Integrand size = 43, antiderivative size = 475 \[ \int \frac {(a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {13}{2}}(c+d x)} \, dx=-\frac {2 \left (7 a^4 B+54 a^2 b^2 B+15 b^4 B+12 a b^3 (3 A+5 C)+4 a^3 b (7 A+9 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {2 \left (220 a^3 b B+308 a b^3 B+77 b^4 (A+3 C)+66 a^2 b^2 (5 A+7 C)+5 a^4 (9 A+11 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{231 d}+\frac {2 a \left (192 A b^3+539 a^3 B+1353 a b^2 B+2 a^2 b (673 A+891 C)\right ) \sin (c+d x)}{3465 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (64 A b^4+660 a^3 b B+682 a b^3 B+15 a^4 (9 A+11 C)+9 a^2 b^2 (101 A+143 C)\right ) \sin (c+d x)}{693 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (7 a^4 B+54 a^2 b^2 B+15 b^4 B+12 a b^3 (3 A+5 C)+4 a^3 b (7 A+9 C)\right ) \sin (c+d x)}{15 d \sqrt {\cos (c+d x)}}+\frac {2 \left (16 A b^2+55 a b B+3 a^2 (9 A+11 C)\right ) (a+b \cos (c+d x))^2 \sin (c+d x)}{231 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 (8 A b+11 a B) (a+b \cos (c+d x))^3 \sin (c+d x)}{99 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 A (a+b \cos (c+d x))^4 \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)} \]
-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))/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))*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)* EllipticF(sin(1/2*d*x+1/2*c),2^(1/2))/d+2/3465*a*(192*A*b^3+539*B*a^3+1353 *B*a*b^2+2*a^2*b*(673*A+891*C))*sin(d*x+c)/d/cos(d*x+c)^(5/2)+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))*sin (d*x+c)/d/cos(d*x+c)^(3/2)+2/231*(16*A*b^2+55*B*a*b+3*a^2*(9*A+11*C))*(a+b *cos(d*x+c))^2*sin(d*x+c)/d/cos(d*x+c)^(7/2)+2/99*(8*A*b+11*B*a)*(a+b*cos( d*x+c))^3*sin(d*x+c)/d/cos(d*x+c)^(9/2)+2/11*A*(a+b*cos(d*x+c))^4*sin(d*x+ c)/d/cos(d*x+c)^(11/2)+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)/d/cos(d*x+c)^(1/2)
Time = 13.16 (sec) , antiderivative size = 381, normalized size of antiderivative = 0.80 \[ \int \frac {(a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {13}{2}}(c+d x)} \, dx=\frac {-154 \left (7 a^4 B+54 a^2 b^2 B+15 b^4 B+12 a b^3 (3 A+5 C)+4 a^3 b (7 A+9 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+10 \left (220 a^3 b B+308 a b^3 B+77 b^4 (A+3 C)+66 a^2 b^2 (5 A+7 C)+5 a^4 (9 A+11 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+\frac {2 \left (385 a^3 (4 A b+a B)+77 a \left (36 A b^3+7 a^3 B+54 a b^2 B+4 a^2 b (7 A+9 C)\right ) \cos ^2(c+d x)+15 \left (77 A b^4+220 a^3 b B+308 a b^3 B+66 a^2 b^2 (5 A+7 C)+5 a^4 (9 A+11 C)\right ) \cos ^3(c+d x)+231 \left (7 a^4 B+54 a^2 b^2 B+15 b^4 B+12 a b^3 (3 A+5 C)+4 a^3 b (7 A+9 C)\right ) \cos ^4(c+d x)\right ) \sin (c+d x)+45 \left (a^2 \left (66 A b^2+44 a b B+a^2 (9 A+11 C)\right ) \sin (2 (c+d x))+14 a^4 A \tan (c+d x)\right )}{3 \cos ^{\frac {9}{2}}(c+d x)}}{1155 d} \]
Integrate[((a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)) /Cos[c + d*x]^(13/2),x]
(-154*(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))*EllipticE[(c + d*x)/2, 2] + 10*(220*a^3*b*B + 308*a*b^3*B + 7 7*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] + (2*(385*a^3*(4*A*b + a*B) + 77*a*(36*A*b^3 + 7*a^3*B + 54 *a*b^2*B + 4*a^2*b*(7*A + 9*C))*Cos[c + d*x]^2 + 15*(77*A*b^4 + 220*a^3*b* B + 308*a*b^3*B + 66*a^2*b^2*(5*A + 7*C) + 5*a^4*(9*A + 11*C))*Cos[c + d*x ]^3 + 231*(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))*Cos[c + d*x]^4)*Sin[c + d*x] + 45*(a^2*(66*A*b^2 + 44*a*b *B + a^2*(9*A + 11*C))*Sin[2*(c + d*x)] + 14*a^4*A*Tan[c + d*x]))/(3*Cos[c + d*x]^(9/2)))/(1155*d)
Time = 2.85 (sec) , antiderivative size = 445, normalized size of antiderivative = 0.94, number of steps used = 22, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.512, Rules used = {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.
| \(\displaystyle \int \frac {(a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {13}{2}}(c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^4 \left (A+B \sin \left (c+d x+\frac {\pi }{2}\right )+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{13/2}}dx\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle \frac {2}{11} \int \frac {(a+b \cos (c+d x))^3 \left (b (A+11 C) \cos ^2(c+d x)+(9 a A+11 b B+11 a C) \cos (c+d x)+8 A b+11 a B\right )}{2 \cos ^{\frac {11}{2}}(c+d x)}dx+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{11} \int \frac {(a+b \cos (c+d x))^3 \left (b (A+11 C) \cos ^2(c+d x)+(9 a A+11 b B+11 a C) \cos (c+d x)+8 A b+11 a B\right )}{\cos ^{\frac {11}{2}}(c+d x)}dx+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^3 \left (b (A+11 C) \sin \left (c+d x+\frac {\pi }{2}\right )^2+(9 a A+11 b B+11 a C) \sin \left (c+d x+\frac {\pi }{2}\right )+8 A b+11 a B\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{11/2}}dx+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle \frac {1}{11} \left (\frac {2}{9} \int \frac {(a+b \cos (c+d x))^2 \left (b (17 A b+99 C b+11 a B) \cos ^2(c+d x)+\left (77 B a^2+146 A b a+198 b C a+99 b^2 B\right ) \cos (c+d x)+3 \left (3 (9 A+11 C) a^2+55 b B a+16 A b^2\right )\right )}{2 \cos ^{\frac {9}{2}}(c+d x)}dx+\frac {2 (11 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \int \frac {(a+b \cos (c+d x))^2 \left (b (17 A b+99 C b+11 a B) \cos ^2(c+d x)+\left (77 B a^2+146 A b a+198 b C a+99 b^2 B\right ) \cos (c+d x)+3 \left (3 (9 A+11 C) a^2+55 b B a+16 A b^2\right )\right )}{\cos ^{\frac {9}{2}}(c+d x)}dx+\frac {2 (11 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^2 \left (b (17 A b+99 C b+11 a B) \sin \left (c+d x+\frac {\pi }{2}\right )^2+\left (77 B a^2+146 A b a+198 b C a+99 b^2 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+3 \left (3 (9 A+11 C) a^2+55 b B a+16 A b^2\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{9/2}}dx+\frac {2 (11 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {2}{7} \int \frac {(a+b \cos (c+d x)) \left (539 B a^3+2 b (673 A+891 C) a^2+1353 b^2 B a+192 A b^3+b \left (9 (9 A+11 C) a^2+242 b B a+b^2 (167 A+693 C)\right ) \cos ^2(c+d x)+\left (45 (9 A+11 C) a^3+1441 b B a^2+b^2 (1381 A+2079 C) a+693 b^3 B\right ) \cos (c+d x)\right )}{2 \cos ^{\frac {7}{2}}(c+d x)}dx+\frac {6 \sin (c+d x) \left (3 a^2 (9 A+11 C)+55 a b B+16 A b^2\right ) (a+b \cos (c+d x))^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 (11 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \int \frac {(a+b \cos (c+d x)) \left (539 B a^3+2 b (673 A+891 C) a^2+1353 b^2 B a+192 A b^3+b \left (9 (9 A+11 C) a^2+242 b B a+b^2 (167 A+693 C)\right ) \cos ^2(c+d x)+\left (45 (9 A+11 C) a^3+1441 b B a^2+b^2 (1381 A+2079 C) a+693 b^3 B\right ) \cos (c+d x)\right )}{\cos ^{\frac {7}{2}}(c+d x)}dx+\frac {6 \sin (c+d x) \left (3 a^2 (9 A+11 C)+55 a b B+16 A b^2\right ) (a+b \cos (c+d x))^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 (11 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right ) \left (539 B a^3+2 b (673 A+891 C) a^2+1353 b^2 B a+192 A b^3+b \left (9 (9 A+11 C) a^2+242 b B a+b^2 (167 A+693 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+\left (45 (9 A+11 C) a^3+1441 b B a^2+b^2 (1381 A+2079 C) a+693 b^3 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{7/2}}dx+\frac {6 \sin (c+d x) \left (3 a^2 (9 A+11 C)+55 a b B+16 A b^2\right ) (a+b \cos (c+d x))^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 (11 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3510 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {2 a \sin (c+d x) \left (539 a^3 B+2 a^2 b (673 A+891 C)+1353 a b^2 B+192 A b^3\right )}{5 d \cos ^{\frac {5}{2}}(c+d x)}-\frac {2}{5} \int -\frac {5 b^2 \left (9 (9 A+11 C) a^2+242 b B a+b^2 (167 A+693 C)\right ) \cos ^2(c+d x)+231 \left (7 B a^4+4 b (7 A+9 C) a^3+54 b^2 B a^2+12 b^3 (3 A+5 C) a+15 b^4 B\right ) \cos (c+d x)+15 \left (15 (9 A+11 C) a^4+660 b B a^3+9 b^2 (101 A+143 C) a^2+682 b^3 B a+64 A b^4\right )}{2 \cos ^{\frac {5}{2}}(c+d x)}dx\right )+\frac {6 \sin (c+d x) \left (3 a^2 (9 A+11 C)+55 a b B+16 A b^2\right ) (a+b \cos (c+d x))^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 (11 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \int \frac {5 b^2 \left (9 (9 A+11 C) a^2+242 b B a+b^2 (167 A+693 C)\right ) \cos ^2(c+d x)+231 \left (7 B a^4+4 b (7 A+9 C) a^3+54 b^2 B a^2+12 b^3 (3 A+5 C) a+15 b^4 B\right ) \cos (c+d x)+15 \left (15 (9 A+11 C) a^4+660 b B a^3+9 b^2 (101 A+143 C) a^2+682 b^3 B a+64 A b^4\right )}{\cos ^{\frac {5}{2}}(c+d x)}dx+\frac {2 a \sin (c+d x) \left (539 a^3 B+2 a^2 b (673 A+891 C)+1353 a b^2 B+192 A b^3\right )}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {6 \sin (c+d x) \left (3 a^2 (9 A+11 C)+55 a b B+16 A b^2\right ) (a+b \cos (c+d x))^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 (11 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \int \frac {5 b^2 \left (9 (9 A+11 C) a^2+242 b B a+b^2 (167 A+693 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+231 \left (7 B a^4+4 b (7 A+9 C) a^3+54 b^2 B a^2+12 b^3 (3 A+5 C) a+15 b^4 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+15 \left (15 (9 A+11 C) a^4+660 b B a^3+9 b^2 (101 A+143 C) a^2+682 b^3 B a+64 A b^4\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{5/2}}dx+\frac {2 a \sin (c+d x) \left (539 a^3 B+2 a^2 b (673 A+891 C)+1353 a b^2 B+192 A b^3\right )}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {6 \sin (c+d x) \left (3 a^2 (9 A+11 C)+55 a b B+16 A b^2\right ) (a+b \cos (c+d x))^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 (11 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3500 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {2}{3} \int \frac {9 \left (77 \left (7 B a^4+4 b (7 A+9 C) a^3+54 b^2 B a^2+12 b^3 (3 A+5 C) a+15 b^4 B\right )+5 \left (5 (9 A+11 C) a^4+220 b B a^3+66 b^2 (5 A+7 C) a^2+308 b^3 B a+77 b^4 (A+3 C)\right ) \cos (c+d x)\right )}{2 \cos ^{\frac {3}{2}}(c+d x)}dx+\frac {10 \sin (c+d x) \left (15 a^4 (9 A+11 C)+660 a^3 b B+9 a^2 b^2 (101 A+143 C)+682 a b^3 B+64 A b^4\right )}{d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 a \sin (c+d x) \left (539 a^3 B+2 a^2 b (673 A+891 C)+1353 a b^2 B+192 A b^3\right )}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {6 \sin (c+d x) \left (3 a^2 (9 A+11 C)+55 a b B+16 A b^2\right ) (a+b \cos (c+d x))^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 (11 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (3 \int \frac {77 \left (7 B a^4+4 b (7 A+9 C) a^3+54 b^2 B a^2+12 b^3 (3 A+5 C) a+15 b^4 B\right )+5 \left (5 (9 A+11 C) a^4+220 b B a^3+66 b^2 (5 A+7 C) a^2+308 b^3 B a+77 b^4 (A+3 C)\right ) \cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x)}dx+\frac {10 \sin (c+d x) \left (15 a^4 (9 A+11 C)+660 a^3 b B+9 a^2 b^2 (101 A+143 C)+682 a b^3 B+64 A b^4\right )}{d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 a \sin (c+d x) \left (539 a^3 B+2 a^2 b (673 A+891 C)+1353 a b^2 B+192 A b^3\right )}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {6 \sin (c+d x) \left (3 a^2 (9 A+11 C)+55 a b B+16 A b^2\right ) (a+b \cos (c+d x))^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 (11 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (3 \int \frac {77 \left (7 B a^4+4 b (7 A+9 C) a^3+54 b^2 B a^2+12 b^3 (3 A+5 C) a+15 b^4 B\right )+5 \left (5 (9 A+11 C) a^4+220 b B a^3+66 b^2 (5 A+7 C) a^2+308 b^3 B a+77 b^4 (A+3 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2}}dx+\frac {10 \sin (c+d x) \left (15 a^4 (9 A+11 C)+660 a^3 b B+9 a^2 b^2 (101 A+143 C)+682 a b^3 B+64 A b^4\right )}{d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 a \sin (c+d x) \left (539 a^3 B+2 a^2 b (673 A+891 C)+1353 a b^2 B+192 A b^3\right )}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {6 \sin (c+d x) \left (3 a^2 (9 A+11 C)+55 a b B+16 A b^2\right ) (a+b \cos (c+d x))^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 (11 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (3 \left (77 \left (7 a^4 B+4 a^3 b (7 A+9 C)+54 a^2 b^2 B+12 a b^3 (3 A+5 C)+15 b^4 B\right ) \int \frac {1}{\cos ^{\frac {3}{2}}(c+d x)}dx+5 \left (5 a^4 (9 A+11 C)+220 a^3 b B+66 a^2 b^2 (5 A+7 C)+308 a b^3 B+77 b^4 (A+3 C)\right ) \int \frac {1}{\sqrt {\cos (c+d x)}}dx\right )+\frac {10 \sin (c+d x) \left (15 a^4 (9 A+11 C)+660 a^3 b B+9 a^2 b^2 (101 A+143 C)+682 a b^3 B+64 A b^4\right )}{d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 a \sin (c+d x) \left (539 a^3 B+2 a^2 b (673 A+891 C)+1353 a b^2 B+192 A b^3\right )}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {6 \sin (c+d x) \left (3 a^2 (9 A+11 C)+55 a b B+16 A b^2\right ) (a+b \cos (c+d x))^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 (11 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (3 \left (77 \left (7 a^4 B+4 a^3 b (7 A+9 C)+54 a^2 b^2 B+12 a b^3 (3 A+5 C)+15 b^4 B\right ) \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2}}dx+5 \left (5 a^4 (9 A+11 C)+220 a^3 b B+66 a^2 b^2 (5 A+7 C)+308 a b^3 B+77 b^4 (A+3 C)\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx\right )+\frac {10 \sin (c+d x) \left (15 a^4 (9 A+11 C)+660 a^3 b B+9 a^2 b^2 (101 A+143 C)+682 a b^3 B+64 A b^4\right )}{d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 a \sin (c+d x) \left (539 a^3 B+2 a^2 b (673 A+891 C)+1353 a b^2 B+192 A b^3\right )}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {6 \sin (c+d x) \left (3 a^2 (9 A+11 C)+55 a b B+16 A b^2\right ) (a+b \cos (c+d x))^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 (11 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3116 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (3 \left (5 \left (5 a^4 (9 A+11 C)+220 a^3 b B+66 a^2 b^2 (5 A+7 C)+308 a b^3 B+77 b^4 (A+3 C)\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+77 \left (7 a^4 B+4 a^3 b (7 A+9 C)+54 a^2 b^2 B+12 a b^3 (3 A+5 C)+15 b^4 B\right ) \left (\frac {2 \sin (c+d x)}{d \sqrt {\cos (c+d x)}}-\int \sqrt {\cos (c+d x)}dx\right )\right )+\frac {10 \sin (c+d x) \left (15 a^4 (9 A+11 C)+660 a^3 b B+9 a^2 b^2 (101 A+143 C)+682 a b^3 B+64 A b^4\right )}{d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 a \sin (c+d x) \left (539 a^3 B+2 a^2 b (673 A+891 C)+1353 a b^2 B+192 A b^3\right )}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {6 \sin (c+d x) \left (3 a^2 (9 A+11 C)+55 a b B+16 A b^2\right ) (a+b \cos (c+d x))^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 (11 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (3 \left (5 \left (5 a^4 (9 A+11 C)+220 a^3 b B+66 a^2 b^2 (5 A+7 C)+308 a b^3 B+77 b^4 (A+3 C)\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+77 \left (7 a^4 B+4 a^3 b (7 A+9 C)+54 a^2 b^2 B+12 a b^3 (3 A+5 C)+15 b^4 B\right ) \left (\frac {2 \sin (c+d x)}{d \sqrt {\cos (c+d x)}}-\int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx\right )\right )+\frac {10 \sin (c+d x) \left (15 a^4 (9 A+11 C)+660 a^3 b B+9 a^2 b^2 (101 A+143 C)+682 a b^3 B+64 A b^4\right )}{d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 a \sin (c+d x) \left (539 a^3 B+2 a^2 b (673 A+891 C)+1353 a b^2 B+192 A b^3\right )}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {6 \sin (c+d x) \left (3 a^2 (9 A+11 C)+55 a b B+16 A b^2\right ) (a+b \cos (c+d x))^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 (11 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (3 \left (5 \left (5 a^4 (9 A+11 C)+220 a^3 b B+66 a^2 b^2 (5 A+7 C)+308 a b^3 B+77 b^4 (A+3 C)\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+77 \left (7 a^4 B+4 a^3 b (7 A+9 C)+54 a^2 b^2 B+12 a b^3 (3 A+5 C)+15 b^4 B\right ) \left (\frac {2 \sin (c+d x)}{d \sqrt {\cos (c+d x)}}-\frac {2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )\right )+\frac {10 \sin (c+d x) \left (15 a^4 (9 A+11 C)+660 a^3 b B+9 a^2 b^2 (101 A+143 C)+682 a b^3 B+64 A b^4\right )}{d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 a \sin (c+d x) \left (539 a^3 B+2 a^2 b (673 A+891 C)+1353 a b^2 B+192 A b^3\right )}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {6 \sin (c+d x) \left (3 a^2 (9 A+11 C)+55 a b B+16 A b^2\right ) (a+b \cos (c+d x))^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 (11 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {6 \sin (c+d x) \left (3 a^2 (9 A+11 C)+55 a b B+16 A b^2\right ) (a+b \cos (c+d x))^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {1}{7} \left (\frac {2 a \sin (c+d x) \left (539 a^3 B+2 a^2 b (673 A+891 C)+1353 a b^2 B+192 A b^3\right )}{5 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {1}{5} \left (\frac {10 \sin (c+d x) \left (15 a^4 (9 A+11 C)+660 a^3 b B+9 a^2 b^2 (101 A+143 C)+682 a b^3 B+64 A b^4\right )}{d \cos ^{\frac {3}{2}}(c+d x)}+3 \left (\frac {10 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (5 a^4 (9 A+11 C)+220 a^3 b B+66 a^2 b^2 (5 A+7 C)+308 a b^3 B+77 b^4 (A+3 C)\right )}{d}+77 \left (7 a^4 B+4 a^3 b (7 A+9 C)+54 a^2 b^2 B+12 a b^3 (3 A+5 C)+15 b^4 B\right ) \left (\frac {2 \sin (c+d x)}{d \sqrt {\cos (c+d x)}}-\frac {2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )\right )\right )\right )\right )+\frac {2 (11 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
(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 + 891*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*Elliptic E[(c + d*x)/2, 2])/d + (2*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]))))/5)/7)/9) /11
3.11.94.3.1 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]
Leaf count of result is larger than twice the leaf count of optimal. \(1522\) vs. \(2(499)=998\).
Time = 12.78 (sec) , antiderivative size = 1523, normalized size of antiderivative = 3.21
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1523\) |
| parts | \(\text {Expression too large to display}\) | \(1790\) |
int((a+b*cos(d*x+c))^4*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/cos(d*x+c)^(13/2),x ,method=_RETURNVERBOSE)
-(-(-2*cos(1/2*d*x+1/2*c)^2+1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*C*b^4*(sin(1 /2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1 /2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))+ 2*A*a^4*(-1/352*cos(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/ 2*c)^2)^(1/2)/(cos(1/2*d*x+1/2*c)^2-1/2)^6-9/616*cos(1/2*d*x+1/2*c)*(-2*si n(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/(cos(1/2*d*x+1/2*c)^2-1/2)^ 4-15/154*cos(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2) ^(1/2)/(cos(1/2*d*x+1/2*c)^2-1/2)^2+15/77*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2 *cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c) ^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2)))+2*b^3*(B*b+4*C*a)/sin(1/2 *d*x+1/2*c)^2/(2*sin(1/2*d*x+1/2*c)^2-1)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2* d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)-(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*a^3*(4*A*b+B*a)*(-1/144*cos(1/2*d*x+1/2*c)*(-2*sin(1/2*d *x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/(cos(1/2*d*x+1/2*c)^2-1/2)^5-7/180 *cos(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/( cos(1/2*d*x+1/2*c)^2-1/2)^3-14/15*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)/ (-(-2*cos(1/2*d*x+1/2*c)^2+1)*sin(1/2*d*x+1/2*c)^2)^(1/2)+7/15*(sin(1/2*d* x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c) ^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-7/...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.18 (sec) , antiderivative size = 588, normalized size of antiderivative = 1.24 \[ \int \frac {(a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\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 )^{6} {\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 )^{6} {\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 )^{6} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 231 \, \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 )^{6} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - 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 )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )}{3465 \, d \cos \left (d x + c\right )^{6}} \]
integrate((a+b*cos(d*x+c))^4*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/cos(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)^6*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)^6*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)^6*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)^6*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))*sqrt(cos(d*x + c))*sin(d*x + c))/(d*cos(d*x + c)^6)
Timed out. \[ \int \frac {(a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {13}{2}}(c+d x)} \, dx=\text {Timed out} \]
\[ \int \frac {(a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {13}{2}}(c+d x)} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{4}}{\cos \left (d x + c\right )^{\frac {13}{2}}} \,d x } \]
integrate((a+b*cos(d*x+c))^4*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/cos(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/c os(d*x + c)^(13/2), x)
\[ \int \frac {(a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {13}{2}}(c+d x)} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{4}}{\cos \left (d x + c\right )^{\frac {13}{2}}} \,d x } \]
integrate((a+b*cos(d*x+c))^4*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/cos(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/c os(d*x + c)^(13/2), x)
Time = 12.68 (sec) , antiderivative size = 1161, normalized size of antiderivative = 2.44 \[ \int \frac {(a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {13}{2}}(c+d x)} \, dx=\text {Too large to display} \]
(8*hypergeom([-5/4, 1/2], -1/4, cos(c + d*x)^2)*((9*A*a*b^3*sin(c + d*x))/ (cos(c + d*x)^(5/2)*(sin(c + d*x)^2)^(1/2)) + (4*A*a^3*b*sin(c + d*x))/(co s(c + d*x)^(5/2)*(sin(c + d*x)^2)^(1/2)) + (5*A*a^3*b*sin(c + d*x))/(cos(c + d*x)^(9/2)*(sin(c + d*x)^2)^(1/2))))/(45*d) + (8*hypergeom([-3/4, 1/2], 1/4, cos(c + d*x)^2)*((7*B*a*b^3*sin(c + d*x))/(cos(c + d*x)^(3/2)*(sin(c + d*x)^2)^(1/2)) + (4*B*a^3*b*sin(c + d*x))/(cos(c + d*x)^(3/2)*(sin(c + d*x)^2)^(1/2)) + (3*B*a^3*b*sin(c + d*x))/(cos(c + d*x)^(7/2)*(sin(c + d*x )^2)^(1/2))))/(21*d) + (8*hypergeom([-3/4, 1/2], 5/4, cos(c + d*x)^2)*((9* A*a^4*sin(c + d*x))/(cos(c + d*x)^(3/2)*(sin(c + d*x)^2)^(1/2)) + (7*A*a^4 *sin(c + d*x))/(cos(c + d*x)^(7/2)*(sin(c + d*x)^2)^(1/2)) + (66*A*a^2*b^2 *sin(c + d*x))/(cos(c + d*x)^(3/2)*(sin(c + d*x)^2)^(1/2))))/(231*d) - (8* hypergeom([-1/4, 1/2], 7/4, cos(c + d*x)^2)*((7*B*a^4*sin(c + d*x))/(cos(c + d*x)^(1/2)*(sin(c + d*x)^2)^(1/2)) + (5*B*a^4*sin(c + d*x))/(cos(c + d* x)^(5/2)*(sin(c + d*x)^2)^(1/2)) + (54*B*a^2*b^2*sin(c + d*x))/(cos(c + d* x)^(1/2)*(sin(c + d*x)^2)^(1/2))))/(135*d) + (2*hypergeom([-3/4, 1/2], 1/4 , cos(c + d*x)^2)*((36*A*a^4*sin(c + d*x))/(cos(c + d*x)^(3/2)*(sin(c + d* x)^2)^(1/2)) + (20*A*a^4*sin(c + d*x))/(cos(c + d*x)^(7/2)*(sin(c + d*x)^2 )^(1/2)) + (21*A*a^4*sin(c + d*x))/(cos(c + d*x)^(11/2)*(sin(c + d*x)^2)^( 1/2)) + (77*A*b^4*sin(c + d*x))/(cos(c + d*x)^(3/2)*(sin(c + d*x)^2)^(1/2) ) + (264*A*a^2*b^2*sin(c + d*x))/(cos(c + d*x)^(3/2)*(sin(c + d*x)^2)^(...