Integrand size = 43, antiderivative size = 274 \[ \int \cos ^{\frac {11}{2}}(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {8 a^4 (16 A+19 B+24 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {8 a^4 (113 A+132 B+187 C) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{231 d}+\frac {4 a^4 (667 A+803 B+913 C) \sqrt {\cos (c+d x)} \sin (c+d x)}{1155 d}+\frac {2 a (8 A+11 B) \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^3 \sin (c+d x)}{99 d}+\frac {2 A \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^4 \sin (c+d x)}{11 d}+\frac {2 (43 A+55 B+33 C) \sqrt {\cos (c+d x)} \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{231 d}+\frac {4 (769 A+946 B+891 C) \sqrt {\cos (c+d x)} \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{3465 d} \] Output:
8/15*a^4*(16*A+19*B+24*C)*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2))/d+8/231*a^ 4*(113*A+132*B+187*C)*InverseJacobiAM(1/2*d*x+1/2*c,2^(1/2))/d+4/1155*a^4* (667*A+803*B+913*C)*cos(d*x+c)^(1/2)*sin(d*x+c)/d+2/99*a*(8*A+11*B)*cos(d* x+c)^(1/2)*(a+a*cos(d*x+c))^3*sin(d*x+c)/d+2/11*A*cos(d*x+c)^(1/2)*(a+a*co s(d*x+c))^4*sin(d*x+c)/d+2/231*(43*A+55*B+33*C)*cos(d*x+c)^(1/2)*(a^2+a^2* cos(d*x+c))^2*sin(d*x+c)/d+4/3465*(769*A+946*B+891*C)*cos(d*x+c)^(1/2)*(a^ 4+a^4*cos(d*x+c))*sin(d*x+c)/d
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 12.43 (sec) , antiderivative size = 1751, normalized size of antiderivative = 6.39 \[ \int \cos ^{\frac {11}{2}}(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx =\text {Too large to display} \] Input:
Integrate[Cos[c + d*x]^(11/2)*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]
Output:
(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)*(-1/15*((16*A + 19*B + 24*C)*Cot[c])/d + (( 4087*A + 4488*B + 4202*C)*Cos[d*x]*Sin[c])/(7392*d) + ((148*A + 127*B + 72 *C)*Cos[2*d*x]*Sin[2*c])/(720*d) + ((321*A + 176*B + 44*C)*Cos[3*d*x]*Sin[ 3*c])/(4928*d) + ((4*A + B)*Cos[4*d*x]*Sin[4*c])/(288*d) + (A*Cos[5*d*x]*S in[5*c])/(704*d) + ((4087*A + 4488*B + 4202*C)*Cos[c]*Sin[d*x])/(7392*d) + ((148*A + 127*B + 72*C)*Cos[2*c]*Sin[2*d*x])/(720*d) + ((321*A + 176*B + 44*C)*Cos[3*c]*Sin[3*d*x])/(4928*d) + ((4*A + B)*Cos[4*c]*Sin[4*d*x])/(288 *d) + (A*Cos[5*c]*Sin[5*d*x])/(704*d)))/(A + 2*C + 2*B*Cos[c + d*x] + A*Co s[2*c + 2*d*x]) - (113*A*Cos[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]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(231*d*(A + 2 *C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (4*B*Cos [c + d*x]^6*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[C ot[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[C ot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt [1 + Sin[d*x - ArcTan[Cot[c]]]])/(7*d*(A + 2*C + 2*B*Cos[c + d*x] + A*C...
Time = 2.14 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.08, 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, 3524, 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.
| \(\displaystyle \int \cos ^{\frac {11}{2}}(c+d x) (a \sec (c+d x)+a)^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \cos (c+d x)^{11/2} (a \sec (c+d x)+a)^4 \left (A+B \sec (c+d x)+C \sec (c+d x)^2\right )dx\) |
| \(\Big \downarrow \) 4600 |
| \(\displaystyle \int \frac {(a \cos (c+d x)+a)^4 \left (A \cos ^2(c+d x)+B \cos (c+d x)+C\right )}{\sqrt {\cos (c+d x)}}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^4 \left (A \sin \left (c+d x+\frac {\pi }{2}\right )^2+B \sin \left (c+d x+\frac {\pi }{2}\right )+C\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx\) |
| \(\Big \downarrow \) 3524 |
| \(\displaystyle \frac {2 \int \frac {(\cos (c+d x) a+a)^4 (a (A+11 C)+a (8 A+11 B) \cos (c+d x))}{2 \sqrt {\cos (c+d x)}}dx}{11 a}+\frac {2 A \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^4}{11 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(\cos (c+d x) a+a)^4 (a (A+11 C)+a (8 A+11 B) \cos (c+d x))}{\sqrt {\cos (c+d x)}}dx}{11 a}+\frac {2 A \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^4}{11 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^4 \left (a (A+11 C)+a (8 A+11 B) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{11 a}+\frac {2 A \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^4}{11 d}\) |
| \(\Big \downarrow \) 3455 |
| \(\displaystyle \frac {\frac {2}{9} \int \frac {(\cos (c+d x) a+a)^3 \left ((17 A+11 B+99 C) a^2+3 (43 A+55 B+33 C) \cos (c+d x) a^2\right )}{2 \sqrt {\cos (c+d x)}}dx+\frac {2 a^2 (8 A+11 B) \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^3}{9 d}}{11 a}+\frac {2 A \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^4}{11 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{9} \int \frac {(\cos (c+d x) a+a)^3 \left ((17 A+11 B+99 C) a^2+3 (43 A+55 B+33 C) \cos (c+d x) a^2\right )}{\sqrt {\cos (c+d x)}}dx+\frac {2 a^2 (8 A+11 B) \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^3}{9 d}}{11 a}+\frac {2 A \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^4}{11 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{9} \int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^3 \left ((17 A+11 B+99 C) a^2+3 (43 A+55 B+33 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^2\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 a^2 (8 A+11 B) \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^3}{9 d}}{11 a}+\frac {2 A \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^4}{11 d}\) |
| \(\Big \downarrow \) 3455 |
| \(\displaystyle \frac {\frac {1}{9} \left (\frac {2}{7} \int \frac {(\cos (c+d x) a+a)^2 \left ((124 A+121 B+396 C) a^3+(769 A+946 B+891 C) \cos (c+d x) a^3\right )}{\sqrt {\cos (c+d x)}}dx+\frac {6 a^3 (43 A+55 B+33 C) \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^2}{7 d}\right )+\frac {2 a^2 (8 A+11 B) \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^3}{9 d}}{11 a}+\frac {2 A \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^4}{11 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{9} \left (\frac {2}{7} \int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2 \left ((124 A+121 B+396 C) a^3+(769 A+946 B+891 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^3\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {6 a^3 (43 A+55 B+33 C) \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^2}{7 d}\right )+\frac {2 a^2 (8 A+11 B) \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^3}{9 d}}{11 a}+\frac {2 A \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^4}{11 d}\) |
| \(\Big \downarrow \) 3455 |
| \(\displaystyle \frac {\frac {1}{9} \left (\frac {2}{7} \left (\frac {2}{5} \int \frac {3 (\cos (c+d x) a+a) \left ((463 A+517 B+957 C) a^4+3 (667 A+803 B+913 C) \cos (c+d x) a^4\right )}{2 \sqrt {\cos (c+d x)}}dx+\frac {2 (769 A+946 B+891 C) \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^5 \cos (c+d x)+a^5\right )}{5 d}\right )+\frac {6 a^3 (43 A+55 B+33 C) \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^2}{7 d}\right )+\frac {2 a^2 (8 A+11 B) \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^3}{9 d}}{11 a}+\frac {2 A \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^4}{11 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{9} \left (\frac {2}{7} \left (\frac {3}{5} \int \frac {(\cos (c+d x) a+a) \left ((463 A+517 B+957 C) a^4+3 (667 A+803 B+913 C) \cos (c+d x) a^4\right )}{\sqrt {\cos (c+d x)}}dx+\frac {2 (769 A+946 B+891 C) \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^5 \cos (c+d x)+a^5\right )}{5 d}\right )+\frac {6 a^3 (43 A+55 B+33 C) \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^2}{7 d}\right )+\frac {2 a^2 (8 A+11 B) \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^3}{9 d}}{11 a}+\frac {2 A \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^4}{11 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{9} \left (\frac {2}{7} \left (\frac {3}{5} \int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right ) \left ((463 A+517 B+957 C) a^4+3 (667 A+803 B+913 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^4\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 (769 A+946 B+891 C) \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^5 \cos (c+d x)+a^5\right )}{5 d}\right )+\frac {6 a^3 (43 A+55 B+33 C) \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^2}{7 d}\right )+\frac {2 a^2 (8 A+11 B) \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^3}{9 d}}{11 a}+\frac {2 A \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^4}{11 d}\) |
| \(\Big \downarrow \) 3447 |
| \(\displaystyle \frac {\frac {1}{9} \left (\frac {2}{7} \left (\frac {3}{5} \int \frac {3 (667 A+803 B+913 C) \cos ^2(c+d x) a^5+(463 A+517 B+957 C) a^5+\left (3 (667 A+803 B+913 C) a^5+(463 A+517 B+957 C) a^5\right ) \cos (c+d x)}{\sqrt {\cos (c+d x)}}dx+\frac {2 (769 A+946 B+891 C) \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^5 \cos (c+d x)+a^5\right )}{5 d}\right )+\frac {6 a^3 (43 A+55 B+33 C) \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^2}{7 d}\right )+\frac {2 a^2 (8 A+11 B) \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^3}{9 d}}{11 a}+\frac {2 A \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^4}{11 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{9} \left (\frac {2}{7} \left (\frac {3}{5} \int \frac {3 (667 A+803 B+913 C) \sin \left (c+d x+\frac {\pi }{2}\right )^2 a^5+(463 A+517 B+957 C) a^5+\left (3 (667 A+803 B+913 C) a^5+(463 A+517 B+957 C) a^5\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 (769 A+946 B+891 C) \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^5 \cos (c+d x)+a^5\right )}{5 d}\right )+\frac {6 a^3 (43 A+55 B+33 C) \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^2}{7 d}\right )+\frac {2 a^2 (8 A+11 B) \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^3}{9 d}}{11 a}+\frac {2 A \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^4}{11 d}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {\frac {1}{9} \left (\frac {2}{7} \left (\frac {3}{5} \left (\frac {2}{3} \int \frac {3 \left (5 (113 A+132 B+187 C) a^5+77 (16 A+19 B+24 C) \cos (c+d x) a^5\right )}{\sqrt {\cos (c+d x)}}dx+\frac {2 a^5 (667 A+803 B+913 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{d}\right )+\frac {2 (769 A+946 B+891 C) \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^5 \cos (c+d x)+a^5\right )}{5 d}\right )+\frac {6 a^3 (43 A+55 B+33 C) \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^2}{7 d}\right )+\frac {2 a^2 (8 A+11 B) \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^3}{9 d}}{11 a}+\frac {2 A \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^4}{11 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{9} \left (\frac {2}{7} \left (\frac {3}{5} \left (2 \int \frac {5 (113 A+132 B+187 C) a^5+77 (16 A+19 B+24 C) \cos (c+d x) a^5}{\sqrt {\cos (c+d x)}}dx+\frac {2 a^5 (667 A+803 B+913 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{d}\right )+\frac {2 (769 A+946 B+891 C) \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^5 \cos (c+d x)+a^5\right )}{5 d}\right )+\frac {6 a^3 (43 A+55 B+33 C) \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^2}{7 d}\right )+\frac {2 a^2 (8 A+11 B) \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^3}{9 d}}{11 a}+\frac {2 A \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^4}{11 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{9} \left (\frac {2}{7} \left (\frac {3}{5} \left (2 \int \frac {5 (113 A+132 B+187 C) a^5+77 (16 A+19 B+24 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^5}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 a^5 (667 A+803 B+913 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{d}\right )+\frac {2 (769 A+946 B+891 C) \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^5 \cos (c+d x)+a^5\right )}{5 d}\right )+\frac {6 a^3 (43 A+55 B+33 C) \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^2}{7 d}\right )+\frac {2 a^2 (8 A+11 B) \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^3}{9 d}}{11 a}+\frac {2 A \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^4}{11 d}\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle \frac {\frac {1}{9} \left (\frac {2}{7} \left (\frac {3}{5} \left (2 \left (5 a^5 (113 A+132 B+187 C) \int \frac {1}{\sqrt {\cos (c+d x)}}dx+77 a^5 (16 A+19 B+24 C) \int \sqrt {\cos (c+d x)}dx\right )+\frac {2 a^5 (667 A+803 B+913 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{d}\right )+\frac {2 (769 A+946 B+891 C) \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^5 \cos (c+d x)+a^5\right )}{5 d}\right )+\frac {6 a^3 (43 A+55 B+33 C) \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^2}{7 d}\right )+\frac {2 a^2 (8 A+11 B) \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^3}{9 d}}{11 a}+\frac {2 A \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^4}{11 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{9} \left (\frac {2}{7} \left (\frac {3}{5} \left (2 \left (5 a^5 (113 A+132 B+187 C) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+77 a^5 (16 A+19 B+24 C) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx\right )+\frac {2 a^5 (667 A+803 B+913 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{d}\right )+\frac {2 (769 A+946 B+891 C) \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^5 \cos (c+d x)+a^5\right )}{5 d}\right )+\frac {6 a^3 (43 A+55 B+33 C) \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^2}{7 d}\right )+\frac {2 a^2 (8 A+11 B) \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^3}{9 d}}{11 a}+\frac {2 A \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^4}{11 d}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {\frac {1}{9} \left (\frac {2}{7} \left (\frac {3}{5} \left (2 \left (5 a^5 (113 A+132 B+187 C) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {154 a^5 (16 A+19 B+24 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )+\frac {2 a^5 (667 A+803 B+913 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{d}\right )+\frac {2 (769 A+946 B+891 C) \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^5 \cos (c+d x)+a^5\right )}{5 d}\right )+\frac {6 a^3 (43 A+55 B+33 C) \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^2}{7 d}\right )+\frac {2 a^2 (8 A+11 B) \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^3}{9 d}}{11 a}+\frac {2 A \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^4}{11 d}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {\frac {2 a^2 (8 A+11 B) \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^3}{9 d}+\frac {1}{9} \left (\frac {2}{7} \left (\frac {2 (769 A+946 B+891 C) \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^5 \cos (c+d x)+a^5\right )}{5 d}+\frac {3}{5} \left (\frac {2 a^5 (667 A+803 B+913 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{d}+2 \left (\frac {10 a^5 (113 A+132 B+187 C) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{d}+\frac {154 a^5 (16 A+19 B+24 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )\right )\right )+\frac {6 a^3 (43 A+55 B+33 C) \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^2}{7 d}\right )}{11 a}+\frac {2 A \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^4}{11 d}\) |
Input:
Int[Cos[c + d*x]^(11/2)*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec [c + d*x]^2),x]
Output:
(2*A*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(11*d) + ((2* a^2*(8*A + 11*B)*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/( 9*d) + ((6*a^3*(43*A + 55*B + 33*C)*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x] )^2*Sin[c + d*x])/(7*d) + (2*((2*(769*A + 946*B + 891*C)*Sqrt[Cos[c + d*x] ]*(a^5 + a^5*Cos[c + d*x])*Sin[c + d*x])/(5*d) + (3*(2*((154*a^5*(16*A + 1 9*B + 24*C)*EllipticE[(c + d*x)/2, 2])/d + (10*a^5*(113*A + 132*B + 187*C) *EllipticF[(c + d*x)/2, 2])/d) + (2*a^5*(667*A + 803*B + 913*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/d))/5))/7)/9)/(11*a)
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)*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]
Leaf count of result is larger than twice the leaf count of optimal. \(544\) vs. \(2(253)=506\).
Time = 500.38 (sec) , antiderivative size = 545, normalized size of antiderivative = 1.99
| 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 (5040 A \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}+\left (-24920 A -3080 B \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{10} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (50740 A +14080 B +1980 C \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{8} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-54886 A -25894 B -8514 C \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (34496 A +24794 B +14784 C \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-8469 A -7491 B -5511 C \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+1695 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 )-3696 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 )+1980 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 )-4389 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 )+2805 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 )-5544 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 )}{3465 \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}\) | \(545\) |
Input:
int(cos(d*x+c)^(11/2)*(a+a*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x ,method=_RETURNVERBOSE)
Output:
-8/3465*((2*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^4*(5040* A*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^12+(-24920*A-3080*B)*sin(1/2*d*x+1 /2*c)^10*cos(1/2*d*x+1/2*c)+(50740*A+14080*B+1980*C)*sin(1/2*d*x+1/2*c)^8* cos(1/2*d*x+1/2*c)+(-54886*A-25894*B-8514*C)*sin(1/2*d*x+1/2*c)^6*cos(1/2* d*x+1/2*c)+(34496*A+24794*B+14784*C)*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2* c)+(-8469*A-7491*B-5511*C)*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)+1695*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(co s(1/2*d*x+1/2*c),2^(1/2))-3696*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))+1980*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))-4389*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))+2805*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))-5544*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 complex when optimal does not.
Time = 0.10 (sec) , antiderivative size = 264, normalized size of antiderivative = 0.96 \[ \int \cos ^{\frac {11}{2}}(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=-\frac {2 \, {\left (30 i \, \sqrt {2} {\left (113 \, A + 132 \, B + 187 \, C\right )} a^{4} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 30 i \, \sqrt {2} {\left (113 \, A + 132 \, B + 187 \, C\right )} a^{4} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 462 i \, \sqrt {2} {\left (16 \, A + 19 \, B + 24 \, C\right )} a^{4} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 462 i \, \sqrt {2} {\left (16 \, A + 19 \, B + 24 \, C\right )} a^{4} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - {\left (315 \, A a^{4} \cos \left (d x + c\right )^{4} + 385 \, {\left (4 \, A + B\right )} a^{4} \cos \left (d x + c\right )^{3} + 45 \, {\left (75 \, A + 44 \, B + 11 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} + 77 \, {\left (64 \, A + 61 \, B + 36 \, C\right )} a^{4} \cos \left (d x + c\right ) + 15 \, {\left (452 \, A + 528 \, B + 517 \, C\right )} a^{4}\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )\right )}}{3465 \, d} \] Input:
integrate(cos(d*x+c)^(11/2)*(a+a*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c )^2),x, algorithm="fricas")
Output:
-2/3465*(30*I*sqrt(2)*(113*A + 132*B + 187*C)*a^4*weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c)) - 30*I*sqrt(2)*(113*A + 132*B + 187*C)*a ^4*weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) - 462*I*sqrt( 2)*(16*A + 19*B + 24*C)*a^4*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) + 462*I*sqrt(2)*(16*A + 19*B + 24*C)*a ^4*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin( d*x + c))) - (315*A*a^4*cos(d*x + c)^4 + 385*(4*A + B)*a^4*cos(d*x + c)^3 + 45*(75*A + 44*B + 11*C)*a^4*cos(d*x + c)^2 + 77*(64*A + 61*B + 36*C)*a^4 *cos(d*x + c) + 15*(452*A + 528*B + 517*C)*a^4)*sqrt(cos(d*x + c))*sin(d*x + c))/d
Timed out. \[ \int \cos ^{\frac {11}{2}}(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \] Input:
integrate(cos(d*x+c)**(11/2)*(a+a*sec(d*x+c))**4*(A+B*sec(d*x+c)+C*sec(d*x +c)**2),x)
Output:
Timed out
Timed out. \[ \int \cos ^{\frac {11}{2}}(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \] Input:
integrate(cos(d*x+c)^(11/2)*(a+a*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c )^2),x, algorithm="maxima")
Output:
Timed out
\[ \int \cos ^{\frac {11}{2}}(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (a \sec \left (d x + c\right ) + a\right )}^{4} \cos \left (d x + c\right )^{\frac {11}{2}} \,d x } \] Input:
integrate(cos(d*x+c)^(11/2)*(a+a*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c )^2),x, algorithm="giac")
Output:
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)^(11/2), x)
Time = 14.82 (sec) , antiderivative size = 601, normalized size of antiderivative = 2.19 \[ \int \cos ^{\frac {11}{2}}(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx =\text {Too large to display} \] Input:
int(cos(c + d*x)^(11/2)*(a + a/cos(c + d*x))^4*(A + B/cos(c + d*x) + C/cos (c + d*x)^2),x)
Output:
(2*(3*B*a^4*ellipticE(c/2 + (d*x)/2, 2) + 4*B*a^4*ellipticF(c/2 + (d*x)/2, 2) + 4*B*a^4*cos(c + d*x)^(1/2)*sin(c + d*x)))/(3*d) + (2*(4*C*a^4*ellipt icE(c/2 + (d*x)/2, 2) + 3*C*a^4*ellipticF(c/2 + (d*x)/2, 2) + 2*C*a^4*cos( c + d*x)^(1/2)*sin(c + d*x)))/d - (2*((66*B*a^4*cos(c + d*x)^(7/2)*sin(c + d*x))/(sin(c + d*x)^2)^(1/2) - (17*B*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) + (A*a^4*((2*cos(c + d*x)^(1/2)*sin(c + d*x))/3 + (2*ellipticF(c/2 + (d*x)/2, 2))/3))/d - (8*A*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*A*a^4*c os(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*A*a^4*cos(c + d*x)^(11/2)*sin(c + d*x) *hypergeom([1/2, 11/4], 15/4, cos(c + d*x)^2))/(11*d*(sin(c + d*x)^2)^(1/2 )) - (2*A*a^4*cos(c + d*x)^(13/2)*sin(c + d*x)*hypergeom([1/2, 13/4], 17/4 , cos(c + d*x)^2))/(13*d*(sin(c + d*x)^2)^(1/2)) - (8*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)) - (208*B*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*C* 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*C*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)...
\[ \int \cos ^{\frac {11}{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=a^{4} \left (\left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{5} \sec \left (d x +c \right )^{6}d x \right ) c +\left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{5} \sec \left (d x +c \right )^{5}d x \right ) b +4 \left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{5} \sec \left (d x +c \right )^{5}d x \right ) c +\left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{5} \sec \left (d x +c \right )^{4}d x \right ) a +4 \left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{5} \sec \left (d x +c \right )^{4}d x \right ) b +6 \left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{5} \sec \left (d x +c \right )^{4}d x \right ) c +4 \left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{5} \sec \left (d x +c \right )^{3}d x \right ) a +6 \left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{5} \sec \left (d x +c \right )^{3}d x \right ) b +4 \left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{5} \sec \left (d x +c \right )^{3}d x \right ) c +6 \left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{5} \sec \left (d x +c \right )^{2}d x \right ) a +4 \left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{5} \sec \left (d x +c \right )^{2}d x \right ) b +\left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{5} \sec \left (d x +c \right )^{2}d x \right ) c +4 \left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{5} \sec \left (d x +c \right )d x \right ) a +\left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{5} \sec \left (d x +c \right )d x \right ) b +\left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{5}d x \right ) a \right ) \] Input:
int(cos(d*x+c)^(11/2)*(a+a*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x )
Output:
a**4*(int(sqrt(cos(c + d*x))*cos(c + d*x)**5*sec(c + d*x)**6,x)*c + int(sq rt(cos(c + d*x))*cos(c + d*x)**5*sec(c + d*x)**5,x)*b + 4*int(sqrt(cos(c + d*x))*cos(c + d*x)**5*sec(c + d*x)**5,x)*c + int(sqrt(cos(c + d*x))*cos(c + d*x)**5*sec(c + d*x)**4,x)*a + 4*int(sqrt(cos(c + d*x))*cos(c + d*x)**5 *sec(c + d*x)**4,x)*b + 6*int(sqrt(cos(c + d*x))*cos(c + d*x)**5*sec(c + d *x)**4,x)*c + 4*int(sqrt(cos(c + d*x))*cos(c + d*x)**5*sec(c + d*x)**3,x)* a + 6*int(sqrt(cos(c + d*x))*cos(c + d*x)**5*sec(c + d*x)**3,x)*b + 4*int( sqrt(cos(c + d*x))*cos(c + d*x)**5*sec(c + d*x)**3,x)*c + 6*int(sqrt(cos(c + d*x))*cos(c + d*x)**5*sec(c + d*x)**2,x)*a + 4*int(sqrt(cos(c + d*x))*c os(c + d*x)**5*sec(c + d*x)**2,x)*b + int(sqrt(cos(c + d*x))*cos(c + d*x)* *5*sec(c + d*x)**2,x)*c + 4*int(sqrt(cos(c + d*x))*cos(c + d*x)**5*sec(c + d*x),x)*a + int(sqrt(cos(c + d*x))*cos(c + d*x)**5*sec(c + d*x),x)*b + in t(sqrt(cos(c + d*x))*cos(c + d*x)**5,x)*a)