Integrand size = 43, antiderivative size = 270 \[ \int \cos ^{\frac {9}{2}}(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {8 a^4 (19 A+24 B+21 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {8 a^4 (12 A+17 B+28 C) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 d}+\frac {4 a^4 (73 A+83 B+7 C) \sqrt {\cos (c+d x)} \sin (c+d x)}{105 d}+\frac {2 a (A-9 C) \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^3 \sin (c+d x)}{9 d}+\frac {2 C (a+a \cos (c+d x))^4 \sin (c+d x)}{d \sqrt {\cos (c+d x)}}+\frac {2 (5 A+3 B-21 C) \sqrt {\cos (c+d x)} \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{21 d}+\frac {4 (86 A+81 B-126 C) \sqrt {\cos (c+d x)} \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{315 d} \]
8/15*a^4*(19*A+24*B+21*C)*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)* EllipticE(sin(1/2*d*x+1/2*c),2^(1/2))/d+8/21*a^4*(12*A+17*B+28*C)*(cos(1/2 *d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticF(sin(1/2*d*x+1/2*c),2^(1/ 2))/d+2*C*(a+a*cos(d*x+c))^4*sin(d*x+c)/d/cos(d*x+c)^(1/2)+4/105*a^4*(73*A +83*B+7*C)*sin(d*x+c)*cos(d*x+c)^(1/2)/d+2/9*a*(A-9*C)*(a+a*cos(d*x+c))^3* sin(d*x+c)*cos(d*x+c)^(1/2)/d+2/21*(5*A+3*B-21*C)*(a^2+a^2*cos(d*x+c))^2*s in(d*x+c)*cos(d*x+c)^(1/2)/d+4/315*(86*A+81*B-126*C)*(a^4+a^4*cos(d*x+c))* sin(d*x+c)*cos(d*x+c)^(1/2)/d
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 15.24 (sec) , antiderivative size = 1742, normalized size of antiderivative = 6.45 \[ \int \cos ^{\frac {9}{2}}(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx =\text {Too large to display} \]
Integrate[Cos[c + d*x]^(9/2)*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]
(Cos[c + d*x]^(13/2)*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + B*Se c[c + d*x] + C*Sec[c + d*x]^2)*(-1/120*((76*A + 96*B + 69*C + 76*A*Cos[2*c ] + 96*B*Cos[2*c] + 99*C*Cos[2*c])*Csc[c]*Sec[c])/d + ((204*A + 191*B + 11 2*C)*Cos[d*x]*Sin[c])/(336*d) + ((127*A + 72*B + 18*C)*Cos[2*d*x]*Sin[2*c] )/(720*d) + ((4*A + B)*Cos[3*d*x]*Sin[3*c])/(112*d) + (A*Cos[4*d*x]*Sin[4* c])/(288*d) + ((204*A + 191*B + 112*C)*Cos[c]*Sin[d*x])/(336*d) + (C*Sec[c ]*Sec[c + d*x]*Sin[d*x])/(4*d) + ((127*A + 72*B + 18*C)*Cos[2*c]*Sin[2*d*x ])/(720*d) + ((4*A + B)*Cos[3*c]*Sin[3*d*x])/(112*d) + (A*Cos[4*c]*Sin[4*d *x])/(288*d)))/(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x]) - (4*A*Co s[c + d*x]^6*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[ Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x ] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[ Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqr t[1 + Sin[d*x - ArcTan[Cot[c]]]])/(7*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos [2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (17*B*Cos[c + d*x]^6*Csc[c]*Hypergeom etricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2 ]^8*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot [c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c] ]]])/(21*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + C...
Time = 2.05 (sec) , antiderivative size = 288, normalized size of antiderivative = 1.07, number of steps used = 23, number of rules used = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.535, Rules used = {3042, 4600, 3042, 3522, 27, 3042, 3455, 27, 3042, 3455, 3042, 3455, 27, 3042, 3447, 3042, 3502, 27, 3042, 3227, 3042, 3119, 3120}
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.
| \(\displaystyle \int \cos ^{\frac {9}{2}}(c+d x) (a \sec (c+d x)+a)^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \cos (c+d x)^{9/2} (a \sec (c+d x)+a)^4 \left (A+B \sec (c+d x)+C \sec (c+d x)^2\right )dx\) |
| \(\Big \downarrow \) 4600 |
| \(\displaystyle \int \frac {(a \cos (c+d x)+a)^4 \left (A \cos ^2(c+d x)+B \cos (c+d x)+C\right )}{\cos ^{\frac {3}{2}}(c+d x)}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^4 \left (A \sin \left (c+d x+\frac {\pi }{2}\right )^2+B \sin \left (c+d x+\frac {\pi }{2}\right )+C\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 3522 |
| \(\displaystyle \frac {2 \int \frac {(\cos (c+d x) a+a)^4 (a (B+8 C)+a (A-9 C) \cos (c+d x))}{2 \sqrt {\cos (c+d x)}}dx}{a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{d \sqrt {\cos (c+d x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(\cos (c+d x) a+a)^4 (a (B+8 C)+a (A-9 C) \cos (c+d x))}{\sqrt {\cos (c+d x)}}dx}{a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{d \sqrt {\cos (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^4 \left (a (B+8 C)+a (A-9 C) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{d \sqrt {\cos (c+d x)}}\) |
| \(\Big \downarrow \) 3455 |
| \(\displaystyle \frac {\frac {2}{9} \int \frac {(\cos (c+d x) a+a)^3 \left ((A+9 B+63 C) a^2+3 (5 A+3 B-21 C) \cos (c+d x) a^2\right )}{2 \sqrt {\cos (c+d x)}}dx+\frac {2 a^2 (A-9 C) \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^3}{9 d}}{a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{d \sqrt {\cos (c+d x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{9} \int \frac {(\cos (c+d x) a+a)^3 \left ((A+9 B+63 C) a^2+3 (5 A+3 B-21 C) \cos (c+d x) a^2\right )}{\sqrt {\cos (c+d x)}}dx+\frac {2 a^2 (A-9 C) \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^3}{9 d}}{a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{d \sqrt {\cos (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{9} \int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^3 \left ((A+9 B+63 C) a^2+3 (5 A+3 B-21 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^2\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 a^2 (A-9 C) \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^3}{9 d}}{a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{d \sqrt {\cos (c+d x)}}\) |
| \(\Big \downarrow \) 3455 |
| \(\displaystyle \frac {\frac {1}{9} \left (\frac {2}{7} \int \frac {(\cos (c+d x) a+a)^2 \left ((11 A+36 B+189 C) a^3+(86 A+81 B-126 C) \cos (c+d x) a^3\right )}{\sqrt {\cos (c+d x)}}dx+\frac {6 a^3 (5 A+3 B-21 C) \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^2}{7 d}\right )+\frac {2 a^2 (A-9 C) \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^3}{9 d}}{a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{d \sqrt {\cos (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{9} \left (\frac {2}{7} \int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2 \left ((11 A+36 B+189 C) a^3+(86 A+81 B-126 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^3\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {6 a^3 (5 A+3 B-21 C) \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^2}{7 d}\right )+\frac {2 a^2 (A-9 C) \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^3}{9 d}}{a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{d \sqrt {\cos (c+d x)}}\) |
| \(\Big \downarrow \) 3455 |
| \(\displaystyle \frac {\frac {1}{9} \left (\frac {2}{7} \left (\frac {2}{5} \int \frac {3 (\cos (c+d x) a+a) \left ((47 A+87 B+273 C) a^4+3 (73 A+83 B+7 C) \cos (c+d x) a^4\right )}{2 \sqrt {\cos (c+d x)}}dx+\frac {2 (86 A+81 B-126 C) \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^5 \cos (c+d x)+a^5\right )}{5 d}\right )+\frac {6 a^3 (5 A+3 B-21 C) \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^2}{7 d}\right )+\frac {2 a^2 (A-9 C) \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^3}{9 d}}{a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{d \sqrt {\cos (c+d x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{9} \left (\frac {2}{7} \left (\frac {3}{5} \int \frac {(\cos (c+d x) a+a) \left ((47 A+87 B+273 C) a^4+3 (73 A+83 B+7 C) \cos (c+d x) a^4\right )}{\sqrt {\cos (c+d x)}}dx+\frac {2 (86 A+81 B-126 C) \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^5 \cos (c+d x)+a^5\right )}{5 d}\right )+\frac {6 a^3 (5 A+3 B-21 C) \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^2}{7 d}\right )+\frac {2 a^2 (A-9 C) \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^3}{9 d}}{a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{d \sqrt {\cos (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{9} \left (\frac {2}{7} \left (\frac {3}{5} \int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right ) \left ((47 A+87 B+273 C) a^4+3 (73 A+83 B+7 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^4\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 (86 A+81 B-126 C) \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^5 \cos (c+d x)+a^5\right )}{5 d}\right )+\frac {6 a^3 (5 A+3 B-21 C) \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^2}{7 d}\right )+\frac {2 a^2 (A-9 C) \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^3}{9 d}}{a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{d \sqrt {\cos (c+d x)}}\) |
| \(\Big \downarrow \) 3447 |
| \(\displaystyle \frac {\frac {1}{9} \left (\frac {2}{7} \left (\frac {3}{5} \int \frac {3 (73 A+83 B+7 C) \cos ^2(c+d x) a^5+(47 A+87 B+273 C) a^5+\left (3 (73 A+83 B+7 C) a^5+(47 A+87 B+273 C) a^5\right ) \cos (c+d x)}{\sqrt {\cos (c+d x)}}dx+\frac {2 (86 A+81 B-126 C) \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^5 \cos (c+d x)+a^5\right )}{5 d}\right )+\frac {6 a^3 (5 A+3 B-21 C) \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^2}{7 d}\right )+\frac {2 a^2 (A-9 C) \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^3}{9 d}}{a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{d \sqrt {\cos (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{9} \left (\frac {2}{7} \left (\frac {3}{5} \int \frac {3 (73 A+83 B+7 C) \sin \left (c+d x+\frac {\pi }{2}\right )^2 a^5+(47 A+87 B+273 C) a^5+\left (3 (73 A+83 B+7 C) a^5+(47 A+87 B+273 C) a^5\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 (86 A+81 B-126 C) \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^5 \cos (c+d x)+a^5\right )}{5 d}\right )+\frac {6 a^3 (5 A+3 B-21 C) \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^2}{7 d}\right )+\frac {2 a^2 (A-9 C) \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^3}{9 d}}{a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{d \sqrt {\cos (c+d x)}}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {\frac {1}{9} \left (\frac {2}{7} \left (\frac {3}{5} \left (\frac {2}{3} \int \frac {3 \left (5 (12 A+17 B+28 C) a^5+7 (19 A+24 B+21 C) \cos (c+d x) a^5\right )}{\sqrt {\cos (c+d x)}}dx+\frac {2 a^5 (73 A+83 B+7 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{d}\right )+\frac {2 (86 A+81 B-126 C) \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^5 \cos (c+d x)+a^5\right )}{5 d}\right )+\frac {6 a^3 (5 A+3 B-21 C) \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^2}{7 d}\right )+\frac {2 a^2 (A-9 C) \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^3}{9 d}}{a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{d \sqrt {\cos (c+d x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{9} \left (\frac {2}{7} \left (\frac {3}{5} \left (2 \int \frac {5 (12 A+17 B+28 C) a^5+7 (19 A+24 B+21 C) \cos (c+d x) a^5}{\sqrt {\cos (c+d x)}}dx+\frac {2 a^5 (73 A+83 B+7 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{d}\right )+\frac {2 (86 A+81 B-126 C) \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^5 \cos (c+d x)+a^5\right )}{5 d}\right )+\frac {6 a^3 (5 A+3 B-21 C) \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^2}{7 d}\right )+\frac {2 a^2 (A-9 C) \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^3}{9 d}}{a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{d \sqrt {\cos (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{9} \left (\frac {2}{7} \left (\frac {3}{5} \left (2 \int \frac {5 (12 A+17 B+28 C) a^5+7 (19 A+24 B+21 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^5}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 a^5 (73 A+83 B+7 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{d}\right )+\frac {2 (86 A+81 B-126 C) \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^5 \cos (c+d x)+a^5\right )}{5 d}\right )+\frac {6 a^3 (5 A+3 B-21 C) \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^2}{7 d}\right )+\frac {2 a^2 (A-9 C) \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^3}{9 d}}{a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{d \sqrt {\cos (c+d x)}}\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle \frac {\frac {1}{9} \left (\frac {2}{7} \left (\frac {3}{5} \left (2 \left (5 a^5 (12 A+17 B+28 C) \int \frac {1}{\sqrt {\cos (c+d x)}}dx+7 a^5 (19 A+24 B+21 C) \int \sqrt {\cos (c+d x)}dx\right )+\frac {2 a^5 (73 A+83 B+7 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{d}\right )+\frac {2 (86 A+81 B-126 C) \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^5 \cos (c+d x)+a^5\right )}{5 d}\right )+\frac {6 a^3 (5 A+3 B-21 C) \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^2}{7 d}\right )+\frac {2 a^2 (A-9 C) \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^3}{9 d}}{a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{d \sqrt {\cos (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{9} \left (\frac {2}{7} \left (\frac {3}{5} \left (2 \left (5 a^5 (12 A+17 B+28 C) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+7 a^5 (19 A+24 B+21 C) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx\right )+\frac {2 a^5 (73 A+83 B+7 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{d}\right )+\frac {2 (86 A+81 B-126 C) \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^5 \cos (c+d x)+a^5\right )}{5 d}\right )+\frac {6 a^3 (5 A+3 B-21 C) \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^2}{7 d}\right )+\frac {2 a^2 (A-9 C) \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^3}{9 d}}{a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{d \sqrt {\cos (c+d x)}}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {\frac {1}{9} \left (\frac {2}{7} \left (\frac {3}{5} \left (2 \left (5 a^5 (12 A+17 B+28 C) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {14 a^5 (19 A+24 B+21 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )+\frac {2 a^5 (73 A+83 B+7 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{d}\right )+\frac {2 (86 A+81 B-126 C) \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^5 \cos (c+d x)+a^5\right )}{5 d}\right )+\frac {6 a^3 (5 A+3 B-21 C) \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^2}{7 d}\right )+\frac {2 a^2 (A-9 C) \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^3}{9 d}}{a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{d \sqrt {\cos (c+d x)}}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {\frac {2 a^2 (A-9 C) \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^3}{9 d}+\frac {1}{9} \left (\frac {2}{7} \left (\frac {2 (86 A+81 B-126 C) \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^5 \cos (c+d x)+a^5\right )}{5 d}+\frac {3}{5} \left (\frac {2 a^5 (73 A+83 B+7 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{d}+2 \left (\frac {10 a^5 (12 A+17 B+28 C) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{d}+\frac {14 a^5 (19 A+24 B+21 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )\right )\right )+\frac {6 a^3 (5 A+3 B-21 C) \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^2}{7 d}\right )}{a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{d \sqrt {\cos (c+d x)}}\) |
(2*C*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) + ((2*a^2 *(A - 9*C)*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(9*d) + ((6*a^3*(5*A + 3*B - 21*C)*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^2*Sin[ c + d*x])/(7*d) + (2*((2*(86*A + 81*B - 126*C)*Sqrt[Cos[c + d*x]]*(a^5 + a ^5*Cos[c + d*x])*Sin[c + d*x])/(5*d) + (3*(2*((14*a^5*(19*A + 24*B + 21*C) *EllipticE[(c + d*x)/2, 2])/d + (10*a^5*(12*A + 17*B + 28*C)*EllipticF[(c + d*x)/2, 2])/d) + (2*a^5*(73*A + 83*B + 7*C)*Sqrt[Cos[c + d*x]]*Sin[c + d *x])/d))/5))/7)/9)/a
3.13.11.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)* (c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(1/2 )*(c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x _)]), x_Symbol] :> Simp[c Int[(b*Sin[e + f*x])^m, x], x] + Simp[d/b Int [(b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Int[(a + b*Sin[e + f*x])^m*(A*c + (B*c + A*d)*Sin[e + f*x] + B*d*Sin[e + f*x]^2), x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim p[(-b)*B*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n + 1))), x] + Simp[1/(d*(m + n + 1)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A*d*(m + n + 1) + B*(a*c*(m - 1 ) + b*d*(n + 1)) + (A*b*d*(m + n + 1) - B*(b*c*m - a*d*(2*m + n)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 1/2] && !LtQ[n, -1 ] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c, 0])
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Co s[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2))), x] + Simp[1/(b*(m + 2)) Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && !LtQ[m, -1]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(c^2*C - B*c*d + A*d^2))*Cos[e + f*x ]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 - d^2))), x] + Simp[1/(b*d*(n + 1)*(c^2 - d^2)) Int[(a + b*Sin[e + f*x])^m* (c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(a*d*m + b*c*(n + 1)) + (c*C - B*d)*( a*c*m + b*d*(n + 1)) + b*(d*(B*c - A*d)*(m + n + 2) - C*(c^2*(m + 1) + d^2* (n + 1)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && !LtQ [m, -2^(-1)] && (LtQ[n, -1] || EqQ[m + n + 2, 0])
Int[(cos[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*((a_) + (b_.)*sec[(e_.) + (f_.)*(x _)])^(m_.)*((A_.) + (B_.)*sec[(e_.) + (f_.)*(x_)] + (C_.)*sec[(e_.) + (f_.) *(x_)]^2), x_Symbol] :> Simp[d^(m + 2) Int[(b + a*Cos[e + f*x])^m*(d*Cos[ e + f*x])^(n - m - 2)*(C + B*Cos[e + f*x] + A*Cos[e + f*x]^2), x], x] /; Fr eeQ[{a, b, d, e, f, A, B, C, n}, x] && !IntegerQ[n] && IntegerQ[m]
Leaf count of result is larger than twice the leaf count of optimal. \(785\) vs. \(2(300)=600\).
Time = 429.85 (sec) , antiderivative size = 786, normalized size of antiderivative = 2.91
int(cos(d*x+c)^(9/2)*(a+a*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, method=_RETURNVERBOSE)
-4/315*a^4*(-560*A*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*co s(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^10+40*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2 *d*x+1/2*c)^2)^(1/2)*(64*A+9*B)*sin(1/2*d*x+1/2*c)^8*cos(1/2*d*x+1/2*c)-4* (-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*(1177*A+387*B+63*C)*s in(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)+28*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2 *d*x+1/2*c)^2)^(1/2)*(161*A+96*B+39*C)*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/ 2*c)-6*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*(227*A+167*B+1 33*C)*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)+360*A*(-2*sin(1/2*d*x+1/2*c) ^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x +1/2*c)^2-1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-798*A*(-2*sin(1/2 *d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2* sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))+510*B* (-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*(sin(1/2*d*x+1/2*c)^2 )^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1 /2))-1008*B*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*(sin(1/2* d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x +1/2*c),2^(1/2))+840*C*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^ 2-1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))*(-2*sin(1/2*d*x+1/2*c)^4+ sin(1/2*d*x+1/2*c)^2)^(1/2)-882*C*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2 *c)^2)^(1/2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.14 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.06 \[ \int \cos ^{\frac {9}{2}}(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=-\frac {2 \, {\left (30 i \, \sqrt {2} {\left (12 \, A + 17 \, B + 28 \, C\right )} a^{4} \cos \left (d x + c\right ) {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 30 i \, \sqrt {2} {\left (12 \, A + 17 \, B + 28 \, C\right )} a^{4} \cos \left (d x + c\right ) {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 42 i \, \sqrt {2} {\left (19 \, A + 24 \, B + 21 \, C\right )} a^{4} \cos \left (d x + c\right ) {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 42 i \, \sqrt {2} {\left (19 \, A + 24 \, B + 21 \, C\right )} a^{4} \cos \left (d x + c\right ) {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - {\left (35 \, A a^{4} \cos \left (d x + c\right )^{4} + 45 \, {\left (4 \, A + B\right )} a^{4} \cos \left (d x + c\right )^{3} + 7 \, {\left (61 \, A + 36 \, B + 9 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} + 15 \, {\left (48 \, A + 47 \, B + 28 \, C\right )} a^{4} \cos \left (d x + c\right ) + 315 \, C a^{4}\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )\right )}}{315 \, d \cos \left (d x + c\right )} \]
integrate(cos(d*x+c)^(9/2)*(a+a*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c) ^2),x, algorithm="fricas")
-2/315*(30*I*sqrt(2)*(12*A + 17*B + 28*C)*a^4*cos(d*x + c)*weierstrassPInv erse(-4, 0, cos(d*x + c) + I*sin(d*x + c)) - 30*I*sqrt(2)*(12*A + 17*B + 2 8*C)*a^4*cos(d*x + c)*weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) - 42*I*sqrt(2)*(19*A + 24*B + 21*C)*a^4*cos(d*x + c)*weierstrassZeta (-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) + 42*I* sqrt(2)*(19*A + 24*B + 21*C)*a^4*cos(d*x + c)*weierstrassZeta(-4, 0, weier strassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c))) - (35*A*a^4*cos(d*x + c)^4 + 45*(4*A + B)*a^4*cos(d*x + c)^3 + 7*(61*A + 36*B + 9*C)*a^4*cos(d *x + c)^2 + 15*(48*A + 47*B + 28*C)*a^4*cos(d*x + c) + 315*C*a^4)*sqrt(cos (d*x + c))*sin(d*x + c))/(d*cos(d*x + c))
Timed out. \[ \int \cos ^{\frac {9}{2}}(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \]
Timed out. \[ \int \cos ^{\frac {9}{2}}(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \]
integrate(cos(d*x+c)^(9/2)*(a+a*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c) ^2),x, algorithm="maxima")
\[ \int \cos ^{\frac {9}{2}}(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (a \sec \left (d x + c\right ) + a\right )}^{4} \cos \left (d x + c\right )^{\frac {9}{2}} \,d x } \]
integrate(cos(d*x+c)^(9/2)*(a+a*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c) ^2),x, algorithm="giac")
integrate((C*sec(d*x + c)^2 + B*sec(d*x + c) + A)*(a*sec(d*x + c) + a)^4*c os(d*x + c)^(9/2), x)
Time = 20.70 (sec) , antiderivative size = 548, normalized size of antiderivative = 2.03 \[ \int \cos ^{\frac {9}{2}}(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {2\,\left (3\,A\,a^4\,\mathrm {E}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )+4\,A\,a^4\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )+4\,A\,a^4\,\sqrt {\cos \left (c+d\,x\right )}\,\sin \left (c+d\,x\right )\right )}{3\,d}+\frac {2\,\left (4\,B\,a^4\,\mathrm {E}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )+3\,B\,a^4\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )+2\,B\,a^4\,\sqrt {\cos \left (c+d\,x\right )}\,\sin \left (c+d\,x\right )\right )}{d}-\frac {2\,\left (\frac {66\,A\,a^4\,{\cos \left (c+d\,x\right )}^{7/2}\,\sin \left (c+d\,x\right )}{\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {17\,A\,a^4\,{\cos \left (c+d\,x\right )}^{11/2}\,\sin \left (c+d\,x\right )}{\sqrt {{\sin \left (c+d\,x\right )}^2}}\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {11}{4};\ \frac {15}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{77\,d}+\frac {4\,C\,a^4\,\left (\frac {2\,\sqrt {\cos \left (c+d\,x\right )}\,\sin \left (c+d\,x\right )}{3}+\frac {2\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{3}\right )}{d}+\frac {12\,C\,a^4\,\mathrm {E}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{d}+\frac {8\,C\,a^4\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{d}-\frac {8\,A\,a^4\,{\cos \left (c+d\,x\right )}^{9/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {9}{4};\ \frac {13}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{9\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {208\,A\,a^4\,{\cos \left (c+d\,x\right )}^{11/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {11}{4};\ \frac {19}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{385\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {8\,B\,a^4\,{\cos \left (c+d\,x\right )}^{7/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {7}{4};\ \frac {11}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{7\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {2\,B\,a^4\,{\cos \left (c+d\,x\right )}^{9/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {9}{4};\ \frac {13}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{9\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}+\frac {2\,C\,a^4\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{4},\frac {1}{2};\ \frac {3}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{d\,\sqrt {\cos \left (c+d\,x\right )}\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {2\,C\,a^4\,{\cos \left (c+d\,x\right )}^{7/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {7}{4};\ \frac {11}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{7\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}} \]
(2*(3*A*a^4*ellipticE(c/2 + (d*x)/2, 2) + 4*A*a^4*ellipticF(c/2 + (d*x)/2, 2) + 4*A*a^4*cos(c + d*x)^(1/2)*sin(c + d*x)))/(3*d) + (2*(4*B*a^4*ellipt icE(c/2 + (d*x)/2, 2) + 3*B*a^4*ellipticF(c/2 + (d*x)/2, 2) + 2*B*a^4*cos( c + d*x)^(1/2)*sin(c + d*x)))/d - (2*((66*A*a^4*cos(c + d*x)^(7/2)*sin(c + d*x))/(sin(c + d*x)^2)^(1/2) - (17*A*a^4*cos(c + d*x)^(11/2)*sin(c + d*x) )/(sin(c + d*x)^2)^(1/2))*hypergeom([1/2, 11/4], 15/4, cos(c + d*x)^2))/(7 7*d) + (4*C*a^4*((2*cos(c + d*x)^(1/2)*sin(c + d*x))/3 + (2*ellipticF(c/2 + (d*x)/2, 2))/3))/d + (12*C*a^4*ellipticE(c/2 + (d*x)/2, 2))/d + (8*C*a^4 *ellipticF(c/2 + (d*x)/2, 2))/d - (8*A*a^4*cos(c + d*x)^(9/2)*sin(c + d*x) *hypergeom([1/2, 9/4], 13/4, cos(c + d*x)^2))/(9*d*(sin(c + d*x)^2)^(1/2)) - (208*A*a^4*cos(c + d*x)^(11/2)*sin(c + d*x)*hypergeom([1/2, 11/4], 19/4 , cos(c + d*x)^2))/(385*d*(sin(c + d*x)^2)^(1/2)) - (8*B*a^4*cos(c + d*x)^ (7/2)*sin(c + d*x)*hypergeom([1/2, 7/4], 11/4, cos(c + d*x)^2))/(7*d*(sin( c + d*x)^2)^(1/2)) - (2*B*a^4*cos(c + d*x)^(9/2)*sin(c + d*x)*hypergeom([1 /2, 9/4], 13/4, cos(c + d*x)^2))/(9*d*(sin(c + d*x)^2)^(1/2)) + (2*C*a^4*s in(c + d*x)*hypergeom([-1/4, 1/2], 3/4, cos(c + d*x)^2))/(d*cos(c + d*x)^( 1/2)*(sin(c + d*x)^2)^(1/2)) - (2*C*a^4*cos(c + d*x)^(7/2)*sin(c + d*x)*hy pergeom([1/2, 7/4], 11/4, cos(c + d*x)^2))/(7*d*(sin(c + d*x)^2)^(1/2))