Integrand size = 35, antiderivative size = 377 \[ \int \frac {(a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {13}{2}}(c+d x)} \, dx=-\frac {8 a b \left (3 b^2 (3 A+5 C)+a^2 (7 A+9 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {2 \left (77 b^4 (A+3 C)+66 a^2 b^2 (5 A+7 C)+5 a^4 (9 A+11 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{231 d}+\frac {4 a b \left (96 A b^2+a^2 (673 A+891 C)\right ) \sin (c+d x)}{3465 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (64 A b^4+15 a^4 (9 A+11 C)+9 a^2 b^2 (101 A+143 C)\right ) \sin (c+d x)}{693 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {8 a b \left (3 b^2 (3 A+5 C)+a^2 (7 A+9 C)\right ) \sin (c+d x)}{15 d \sqrt {\cos (c+d x)}}+\frac {2 \left (16 A b^2+3 a^2 (9 A+11 C)\right ) (a+b \cos (c+d x))^2 \sin (c+d x)}{231 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {16 A b (a+b \cos (c+d x))^3 \sin (c+d x)}{99 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 A (a+b \cos (c+d x))^4 \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)} \] Output:
-8/15*a*b*(3*b^2*(3*A+5*C)+a^2*(7*A+9*C))*EllipticE(sin(1/2*d*x+1/2*c),2^( 1/2))/d+2/231*(77*b^4*(A+3*C)+66*a^2*b^2*(5*A+7*C)+5*a^4*(9*A+11*C))*Inver seJacobiAM(1/2*d*x+1/2*c,2^(1/2))/d+4/3465*a*b*(96*A*b^2+a^2*(673*A+891*C) )*sin(d*x+c)/d/cos(d*x+c)^(5/2)+2/693*(64*A*b^4+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)+8/15*a*b*(3*b^2*(3*A+5*C)+a ^2*(7*A+9*C))*sin(d*x+c)/d/cos(d*x+c)^(1/2)+2/231*(16*A*b^2+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)+16/99*A*b*(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)
Time = 10.81 (sec) , antiderivative size = 284, normalized size of antiderivative = 0.75 \[ \int \frac {(a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {13}{2}}(c+d x)} \, dx=\frac {-616 \left (3 a b^3 (3 A+5 C)+a^3 b (7 A+9 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+10 \left (77 b^4 (A+3 C)+66 a^2 b^2 (5 A+7 C)+5 a^4 (9 A+11 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+\frac {2 \left (1540 a^3 A b+308 a b \left (9 A b^2+a^2 (7 A+9 C)\right ) \cos ^2(c+d x)+15 \left (77 A b^4+66 a^2 b^2 (5 A+7 C)+5 a^4 (9 A+11 C)\right ) \cos ^3(c+d x)+924 a b \left (3 b^2 (3 A+5 C)+a^2 (7 A+9 C)\right ) \cos ^4(c+d x)\right ) \sin (c+d x)+45 \left (\left (66 a^2 A b^2+a^4 (9 A+11 C)\right ) \sin (2 (c+d x))+14 a^4 A \tan (c+d x)\right )}{3 \cos ^{\frac {9}{2}}(c+d x)}}{1155 d} \] Input:
Integrate[((a + b*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(13 /2),x]
Output:
(-616*(3*a*b^3*(3*A + 5*C) + a^3*b*(7*A + 9*C))*EllipticE[(c + d*x)/2, 2] + 10*(77*b^4*(A + 3*C) + 66*a^2*b^2*(5*A + 7*C) + 5*a^4*(9*A + 11*C))*Elli pticF[(c + d*x)/2, 2] + (2*(1540*a^3*A*b + 308*a*b*(9*A*b^2 + a^2*(7*A + 9 *C))*Cos[c + d*x]^2 + 15*(77*A*b^4 + 66*a^2*b^2*(5*A + 7*C) + 5*a^4*(9*A + 11*C))*Cos[c + d*x]^3 + 924*a*b*(3*b^2*(3*A + 5*C) + a^2*(7*A + 9*C))*Cos [c + d*x]^4)*Sin[c + d*x] + 45*((66*a^2*A*b^2 + a^4*(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 = 2.47 (sec) , antiderivative size = 369, normalized size of antiderivative = 0.98, number of steps used = 22, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.629, Rules used = {3042, 3527, 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.
| \(\displaystyle \int \frac {(a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {13}{2}}(c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^4 \left (A+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{13/2}}dx\) |
| \(\Big \downarrow \) 3527 |
| \(\displaystyle \frac {2}{11} \int \frac {(a+b \cos (c+d x))^3 \left (b (A+11 C) \cos ^2(c+d x)+a (9 A+11 C) \cos (c+d x)+8 A b\right )}{2 \cos ^{\frac {11}{2}}(c+d x)}dx+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{11} \int \frac {(a+b \cos (c+d x))^3 \left (b (A+11 C) \cos ^2(c+d x)+a (9 A+11 C) \cos (c+d x)+8 A b\right )}{\cos ^{\frac {11}{2}}(c+d x)}dx+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^3 \left (b (A+11 C) \sin \left (c+d x+\frac {\pi }{2}\right )^2+a (9 A+11 C) \sin \left (c+d x+\frac {\pi }{2}\right )+8 A b\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{11/2}}dx+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle \frac {1}{11} \left (\frac {2}{9} \int \frac {(a+b \cos (c+d x))^2 \left (b^2 (17 A+99 C) \cos ^2(c+d x)+2 a b (73 A+99 C) \cos (c+d x)+3 \left (3 (9 A+11 C) a^2+16 A b^2\right )\right )}{2 \cos ^{\frac {9}{2}}(c+d x)}dx+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \int \frac {(a+b \cos (c+d x))^2 \left (b^2 (17 A+99 C) \cos ^2(c+d x)+2 a b (73 A+99 C) \cos (c+d x)+3 \left (3 (9 A+11 C) a^2+16 A b^2\right )\right )}{\cos ^{\frac {9}{2}}(c+d x)}dx+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^2 \left (b^2 (17 A+99 C) \sin \left (c+d x+\frac {\pi }{2}\right )^2+2 a b (73 A+99 C) \sin \left (c+d x+\frac {\pi }{2}\right )+3 \left (3 (9 A+11 C) a^2+16 A b^2\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{9/2}}dx+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {2}{7} \int \frac {(a+b \cos (c+d x)) \left (b \left (9 (9 A+11 C) a^2+b^2 (167 A+693 C)\right ) \cos ^2(c+d x)+a \left (45 (9 A+11 C) a^2+b^2 (1381 A+2079 C)\right ) \cos (c+d x)+2 b \left ((673 A+891 C) a^2+96 A b^2\right )\right )}{2 \cos ^{\frac {7}{2}}(c+d x)}dx+\frac {6 \left (3 a^2 (9 A+11 C)+16 A b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \int \frac {(a+b \cos (c+d x)) \left (b \left (9 (9 A+11 C) a^2+b^2 (167 A+693 C)\right ) \cos ^2(c+d x)+a \left (45 (9 A+11 C) a^2+b^2 (1381 A+2079 C)\right ) \cos (c+d x)+2 b \left ((673 A+891 C) a^2+96 A b^2\right )\right )}{\cos ^{\frac {7}{2}}(c+d x)}dx+\frac {6 \left (3 a^2 (9 A+11 C)+16 A b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right ) \left (b \left (9 (9 A+11 C) a^2+b^2 (167 A+693 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+a \left (45 (9 A+11 C) a^2+b^2 (1381 A+2079 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+2 b \left ((673 A+891 C) a^2+96 A b^2\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{7/2}}dx+\frac {6 \left (3 a^2 (9 A+11 C)+16 A b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3510 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {4 a b \left (a^2 (673 A+891 C)+96 A b^2\right ) \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x)}-\frac {2}{5} \int -\frac {5 b^2 \left (9 (9 A+11 C) a^2+b^2 (167 A+693 C)\right ) \cos ^2(c+d x)+924 a b \left ((7 A+9 C) a^2+3 b^2 (3 A+5 C)\right ) \cos (c+d x)+15 \left (15 (9 A+11 C) a^4+9 b^2 (101 A+143 C) a^2+64 A b^4\right )}{2 \cos ^{\frac {5}{2}}(c+d x)}dx\right )+\frac {6 \left (3 a^2 (9 A+11 C)+16 A b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \int \frac {5 b^2 \left (9 (9 A+11 C) a^2+b^2 (167 A+693 C)\right ) \cos ^2(c+d x)+924 a b \left ((7 A+9 C) a^2+3 b^2 (3 A+5 C)\right ) \cos (c+d x)+15 \left (15 (9 A+11 C) a^4+9 b^2 (101 A+143 C) a^2+64 A b^4\right )}{\cos ^{\frac {5}{2}}(c+d x)}dx+\frac {4 a b \left (a^2 (673 A+891 C)+96 A b^2\right ) \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {6 \left (3 a^2 (9 A+11 C)+16 A b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \int \frac {5 b^2 \left (9 (9 A+11 C) a^2+b^2 (167 A+693 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+924 a b \left ((7 A+9 C) a^2+3 b^2 (3 A+5 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+15 \left (15 (9 A+11 C) a^4+9 b^2 (101 A+143 C) a^2+64 A b^4\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{5/2}}dx+\frac {4 a b \left (a^2 (673 A+891 C)+96 A b^2\right ) \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {6 \left (3 a^2 (9 A+11 C)+16 A b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3500 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {2}{3} \int \frac {9 \left (308 a b \left ((7 A+9 C) a^2+3 b^2 (3 A+5 C)\right )+5 \left (5 (9 A+11 C) a^4+66 b^2 (5 A+7 C) a^2+77 b^4 (A+3 C)\right ) \cos (c+d x)\right )}{2 \cos ^{\frac {3}{2}}(c+d x)}dx+\frac {10 \left (15 a^4 (9 A+11 C)+9 a^2 b^2 (101 A+143 C)+64 A b^4\right ) \sin (c+d x)}{d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {4 a b \left (a^2 (673 A+891 C)+96 A b^2\right ) \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {6 \left (3 a^2 (9 A+11 C)+16 A b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (3 \int \frac {308 a b \left ((7 A+9 C) a^2+3 b^2 (3 A+5 C)\right )+5 \left (5 (9 A+11 C) a^4+66 b^2 (5 A+7 C) a^2+77 b^4 (A+3 C)\right ) \cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x)}dx+\frac {10 \left (15 a^4 (9 A+11 C)+9 a^2 b^2 (101 A+143 C)+64 A b^4\right ) \sin (c+d x)}{d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {4 a b \left (a^2 (673 A+891 C)+96 A b^2\right ) \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {6 \left (3 a^2 (9 A+11 C)+16 A b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (3 \int \frac {308 a b \left ((7 A+9 C) a^2+3 b^2 (3 A+5 C)\right )+5 \left (5 (9 A+11 C) a^4+66 b^2 (5 A+7 C) a^2+77 b^4 (A+3 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2}}dx+\frac {10 \left (15 a^4 (9 A+11 C)+9 a^2 b^2 (101 A+143 C)+64 A b^4\right ) \sin (c+d x)}{d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {4 a b \left (a^2 (673 A+891 C)+96 A b^2\right ) \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {6 \left (3 a^2 (9 A+11 C)+16 A b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (3 \left (308 a b \left (a^2 (7 A+9 C)+3 b^2 (3 A+5 C)\right ) \int \frac {1}{\cos ^{\frac {3}{2}}(c+d x)}dx+5 \left (5 a^4 (9 A+11 C)+66 a^2 b^2 (5 A+7 C)+77 b^4 (A+3 C)\right ) \int \frac {1}{\sqrt {\cos (c+d x)}}dx\right )+\frac {10 \left (15 a^4 (9 A+11 C)+9 a^2 b^2 (101 A+143 C)+64 A b^4\right ) \sin (c+d x)}{d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {4 a b \left (a^2 (673 A+891 C)+96 A b^2\right ) \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {6 \left (3 a^2 (9 A+11 C)+16 A b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (3 \left (308 a b \left (a^2 (7 A+9 C)+3 b^2 (3 A+5 C)\right ) \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2}}dx+5 \left (5 a^4 (9 A+11 C)+66 a^2 b^2 (5 A+7 C)+77 b^4 (A+3 C)\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx\right )+\frac {10 \left (15 a^4 (9 A+11 C)+9 a^2 b^2 (101 A+143 C)+64 A b^4\right ) \sin (c+d x)}{d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {4 a b \left (a^2 (673 A+891 C)+96 A b^2\right ) \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {6 \left (3 a^2 (9 A+11 C)+16 A b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3116 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (3 \left (308 a b \left (a^2 (7 A+9 C)+3 b^2 (3 A+5 C)\right ) \left (\frac {2 \sin (c+d x)}{d \sqrt {\cos (c+d x)}}-\int \sqrt {\cos (c+d x)}dx\right )+5 \left (5 a^4 (9 A+11 C)+66 a^2 b^2 (5 A+7 C)+77 b^4 (A+3 C)\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx\right )+\frac {10 \left (15 a^4 (9 A+11 C)+9 a^2 b^2 (101 A+143 C)+64 A b^4\right ) \sin (c+d x)}{d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {4 a b \left (a^2 (673 A+891 C)+96 A b^2\right ) \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {6 \left (3 a^2 (9 A+11 C)+16 A b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (3 \left (308 a b \left (a^2 (7 A+9 C)+3 b^2 (3 A+5 C)\right ) \left (\frac {2 \sin (c+d x)}{d \sqrt {\cos (c+d x)}}-\int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx\right )+5 \left (5 a^4 (9 A+11 C)+66 a^2 b^2 (5 A+7 C)+77 b^4 (A+3 C)\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx\right )+\frac {10 \left (15 a^4 (9 A+11 C)+9 a^2 b^2 (101 A+143 C)+64 A b^4\right ) \sin (c+d x)}{d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {4 a b \left (a^2 (673 A+891 C)+96 A b^2\right ) \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {6 \left (3 a^2 (9 A+11 C)+16 A b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (3 \left (5 \left (5 a^4 (9 A+11 C)+66 a^2 b^2 (5 A+7 C)+77 b^4 (A+3 C)\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+308 a b \left (a^2 (7 A+9 C)+3 b^2 (3 A+5 C)\right ) \left (\frac {2 \sin (c+d x)}{d \sqrt {\cos (c+d x)}}-\frac {2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )\right )+\frac {10 \left (15 a^4 (9 A+11 C)+9 a^2 b^2 (101 A+143 C)+64 A b^4\right ) \sin (c+d x)}{d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {4 a b \left (a^2 (673 A+891 C)+96 A b^2\right ) \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {6 \left (3 a^2 (9 A+11 C)+16 A b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {6 \left (3 a^2 (9 A+11 C)+16 A b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {1}{7} \left (\frac {4 a b \left (a^2 (673 A+891 C)+96 A b^2\right ) \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {1}{5} \left (\frac {10 \left (15 a^4 (9 A+11 C)+9 a^2 b^2 (101 A+143 C)+64 A b^4\right ) \sin (c+d x)}{d \cos ^{\frac {3}{2}}(c+d x)}+3 \left (308 a b \left (a^2 (7 A+9 C)+3 b^2 (3 A+5 C)\right ) \left (\frac {2 \sin (c+d x)}{d \sqrt {\cos (c+d x)}}-\frac {2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )+\frac {10 \left (5 a^4 (9 A+11 C)+66 a^2 b^2 (5 A+7 C)+77 b^4 (A+3 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{d}\right )\right )\right )\right )+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
Input:
Int[((a + b*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(13/2),x]
Output:
(2*A*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(11*d*Cos[c + d*x]^(11/2)) + ((1 6*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 + 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)) + ((4*a*b*(96*A*b^2 + a^2*(673*A + 891*C))*Sin[c + d*x ])/(5*d*Cos[c + d*x]^(5/2)) + ((10*(64*A*b^4 + 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*(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 + 308*a*b*(3*b^2*(3*A + 5*C) + a^2*(7*A + 9*C))*((-2*Ellipti cE[(c + d*x)/2, 2])/d + (2*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]))))/5)/7)/9 )/11
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[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(c^2*C + A*d^2))*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 - d^2))), x] + Simp[1/(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*c*m + a*d*(n + 1)) - (A*d*(a*d*(n + 2) - b *c*(n + 1)) - C*(b*c*d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x] - b*( A*d^2*(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, 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. \(1493\) vs. \(2(352)=704\).
Time = 39.38 (sec) , antiderivative size = 1494, normalized size of antiderivative = 3.96
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1494\) |
| parts | \(\text {Expression too large to display}\) | \(1751\) |
Input:
int((a+b*cos(d*x+c))^4*(A+C*cos(d*x+c)^2)/cos(d*x+c)^(13/2),x,method=_RETU RNVERBOSE)
Output:
-(-(1-2*cos(1/2*d*x+1/2*c)^2)*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)*(1-2*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)*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*sin( 1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/(cos(1/2*d*x+1/2*c)^2-1/2)^4- 15/154*cos(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^( 1/2)/(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)*(1-2* 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)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2)))+2*a^2*(6*A*b^2+C*a^2)*(-1/56 *cos(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/( cos(1/2*d*x+1/2*c)^2-1/2)^4-5/42*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+5/21*(sin(1/2* d*x+1/2*c)^2)^(1/2)*(1-2*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)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2)))+2*b ^2*(A*b^2+6*C*a^2)*(-1/6*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+1/3*(sin(1/2*d*x+1/2*c )^2)^(1/2)*(1-2*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)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2)))+8*A*a^3*b*(- 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)...
Result contains complex when optimal does not.
Time = 0.13 (sec) , antiderivative size = 451, normalized size of antiderivative = 1.20 \[ \int \frac {(a+b \cos (c+d x))^4 \left (A+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} + 66 i \, {\left (5 \, A + 7 \, C\right )} a^{2} b^{2} + 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} - 66 i \, {\left (5 \, A + 7 \, C\right )} a^{2} b^{2} - 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 ) + 924 \, \sqrt {2} {\left (i \, {\left (7 \, A + 9 \, C\right )} a^{3} b + 3 i \, {\left (3 \, A + 5 \, C\right )} a b^{3}\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 ) + 924 \, \sqrt {2} {\left (-i \, {\left (7 \, A + 9 \, C\right )} a^{3} b - 3 i \, {\left (3 \, A + 5 \, C\right )} a b^{3}\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 (1540 \, A a^{3} b \cos \left (d x + c\right ) + 924 \, {\left ({\left (7 \, A + 9 \, C\right )} a^{3} b + 3 \, {\left (3 \, A + 5 \, C\right )} a b^{3}\right )} \cos \left (d x + c\right )^{5} + 315 \, A a^{4} + 15 \, {\left (5 \, {\left (9 \, A + 11 \, C\right )} a^{4} + 66 \, {\left (5 \, A + 7 \, C\right )} a^{2} b^{2} + 77 \, A b^{4}\right )} \cos \left (d x + c\right )^{4} + 308 \, {\left ({\left (7 \, A + 9 \, C\right )} a^{3} b + 9 \, A a b^{3}\right )} \cos \left (d x + c\right )^{3} + 45 \, {\left ({\left (9 \, A + 11 \, C\right )} a^{4} + 66 \, A a^{2} b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )}{3465 \, d \cos \left (d x + c\right )^{6}} \] Input:
integrate((a+b*cos(d*x+c))^4*(A+C*cos(d*x+c)^2)/cos(d*x+c)^(13/2),x, algor ithm="fricas")
Output:
-1/3465*(15*sqrt(2)*(5*I*(9*A + 11*C)*a^4 + 66*I*(5*A + 7*C)*a^2*b^2 + 77* I*(A + 3*C)*b^4)*cos(d*x + c)^6*weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c)) + 15*sqrt(2)*(-5*I*(9*A + 11*C)*a^4 - 66*I*(5*A + 7*C)*a^2 *b^2 - 77*I*(A + 3*C)*b^4)*cos(d*x + c)^6*weierstrassPInverse(-4, 0, cos(d *x + c) - I*sin(d*x + c)) + 924*sqrt(2)*(I*(7*A + 9*C)*a^3*b + 3*I*(3*A + 5*C)*a*b^3)*cos(d*x + c)^6*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) + 924*sqrt(2)*(-I*(7*A + 9*C)*a^3*b - 3 *I*(3*A + 5*C)*a*b^3)*cos(d*x + c)^6*weierstrassZeta(-4, 0, weierstrassPIn verse(-4, 0, cos(d*x + c) - I*sin(d*x + c))) - 2*(1540*A*a^3*b*cos(d*x + c ) + 924*((7*A + 9*C)*a^3*b + 3*(3*A + 5*C)*a*b^3)*cos(d*x + c)^5 + 315*A*a ^4 + 15*(5*(9*A + 11*C)*a^4 + 66*(5*A + 7*C)*a^2*b^2 + 77*A*b^4)*cos(d*x + c)^4 + 308*((7*A + 9*C)*a^3*b + 9*A*a*b^3)*cos(d*x + c)^3 + 45*((9*A + 11 *C)*a^4 + 66*A*a^2*b^2)*cos(d*x + c)^2)*sqrt(cos(d*x + c))*sin(d*x + c))/( d*cos(d*x + c)^6)
Timed out. \[ \int \frac {(a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {13}{2}}(c+d x)} \, dx=\text {Timed out} \] Input:
integrate((a+b*cos(d*x+c))**4*(A+C*cos(d*x+c)**2)/cos(d*x+c)**(13/2),x)
Output:
Timed out
\[ \int \frac {(a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {13}{2}}(c+d x)} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{4}}{\cos \left (d x + c\right )^{\frac {13}{2}}} \,d x } \] Input:
integrate((a+b*cos(d*x+c))^4*(A+C*cos(d*x+c)^2)/cos(d*x+c)^(13/2),x, algor ithm="maxima")
Output:
integrate((C*cos(d*x + c)^2 + A)*(b*cos(d*x + c) + a)^4/cos(d*x + c)^(13/2 ), x)
\[ \int \frac {(a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {13}{2}}(c+d x)} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{4}}{\cos \left (d x + c\right )^{\frac {13}{2}}} \,d x } \] Input:
integrate((a+b*cos(d*x+c))^4*(A+C*cos(d*x+c)^2)/cos(d*x+c)^(13/2),x, algor ithm="giac")
Output:
integrate((C*cos(d*x + c)^2 + A)*(b*cos(d*x + c) + a)^4/cos(d*x + c)^(13/2 ), x)
Time = 3.89 (sec) , antiderivative size = 685, normalized size of antiderivative = 1.82 \[ \int \frac {(a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {13}{2}}(c+d x)} \, dx =\text {Too large to display} \] Input:
int(((A + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^4)/cos(c + d*x)^(13/2),x)
Output:
(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], 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) + (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*s in(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)^(1/2)) + (198*A*a^2*b^2*si n(c + d*x))/(cos(c + d*x)^(7/2)*(sin(c + d*x)^2)^(1/2))))/(231*d) + (2*C*b ^4*ellipticF(c/2 + (d*x)/2, 2))/d + (2*C*a^4*sin(c + d*x)*hypergeom([-7/4, 1/2], -3/4, cos(c + d*x)^2))/(7*d*cos(c + d*x)^(7/2)*(sin(c + d*x)^2)^(1/ 2)) - (32*A*a^3*b*sin(c + d*x)*hypergeom([-5/4, 1/2], 3/4, cos(c + d*x)^2) )/(15*d*cos(c + d*x)^(5/2)*(sin(c + d*x)^2)^(1/2)) + (8*C*a*b^3*sin(c + d* x)*hypergeom([-1/4, 1/2], 3/4, cos(c + d*x)^2))/(d*cos(c + d*x)^(1/2)*(sin (c + d*x)^2)^(1/2)) + (8*C*a^3*b*sin(c + d*x)*hypergeom([-5/4, 1/2], -1...
\[ \int \frac {(a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {13}{2}}(c+d x)} \, dx=\left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )}d x \right ) b^{4} c +\left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{7}}d x \right ) a^{5}+4 \left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{6}}d x \right ) a^{4} b +\left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{5}}d x \right ) a^{4} c +6 \left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{5}}d x \right ) a^{3} b^{2}+4 \left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{4}}d x \right ) a^{3} b c +4 \left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{4}}d x \right ) a^{2} b^{3}+6 \left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{3}}d x \right ) a^{2} b^{2} c +\left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{3}}d x \right ) a \,b^{4}+4 \left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{2}}d x \right ) a \,b^{3} c \] Input:
int((a+b*cos(d*x+c))^4*(A+C*cos(d*x+c)^2)/cos(d*x+c)^(13/2),x)
Output:
int(sqrt(cos(c + d*x))/cos(c + d*x),x)*b**4*c + int(sqrt(cos(c + d*x))/cos (c + d*x)**7,x)*a**5 + 4*int(sqrt(cos(c + d*x))/cos(c + d*x)**6,x)*a**4*b + int(sqrt(cos(c + d*x))/cos(c + d*x)**5,x)*a**4*c + 6*int(sqrt(cos(c + d* x))/cos(c + d*x)**5,x)*a**3*b**2 + 4*int(sqrt(cos(c + d*x))/cos(c + d*x)** 4,x)*a**3*b*c + 4*int(sqrt(cos(c + d*x))/cos(c + d*x)**4,x)*a**2*b**3 + 6* int(sqrt(cos(c + d*x))/cos(c + d*x)**3,x)*a**2*b**2*c + int(sqrt(cos(c + d *x))/cos(c + d*x)**3,x)*a*b**4 + 4*int(sqrt(cos(c + d*x))/cos(c + d*x)**2, x)*a*b**3*c