Integrand size = 35, antiderivative size = 279 \[ \int \frac {(a+a \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right )}{\cos ^{\frac {3}{2}}(c+d x)} \, dx=-\frac {4 a^3 (7 A+5 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {4 a^3 (143 A+105 C) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{231 d}+\frac {8 a^3 (44 A+35 C) \sin (c+d x)}{385 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {4 a^3 (143 A+105 C) \sin (c+d x)}{231 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {4 a^3 (7 A+5 C) \sin (c+d x)}{5 d \sqrt {\cos (c+d x)}}+\frac {2 C (a+a \cos (c+d x))^3 \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)}+\frac {4 C \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{33 a d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 (33 A+35 C) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{231 d \cos ^{\frac {7}{2}}(c+d x)} \]
-4/5*a^3*(7*A+5*C)*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*Ellipti cE(sin(1/2*d*x+1/2*c),2^(1/2))/d+4/231*a^3*(143*A+105*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+8/3 85*a^3*(44*A+35*C)*sin(d*x+c)/d/cos(d*x+c)^(5/2)+4/231*a^3*(143*A+105*C)*s in(d*x+c)/d/cos(d*x+c)^(3/2)+2/11*C*(a+a*cos(d*x+c))^3*sin(d*x+c)/d/cos(d* x+c)^(11/2)+4/33*C*(a^2+a^2*cos(d*x+c))^2*sin(d*x+c)/a/d/cos(d*x+c)^(9/2)+ 2/231*(33*A+35*C)*(a^3+a^3*cos(d*x+c))*sin(d*x+c)/d/cos(d*x+c)^(7/2)+4/5*a ^3*(7*A+5*C)*sin(d*x+c)/d/cos(d*x+c)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 13.06 (sec) , antiderivative size = 1179, normalized size of antiderivative = 4.23 \[ \int \frac {(a+a \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right )}{\cos ^{\frac {3}{2}}(c+d x)} \, dx =\text {Too large to display} \]
(Cos[c + d*x]^(11/2)*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + C*Se c[c + d*x]^2)*(((7*A + 5*C)*Csc[c]*Sec[c])/(5*d) + (C*Sec[c]*Sec[c + d*x]^ 6*Sin[d*x])/(22*d) + (Sec[c]*Sec[c + d*x]^5*(3*C*Sin[c] + 11*C*Sin[d*x]))/ (66*d) + (Sec[c]*Sec[c + d*x]^4*(77*C*Sin[c] + 33*A*Sin[d*x] + 126*C*Sin[d *x]))/(462*d) + (Sec[c]*Sec[c + d*x]^3*(165*A*Sin[c] + 630*C*Sin[c] + 693* A*Sin[d*x] + 770*C*Sin[d*x]))/(2310*d) + (Sec[c]*Sec[c + d*x]^2*(693*A*Sin [c] + 770*C*Sin[c] + 1430*A*Sin[d*x] + 1050*C*Sin[d*x]))/(2310*d) + (Sec[c ]*Sec[c + d*x]*(715*A*Sin[c] + 525*C*Sin[c] + 1617*A*Sin[d*x] + 1155*C*Sin [d*x]))/(1155*d)))/(A + 2*C + A*Cos[2*c + 2*d*x]) - (13*A*Cos[c + d*x]^5*C sc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Se c[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + 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 + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (5*C*Cos[c + d*x]^5*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c ]]]^2]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + 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]]]])/(11*d*(A + 2*C + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) + (7* A*Cos[c + d*x]^5*Csc[c]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A ...
Time = 1.76 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.01, number of steps used = 23, number of rules used = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.657, Rules used = {3042, 4602, 3042, 3523, 27, 3042, 3454, 27, 3042, 3454, 27, 3042, 3447, 3042, 3500, 27, 3042, 3227, 3042, 3116, 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.
| \(\displaystyle \int \frac {(a \sec (c+d x)+a)^3 \left (A+C \sec ^2(c+d x)\right )}{\cos ^{\frac {3}{2}}(c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a \sec (c+d x)+a)^3 \left (A+C \sec (c+d x)^2\right )}{\cos (c+d x)^{3/2}}dx\) |
| \(\Big \downarrow \) 4602 |
| \(\displaystyle \int \frac {(a \cos (c+d x)+a)^3 \left (A \cos ^2(c+d x)+C\right )}{\cos ^{\frac {13}{2}}(c+d x)}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^3 \left (A \sin \left (c+d x+\frac {\pi }{2}\right )^2+C\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{13/2}}dx\) |
| \(\Big \downarrow \) 3523 |
| \(\displaystyle \frac {2 \int \frac {(\cos (c+d x) a+a)^3 (6 a C+a (11 A+3 C) \cos (c+d x))}{2 \cos ^{\frac {11}{2}}(c+d x)}dx}{11 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^3}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(\cos (c+d x) a+a)^3 (6 a C+a (11 A+3 C) \cos (c+d x))}{\cos ^{\frac {11}{2}}(c+d x)}dx}{11 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^3}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^3 \left (6 a C+a (11 A+3 C) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{11/2}}dx}{11 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^3}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3454 |
| \(\displaystyle \frac {\frac {2}{9} \int \frac {3 (\cos (c+d x) a+a)^2 \left ((33 A+35 C) a^2+3 (11 A+5 C) \cos (c+d x) a^2\right )}{2 \cos ^{\frac {9}{2}}(c+d x)}dx+\frac {4 C \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{3 d \cos ^{\frac {9}{2}}(c+d x)}}{11 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^3}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{3} \int \frac {(\cos (c+d x) a+a)^2 \left ((33 A+35 C) a^2+3 (11 A+5 C) \cos (c+d x) a^2\right )}{\cos ^{\frac {9}{2}}(c+d x)}dx+\frac {4 C \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{3 d \cos ^{\frac {9}{2}}(c+d x)}}{11 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^3}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{3} \int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2 \left ((33 A+35 C) a^2+3 (11 A+5 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{9/2}}dx+\frac {4 C \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{3 d \cos ^{\frac {9}{2}}(c+d x)}}{11 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^3}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3454 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {2}{7} \int \frac {3 (\cos (c+d x) a+a) \left (2 (44 A+35 C) a^3+5 (11 A+7 C) \cos (c+d x) a^3\right )}{\cos ^{\frac {7}{2}}(c+d x)}dx+\frac {2 (33 A+35 C) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {4 C \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{3 d \cos ^{\frac {9}{2}}(c+d x)}}{11 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^3}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {6}{7} \int \frac {(\cos (c+d x) a+a) \left (2 (44 A+35 C) a^3+5 (11 A+7 C) \cos (c+d x) a^3\right )}{\cos ^{\frac {7}{2}}(c+d x)}dx+\frac {2 (33 A+35 C) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {4 C \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{3 d \cos ^{\frac {9}{2}}(c+d x)}}{11 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^3}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {6}{7} \int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right ) \left (2 (44 A+35 C) a^3+5 (11 A+7 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^3\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{7/2}}dx+\frac {2 (33 A+35 C) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {4 C \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{3 d \cos ^{\frac {9}{2}}(c+d x)}}{11 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^3}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3447 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {6}{7} \int \frac {5 (11 A+7 C) \cos ^2(c+d x) a^4+2 (44 A+35 C) a^4+\left (5 (11 A+7 C) a^4+2 (44 A+35 C) a^4\right ) \cos (c+d x)}{\cos ^{\frac {7}{2}}(c+d x)}dx+\frac {2 (33 A+35 C) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {4 C \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{3 d \cos ^{\frac {9}{2}}(c+d x)}}{11 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^3}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {6}{7} \int \frac {5 (11 A+7 C) \sin \left (c+d x+\frac {\pi }{2}\right )^2 a^4+2 (44 A+35 C) a^4+\left (5 (11 A+7 C) a^4+2 (44 A+35 C) a^4\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{7/2}}dx+\frac {2 (33 A+35 C) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {4 C \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{3 d \cos ^{\frac {9}{2}}(c+d x)}}{11 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^3}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3500 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {6}{7} \left (\frac {2}{5} \int \frac {5 (143 A+105 C) a^4+77 (7 A+5 C) \cos (c+d x) a^4}{2 \cos ^{\frac {5}{2}}(c+d x)}dx+\frac {4 a^4 (44 A+35 C) \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (33 A+35 C) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {4 C \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{3 d \cos ^{\frac {9}{2}}(c+d x)}}{11 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^3}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {6}{7} \left (\frac {1}{5} \int \frac {5 (143 A+105 C) a^4+77 (7 A+5 C) \cos (c+d x) a^4}{\cos ^{\frac {5}{2}}(c+d x)}dx+\frac {4 a^4 (44 A+35 C) \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (33 A+35 C) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {4 C \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{3 d \cos ^{\frac {9}{2}}(c+d x)}}{11 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^3}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {6}{7} \left (\frac {1}{5} \int \frac {5 (143 A+105 C) a^4+77 (7 A+5 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^4}{\sin \left (c+d x+\frac {\pi }{2}\right )^{5/2}}dx+\frac {4 a^4 (44 A+35 C) \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (33 A+35 C) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {4 C \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{3 d \cos ^{\frac {9}{2}}(c+d x)}}{11 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^3}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {6}{7} \left (\frac {1}{5} \left (5 a^4 (143 A+105 C) \int \frac {1}{\cos ^{\frac {5}{2}}(c+d x)}dx+77 a^4 (7 A+5 C) \int \frac {1}{\cos ^{\frac {3}{2}}(c+d x)}dx\right )+\frac {4 a^4 (44 A+35 C) \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (33 A+35 C) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {4 C \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{3 d \cos ^{\frac {9}{2}}(c+d x)}}{11 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^3}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {6}{7} \left (\frac {1}{5} \left (5 a^4 (143 A+105 C) \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right )^{5/2}}dx+77 a^4 (7 A+5 C) \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2}}dx\right )+\frac {4 a^4 (44 A+35 C) \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (33 A+35 C) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {4 C \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{3 d \cos ^{\frac {9}{2}}(c+d x)}}{11 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^3}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3116 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {6}{7} \left (\frac {1}{5} \left (5 a^4 (143 A+105 C) \left (\frac {1}{3} \int \frac {1}{\sqrt {\cos (c+d x)}}dx+\frac {2 \sin (c+d x)}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+77 a^4 (7 A+5 C) \left (\frac {2 \sin (c+d x)}{d \sqrt {\cos (c+d x)}}-\int \sqrt {\cos (c+d x)}dx\right )\right )+\frac {4 a^4 (44 A+35 C) \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (33 A+35 C) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {4 C \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{3 d \cos ^{\frac {9}{2}}(c+d x)}}{11 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^3}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {6}{7} \left (\frac {1}{5} \left (5 a^4 (143 A+105 C) \left (\frac {1}{3} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sin (c+d x)}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+77 a^4 (7 A+5 C) \left (\frac {2 \sin (c+d x)}{d \sqrt {\cos (c+d x)}}-\int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx\right )\right )+\frac {4 a^4 (44 A+35 C) \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (33 A+35 C) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {4 C \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{3 d \cos ^{\frac {9}{2}}(c+d x)}}{11 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^3}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {6}{7} \left (\frac {1}{5} \left (5 a^4 (143 A+105 C) \left (\frac {1}{3} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sin (c+d x)}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+77 a^4 (7 A+5 C) \left (\frac {2 \sin (c+d x)}{d \sqrt {\cos (c+d x)}}-\frac {2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )\right )+\frac {4 a^4 (44 A+35 C) \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (33 A+35 C) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {4 C \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{3 d \cos ^{\frac {9}{2}}(c+d x)}}{11 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^3}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {2 (33 A+35 C) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{7 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {6}{7} \left (\frac {4 a^4 (44 A+35 C) \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {1}{5} \left (5 a^4 (143 A+105 C) \left (\frac {2 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d}+\frac {2 \sin (c+d x)}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+77 a^4 (7 A+5 C) \left (\frac {2 \sin (c+d x)}{d \sqrt {\cos (c+d x)}}-\frac {2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )\right )\right )\right )+\frac {4 C \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{3 d \cos ^{\frac {9}{2}}(c+d x)}}{11 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^3}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
(2*C*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(11*d*Cos[c + d*x]^(11/2)) + ((4 *C*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(3*d*Cos[c + d*x]^(9/2)) + ((2 *(33*A + 35*C)*(a^4 + a^4*Cos[c + d*x])*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7 /2)) + (6*((4*a^4*(44*A + 35*C)*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + ( 5*a^4*(143*A + 105*C)*((2*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*Sin[c + d* x])/(3*d*Cos[c + d*x]^(3/2))) + 77*a^4*(7*A + 5*C)*((-2*EllipticE[(c + d*x )/2, 2])/d + (2*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])))/5))/7)/3)/(11*a)
3.12.6.3.1 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[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[Cos[c + d*x]*(( b*Sin[c + d*x])^(n + 1)/(b*d*(n + 1))), x] + Simp[(n + 2)/(b^2*(n + 1)) I nt[(b*Sin[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1] && IntegerQ[2*n]
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^2)*(B*c - A*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*((c + d*Sin[ e + f*x])^(n + 1)/(d*f*(n + 1)*(b*c + a*d))), x] - Simp[b/(d*(n + 1)*(b*c + a*d)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp [a*A*d*(m - n - 2) - B*(a*c*(m - 1) + b*d*(n + 1)) - (A*b*d*(m + n + 1) - B *(b*c*m - a*d*(n + 1)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f , A, B}, 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[(-(A*b^2 - a*b*B + a^2*C))*Cos[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 1)* (a^2 - b^2))), x] + Simp[1/(b*(m + 1)*(a^2 - b^2)) Int[(a + b*Sin[e + f*x ])^(m + 1)*Simp[b*(a*A - b*B + a*C)*(m + 1) - (A*b^2 - a*b*B + a^2*C + b*(A *b - a*B + b*C)*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C}, x] && LtQ[m, -1] && NeQ[a^2 - b^2, 0]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(c^2*C + 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*(a*c*m + b*d*(n + 1)) - b*(A*d^2*(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, 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_.) + (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 + A*Cos [e + f*x]^2), x], x] /; FreeQ[{a, b, d, e, f, A, C, n}, x] && !IntegerQ[n] && IntegerQ[m]
Leaf count of result is larger than twice the leaf count of optimal. \(1380\) vs. \(2(303)=606\).
Time = 5.66 (sec) , antiderivative size = 1381, normalized size of antiderivative = 4.95
-16*(-(-2*cos(1/2*d*x+1/2*c)^2+1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^3*(1/8*C*( -1/352*cos(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^( 1/2)/(cos(1/2*d*x+1/2*c)^2-1/2)^6-9/616*cos(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x +1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/(cos(1/2*d*x+1/2*c)^2-1/2)^4-15/154* cos(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/(c os(1/2*d*x+1/2*c)^2-1/2)^2+15/77*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2* d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2) *EllipticF(cos(1/2*d*x+1/2*c),2^(1/2)))+3/8*C*(-1/144*cos(1/2*d*x+1/2*c)*( -2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/(cos(1/2*d*x+1/2*c)^2- 1/2)^5-7/180*cos(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c )^2)^(1/2)/(cos(1/2*d*x+1/2*c)^2-1/2)^3-14/15*sin(1/2*d*x+1/2*c)^2*cos(1/2 *d*x+1/2*c)/(-(-2*cos(1/2*d*x+1/2*c)^2+1)*sin(1/2*d*x+1/2*c)^2)^(1/2)+7/15 *(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/ 2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^ (1/2))-7/15*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2) /(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*(EllipticF(cos(1/2*d *x+1/2*c),2^(1/2))-EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))))+1/8*A/sin(1/2*d *x+1/2*c)^2/(2*sin(1/2*d*x+1/2*c)^2-1)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d* x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)-EllipticE(cos (1/2*d*x+1/2*c),2^(1/2))*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*(sin(1/2*d*x+...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.14 (sec) , antiderivative size = 299, normalized size of antiderivative = 1.07 \[ \int \frac {(a+a \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right )}{\cos ^{\frac {3}{2}}(c+d x)} \, dx=-\frac {2 \, {\left (5 i \, \sqrt {2} {\left (143 \, A + 105 \, C\right )} a^{3} \cos \left (d x + c\right )^{6} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 5 i \, \sqrt {2} {\left (143 \, A + 105 \, C\right )} a^{3} \cos \left (d x + c\right )^{6} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 231 i \, \sqrt {2} {\left (7 \, A + 5 \, C\right )} a^{3} \cos \left (d x + c\right )^{6} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - 231 i \, \sqrt {2} {\left (7 \, A + 5 \, C\right )} a^{3} \cos \left (d x + c\right )^{6} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - {\left (462 \, {\left (7 \, A + 5 \, C\right )} a^{3} \cos \left (d x + c\right )^{5} + 10 \, {\left (143 \, A + 105 \, C\right )} a^{3} \cos \left (d x + c\right )^{4} + 77 \, {\left (9 \, A + 10 \, C\right )} a^{3} \cos \left (d x + c\right )^{3} + 15 \, {\left (11 \, A + 42 \, C\right )} a^{3} \cos \left (d x + c\right )^{2} + 385 \, C a^{3} \cos \left (d x + c\right ) + 105 \, C a^{3}\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )\right )}}{1155 \, d \cos \left (d x + c\right )^{6}} \]
-2/1155*(5*I*sqrt(2)*(143*A + 105*C)*a^3*cos(d*x + c)^6*weierstrassPInvers e(-4, 0, cos(d*x + c) + I*sin(d*x + c)) - 5*I*sqrt(2)*(143*A + 105*C)*a^3* cos(d*x + c)^6*weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) + 231*I*sqrt(2)*(7*A + 5*C)*a^3*cos(d*x + c)^6*weierstrassZeta(-4, 0, weier strassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) - 231*I*sqrt(2)*(7*A + 5*C)*a^3*cos(d*x + c)^6*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c))) - (462*(7*A + 5*C)*a^3*cos(d*x + c)^5 + 10*(143*A + 105*C)*a^3*cos(d*x + c)^4 + 77*(9*A + 10*C)*a^3*cos(d*x + c)^ 3 + 15*(11*A + 42*C)*a^3*cos(d*x + c)^2 + 385*C*a^3*cos(d*x + c) + 105*C*a ^3)*sqrt(cos(d*x + c))*sin(d*x + c))/(d*cos(d*x + c)^6)
\[ \int \frac {(a+a \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right )}{\cos ^{\frac {3}{2}}(c+d x)} \, dx=a^{3} \left (\int \frac {A}{\cos ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx + \int \frac {3 A \sec {\left (c + d x \right )}}{\cos ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx + \int \frac {3 A \sec ^{2}{\left (c + d x \right )}}{\cos ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx + \int \frac {A \sec ^{3}{\left (c + d x \right )}}{\cos ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx + \int \frac {C \sec ^{2}{\left (c + d x \right )}}{\cos ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx + \int \frac {3 C \sec ^{3}{\left (c + d x \right )}}{\cos ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx + \int \frac {3 C \sec ^{4}{\left (c + d x \right )}}{\cos ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx + \int \frac {C \sec ^{5}{\left (c + d x \right )}}{\cos ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx\right ) \]
a**3*(Integral(A/cos(c + d*x)**(3/2), x) + Integral(3*A*sec(c + d*x)/cos(c + d*x)**(3/2), x) + Integral(3*A*sec(c + d*x)**2/cos(c + d*x)**(3/2), x) + Integral(A*sec(c + d*x)**3/cos(c + d*x)**(3/2), x) + Integral(C*sec(c + d*x)**2/cos(c + d*x)**(3/2), x) + Integral(3*C*sec(c + d*x)**3/cos(c + d*x )**(3/2), x) + Integral(3*C*sec(c + d*x)**4/cos(c + d*x)**(3/2), x) + Inte gral(C*sec(c + d*x)**5/cos(c + d*x)**(3/2), x))
Timed out. \[ \int \frac {(a+a \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right )}{\cos ^{\frac {3}{2}}(c+d x)} \, dx=\text {Timed out} \]
\[ \int \frac {(a+a \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right )}{\cos ^{\frac {3}{2}}(c+d x)} \, dx=\int { \frac {{\left (C \sec \left (d x + c\right )^{2} + A\right )} {\left (a \sec \left (d x + c\right ) + a\right )}^{3}}{\cos \left (d x + c\right )^{\frac {3}{2}}} \,d x } \]
Time = 21.56 (sec) , antiderivative size = 621, normalized size of antiderivative = 2.23 \[ \int \frac {(a+a \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right )}{\cos ^{\frac {3}{2}}(c+d x)} \, dx=\frac {8\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{4},\frac {1}{2};\ \frac {5}{4};\ {\cos \left (c+d\,x\right )}^2\right )\,\left (\frac {11\,A\,a^3\,\sin \left (c+d\,x\right )}{{\cos \left (c+d\,x\right )}^{3/2}\,\sqrt {1-{\cos \left (c+d\,x\right )}^2}}+\frac {42\,C\,a^3\,\sin \left (c+d\,x\right )}{{\cos \left (c+d\,x\right )}^{3/2}\,\sqrt {1-{\cos \left (c+d\,x\right )}^2}}+\frac {7\,C\,a^3\,\sin \left (c+d\,x\right )}{{\cos \left (c+d\,x\right )}^{7/2}\,\sqrt {1-{\cos \left (c+d\,x\right )}^2}}\right )}{231\,d}-\frac {8\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{4},\frac {1}{2};\ \frac {7}{4};\ {\cos \left (c+d\,x\right )}^2\right )\,\left (\frac {9\,A\,a^3\,\sin \left (c+d\,x\right )}{\sqrt {\cos \left (c+d\,x\right )}\,\sqrt {1-{\cos \left (c+d\,x\right )}^2}}+\frac {10\,C\,a^3\,\sin \left (c+d\,x\right )}{\sqrt {\cos \left (c+d\,x\right )}\,\sqrt {1-{\cos \left (c+d\,x\right )}^2}}+\frac {5\,C\,a^3\,\sin \left (c+d\,x\right )}{{\cos \left (c+d\,x\right )}^{5/2}\,\sqrt {1-{\cos \left (c+d\,x\right )}^2}}\right )}{45\,d}+\frac {2\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{4},\frac {1}{2};\ \frac {3}{4};\ {\cos \left (c+d\,x\right )}^2\right )\,\left (\frac {51\,A\,a^3\,\sin \left (c+d\,x\right )}{\sqrt {\cos \left (c+d\,x\right )}\,\sqrt {1-{\cos \left (c+d\,x\right )}^2}}+\frac {9\,A\,a^3\,\sin \left (c+d\,x\right )}{{\cos \left (c+d\,x\right )}^{5/2}\,\sqrt {1-{\cos \left (c+d\,x\right )}^2}}+\frac {40\,C\,a^3\,\sin \left (c+d\,x\right )}{\sqrt {\cos \left (c+d\,x\right )}\,\sqrt {1-{\cos \left (c+d\,x\right )}^2}}+\frac {15\,C\,a^3\,\sin \left (c+d\,x\right )}{{\cos \left (c+d\,x\right )}^{5/2}\,\sqrt {1-{\cos \left (c+d\,x\right )}^2}}+\frac {5\,C\,a^3\,\sin \left (c+d\,x\right )}{{\cos \left (c+d\,x\right )}^{9/2}\,\sqrt {1-{\cos \left (c+d\,x\right )}^2}}\right )}{15\,d}+\frac {2\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{4},\frac {1}{2};\ \frac {1}{4};\ {\cos \left (c+d\,x\right )}^2\right )\,\left (\frac {275\,A\,a^3\,\sin \left (c+d\,x\right )}{{\cos \left (c+d\,x\right )}^{3/2}\,\sqrt {1-{\cos \left (c+d\,x\right )}^2}}+\frac {33\,A\,a^3\,\sin \left (c+d\,x\right )}{{\cos \left (c+d\,x\right )}^{7/2}\,\sqrt {1-{\cos \left (c+d\,x\right )}^2}}+\frac {168\,C\,a^3\,\sin \left (c+d\,x\right )}{{\cos \left (c+d\,x\right )}^{3/2}\,\sqrt {1-{\cos \left (c+d\,x\right )}^2}}+\frac {119\,C\,a^3\,\sin \left (c+d\,x\right )}{{\cos \left (c+d\,x\right )}^{7/2}\,\sqrt {1-{\cos \left (c+d\,x\right )}^2}}+\frac {21\,C\,a^3\,\sin \left (c+d\,x\right )}{{\cos \left (c+d\,x\right )}^{11/2}\,\sqrt {1-{\cos \left (c+d\,x\right )}^2}}\right )}{231\,d} \]
(8*hypergeom([-3/4, 1/2], 5/4, cos(c + d*x)^2)*((11*A*a^3*sin(c + d*x))/(c os(c + d*x)^(3/2)*(1 - cos(c + d*x)^2)^(1/2)) + (42*C*a^3*sin(c + d*x))/(c os(c + d*x)^(3/2)*(1 - cos(c + d*x)^2)^(1/2)) + (7*C*a^3*sin(c + d*x))/(co s(c + d*x)^(7/2)*(1 - cos(c + d*x)^2)^(1/2))))/(231*d) - (8*hypergeom([-1/ 4, 1/2], 7/4, cos(c + d*x)^2)*((9*A*a^3*sin(c + d*x))/(cos(c + d*x)^(1/2)* (1 - cos(c + d*x)^2)^(1/2)) + (10*C*a^3*sin(c + d*x))/(cos(c + d*x)^(1/2)* (1 - cos(c + d*x)^2)^(1/2)) + (5*C*a^3*sin(c + d*x))/(cos(c + d*x)^(5/2)*( 1 - cos(c + d*x)^2)^(1/2))))/(45*d) + (2*hypergeom([-1/4, 1/2], 3/4, cos(c + d*x)^2)*((51*A*a^3*sin(c + d*x))/(cos(c + d*x)^(1/2)*(1 - cos(c + d*x)^ 2)^(1/2)) + (9*A*a^3*sin(c + d*x))/(cos(c + d*x)^(5/2)*(1 - cos(c + d*x)^2 )^(1/2)) + (40*C*a^3*sin(c + d*x))/(cos(c + d*x)^(1/2)*(1 - cos(c + d*x)^2 )^(1/2)) + (15*C*a^3*sin(c + d*x))/(cos(c + d*x)^(5/2)*(1 - cos(c + d*x)^2 )^(1/2)) + (5*C*a^3*sin(c + d*x))/(cos(c + d*x)^(9/2)*(1 - cos(c + d*x)^2) ^(1/2))))/(15*d) + (2*hypergeom([-3/4, 1/2], 1/4, cos(c + d*x)^2)*((275*A* a^3*sin(c + d*x))/(cos(c + d*x)^(3/2)*(1 - cos(c + d*x)^2)^(1/2)) + (33*A* a^3*sin(c + d*x))/(cos(c + d*x)^(7/2)*(1 - cos(c + d*x)^2)^(1/2)) + (168*C *a^3*sin(c + d*x))/(cos(c + d*x)^(3/2)*(1 - cos(c + d*x)^2)^(1/2)) + (119* C*a^3*sin(c + d*x))/(cos(c + d*x)^(7/2)*(1 - cos(c + d*x)^2)^(1/2)) + (21* C*a^3*sin(c + d*x))/(cos(c + d*x)^(11/2)*(1 - cos(c + d*x)^2)^(1/2))))/(23 1*d)