Integrand size = 43, antiderivative size = 310 \[ \int \cos ^{\frac {13}{2}}(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {8 a^4 (185 A+208 B+247 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{195 d}+\frac {8 a^4 (100 A+113 B+132 C) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{231 d}+\frac {8 a^4 (100 A+113 B+132 C) \sqrt {\cos (c+d x)} \sin (c+d x)}{231 d}+\frac {4 a^4 (5255 A+6019 B+6721 C) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{15015 d}+\frac {2 a (8 A+13 B) \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{143 d}+\frac {2 A \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^4 \sin (c+d x)}{13 d}+\frac {2 (13 A+17 B+11 C) \cos ^{\frac {3}{2}}(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{99 d}+\frac {4 (1355 A+1612 B+1573 C) \cos ^{\frac {3}{2}}(c+d x) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{9009 d} \]
8/195*a^4*(185*A+208*B+247*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+8/231*a^4*(100*A+113*B+132*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+4/15015*a^4*(5255*A+6019*B+6721*C)*cos(d*x+c)^(3/2)*sin(d*x+ c)/d+2/143*a*(8*A+13*B)*cos(d*x+c)^(3/2)*(a+a*cos(d*x+c))^3*sin(d*x+c)/d+2 /13*A*cos(d*x+c)^(3/2)*(a+a*cos(d*x+c))^4*sin(d*x+c)/d+2/99*(13*A+17*B+11* C)*cos(d*x+c)^(3/2)*(a^2+a^2*cos(d*x+c))^2*sin(d*x+c)/d+4/9009*(1355*A+161 2*B+1573*C)*cos(d*x+c)^(3/2)*(a^4+a^4*cos(d*x+c))*sin(d*x+c)/d+8/231*a^4*( 100*A+113*B+132*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 = 10.54 (sec) , antiderivative size = 1416, normalized size of antiderivative = 4.57 \[ \int \cos ^{\frac {13}{2}}(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx =\text {Too large to display} \]
Integrate[Cos[c + d*x]^(13/2)*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]
a^4*(Sqrt[Cos[c + d*x]]*(1 + Cos[c + d*x])^4*Sec[c/2 + (d*x)/2]^8*(-1/390* ((185*A + 208*B + 247*C)*Cot[c])/d + ((3764*A + 4087*B + 4488*C)*Cos[d*x]* Sin[c])/(14784*d) + ((15625*A + 15392*B + 13208*C)*Cos[2*d*x]*Sin[2*c])/(1 49760*d) + ((404*A + 321*B + 176*C)*Cos[3*d*x]*Sin[3*c])/(9856*d) + ((98*A + 52*B + 13*C)*Cos[4*d*x]*Sin[4*c])/(7488*d) + ((4*A + B)*Cos[5*d*x]*Sin[ 5*c])/(1408*d) + (A*Cos[6*d*x]*Sin[6*c])/(3328*d) + ((3764*A + 4087*B + 44 88*C)*Cos[c]*Sin[d*x])/(14784*d) + ((15625*A + 15392*B + 13208*C)*Cos[2*c] *Sin[2*d*x])/(149760*d) + ((404*A + 321*B + 176*C)*Cos[3*c]*Sin[3*d*x])/(9 856*d) + ((98*A + 52*B + 13*C)*Cos[4*c]*Sin[4*d*x])/(7488*d) + ((4*A + B)* Cos[5*c]*Sin[5*d*x])/(1408*d) + (A*Cos[6*c]*Sin[6*d*x])/(3328*d)) - (50*A* (1 + Cos[c + d*x])^4*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^8*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 - Ar cTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(231*d*Sqrt[1 + Cot[c ]^2]) - (113*B*(1 + Cos[c + d*x])^4*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, { 5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^8*Sec[d*x - ArcTan[C ot[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]]]])/(462*d *Sqrt[1 + Cot[c]^2]) - (2*C*(1 + Cos[c + d*x])^4*Csc[c]*HypergeometricPFQ[ {1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^8*Se...
Time = 2.26 (sec) , antiderivative size = 327, normalized size of antiderivative = 1.05, number of steps used = 24, number of rules used = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.558, Rules used = {3042, 4600, 3042, 3524, 27, 3042, 3455, 27, 3042, 3455, 3042, 3455, 27, 3042, 3447, 3042, 3502, 3042, 3227, 3042, 3115, 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.
| \(\displaystyle \int \cos ^{\frac {13}{2}}(c+d x) (a \sec (c+d x)+a)^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \cos (c+d x)^{13/2} (a \sec (c+d x)+a)^4 \left (A+B \sec (c+d x)+C \sec (c+d x)^2\right )dx\) |
| \(\Big \downarrow \) 4600 |
| \(\displaystyle \int \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^4 \left (A \cos ^2(c+d x)+B \cos (c+d x)+C\right )dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^4 \left (A \sin \left (c+d x+\frac {\pi }{2}\right )^2+B \sin \left (c+d x+\frac {\pi }{2}\right )+C\right )dx\) |
| \(\Big \downarrow \) 3524 |
| \(\displaystyle \frac {2 \int \frac {1}{2} \sqrt {\cos (c+d x)} (\cos (c+d x) a+a)^4 (a (3 A+13 C)+a (8 A+13 B) \cos (c+d x))dx}{13 a}+\frac {2 A \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^4}{13 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \sqrt {\cos (c+d x)} (\cos (c+d x) a+a)^4 (a (3 A+13 C)+a (8 A+13 B) \cos (c+d x))dx}{13 a}+\frac {2 A \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^4}{13 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^4 \left (a (3 A+13 C)+a (8 A+13 B) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx}{13 a}+\frac {2 A \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^4}{13 d}\) |
| \(\Big \downarrow \) 3455 |
| \(\displaystyle \frac {\frac {2}{11} \int \frac {1}{2} \sqrt {\cos (c+d x)} (\cos (c+d x) a+a)^3 \left ((57 A+39 B+143 C) a^2+13 (13 A+17 B+11 C) \cos (c+d x) a^2\right )dx+\frac {2 a^2 (8 A+13 B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^3}{11 d}}{13 a}+\frac {2 A \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^4}{13 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{11} \int \sqrt {\cos (c+d x)} (\cos (c+d x) a+a)^3 \left ((57 A+39 B+143 C) a^2+13 (13 A+17 B+11 C) \cos (c+d x) a^2\right )dx+\frac {2 a^2 (8 A+13 B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^3}{11 d}}{13 a}+\frac {2 A \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^4}{13 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{11} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^3 \left ((57 A+39 B+143 C) a^2+13 (13 A+17 B+11 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^2\right )dx+\frac {2 a^2 (8 A+13 B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^3}{11 d}}{13 a}+\frac {2 A \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^4}{13 d}\) |
| \(\Big \downarrow \) 3455 |
| \(\displaystyle \frac {\frac {1}{11} \left (\frac {2}{9} \int \sqrt {\cos (c+d x)} (\cos (c+d x) a+a)^2 \left (3 (170 A+169 B+286 C) a^3+(1355 A+1612 B+1573 C) \cos (c+d x) a^3\right )dx+\frac {26 a^3 (13 A+17 B+11 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^2}{9 d}\right )+\frac {2 a^2 (8 A+13 B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^3}{11 d}}{13 a}+\frac {2 A \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^4}{13 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{11} \left (\frac {2}{9} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2 \left (3 (170 A+169 B+286 C) a^3+(1355 A+1612 B+1573 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^3\right )dx+\frac {26 a^3 (13 A+17 B+11 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^2}{9 d}\right )+\frac {2 a^2 (8 A+13 B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^3}{11 d}}{13 a}+\frac {2 A \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^4}{13 d}\) |
| \(\Big \downarrow \) 3455 |
| \(\displaystyle \frac {\frac {1}{11} \left (\frac {2}{9} \left (\frac {2}{7} \int \frac {3}{2} \sqrt {\cos (c+d x)} (\cos (c+d x) a+a) \left (5 (509 A+559 B+715 C) a^4+(5255 A+6019 B+6721 C) \cos (c+d x) a^4\right )dx+\frac {2 (1355 A+1612 B+1573 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{7 d}\right )+\frac {26 a^3 (13 A+17 B+11 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^2}{9 d}\right )+\frac {2 a^2 (8 A+13 B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^3}{11 d}}{13 a}+\frac {2 A \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^4}{13 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{11} \left (\frac {2}{9} \left (\frac {3}{7} \int \sqrt {\cos (c+d x)} (\cos (c+d x) a+a) \left (5 (509 A+559 B+715 C) a^4+(5255 A+6019 B+6721 C) \cos (c+d x) a^4\right )dx+\frac {2 (1355 A+1612 B+1573 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{7 d}\right )+\frac {26 a^3 (13 A+17 B+11 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^2}{9 d}\right )+\frac {2 a^2 (8 A+13 B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^3}{11 d}}{13 a}+\frac {2 A \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^4}{13 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{11} \left (\frac {2}{9} \left (\frac {3}{7} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right ) \left (5 (509 A+559 B+715 C) a^4+(5255 A+6019 B+6721 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^4\right )dx+\frac {2 (1355 A+1612 B+1573 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{7 d}\right )+\frac {26 a^3 (13 A+17 B+11 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^2}{9 d}\right )+\frac {2 a^2 (8 A+13 B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^3}{11 d}}{13 a}+\frac {2 A \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^4}{13 d}\) |
| \(\Big \downarrow \) 3447 |
| \(\displaystyle \frac {\frac {1}{11} \left (\frac {2}{9} \left (\frac {3}{7} \int \sqrt {\cos (c+d x)} \left ((5255 A+6019 B+6721 C) \cos ^2(c+d x) a^5+5 (509 A+559 B+715 C) a^5+\left (5 (509 A+559 B+715 C) a^5+(5255 A+6019 B+6721 C) a^5\right ) \cos (c+d x)\right )dx+\frac {2 (1355 A+1612 B+1573 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{7 d}\right )+\frac {26 a^3 (13 A+17 B+11 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^2}{9 d}\right )+\frac {2 a^2 (8 A+13 B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^3}{11 d}}{13 a}+\frac {2 A \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^4}{13 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{11} \left (\frac {2}{9} \left (\frac {3}{7} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left ((5255 A+6019 B+6721 C) \sin \left (c+d x+\frac {\pi }{2}\right )^2 a^5+5 (509 A+559 B+715 C) a^5+\left (5 (509 A+559 B+715 C) a^5+(5255 A+6019 B+6721 C) a^5\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx+\frac {2 (1355 A+1612 B+1573 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{7 d}\right )+\frac {26 a^3 (13 A+17 B+11 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^2}{9 d}\right )+\frac {2 a^2 (8 A+13 B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^3}{11 d}}{13 a}+\frac {2 A \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^4}{13 d}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {\frac {1}{11} \left (\frac {2}{9} \left (\frac {3}{7} \left (\frac {2}{5} \int \sqrt {\cos (c+d x)} \left (77 (185 A+208 B+247 C) a^5+195 (100 A+113 B+132 C) \cos (c+d x) a^5\right )dx+\frac {2 a^5 (5255 A+6019 B+6721 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+\frac {2 (1355 A+1612 B+1573 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{7 d}\right )+\frac {26 a^3 (13 A+17 B+11 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^2}{9 d}\right )+\frac {2 a^2 (8 A+13 B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^3}{11 d}}{13 a}+\frac {2 A \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^4}{13 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{11} \left (\frac {2}{9} \left (\frac {3}{7} \left (\frac {2}{5} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (77 (185 A+208 B+247 C) a^5+195 (100 A+113 B+132 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^5\right )dx+\frac {2 a^5 (5255 A+6019 B+6721 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+\frac {2 (1355 A+1612 B+1573 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{7 d}\right )+\frac {26 a^3 (13 A+17 B+11 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^2}{9 d}\right )+\frac {2 a^2 (8 A+13 B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^3}{11 d}}{13 a}+\frac {2 A \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^4}{13 d}\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle \frac {\frac {1}{11} \left (\frac {2}{9} \left (\frac {3}{7} \left (\frac {2}{5} \left (195 a^5 (100 A+113 B+132 C) \int \cos ^{\frac {3}{2}}(c+d x)dx+77 a^5 (185 A+208 B+247 C) \int \sqrt {\cos (c+d x)}dx\right )+\frac {2 a^5 (5255 A+6019 B+6721 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+\frac {2 (1355 A+1612 B+1573 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{7 d}\right )+\frac {26 a^3 (13 A+17 B+11 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^2}{9 d}\right )+\frac {2 a^2 (8 A+13 B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^3}{11 d}}{13 a}+\frac {2 A \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^4}{13 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{11} \left (\frac {2}{9} \left (\frac {3}{7} \left (\frac {2}{5} \left (77 a^5 (185 A+208 B+247 C) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx+195 a^5 (100 A+113 B+132 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )^{3/2}dx\right )+\frac {2 a^5 (5255 A+6019 B+6721 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+\frac {2 (1355 A+1612 B+1573 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{7 d}\right )+\frac {26 a^3 (13 A+17 B+11 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^2}{9 d}\right )+\frac {2 a^2 (8 A+13 B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^3}{11 d}}{13 a}+\frac {2 A \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^4}{13 d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {\frac {1}{11} \left (\frac {2}{9} \left (\frac {3}{7} \left (\frac {2}{5} \left (77 a^5 (185 A+208 B+247 C) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx+195 a^5 (100 A+113 B+132 C) \left (\frac {1}{3} \int \frac {1}{\sqrt {\cos (c+d x)}}dx+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )\right )+\frac {2 a^5 (5255 A+6019 B+6721 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+\frac {2 (1355 A+1612 B+1573 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{7 d}\right )+\frac {26 a^3 (13 A+17 B+11 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^2}{9 d}\right )+\frac {2 a^2 (8 A+13 B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^3}{11 d}}{13 a}+\frac {2 A \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^4}{13 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{11} \left (\frac {2}{9} \left (\frac {3}{7} \left (\frac {2}{5} \left (77 a^5 (185 A+208 B+247 C) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx+195 a^5 (100 A+113 B+132 C) \left (\frac {1}{3} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )\right )+\frac {2 a^5 (5255 A+6019 B+6721 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+\frac {2 (1355 A+1612 B+1573 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{7 d}\right )+\frac {26 a^3 (13 A+17 B+11 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^2}{9 d}\right )+\frac {2 a^2 (8 A+13 B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^3}{11 d}}{13 a}+\frac {2 A \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^4}{13 d}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {\frac {1}{11} \left (\frac {2}{9} \left (\frac {3}{7} \left (\frac {2}{5} \left (195 a^5 (100 A+113 B+132 C) \left (\frac {1}{3} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )+\frac {154 a^5 (185 A+208 B+247 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )+\frac {2 a^5 (5255 A+6019 B+6721 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+\frac {2 (1355 A+1612 B+1573 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{7 d}\right )+\frac {26 a^3 (13 A+17 B+11 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^2}{9 d}\right )+\frac {2 a^2 (8 A+13 B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^3}{11 d}}{13 a}+\frac {2 A \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^4}{13 d}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {\frac {2 a^2 (8 A+13 B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^3}{11 d}+\frac {1}{11} \left (\frac {2}{9} \left (\frac {2 (1355 A+1612 B+1573 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{7 d}+\frac {3}{7} \left (\frac {2 a^5 (5255 A+6019 B+6721 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}+\frac {2}{5} \left (\frac {154 a^5 (185 A+208 B+247 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}+195 a^5 (100 A+113 B+132 C) \left (\frac {2 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d}+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )\right )\right )\right )+\frac {26 a^3 (13 A+17 B+11 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^2}{9 d}\right )}{13 a}+\frac {2 A \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^4}{13 d}\) |
(2*A*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(13*d) + ((2* a^2*(8*A + 13*B)*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/( 11*d) + ((26*a^3*(13*A + 17*B + 11*C)*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d* x])^2*Sin[c + d*x])/(9*d) + (2*((2*(1355*A + 1612*B + 1573*C)*Cos[c + d*x] ^(3/2)*(a^5 + a^5*Cos[c + d*x])*Sin[c + d*x])/(7*d) + (3*((2*a^5*(5255*A + 6019*B + 6721*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d) + (2*((154*a^5*(1 85*A + 208*B + 247*C)*EllipticE[(c + d*x)/2, 2])/d + 195*a^5*(100*A + 113* B + 132*C)*((2*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*Sqrt[Cos[c + d*x]]*Si n[c + d*x])/(3*d))))/5))/7))/9)/11)/(13*a)
3.13.9.3.1 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[(-b)*Cos[c + d* x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n) Int[(b*Sin [c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[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)*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)*Cos[e + f*x]*(a + b*Sin[e + f*x] )^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n + 2))), x] + Simp[1/(b*d*(m + n + 2)) Int[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n*Simp[A*b*d*(m + n + 2) + C*(a*c*m + b*d*(n + 1)) + (C*(a*d*m - b*c*(m + 1)) + b*B*d*(m + n + 2))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m, n} , x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && !Lt Q[m, -2^(-1)] && NeQ[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]
Time = 434.55 (sec) , antiderivative size = 576, normalized size of antiderivative = 1.86
| method | result | size |
| default | \(-\frac {8 \sqrt {\left (2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, a^{4} \left (-110880 A \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}+\left (594720 A +65520 B \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{12} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-1345120 A -323960 B -40040 C \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{10} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (1667840 A +659620 B +183040 C \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{8} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-1237490 A -713518 B -336622 C \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (572110 A +448448 B +322322 C \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-117945 A -110097 B -97383 C \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+19500 A \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-42735 A \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+22035 B \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-48048 B \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+25740 C \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-57057 C \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )\right )}{45045 \sqrt {-2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, d}\) | \(576\) |
int(cos(d*x+c)^(13/2)*(a+a*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x ,method=_RETURNVERBOSE)
-8/45045*((2*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^4*(-110 880*A*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^14+(594720*A+65520*B)*sin(1/2* d*x+1/2*c)^12*cos(1/2*d*x+1/2*c)+(-1345120*A-323960*B-40040*C)*sin(1/2*d*x +1/2*c)^10*cos(1/2*d*x+1/2*c)+(1667840*A+659620*B+183040*C)*sin(1/2*d*x+1/ 2*c)^8*cos(1/2*d*x+1/2*c)+(-1237490*A-713518*B-336622*C)*sin(1/2*d*x+1/2*c )^6*cos(1/2*d*x+1/2*c)+(572110*A+448448*B+322322*C)*sin(1/2*d*x+1/2*c)^4*c os(1/2*d*x+1/2*c)+(-117945*A-110097*B-97383*C)*sin(1/2*d*x+1/2*c)^2*cos(1/ 2*d*x+1/2*c)+19500*A*(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))-42735*A*(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))+22035*B*(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))-48048*B*(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)) +25740*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)*Ell ipticF(cos(1/2*d*x+1/2*c),2^(1/2))-57057*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)*EllipticE(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)/sin(1/2*d*x+1/2*c)/(2*cos (1/2*d*x+1/2*c)^2-1)^(1/2)/d
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.13 (sec) , antiderivative size = 287, normalized size of antiderivative = 0.93 \[ \int \cos ^{\frac {13}{2}}(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=-\frac {2 \, {\left (390 i \, \sqrt {2} {\left (100 \, A + 113 \, B + 132 \, C\right )} a^{4} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 390 i \, \sqrt {2} {\left (100 \, A + 113 \, B + 132 \, C\right )} a^{4} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 462 i \, \sqrt {2} {\left (185 \, A + 208 \, B + 247 \, C\right )} a^{4} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 462 i \, \sqrt {2} {\left (185 \, A + 208 \, B + 247 \, C\right )} a^{4} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - {\left (3465 \, A a^{4} \cos \left (d x + c\right )^{5} + 4095 \, {\left (4 \, A + B\right )} a^{4} \cos \left (d x + c\right )^{4} + 385 \, {\left (89 \, A + 52 \, B + 13 \, C\right )} a^{4} \cos \left (d x + c\right )^{3} + 585 \, {\left (80 \, A + 75 \, B + 44 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} + 77 \, {\left (740 \, A + 832 \, B + 793 \, C\right )} a^{4} \cos \left (d x + c\right ) + 780 \, {\left (100 \, A + 113 \, B + 132 \, C\right )} a^{4}\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )\right )}}{45045 \, d} \]
integrate(cos(d*x+c)^(13/2)*(a+a*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c )^2),x, algorithm="fricas")
-2/45045*(390*I*sqrt(2)*(100*A + 113*B + 132*C)*a^4*weierstrassPInverse(-4 , 0, cos(d*x + c) + I*sin(d*x + c)) - 390*I*sqrt(2)*(100*A + 113*B + 132*C )*a^4*weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) - 462*I*sq rt(2)*(185*A + 208*B + 247*C)*a^4*weierstrassZeta(-4, 0, weierstrassPInver se(-4, 0, cos(d*x + c) + I*sin(d*x + c))) + 462*I*sqrt(2)*(185*A + 208*B + 247*C)*a^4*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c))) - (3465*A*a^4*cos(d*x + c)^5 + 4095*(4*A + B)*a^4*cos( d*x + c)^4 + 385*(89*A + 52*B + 13*C)*a^4*cos(d*x + c)^3 + 585*(80*A + 75* B + 44*C)*a^4*cos(d*x + c)^2 + 77*(740*A + 832*B + 793*C)*a^4*cos(d*x + c) + 780*(100*A + 113*B + 132*C)*a^4)*sqrt(cos(d*x + c))*sin(d*x + c))/d
Timed out. \[ \int \cos ^{\frac {13}{2}}(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \]
Timed out. \[ \int \cos ^{\frac {13}{2}}(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \]
integrate(cos(d*x+c)^(13/2)*(a+a*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c )^2),x, algorithm="maxima")
\[ \int \cos ^{\frac {13}{2}}(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (a \sec \left (d x + c\right ) + a\right )}^{4} \cos \left (d x + c\right )^{\frac {13}{2}} \,d x } \]
integrate(cos(d*x+c)^(13/2)*(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)^(13/2), x)
Time = 21.05 (sec) , antiderivative size = 764, normalized size of antiderivative = 2.46 \[ \int \cos ^{\frac {13}{2}}(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Too large to display} \]
(2*(3*C*a^4*ellipticE(c/2 + (d*x)/2, 2) + 4*C*a^4*ellipticF(c/2 + (d*x)/2, 2) + 4*C*a^4*cos(c + d*x)^(1/2)*sin(c + d*x)))/(3*d) - (8*((13*A*a^4*cos( c + d*x)^(9/2)*sin(c + d*x))/(sin(c + d*x)^2)^(1/2) + (5*A*a^4*cos(c + d*x )^(13/2)*sin(c + d*x))/(sin(c + d*x)^2)^(1/2))*hypergeom([1/2, 13/4], 17/4 , cos(c + d*x)^2))/(117*d) - (136*((11*A*a^4*cos(c + d*x)^(11/2)*sin(c + d *x))/(sin(c + d*x)^2)^(1/2) + (51*A*a^4*cos(c + d*x)^(15/2)*sin(c + d*x))/ (sin(c + d*x)^2)^(1/2))*hypergeom([1/2, 15/4], 23/4, cos(c + d*x)^2))/(219 45*d) - (2*((66*C*a^4*cos(c + d*x)^(7/2)*sin(c + d*x))/(sin(c + d*x)^2)^(1 /2) - (17*C*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))/(77*d) - (2*hypergeom([1/2, 15/4], 19/4, cos(c + d*x)^2)*((165*A*a^4*cos(c + d*x)^(7/2)*sin(c + d*x))/ (sin(c + d*x)^2)^(1/2) + (578*A*a^4*cos(c + d*x)^(11/2)*sin(c + d*x))/(sin (c + d*x)^2)^(1/2) - (127*A*a^4*cos(c + d*x)^(15/2)*sin(c + d*x))/(sin(c + d*x)^2)^(1/2)))/(1155*d) + (B*a^4*((2*cos(c + d*x)^(1/2)*sin(c + d*x))/3 + (2*ellipticF(c/2 + (d*x)/2, 2))/3))/d - (160*A*a^4*cos(c + d*x)^(13/2)*s in(c + d*x)*hypergeom([1/2, 13/4], 21/4, cos(c + d*x)^2))/(663*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)) - (4*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))/(3 *d*(sin(c + d*x)^2)^(1/2)) - (8*B*a^4*cos(c + d*x)^(11/2)*sin(c + d*x)*...