Integrand size = 43, antiderivative size = 515 \[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {13}{2}}(c+d x) \, dx=-\frac {2 \left (7 a^4 B+54 a^2 b^2 B+15 b^4 B+12 a b^3 (3 A+5 C)+4 a^3 b (7 A+9 C)\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{15 d}+\frac {2 \left (220 a^3 b B+308 a b^3 B+77 b^4 (A+3 C)+66 a^2 b^2 (5 A+7 C)+5 a^4 (9 A+11 C)\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{231 d}+\frac {2 \left (7 a^4 B+54 a^2 b^2 B+15 b^4 B+12 a b^3 (3 A+5 C)+4 a^3 b (7 A+9 C)\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{15 d}+\frac {2 \left (64 A b^4+660 a^3 b B+682 a b^3 B+15 a^4 (9 A+11 C)+9 a^2 b^2 (101 A+143 C)\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{693 d}+\frac {2 a \left (192 A b^3+539 a^3 B+1353 a b^2 B+2 a^2 b (673 A+891 C)\right ) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{3465 d}+\frac {2 \left (16 A b^2+55 a b B+3 a^2 (9 A+11 C)\right ) (a+b \cos (c+d x))^2 \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{231 d}+\frac {2 (8 A b+11 a B) (a+b \cos (c+d x))^3 \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{99 d}+\frac {2 A (a+b \cos (c+d x))^4 \sec ^{\frac {11}{2}}(c+d x) \sin (c+d x)}{11 d} \]
2/693*(64*A*b^4+660*B*a^3*b+682*B*a*b^3+15*a^4*(9*A+11*C)+9*a^2*b^2*(101*A +143*C))*sec(d*x+c)^(3/2)*sin(d*x+c)/d+2/3465*a*(192*A*b^3+539*B*a^3+1353* B*a*b^2+2*a^2*b*(673*A+891*C))*sec(d*x+c)^(5/2)*sin(d*x+c)/d+2/231*(16*A*b ^2+55*B*a*b+3*a^2*(9*A+11*C))*(a+b*cos(d*x+c))^2*sec(d*x+c)^(7/2)*sin(d*x+ c)/d+2/99*(8*A*b+11*B*a)*(a+b*cos(d*x+c))^3*sec(d*x+c)^(9/2)*sin(d*x+c)/d+ 2/11*A*(a+b*cos(d*x+c))^4*sec(d*x+c)^(11/2)*sin(d*x+c)/d+2/15*(7*B*a^4+54* B*a^2*b^2+15*B*b^4+12*a*b^3*(3*A+5*C)+4*a^3*b*(7*A+9*C))*sin(d*x+c)*sec(d* x+c)^(1/2)/d-2/15*(7*B*a^4+54*B*a^2*b^2+15*B*b^4+12*a*b^3*(3*A+5*C)+4*a^3* b*(7*A+9*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))*cos(d*x+c)^(1/2)*sec(d*x+c)^(1/2)/d+2/231*(220*B* a^3*b+308*B*a*b^3+77*b^4*(A+3*C)+66*a^2*b^2*(5*A+7*C)+5*a^4*(9*A+11*C))*(c os(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))*cos(d*x+c)^(1/2)*sec(d*x+c)^(1/2)/d
Time = 9.54 (sec) , antiderivative size = 563, normalized size of antiderivative = 1.09 \[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {13}{2}}(c+d x) \, dx=\frac {\frac {2 \left (-2156 a^3 A b-2772 a A b^3-539 a^4 B-4158 a^2 b^2 B-1155 b^4 B-2772 a^3 b C-4620 a b^3 C\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}}+2 \left (225 a^4 A+1650 a^2 A b^2+385 A b^4+1100 a^3 b B+1540 a b^3 B+275 a^4 C+2310 a^2 b^2 C+1155 b^4 C\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{1155 d}+\frac {\sqrt {\sec (c+d x)} \left (\frac {2}{15} \left (28 a^3 A b+36 a A b^3+7 a^4 B+54 a^2 b^2 B+15 b^4 B+36 a^3 b C+60 a b^3 C\right ) \sin (c+d x)+\frac {2}{9} \sec ^4(c+d x) \left (4 a^3 A b \sin (c+d x)+a^4 B \sin (c+d x)\right )+\frac {2}{77} \sec ^3(c+d x) \left (9 a^4 A \sin (c+d x)+66 a^2 A b^2 \sin (c+d x)+44 a^3 b B \sin (c+d x)+11 a^4 C \sin (c+d x)\right )+\frac {2}{45} \sec ^2(c+d x) \left (28 a^3 A b \sin (c+d x)+36 a A b^3 \sin (c+d x)+7 a^4 B \sin (c+d x)+54 a^2 b^2 B \sin (c+d x)+36 a^3 b C \sin (c+d x)\right )+\frac {2}{231} \sec (c+d x) \left (45 a^4 A \sin (c+d x)+330 a^2 A b^2 \sin (c+d x)+77 A b^4 \sin (c+d x)+220 a^3 b B \sin (c+d x)+308 a b^3 B \sin (c+d x)+55 a^4 C \sin (c+d x)+462 a^2 b^2 C \sin (c+d x)\right )+\frac {2}{11} a^4 A \sec ^4(c+d x) \tan (c+d x)\right )}{d} \]
Integrate[(a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*S ec[c + d*x]^(13/2),x]
((2*(-2156*a^3*A*b - 2772*a*A*b^3 - 539*a^4*B - 4158*a^2*b^2*B - 1155*b^4* B - 2772*a^3*b*C - 4620*a*b^3*C)*EllipticE[(c + d*x)/2, 2])/(Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + 2*(225*a^4*A + 1650*a^2*A*b^2 + 385*A*b^4 + 11 00*a^3*b*B + 1540*a*b^3*B + 275*a^4*C + 2310*a^2*b^2*C + 1155*b^4*C)*Sqrt[ Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(1155*d) + (Sq rt[Sec[c + d*x]]*((2*(28*a^3*A*b + 36*a*A*b^3 + 7*a^4*B + 54*a^2*b^2*B + 1 5*b^4*B + 36*a^3*b*C + 60*a*b^3*C)*Sin[c + d*x])/15 + (2*Sec[c + d*x]^4*(4 *a^3*A*b*Sin[c + d*x] + a^4*B*Sin[c + d*x]))/9 + (2*Sec[c + d*x]^3*(9*a^4* A*Sin[c + d*x] + 66*a^2*A*b^2*Sin[c + d*x] + 44*a^3*b*B*Sin[c + d*x] + 11* a^4*C*Sin[c + d*x]))/77 + (2*Sec[c + d*x]^2*(28*a^3*A*b*Sin[c + d*x] + 36* a*A*b^3*Sin[c + d*x] + 7*a^4*B*Sin[c + d*x] + 54*a^2*b^2*B*Sin[c + d*x] + 36*a^3*b*C*Sin[c + d*x]))/45 + (2*Sec[c + d*x]*(45*a^4*A*Sin[c + d*x] + 33 0*a^2*A*b^2*Sin[c + d*x] + 77*A*b^4*Sin[c + d*x] + 220*a^3*b*B*Sin[c + d*x ] + 308*a*b^3*B*Sin[c + d*x] + 55*a^4*C*Sin[c + d*x] + 462*a^2*b^2*C*Sin[c + d*x]))/231 + (2*a^4*A*Sec[c + d*x]^4*Tan[c + d*x])/11))/d
Time = 3.05 (sec) , antiderivative size = 466, normalized size of antiderivative = 0.90, number of steps used = 24, number of rules used = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.558, Rules used = {3042, 4709, 3042, 3526, 27, 3042, 3526, 27, 3042, 3526, 27, 3042, 3510, 27, 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.
\(\displaystyle \int \sec ^{\frac {13}{2}}(c+d x) (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \sec (c+d x)^{13/2} (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos (c+d x)^2\right )dx\) |
\(\Big \downarrow \) 4709 |
\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {(a+b \cos (c+d x))^4 \left (C \cos ^2(c+d x)+B \cos (c+d x)+A\right )}{\cos ^{\frac {13}{2}}(c+d x)}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^4 \left (C \sin \left (c+d x+\frac {\pi }{2}\right )^2+B \sin \left (c+d x+\frac {\pi }{2}\right )+A\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{13/2}}dx\) |
\(\Big \downarrow \) 3526 |
\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {2}{11} \int \frac {(a+b \cos (c+d x))^3 \left (b (A+11 C) \cos ^2(c+d x)+(9 a A+11 b B+11 a C) \cos (c+d x)+8 A b+11 a B\right )}{2 \cos ^{\frac {11}{2}}(c+d x)}dx+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{11 d \cos ^{\frac {11}{2}}(c+d x)}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \int \frac {(a+b \cos (c+d x))^3 \left (b (A+11 C) \cos ^2(c+d x)+(9 a A+11 b B+11 a C) \cos (c+d x)+8 A b+11 a B\right )}{\cos ^{\frac {11}{2}}(c+d x)}dx+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{11 d \cos ^{\frac {11}{2}}(c+d x)}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^3 \left (b (A+11 C) \sin \left (c+d x+\frac {\pi }{2}\right )^2+(9 a A+11 b B+11 a C) \sin \left (c+d x+\frac {\pi }{2}\right )+8 A b+11 a B\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{11/2}}dx+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{11 d \cos ^{\frac {11}{2}}(c+d x)}\right )\) |
\(\Big \downarrow \) 3526 |
\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {2}{9} \int \frac {(a+b \cos (c+d x))^2 \left (b (17 A b+99 C b+11 a B) \cos ^2(c+d x)+\left (77 B a^2+146 A b a+198 b C a+99 b^2 B\right ) \cos (c+d x)+3 \left (3 (9 A+11 C) a^2+55 b B a+16 A b^2\right )\right )}{2 \cos ^{\frac {9}{2}}(c+d x)}dx+\frac {2 (11 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{11 d \cos ^{\frac {11}{2}}(c+d x)}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \int \frac {(a+b \cos (c+d x))^2 \left (b (17 A b+99 C b+11 a B) \cos ^2(c+d x)+\left (77 B a^2+146 A b a+198 b C a+99 b^2 B\right ) \cos (c+d x)+3 \left (3 (9 A+11 C) a^2+55 b B a+16 A b^2\right )\right )}{\cos ^{\frac {9}{2}}(c+d x)}dx+\frac {2 (11 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{11 d \cos ^{\frac {11}{2}}(c+d x)}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^2 \left (b (17 A b+99 C b+11 a B) \sin \left (c+d x+\frac {\pi }{2}\right )^2+\left (77 B a^2+146 A b a+198 b C a+99 b^2 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+3 \left (3 (9 A+11 C) a^2+55 b B a+16 A b^2\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{9/2}}dx+\frac {2 (11 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{11 d \cos ^{\frac {11}{2}}(c+d x)}\right )\) |
\(\Big \downarrow \) 3526 |
\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {2}{7} \int \frac {(a+b \cos (c+d x)) \left (539 B a^3+2 b (673 A+891 C) a^2+1353 b^2 B a+192 A b^3+b \left (9 (9 A+11 C) a^2+242 b B a+b^2 (167 A+693 C)\right ) \cos ^2(c+d x)+\left (45 (9 A+11 C) a^3+1441 b B a^2+b^2 (1381 A+2079 C) a+693 b^3 B\right ) \cos (c+d x)\right )}{2 \cos ^{\frac {7}{2}}(c+d x)}dx+\frac {6 \sin (c+d x) \left (3 a^2 (9 A+11 C)+55 a b B+16 A b^2\right ) (a+b \cos (c+d x))^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 (11 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{11 d \cos ^{\frac {11}{2}}(c+d x)}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \int \frac {(a+b \cos (c+d x)) \left (539 B a^3+2 b (673 A+891 C) a^2+1353 b^2 B a+192 A b^3+b \left (9 (9 A+11 C) a^2+242 b B a+b^2 (167 A+693 C)\right ) \cos ^2(c+d x)+\left (45 (9 A+11 C) a^3+1441 b B a^2+b^2 (1381 A+2079 C) a+693 b^3 B\right ) \cos (c+d x)\right )}{\cos ^{\frac {7}{2}}(c+d x)}dx+\frac {6 \sin (c+d x) \left (3 a^2 (9 A+11 C)+55 a b B+16 A b^2\right ) (a+b \cos (c+d x))^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 (11 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{11 d \cos ^{\frac {11}{2}}(c+d x)}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right ) \left (539 B a^3+2 b (673 A+891 C) a^2+1353 b^2 B a+192 A b^3+b \left (9 (9 A+11 C) a^2+242 b B a+b^2 (167 A+693 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+\left (45 (9 A+11 C) a^3+1441 b B a^2+b^2 (1381 A+2079 C) a+693 b^3 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{7/2}}dx+\frac {6 \sin (c+d x) \left (3 a^2 (9 A+11 C)+55 a b B+16 A b^2\right ) (a+b \cos (c+d x))^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 (11 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{11 d \cos ^{\frac {11}{2}}(c+d x)}\right )\) |
\(\Big \downarrow \) 3510 |
\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {2 a \sin (c+d x) \left (539 a^3 B+2 a^2 b (673 A+891 C)+1353 a b^2 B+192 A b^3\right )}{5 d \cos ^{\frac {5}{2}}(c+d x)}-\frac {2}{5} \int -\frac {5 b^2 \left (9 (9 A+11 C) a^2+242 b B a+b^2 (167 A+693 C)\right ) \cos ^2(c+d x)+231 \left (7 B a^4+4 b (7 A+9 C) a^3+54 b^2 B a^2+12 b^3 (3 A+5 C) a+15 b^4 B\right ) \cos (c+d x)+15 \left (15 (9 A+11 C) a^4+660 b B a^3+9 b^2 (101 A+143 C) a^2+682 b^3 B a+64 A b^4\right )}{2 \cos ^{\frac {5}{2}}(c+d x)}dx\right )+\frac {6 \sin (c+d x) \left (3 a^2 (9 A+11 C)+55 a b B+16 A b^2\right ) (a+b \cos (c+d x))^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 (11 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{11 d \cos ^{\frac {11}{2}}(c+d x)}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \int \frac {5 b^2 \left (9 (9 A+11 C) a^2+242 b B a+b^2 (167 A+693 C)\right ) \cos ^2(c+d x)+231 \left (7 B a^4+4 b (7 A+9 C) a^3+54 b^2 B a^2+12 b^3 (3 A+5 C) a+15 b^4 B\right ) \cos (c+d x)+15 \left (15 (9 A+11 C) a^4+660 b B a^3+9 b^2 (101 A+143 C) a^2+682 b^3 B a+64 A b^4\right )}{\cos ^{\frac {5}{2}}(c+d x)}dx+\frac {2 a \sin (c+d x) \left (539 a^3 B+2 a^2 b (673 A+891 C)+1353 a b^2 B+192 A b^3\right )}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {6 \sin (c+d x) \left (3 a^2 (9 A+11 C)+55 a b B+16 A b^2\right ) (a+b \cos (c+d x))^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 (11 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{11 d \cos ^{\frac {11}{2}}(c+d x)}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \int \frac {5 b^2 \left (9 (9 A+11 C) a^2+242 b B a+b^2 (167 A+693 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+231 \left (7 B a^4+4 b (7 A+9 C) a^3+54 b^2 B a^2+12 b^3 (3 A+5 C) a+15 b^4 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+15 \left (15 (9 A+11 C) a^4+660 b B a^3+9 b^2 (101 A+143 C) a^2+682 b^3 B a+64 A b^4\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{5/2}}dx+\frac {2 a \sin (c+d x) \left (539 a^3 B+2 a^2 b (673 A+891 C)+1353 a b^2 B+192 A b^3\right )}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {6 \sin (c+d x) \left (3 a^2 (9 A+11 C)+55 a b B+16 A b^2\right ) (a+b \cos (c+d x))^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 (11 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{11 d \cos ^{\frac {11}{2}}(c+d x)}\right )\) |
\(\Big \downarrow \) 3500 |
\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {2}{3} \int \frac {9 \left (77 \left (7 B a^4+4 b (7 A+9 C) a^3+54 b^2 B a^2+12 b^3 (3 A+5 C) a+15 b^4 B\right )+5 \left (5 (9 A+11 C) a^4+220 b B a^3+66 b^2 (5 A+7 C) a^2+308 b^3 B a+77 b^4 (A+3 C)\right ) \cos (c+d x)\right )}{2 \cos ^{\frac {3}{2}}(c+d x)}dx+\frac {10 \sin (c+d x) \left (15 a^4 (9 A+11 C)+660 a^3 b B+9 a^2 b^2 (101 A+143 C)+682 a b^3 B+64 A b^4\right )}{d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 a \sin (c+d x) \left (539 a^3 B+2 a^2 b (673 A+891 C)+1353 a b^2 B+192 A b^3\right )}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {6 \sin (c+d x) \left (3 a^2 (9 A+11 C)+55 a b B+16 A b^2\right ) (a+b \cos (c+d x))^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 (11 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{11 d \cos ^{\frac {11}{2}}(c+d x)}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (3 \int \frac {77 \left (7 B a^4+4 b (7 A+9 C) a^3+54 b^2 B a^2+12 b^3 (3 A+5 C) a+15 b^4 B\right )+5 \left (5 (9 A+11 C) a^4+220 b B a^3+66 b^2 (5 A+7 C) a^2+308 b^3 B a+77 b^4 (A+3 C)\right ) \cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x)}dx+\frac {10 \sin (c+d x) \left (15 a^4 (9 A+11 C)+660 a^3 b B+9 a^2 b^2 (101 A+143 C)+682 a b^3 B+64 A b^4\right )}{d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 a \sin (c+d x) \left (539 a^3 B+2 a^2 b (673 A+891 C)+1353 a b^2 B+192 A b^3\right )}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {6 \sin (c+d x) \left (3 a^2 (9 A+11 C)+55 a b B+16 A b^2\right ) (a+b \cos (c+d x))^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 (11 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{11 d \cos ^{\frac {11}{2}}(c+d x)}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (3 \int \frac {77 \left (7 B a^4+4 b (7 A+9 C) a^3+54 b^2 B a^2+12 b^3 (3 A+5 C) a+15 b^4 B\right )+5 \left (5 (9 A+11 C) a^4+220 b B a^3+66 b^2 (5 A+7 C) a^2+308 b^3 B a+77 b^4 (A+3 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2}}dx+\frac {10 \sin (c+d x) \left (15 a^4 (9 A+11 C)+660 a^3 b B+9 a^2 b^2 (101 A+143 C)+682 a b^3 B+64 A b^4\right )}{d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 a \sin (c+d x) \left (539 a^3 B+2 a^2 b (673 A+891 C)+1353 a b^2 B+192 A b^3\right )}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {6 \sin (c+d x) \left (3 a^2 (9 A+11 C)+55 a b B+16 A b^2\right ) (a+b \cos (c+d x))^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 (11 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{11 d \cos ^{\frac {11}{2}}(c+d x)}\right )\) |
\(\Big \downarrow \) 3227 |
\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (3 \left (77 \left (7 a^4 B+4 a^3 b (7 A+9 C)+54 a^2 b^2 B+12 a b^3 (3 A+5 C)+15 b^4 B\right ) \int \frac {1}{\cos ^{\frac {3}{2}}(c+d x)}dx+5 \left (5 a^4 (9 A+11 C)+220 a^3 b B+66 a^2 b^2 (5 A+7 C)+308 a b^3 B+77 b^4 (A+3 C)\right ) \int \frac {1}{\sqrt {\cos (c+d x)}}dx\right )+\frac {10 \sin (c+d x) \left (15 a^4 (9 A+11 C)+660 a^3 b B+9 a^2 b^2 (101 A+143 C)+682 a b^3 B+64 A b^4\right )}{d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 a \sin (c+d x) \left (539 a^3 B+2 a^2 b (673 A+891 C)+1353 a b^2 B+192 A b^3\right )}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {6 \sin (c+d x) \left (3 a^2 (9 A+11 C)+55 a b B+16 A b^2\right ) (a+b \cos (c+d x))^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 (11 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{11 d \cos ^{\frac {11}{2}}(c+d x)}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (3 \left (77 \left (7 a^4 B+4 a^3 b (7 A+9 C)+54 a^2 b^2 B+12 a b^3 (3 A+5 C)+15 b^4 B\right ) \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2}}dx+5 \left (5 a^4 (9 A+11 C)+220 a^3 b B+66 a^2 b^2 (5 A+7 C)+308 a b^3 B+77 b^4 (A+3 C)\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx\right )+\frac {10 \sin (c+d x) \left (15 a^4 (9 A+11 C)+660 a^3 b B+9 a^2 b^2 (101 A+143 C)+682 a b^3 B+64 A b^4\right )}{d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 a \sin (c+d x) \left (539 a^3 B+2 a^2 b (673 A+891 C)+1353 a b^2 B+192 A b^3\right )}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {6 \sin (c+d x) \left (3 a^2 (9 A+11 C)+55 a b B+16 A b^2\right ) (a+b \cos (c+d x))^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 (11 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{11 d \cos ^{\frac {11}{2}}(c+d x)}\right )\) |
\(\Big \downarrow \) 3116 |
\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (3 \left (5 \left (5 a^4 (9 A+11 C)+220 a^3 b B+66 a^2 b^2 (5 A+7 C)+308 a b^3 B+77 b^4 (A+3 C)\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+77 \left (7 a^4 B+4 a^3 b (7 A+9 C)+54 a^2 b^2 B+12 a b^3 (3 A+5 C)+15 b^4 B\right ) \left (\frac {2 \sin (c+d x)}{d \sqrt {\cos (c+d x)}}-\int \sqrt {\cos (c+d x)}dx\right )\right )+\frac {10 \sin (c+d x) \left (15 a^4 (9 A+11 C)+660 a^3 b B+9 a^2 b^2 (101 A+143 C)+682 a b^3 B+64 A b^4\right )}{d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 a \sin (c+d x) \left (539 a^3 B+2 a^2 b (673 A+891 C)+1353 a b^2 B+192 A b^3\right )}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {6 \sin (c+d x) \left (3 a^2 (9 A+11 C)+55 a b B+16 A b^2\right ) (a+b \cos (c+d x))^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 (11 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{11 d \cos ^{\frac {11}{2}}(c+d x)}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (3 \left (5 \left (5 a^4 (9 A+11 C)+220 a^3 b B+66 a^2 b^2 (5 A+7 C)+308 a b^3 B+77 b^4 (A+3 C)\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+77 \left (7 a^4 B+4 a^3 b (7 A+9 C)+54 a^2 b^2 B+12 a b^3 (3 A+5 C)+15 b^4 B\right ) \left (\frac {2 \sin (c+d x)}{d \sqrt {\cos (c+d x)}}-\int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx\right )\right )+\frac {10 \sin (c+d x) \left (15 a^4 (9 A+11 C)+660 a^3 b B+9 a^2 b^2 (101 A+143 C)+682 a b^3 B+64 A b^4\right )}{d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 a \sin (c+d x) \left (539 a^3 B+2 a^2 b (673 A+891 C)+1353 a b^2 B+192 A b^3\right )}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {6 \sin (c+d x) \left (3 a^2 (9 A+11 C)+55 a b B+16 A b^2\right ) (a+b \cos (c+d x))^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 (11 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{11 d \cos ^{\frac {11}{2}}(c+d x)}\right )\) |
\(\Big \downarrow \) 3119 |
\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (3 \left (5 \left (5 a^4 (9 A+11 C)+220 a^3 b B+66 a^2 b^2 (5 A+7 C)+308 a b^3 B+77 b^4 (A+3 C)\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+77 \left (7 a^4 B+4 a^3 b (7 A+9 C)+54 a^2 b^2 B+12 a b^3 (3 A+5 C)+15 b^4 B\right ) \left (\frac {2 \sin (c+d x)}{d \sqrt {\cos (c+d x)}}-\frac {2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )\right )+\frac {10 \sin (c+d x) \left (15 a^4 (9 A+11 C)+660 a^3 b B+9 a^2 b^2 (101 A+143 C)+682 a b^3 B+64 A b^4\right )}{d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 a \sin (c+d x) \left (539 a^3 B+2 a^2 b (673 A+891 C)+1353 a b^2 B+192 A b^3\right )}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {6 \sin (c+d x) \left (3 a^2 (9 A+11 C)+55 a b B+16 A b^2\right ) (a+b \cos (c+d x))^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 (11 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{11 d \cos ^{\frac {11}{2}}(c+d x)}\right )\) |
\(\Big \downarrow \) 3120 |
\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {6 \sin (c+d x) \left (3 a^2 (9 A+11 C)+55 a b B+16 A b^2\right ) (a+b \cos (c+d x))^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {1}{7} \left (\frac {2 a \sin (c+d x) \left (539 a^3 B+2 a^2 b (673 A+891 C)+1353 a b^2 B+192 A b^3\right )}{5 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {1}{5} \left (\frac {10 \sin (c+d x) \left (15 a^4 (9 A+11 C)+660 a^3 b B+9 a^2 b^2 (101 A+143 C)+682 a b^3 B+64 A b^4\right )}{d \cos ^{\frac {3}{2}}(c+d x)}+3 \left (\frac {10 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (5 a^4 (9 A+11 C)+220 a^3 b B+66 a^2 b^2 (5 A+7 C)+308 a b^3 B+77 b^4 (A+3 C)\right )}{d}+77 \left (7 a^4 B+4 a^3 b (7 A+9 C)+54 a^2 b^2 B+12 a b^3 (3 A+5 C)+15 b^4 B\right ) \left (\frac {2 \sin (c+d x)}{d \sqrt {\cos (c+d x)}}-\frac {2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )\right )\right )\right )\right )+\frac {2 (11 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{11 d \cos ^{\frac {11}{2}}(c+d x)}\right )\) |
Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((2*A*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(11*d*Cos[c + d*x]^(11/2)) + ((2*(8*A*b + 11*a*B)*(a + b*Cos[c + d* x])^3*Sin[c + d*x])/(9*d*Cos[c + d*x]^(9/2)) + ((6*(16*A*b^2 + 55*a*b*B + 3*a^2*(9*A + 11*C))*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(7*d*Cos[c + d*x] ^(7/2)) + ((2*a*(192*A*b^3 + 539*a^3*B + 1353*a*b^2*B + 2*a^2*b*(673*A + 8 91*C))*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + ((10*(64*A*b^4 + 660*a^3*b *B + 682*a*b^3*B + 15*a^4*(9*A + 11*C) + 9*a^2*b^2*(101*A + 143*C))*Sin[c + d*x])/(d*Cos[c + d*x]^(3/2)) + 3*((10*(220*a^3*b*B + 308*a*b^3*B + 77*b^ 4*(A + 3*C) + 66*a^2*b^2*(5*A + 7*C) + 5*a^4*(9*A + 11*C))*EllipticF[(c + d*x)/2, 2])/d + 77*(7*a^4*B + 54*a^2*b^2*B + 15*b^4*B + 12*a*b^3*(3*A + 5* C) + 4*a^3*b*(7*A + 9*C))*((-2*EllipticE[(c + d*x)/2, 2])/d + (2*Sin[c + d *x])/(d*Sqrt[Cos[c + d*x]]))))/5)/7)/9)/11)
3.15.74.3.1 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_.)*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_)])*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f _.)*(x_)]^2), x_Symbol] :> Simp[(-(b*c - a*d))*(A*b^2 - a*b*B + a^2*C)*Cos[ e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b^2*f*(m + 1)*(a^2 - b^2))), x] - S imp[1/(b^2*(m + 1)*(a^2 - b^2)) Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[b*( m + 1)*((b*B - a*C)*(b*c - a*d) - A*b*(a*c - b*d)) + (b*B*(a^2*d + b^2*d*(m + 1) - a*b*c*(m + 2)) + (b*c - a*d)*(A*b^2*(m + 2) + C*(a^2 + b^2*(m + 1)) ))*Sin[e + f*x] - b*C*d*(m + 1)*(a^2 - b^2)*Sin[e + f*x]^2, x], x], x] /; F reeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && LtQ[m, -1]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(c^2*C - B*c*d + A*d^2))*Cos[e + f*x ]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 - d^2))), x] + Simp[1/(d*(n + 1)*(c^2 - d^2)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(b*d*m + a*c*(n + 1)) + (c*C - B* d)*(b*c*m + a*d*(n + 1)) - (d*(A*(a*d*(n + 2) - b*c*(n + 1)) + B*(b*d*(n + 1) - a*c*(n + 2))) - C*(b*c*d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x ] + b*(d*(B*c - A*d)*(m + n + 2) - C*(c^2*(m + 1) + d^2*(n + 1)))*Sin[e + f *x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d , 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 0] && LtQ[n, -1]
Int[(u_)*((c_.)*sec[(a_.) + (b_.)*(x_)])^(m_.), x_Symbol] :> Simp[(c*Sec[a + b*x])^m*(c*Cos[a + b*x])^m Int[ActivateTrig[u]/(c*Cos[a + b*x])^m, x], x] /; FreeQ[{a, b, c, m}, x] && !IntegerQ[m] && KnownSineIntegrandQ[u, x]
Timed out.
hanged
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.17 (sec) , antiderivative size = 588, normalized size of antiderivative = 1.14 \[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {13}{2}}(c+d x) \, dx=-\frac {15 \, \sqrt {2} {\left (5 i \, {\left (9 \, A + 11 \, C\right )} a^{4} + 220 i \, B a^{3} b + 66 i \, {\left (5 \, A + 7 \, C\right )} a^{2} b^{2} + 308 i \, B a b^{3} + 77 i \, {\left (A + 3 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{5} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 15 \, \sqrt {2} {\left (-5 i \, {\left (9 \, A + 11 \, C\right )} a^{4} - 220 i \, B a^{3} b - 66 i \, {\left (5 \, A + 7 \, C\right )} a^{2} b^{2} - 308 i \, B a b^{3} - 77 i \, {\left (A + 3 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{5} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 231 \, \sqrt {2} {\left (7 i \, B a^{4} + 4 i \, {\left (7 \, A + 9 \, C\right )} a^{3} b + 54 i \, B a^{2} b^{2} + 12 i \, {\left (3 \, A + 5 \, C\right )} a b^{3} + 15 i \, B b^{4}\right )} \cos \left (d x + c\right )^{5} {\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 \, \sqrt {2} {\left (-7 i \, B a^{4} - 4 i \, {\left (7 \, A + 9 \, C\right )} a^{3} b - 54 i \, B a^{2} b^{2} - 12 i \, {\left (3 \, A + 5 \, C\right )} a b^{3} - 15 i \, B b^{4}\right )} \cos \left (d x + c\right )^{5} {\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 ) - \frac {2 \, {\left (231 \, {\left (7 \, B a^{4} + 4 \, {\left (7 \, A + 9 \, C\right )} a^{3} b + 54 \, B a^{2} b^{2} + 12 \, {\left (3 \, A + 5 \, C\right )} a b^{3} + 15 \, B b^{4}\right )} \cos \left (d x + c\right )^{5} + 315 \, A a^{4} + 15 \, {\left (5 \, {\left (9 \, A + 11 \, C\right )} a^{4} + 220 \, B a^{3} b + 66 \, {\left (5 \, A + 7 \, C\right )} a^{2} b^{2} + 308 \, B a b^{3} + 77 \, A b^{4}\right )} \cos \left (d x + c\right )^{4} + 77 \, {\left (7 \, B a^{4} + 4 \, {\left (7 \, A + 9 \, C\right )} a^{3} b + 54 \, B a^{2} b^{2} + 36 \, A a b^{3}\right )} \cos \left (d x + c\right )^{3} + 45 \, {\left ({\left (9 \, A + 11 \, C\right )} a^{4} + 44 \, B a^{3} b + 66 \, A a^{2} b^{2}\right )} \cos \left (d x + c\right )^{2} + 385 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{3465 \, d \cos \left (d x + c\right )^{5}} \]
integrate((a+b*cos(d*x+c))^4*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^(1 3/2),x, algorithm="fricas")
-1/3465*(15*sqrt(2)*(5*I*(9*A + 11*C)*a^4 + 220*I*B*a^3*b + 66*I*(5*A + 7* C)*a^2*b^2 + 308*I*B*a*b^3 + 77*I*(A + 3*C)*b^4)*cos(d*x + c)^5*weierstras sPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c)) + 15*sqrt(2)*(-5*I*(9*A + 11*C)*a^4 - 220*I*B*a^3*b - 66*I*(5*A + 7*C)*a^2*b^2 - 308*I*B*a*b^3 - 77* I*(A + 3*C)*b^4)*cos(d*x + c)^5*weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) + 231*sqrt(2)*(7*I*B*a^4 + 4*I*(7*A + 9*C)*a^3*b + 54*I*B* a^2*b^2 + 12*I*(3*A + 5*C)*a*b^3 + 15*I*B*b^4)*cos(d*x + c)^5*weierstrassZ eta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) + 23 1*sqrt(2)*(-7*I*B*a^4 - 4*I*(7*A + 9*C)*a^3*b - 54*I*B*a^2*b^2 - 12*I*(3*A + 5*C)*a*b^3 - 15*I*B*b^4)*cos(d*x + c)^5*weierstrassZeta(-4, 0, weierstr assPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c))) - 2*(231*(7*B*a^4 + 4*( 7*A + 9*C)*a^3*b + 54*B*a^2*b^2 + 12*(3*A + 5*C)*a*b^3 + 15*B*b^4)*cos(d*x + c)^5 + 315*A*a^4 + 15*(5*(9*A + 11*C)*a^4 + 220*B*a^3*b + 66*(5*A + 7*C )*a^2*b^2 + 308*B*a*b^3 + 77*A*b^4)*cos(d*x + c)^4 + 77*(7*B*a^4 + 4*(7*A + 9*C)*a^3*b + 54*B*a^2*b^2 + 36*A*a*b^3)*cos(d*x + c)^3 + 45*((9*A + 11*C )*a^4 + 44*B*a^3*b + 66*A*a^2*b^2)*cos(d*x + c)^2 + 385*(B*a^4 + 4*A*a^3*b )*cos(d*x + c))*sin(d*x + c)/sqrt(cos(d*x + c)))/(d*cos(d*x + c)^5)
Timed out. \[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {13}{2}}(c+d x) \, dx=\text {Timed out} \]
\[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {13}{2}}(c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{4} \sec \left (d x + c\right )^{\frac {13}{2}} \,d x } \]
integrate((a+b*cos(d*x+c))^4*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^(1 3/2),x, algorithm="maxima")
integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)*(b*cos(d*x + c) + a)^4*s ec(d*x + c)^(13/2), x)
\[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {13}{2}}(c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{4} \sec \left (d x + c\right )^{\frac {13}{2}} \,d x } \]
integrate((a+b*cos(d*x+c))^4*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^(1 3/2),x, algorithm="giac")
integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)*(b*cos(d*x + c) + a)^4*s ec(d*x + c)^(13/2), x)
Timed out. \[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {13}{2}}(c+d x) \, dx=\int {\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{13/2}\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^4\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right ) \,d x \]
int((1/cos(c + d*x))^(13/2)*(a + b*cos(c + d*x))^4*(A + B*cos(c + d*x) + C *cos(c + d*x)^2),x)