 \sqrt {a b}\, \sqrt {\left (\sqrt {a b}-b \right ) b}}\right )}{a}}{d}\) | \(152\) |
| default | \(\frac {\frac {1}{4 a \left (\cos \left (d x +c \right )-1\right )}+\frac {\ln \left (\cos \left (d x +c \right )-1\right )}{4 a}+\frac {1}{4 a \left (1+\cos \left (d x +c \right )\right )}-\frac {\ln \left (1+\cos \left (d x +c \right )\right )}{4 a}+\frac {b^{2} \left (-\frac {\operatorname {arctanh}\left (\frac {\cos \left (d x +c \right ) b}{\sqrt {\left (\sqrt {a b}+b \right ) b}}\right )}{2 \sqrt {a b}\, \sqrt {\left (\sqrt {a b}+b \right ) b}}-\frac {\arctan \left (\frac {\cos \left (d x +c \right ) b}{\sqrt {\left (\sqrt {a b}-b \right ) b}}\right )}{2 \sqrt {a b}\, \sqrt {\left (\sqrt {a b}-b \right ) b}}\right )}{a}}{d}\) | \(152\) |
| risch | \(\frac {{\mathrm e}^{3 i \left (d x +c \right )}+{\mathrm e}^{i \left (d x +c \right )}}{d a \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2}}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{2 d a}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{2 d a}-8 i \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (1048576 a^{7} d^{4}-1048576 a^{6} b \,d^{4}\right ) \textit {\_Z}^{4}-2048 a^{3} b^{2} d^{2} \textit {\_Z}^{2}-b^{3}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\left (\left (-\frac {65536 i d^{3} a^{5}}{b^{2}}+\frac {65536 i d^{3} a^{4}}{b}\right ) \textit {\_R}^{3}+\left (\frac {64 i a^{2} d}{b}+64 i a d \right ) \textit {\_R} \right ) {\mathrm e}^{i \left (d x +c \right )}+1\right )\right )\) | \(202\) |
1/d*(1/4/a/(cos(d*x+c)-1)+1/4/a*ln(cos(d*x+c)-1)+1/4/a/(1+cos(d*x+c))-1/4/ a*ln(1+cos(d*x+c))+1/a*b^2*(-1/2/(a*b)^(1/2)/(((a*b)^(1/2)+b)*b)^(1/2)*arc tanh(cos(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. 924 vs. \(2 (136) = 272\).
Time = 0.41 (sec) , antiderivative size = 924, normalized size of antiderivative = 5.02 \[ \int \frac {\csc ^3(c+d x)}{a-b \sin ^4(c+d x)} \, dx=-\frac {{\left (a d \cos \left (d x + c\right )^{2} - a d\right )} \sqrt {-\frac {{\left (a^{4} - a^{3} b\right )} d^{2} \sqrt {\frac {b^{3}}{{\left (a^{7} - 2 \, a^{6} b + a^{5} b^{2}\right )} d^{4}}} + b^{2}}{{\left (a^{4} - a^{3} b\right )} d^{2}}} \log \left (b^{2} \cos \left (d x + c\right ) - {\left ({\left (a^{5} - a^{4} b\right )} d^{3} \sqrt {\frac {b^{3}}{{\left (a^{7} - 2 \, a^{6} b + a^{5} b^{2}\right )} d^{4}}} - a^{2} b d\right )} \sqrt {-\frac {{\left (a^{4} - a^{3} b\right )} d^{2} \sqrt {\frac {b^{3}}{{\left (a^{7} - 2 \, a^{6} b + a^{5} b^{2}\right )} d^{4}}} + b^{2}}{{\left (a^{4} - a^{3} b\right )} d^{2}}}\right ) - {\left (a d \cos \left (d x + c\right )^{2} - a d\right )} \sqrt {\frac {{\left (a^{4} - a^{3} b\right )} d^{2} \sqrt {\frac {b^{3}}{{\left (a^{7} - 2 \, a^{6} b + a^{5} b^{2}\right )} d^{4}}} - b^{2}}{{\left (a^{4} - a^{3} b\right )} d^{2}}} \log \left (b^{2} \cos \left (d x + c\right ) - {\left ({\left (a^{5} - a^{4} b\right )} d^{3} \sqrt {\frac {b^{3}}{{\left (a^{7} - 2 \, a^{6} b + a^{5} b^{2}\right )} d^{4}}} + a^{2} b d\right )} \sqrt {\frac {{\left (a^{4} - a^{3} b\right )} d^{2} \sqrt {\frac {b^{3}}{{\left (a^{7} - 2 \, a^{6} b + a^{5} b^{2}\right )} d^{4}}} - b^{2}}{{\left (a^{4} - a^{3} b\right )} d^{2}}}\right ) - {\left (a d \cos \left (d x + c\right )^{2} - a d\right )} \sqrt {-\frac {{\left (a^{4} - a^{3} b\right )} d^{2} \sqrt {\frac {b^{3}}{{\left (a^{7} - 2 \, a^{6} b + a^{5} b^{2}\right )} d^{4}}} + b^{2}}{{\left (a^{4} - a^{3} b\right )} d^{2}}} \log \left (-b^{2} \cos \left (d x + c\right ) - {\left ({\left (a^{5} - a^{4} b\right )} d^{3} \sqrt {\frac {b^{3}}{{\left (a^{7} - 2 \, a^{6} b + a^{5} b^{2}\right )} d^{4}}} - a^{2} b d\right )} \sqrt {-\frac {{\left (a^{4} - a^{3} b\right )} d^{2} \sqrt {\frac {b^{3}}{{\left (a^{7} - 2 \, a^{6} b + a^{5} b^{2}\right )} d^{4}}} + b^{2}}{{\left (a^{4} - a^{3} b\right )} d^{2}}}\right ) + {\left (a d \cos \left (d x + c\right )^{2} - a d\right )} \sqrt {\frac {{\left (a^{4} - a^{3} b\right )} d^{2} \sqrt {\frac {b^{3}}{{\left (a^{7} - 2 \, a^{6} b + a^{5} b^{2}\right )} d^{4}}} - b^{2}}{{\left (a^{4} - a^{3} b\right )} d^{2}}} \log \left (-b^{2} \cos \left (d x + c\right ) - {\left ({\left (a^{5} - a^{4} b\right )} d^{3} \sqrt {\frac {b^{3}}{{\left (a^{7} - 2 \, a^{6} b + a^{5} b^{2}\right )} d^{4}}} + a^{2} b d\right )} \sqrt {\frac {{\left (a^{4} - a^{3} b\right )} d^{2} \sqrt {\frac {b^{3}}{{\left (a^{7} - 2 \, a^{6} b + a^{5} b^{2}\right )} d^{4}}} - b^{2}}{{\left (a^{4} - a^{3} b\right )} d^{2}}}\right ) + {\left (\cos \left (d x + c\right )^{2} - 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - {\left (\cos \left (d x + c\right )^{2} - 1\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 2 \, \cos \left (d x + c\right )}{4 \, {\left (a d \cos \left (d x + c\right )^{2} - a d\right )}} \]
-1/4*((a*d*cos(d*x + c)^2 - a*d)*sqrt(-((a^4 - a^3*b)*d^2*sqrt(b^3/((a^7 - 2*a^6*b + a^5*b^2)*d^4)) + b^2)/((a^4 - a^3*b)*d^2))*log(b^2*cos(d*x + c) - ((a^5 - a^4*b)*d^3*sqrt(b^3/((a^7 - 2*a^6*b + a^5*b^2)*d^4)) - a^2*b*d) *sqrt(-((a^4 - a^3*b)*d^2*sqrt(b^3/((a^7 - 2*a^6*b + a^5*b^2)*d^4)) + b^2) /((a^4 - a^3*b)*d^2))) - (a*d*cos(d*x + c)^2 - a*d)*sqrt(((a^4 - a^3*b)*d^ 2*sqrt(b^3/((a^7 - 2*a^6*b + a^5*b^2)*d^4)) - b^2)/((a^4 - a^3*b)*d^2))*lo g(b^2*cos(d*x + c) - ((a^5 - a^4*b)*d^3*sqrt(b^3/((a^7 - 2*a^6*b + a^5*b^2 )*d^4)) + a^2*b*d)*sqrt(((a^4 - a^3*b)*d^2*sqrt(b^3/((a^7 - 2*a^6*b + a^5* b^2)*d^4)) - b^2)/((a^4 - a^3*b)*d^2))) - (a*d*cos(d*x + c)^2 - a*d)*sqrt( -((a^4 - a^3*b)*d^2*sqrt(b^3/((a^7 - 2*a^6*b + a^5*b^2)*d^4)) + b^2)/((a^4 - a^3*b)*d^2))*log(-b^2*cos(d*x + c) - ((a^5 - a^4*b)*d^3*sqrt(b^3/((a^7 - 2*a^6*b + a^5*b^2)*d^4)) - a^2*b*d)*sqrt(-((a^4 - a^3*b)*d^2*sqrt(b^3/(( a^7 - 2*a^6*b + a^5*b^2)*d^4)) + b^2)/((a^4 - a^3*b)*d^2))) + (a*d*cos(d*x + c)^2 - a*d)*sqrt(((a^4 - a^3*b)*d^2*sqrt(b^3/((a^7 - 2*a^6*b + a^5*b^2) *d^4)) - b^2)/((a^4 - a^3*b)*d^2))*log(-b^2*cos(d*x + c) - ((a^5 - a^4*b)* d^3*sqrt(b^3/((a^7 - 2*a^6*b + a^5*b^2)*d^4)) + a^2*b*d)*sqrt(((a^4 - a^3* b)*d^2*sqrt(b^3/((a^7 - 2*a^6*b + a^5*b^2)*d^4)) - b^2)/((a^4 - a^3*b)*d^2 ))) + (cos(d*x + c)^2 - 1)*log(1/2*cos(d*x + c) + 1/2) - (cos(d*x + c)^2 - 1)*log(-1/2*cos(d*x + c) + 1/2) - 2*cos(d*x + c))/(a*d*cos(d*x + c)^2 - a *d)
\[ \int \frac {\csc ^3(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\int \frac {\csc ^{3}{\left (c + d x \right )}}{a - b \sin ^{4}{\left (c + d x \right )}}\, dx \]
\[ \int \frac {\csc ^3(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\int { -\frac {\csc \left (d x + c\right )^{3}}{b \sin \left (d x + c\right )^{4} - a} \,d x } \]
1/4*(4*(cos(3*d*x + 3*c) + cos(d*x + c))*cos(4*d*x + 4*c) - 4*(2*cos(2*d*x + 2*c) - 1)*cos(3*d*x + 3*c) - 8*cos(2*d*x + 2*c)*cos(d*x + c) + 4*(a*d*c os(4*d*x + 4*c)^2 + 4*a*d*cos(2*d*x + 2*c)^2 + a*d*sin(4*d*x + 4*c)^2 - 4* a*d*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 4*a*d*sin(2*d*x + 2*c)^2 - 4*a*d*c os(2*d*x + 2*c) + a*d - 2*(2*a*d*cos(2*d*x + 2*c) - a*d)*cos(4*d*x + 4*c)) *integrate(8*(4*b^2*cos(3*d*x + 3*c)*sin(2*d*x + 2*c) + 2*(8*a*b - 3*b^2)* cos(3*d*x + 3*c)*sin(4*d*x + 4*c) - 2*(8*a*b - 3*b^2)*cos(4*d*x + 4*c)*sin (3*d*x + 3*c) - (b^2*sin(5*d*x + 5*c) - b^2*sin(3*d*x + 3*c))*cos(8*d*x + 8*c) + 4*(b^2*sin(5*d*x + 5*c) - b^2*sin(3*d*x + 3*c))*cos(6*d*x + 6*c) - 2*(2*b^2*sin(2*d*x + 2*c) + (8*a*b - 3*b^2)*sin(4*d*x + 4*c))*cos(5*d*x + 5*c) + (b^2*cos(5*d*x + 5*c) - b^2*cos(3*d*x + 3*c))*sin(8*d*x + 8*c) - 4* (b^2*cos(5*d*x + 5*c) - b^2*cos(3*d*x + 3*c))*sin(6*d*x + 6*c) + (4*b^2*co s(2*d*x + 2*c) - b^2 + 2*(8*a*b - 3*b^2)*cos(4*d*x + 4*c))*sin(5*d*x + 5*c ) - (4*b^2*cos(2*d*x + 2*c) - b^2)*sin(3*d*x + 3*c))/(a*b^2*cos(8*d*x + 8* c)^2 + 16*a*b^2*cos(6*d*x + 6*c)^2 + 16*a*b^2*cos(2*d*x + 2*c)^2 + a*b^2*s in(8*d*x + 8*c)^2 + 16*a*b^2*sin(6*d*x + 6*c)^2 + 16*a*b^2*sin(2*d*x + 2*c )^2 - 8*a*b^2*cos(2*d*x + 2*c) + a*b^2 + 4*(64*a^3 - 48*a^2*b + 9*a*b^2)*c os(4*d*x + 4*c)^2 + 4*(64*a^3 - 48*a^2*b + 9*a*b^2)*sin(4*d*x + 4*c)^2 + 1 6*(8*a^2*b - 3*a*b^2)*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) - 2*(4*a*b^2*cos(6 *d*x + 6*c) + 4*a*b^2*cos(2*d*x + 2*c) - a*b^2 + 2*(8*a^2*b - 3*a*b^2)*...
\[ \int \frac {\csc ^3(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\int { -\frac {\csc \left (d x + c\right )^{3}}{b \sin \left (d x + c\right )^{4} - a} \,d x } \]
Time = 15.16 (sec) , antiderivative size = 2779, normalized size of antiderivative = 15.10 \[ \int \frac {\csc ^3(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\text {Too large to display} \]
(atan(cos(c + d*x)*1i)*1i)/(d*(2*a - 2*a*cos(c + d*x)^2)) - cos(c + d*x)/( d*(2*a - 2*a*cos(c + d*x)^2)) - (atan(cos(c + d*x)*1i)*cos(c + d*x)^2*1i)/ (d*(2*a - 2*a*cos(c + d*x)^2)) + (a*atan((a^13*cos(c + d*x)*(((a^7*b^3)^(1 /2) + a^3*b^2)/(16*a^6*b - 16*a^7))^(5/2)*2048i + a^10*b*cos(c + d*x)*(((a ^7*b^3)^(1/2) + a^3*b^2)/(16*a^6*b - 16*a^7))^(3/2)*64i - a^12*b*cos(c + d *x)*(((a^7*b^3)^(1/2) + a^3*b^2)/(16*a^6*b - 16*a^7))^(5/2)*7168i - a^4*b^ 5*cos(c + d*x)*(((a^7*b^3)^(1/2) + a^3*b^2)/(16*a^6*b - 16*a^7))^(1/2)*8i + a^5*b^4*cos(c + d*x)*(((a^7*b^3)^(1/2) + a^3*b^2)/(16*a^6*b - 16*a^7))^( 1/2)*12i - a^7*b^2*cos(c + d*x)*(((a^7*b^3)^(1/2) + a^3*b^2)/(16*a^6*b - 1 6*a^7))^(1/2)*4i + a^7*b^4*cos(c + d*x)*(((a^7*b^3)^(1/2) + a^3*b^2)/(16*a ^6*b - 16*a^7))^(3/2)*320i - a^8*b^3*cos(c + d*x)*(((a^7*b^3)^(1/2) + a^3* b^2)/(16*a^6*b - 16*a^7))^(3/2)*576i + a^9*b^2*cos(c + d*x)*(((a^7*b^3)^(1 /2) + a^3*b^2)/(16*a^6*b - 16*a^7))^(3/2)*192i - a^10*b^3*cos(c + d*x)*((( a^7*b^3)^(1/2) + a^3*b^2)/(16*a^6*b - 16*a^7))^(5/2)*3072i + a^11*b^2*cos( c + d*x)*(((a^7*b^3)^(1/2) + a^3*b^2)/(16*a^6*b - 16*a^7))^(5/2)*8192i)/(2 *b^3*(a^7*b^3)^(1/2) + a^3*b^5 + a^5*b^3 - a*b^2*(a^7*b^3)^(1/2) + a^2*b*( a^7*b^3)^(1/2)))*(((a^7*b^3)^(1/2) + a^3*b^2)/(16*a^6*b - 16*a^7))^(1/2)*4 i)/(d*(2*a - 2*a*cos(c + d*x)^2)) + (a*atan((a^13*cos(c + d*x)*(-((a^7*b^3 )^(1/2) - a^3*b^2)/(16*a^6*b - 16*a^7))^(5/2)*2048i + a^10*b*cos(c + d*x)* (-((a^7*b^3)^(1/2) - a^3*b^2)/(16*a^6*b - 16*a^7))^(3/2)*64i - a^12*b*c...