Integrand size = 43, antiderivative size = 401 \[ \int \frac {(a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {11}{2}}(c+d x)} \, dx=-\frac {2 \left (36 a^3 b B+60 a b^3 B+15 b^4 (A-C)+18 a^2 b^2 (3 A+5 C)+a^4 (7 A+9 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {2 \left (5 a^4 B+42 a^2 b^2 B+21 b^4 B+28 a b^3 (A+3 C)+4 a^3 b (5 A+7 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 d}+\frac {2 a \left (64 A b^3+75 a^3 B+261 a b^2 B+a^2 (202 A b+294 b C)\right ) \sin (c+d x)}{315 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (192 A b^4+756 a^3 b B+1098 a b^3 B+21 a^4 (7 A+9 C)+7 a^2 b^2 (155 A+261 C)\right ) \sin (c+d x)}{315 d \sqrt {\cos (c+d x)}}+\frac {2 \left (48 A b^2+117 a b B+7 a^2 (7 A+9 C)\right ) (a+b \cos (c+d x))^2 \sin (c+d x)}{315 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 (8 A b+9 a B) (a+b \cos (c+d x))^3 \sin (c+d x)}{63 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 A (a+b \cos (c+d x))^4 \sin (c+d x)}{9 d \cos ^{\frac {9}{2}}(c+d x)} \] Output:
-2/15*(36*B*a^3*b+60*B*a*b^3+15*b^4*(A-C)+18*a^2*b^2*(3*A+5*C)+a^4*(7*A+9* C))*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2))/d+2/21*(5*B*a^4+42*B*a^2*b^2+21* B*b^4+28*a*b^3*(A+3*C)+4*a^3*b*(5*A+7*C))*InverseJacobiAM(1/2*d*x+1/2*c,2^ (1/2))/d+2/315*a*(64*A*b^3+75*B*a^3+261*B*a*b^2+a^2*(202*A*b+294*C*b))*sin (d*x+c)/d/cos(d*x+c)^(3/2)+2/315*(192*A*b^4+756*B*a^3*b+1098*B*a*b^3+21*a^ 4*(7*A+9*C)+7*a^2*b^2*(155*A+261*C))*sin(d*x+c)/d/cos(d*x+c)^(1/2)+2/315*( 48*A*b^2+117*B*a*b+7*a^2*(7*A+9*C))*(a+b*cos(d*x+c))^2*sin(d*x+c)/d/cos(d* x+c)^(5/2)+2/63*(8*A*b+9*B*a)*(a+b*cos(d*x+c))^3*sin(d*x+c)/d/cos(d*x+c)^( 7/2)+2/9*A*(a+b*cos(d*x+c))^4*sin(d*x+c)/d/cos(d*x+c)^(9/2)
Time = 13.35 (sec) , antiderivative size = 463, normalized size of antiderivative = 1.15 \[ \int \frac {(a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {11}{2}}(c+d x)} \, dx=\frac {2 \left (-49 a^4 A-378 a^2 A b^2-105 A b^4-252 a^3 b B-420 a b^3 B-63 a^4 C-630 a^2 b^2 C+105 b^4 C\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+2 \left (100 a^3 A b+140 a A b^3+25 a^4 B+210 a^2 b^2 B+105 b^4 B+140 a^3 b C+420 a b^3 C\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{105 d}+\frac {\sqrt {\cos (c+d x)} \left (\frac {2}{7} \sec ^4(c+d x) \left (4 a^3 A b \sin (c+d x)+a^4 B \sin (c+d x)\right )+\frac {2}{45} \sec ^3(c+d x) \left (7 a^4 A \sin (c+d x)+54 a^2 A b^2 \sin (c+d x)+36 a^3 b B \sin (c+d x)+9 a^4 C \sin (c+d x)\right )+\frac {2}{21} \sec ^2(c+d x) \left (20 a^3 A b \sin (c+d x)+28 a A b^3 \sin (c+d x)+5 a^4 B \sin (c+d x)+42 a^2 b^2 B \sin (c+d x)+28 a^3 b C \sin (c+d x)\right )+\frac {2}{15} \sec (c+d x) \left (7 a^4 A \sin (c+d x)+54 a^2 A b^2 \sin (c+d x)+15 A b^4 \sin (c+d x)+36 a^3 b B \sin (c+d x)+60 a b^3 B \sin (c+d x)+9 a^4 C \sin (c+d x)+90 a^2 b^2 C \sin (c+d x)\right )+\frac {2}{9} a^4 A \sec ^4(c+d x) \tan (c+d x)\right )}{d} \] Input:
Integrate[((a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)) /Cos[c + d*x]^(11/2),x]
Output:
(2*(-49*a^4*A - 378*a^2*A*b^2 - 105*A*b^4 - 252*a^3*b*B - 420*a*b^3*B - 63 *a^4*C - 630*a^2*b^2*C + 105*b^4*C)*EllipticE[(c + d*x)/2, 2] + 2*(100*a^3 *A*b + 140*a*A*b^3 + 25*a^4*B + 210*a^2*b^2*B + 105*b^4*B + 140*a^3*b*C + 420*a*b^3*C)*EllipticF[(c + d*x)/2, 2])/(105*d) + (Sqrt[Cos[c + d*x]]*((2* Sec[c + d*x]^4*(4*a^3*A*b*Sin[c + d*x] + a^4*B*Sin[c + d*x]))/7 + (2*Sec[c + d*x]^3*(7*a^4*A*Sin[c + d*x] + 54*a^2*A*b^2*Sin[c + d*x] + 36*a^3*b*B*S in[c + d*x] + 9*a^4*C*Sin[c + d*x]))/45 + (2*Sec[c + d*x]^2*(20*a^3*A*b*Si n[c + d*x] + 28*a*A*b^3*Sin[c + d*x] + 5*a^4*B*Sin[c + d*x] + 42*a^2*b^2*B *Sin[c + d*x] + 28*a^3*b*C*Sin[c + d*x]))/21 + (2*Sec[c + d*x]*(7*a^4*A*Si n[c + d*x] + 54*a^2*A*b^2*Sin[c + d*x] + 15*A*b^4*Sin[c + d*x] + 36*a^3*b* B*Sin[c + d*x] + 60*a*b^3*B*Sin[c + d*x] + 9*a^4*C*Sin[c + d*x] + 90*a^2*b ^2*C*Sin[c + d*x]))/15 + (2*a^4*A*Sec[c + d*x]^4*Tan[c + d*x])/9))/d
Time = 2.64 (sec) , antiderivative size = 411, normalized size of antiderivative = 1.02, number of steps used = 20, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.465, Rules used = {3042, 3526, 27, 3042, 3526, 27, 3042, 3526, 27, 3042, 3510, 27, 3042, 3500, 27, 3042, 3227, 3042, 3119, 3120}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {(a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {11}{2}}(c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^4 \left (A+B \sin \left (c+d x+\frac {\pi }{2}\right )+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{11/2}}dx\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle \frac {2}{9} \int \frac {(a+b \cos (c+d x))^3 \left (-b (A-9 C) \cos ^2(c+d x)+(7 a A+9 b B+9 a C) \cos (c+d x)+8 A b+9 a B\right )}{2 \cos ^{\frac {9}{2}}(c+d x)}dx+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \int \frac {(a+b \cos (c+d x))^3 \left (-b (A-9 C) \cos ^2(c+d x)+(7 a A+9 b B+9 a C) \cos (c+d x)+8 A b+9 a B\right )}{\cos ^{\frac {9}{2}}(c+d x)}dx+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^3 \left (-b (A-9 C) \sin \left (c+d x+\frac {\pi }{2}\right )^2+(7 a A+9 b B+9 a C) \sin \left (c+d x+\frac {\pi }{2}\right )+8 A b+9 a B\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{9/2}}dx+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle \frac {1}{9} \left (\frac {2}{7} \int \frac {(a+b \cos (c+d x))^2 \left (7 (7 A+9 C) a^2+117 b B a+48 A b^2-3 b (5 A b-21 C b+3 a B) \cos ^2(c+d x)+\left (45 B a^2+82 A b a+126 b C a+63 b^2 B\right ) \cos (c+d x)\right )}{2 \cos ^{\frac {7}{2}}(c+d x)}dx+\frac {2 (9 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \int \frac {(a+b \cos (c+d x))^2 \left (7 (7 A+9 C) a^2+117 b B a+48 A b^2-3 b (5 A b-21 C b+3 a B) \cos ^2(c+d x)+\left (45 B a^2+82 A b a+126 b C a+63 b^2 B\right ) \cos (c+d x)\right )}{\cos ^{\frac {7}{2}}(c+d x)}dx+\frac {2 (9 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^2 \left (7 (7 A+9 C) a^2+117 b B a+48 A b^2-3 b (5 A b-21 C b+3 a B) \sin \left (c+d x+\frac {\pi }{2}\right )^2+\left (45 B a^2+82 A b a+126 b C a+63 b^2 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{7/2}}dx+\frac {2 (9 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {2}{5} \int \frac {(a+b \cos (c+d x)) \left (-b \left (7 (7 A+9 C) a^2+162 b B a+3 b^2 (41 A-105 C)\right ) \cos ^2(c+d x)+\left (21 (7 A+9 C) a^3+531 b B a^2+b^2 (479 A+945 C) a+315 b^3 B\right ) \cos (c+d x)+3 \left (75 B a^3+(202 A b+294 C b) a^2+261 b^2 B a+64 A b^3\right )\right )}{2 \cos ^{\frac {5}{2}}(c+d x)}dx+\frac {2 \sin (c+d x) \left (7 a^2 (7 A+9 C)+117 a b B+48 A b^2\right ) (a+b \cos (c+d x))^2}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (9 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \int \frac {(a+b \cos (c+d x)) \left (-b \left (7 (7 A+9 C) a^2+162 b B a+3 b^2 (41 A-105 C)\right ) \cos ^2(c+d x)+\left (21 (7 A+9 C) a^3+531 b B a^2+b^2 (479 A+945 C) a+315 b^3 B\right ) \cos (c+d x)+3 \left (75 B a^3+(202 A b+294 C b) a^2+261 b^2 B a+64 A b^3\right )\right )}{\cos ^{\frac {5}{2}}(c+d x)}dx+\frac {2 \sin (c+d x) \left (7 a^2 (7 A+9 C)+117 a b B+48 A b^2\right ) (a+b \cos (c+d x))^2}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (9 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right ) \left (-b \left (7 (7 A+9 C) a^2+162 b B a+3 b^2 (41 A-105 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+\left (21 (7 A+9 C) a^3+531 b B a^2+b^2 (479 A+945 C) a+315 b^3 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+3 \left (75 B a^3+(202 A b+294 C b) a^2+261 b^2 B a+64 A b^3\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{5/2}}dx+\frac {2 \sin (c+d x) \left (7 a^2 (7 A+9 C)+117 a b B+48 A b^2\right ) (a+b \cos (c+d x))^2}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (9 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3510 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {2 a \sin (c+d x) \left (75 a^3 B+a^2 (202 A b+294 b C)+261 a b^2 B+64 A b^3\right )}{d \cos ^{\frac {3}{2}}(c+d x)}-\frac {2}{3} \int -\frac {3 \left (21 (7 A+9 C) a^4+756 b B a^3+7 b^2 (155 A+261 C) a^2+1098 b^3 B a+192 A b^4-b^2 \left (7 (7 A+9 C) a^2+162 b B a+3 b^2 (41 A-105 C)\right ) \cos ^2(c+d x)+15 \left (5 B a^4+4 b (5 A+7 C) a^3+42 b^2 B a^2+28 b^3 (A+3 C) a+21 b^4 B\right ) \cos (c+d x)\right )}{2 \cos ^{\frac {3}{2}}(c+d x)}dx\right )+\frac {2 \sin (c+d x) \left (7 a^2 (7 A+9 C)+117 a b B+48 A b^2\right ) (a+b \cos (c+d x))^2}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (9 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (\int \frac {21 (7 A+9 C) a^4+756 b B a^3+7 b^2 (155 A+261 C) a^2+1098 b^3 B a+192 A b^4-b^2 \left (7 (7 A+9 C) a^2+162 b B a+3 b^2 (41 A-105 C)\right ) \cos ^2(c+d x)+15 \left (5 B a^4+4 b (5 A+7 C) a^3+42 b^2 B a^2+28 b^3 (A+3 C) a+21 b^4 B\right ) \cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x)}dx+\frac {2 a \sin (c+d x) \left (75 a^3 B+a^2 (202 A b+294 b C)+261 a b^2 B+64 A b^3\right )}{d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 \sin (c+d x) \left (7 a^2 (7 A+9 C)+117 a b B+48 A b^2\right ) (a+b \cos (c+d x))^2}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (9 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (\int \frac {21 (7 A+9 C) a^4+756 b B a^3+7 b^2 (155 A+261 C) a^2+1098 b^3 B a+192 A b^4-b^2 \left (7 (7 A+9 C) a^2+162 b B a+3 b^2 (41 A-105 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+15 \left (5 B a^4+4 b (5 A+7 C) a^3+42 b^2 B a^2+28 b^3 (A+3 C) a+21 b^4 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2}}dx+\frac {2 a \sin (c+d x) \left (75 a^3 B+a^2 (202 A b+294 b C)+261 a b^2 B+64 A b^3\right )}{d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 \sin (c+d x) \left (7 a^2 (7 A+9 C)+117 a b B+48 A b^2\right ) (a+b \cos (c+d x))^2}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (9 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3500 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (2 \int \frac {3 \left (5 \left (5 B a^4+4 b (5 A+7 C) a^3+42 b^2 B a^2+28 b^3 (A+3 C) a+21 b^4 B\right )-7 \left ((7 A+9 C) a^4+36 b B a^3+18 b^2 (3 A+5 C) a^2+60 b^3 B a+15 b^4 (A-C)\right ) \cos (c+d x)\right )}{2 \sqrt {\cos (c+d x)}}dx+\frac {2 a \sin (c+d x) \left (75 a^3 B+a^2 (202 A b+294 b C)+261 a b^2 B+64 A b^3\right )}{d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 \sin (c+d x) \left (21 a^4 (7 A+9 C)+756 a^3 b B+7 a^2 b^2 (155 A+261 C)+1098 a b^3 B+192 A b^4\right )}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 \sin (c+d x) \left (7 a^2 (7 A+9 C)+117 a b B+48 A b^2\right ) (a+b \cos (c+d x))^2}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (9 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (3 \int \frac {5 \left (5 B a^4+4 b (5 A+7 C) a^3+42 b^2 B a^2+28 b^3 (A+3 C) a+21 b^4 B\right )-7 \left ((7 A+9 C) a^4+36 b B a^3+18 b^2 (3 A+5 C) a^2+60 b^3 B a+15 b^4 (A-C)\right ) \cos (c+d x)}{\sqrt {\cos (c+d x)}}dx+\frac {2 a \sin (c+d x) \left (75 a^3 B+a^2 (202 A b+294 b C)+261 a b^2 B+64 A b^3\right )}{d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 \sin (c+d x) \left (21 a^4 (7 A+9 C)+756 a^3 b B+7 a^2 b^2 (155 A+261 C)+1098 a b^3 B+192 A b^4\right )}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 \sin (c+d x) \left (7 a^2 (7 A+9 C)+117 a b B+48 A b^2\right ) (a+b \cos (c+d x))^2}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (9 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (3 \int \frac {5 \left (5 B a^4+4 b (5 A+7 C) a^3+42 b^2 B a^2+28 b^3 (A+3 C) a+21 b^4 B\right )-7 \left ((7 A+9 C) a^4+36 b B a^3+18 b^2 (3 A+5 C) a^2+60 b^3 B a+15 b^4 (A-C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 a \sin (c+d x) \left (75 a^3 B+a^2 (202 A b+294 b C)+261 a b^2 B+64 A b^3\right )}{d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 \sin (c+d x) \left (21 a^4 (7 A+9 C)+756 a^3 b B+7 a^2 b^2 (155 A+261 C)+1098 a b^3 B+192 A b^4\right )}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 \sin (c+d x) \left (7 a^2 (7 A+9 C)+117 a b B+48 A b^2\right ) (a+b \cos (c+d x))^2}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (9 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (3 \left (5 \left (5 a^4 B+4 a^3 b (5 A+7 C)+42 a^2 b^2 B+28 a b^3 (A+3 C)+21 b^4 B\right ) \int \frac {1}{\sqrt {\cos (c+d x)}}dx-7 \left (a^4 (7 A+9 C)+36 a^3 b B+18 a^2 b^2 (3 A+5 C)+60 a b^3 B+15 b^4 (A-C)\right ) \int \sqrt {\cos (c+d x)}dx\right )+\frac {2 a \sin (c+d x) \left (75 a^3 B+a^2 (202 A b+294 b C)+261 a b^2 B+64 A b^3\right )}{d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 \sin (c+d x) \left (21 a^4 (7 A+9 C)+756 a^3 b B+7 a^2 b^2 (155 A+261 C)+1098 a b^3 B+192 A b^4\right )}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 \sin (c+d x) \left (7 a^2 (7 A+9 C)+117 a b B+48 A b^2\right ) (a+b \cos (c+d x))^2}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (9 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (3 \left (5 \left (5 a^4 B+4 a^3 b (5 A+7 C)+42 a^2 b^2 B+28 a b^3 (A+3 C)+21 b^4 B\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx-7 \left (a^4 (7 A+9 C)+36 a^3 b B+18 a^2 b^2 (3 A+5 C)+60 a b^3 B+15 b^4 (A-C)\right ) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx\right )+\frac {2 a \sin (c+d x) \left (75 a^3 B+a^2 (202 A b+294 b C)+261 a b^2 B+64 A b^3\right )}{d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 \sin (c+d x) \left (21 a^4 (7 A+9 C)+756 a^3 b B+7 a^2 b^2 (155 A+261 C)+1098 a b^3 B+192 A b^4\right )}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 \sin (c+d x) \left (7 a^2 (7 A+9 C)+117 a b B+48 A b^2\right ) (a+b \cos (c+d x))^2}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (9 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (3 \left (5 \left (5 a^4 B+4 a^3 b (5 A+7 C)+42 a^2 b^2 B+28 a b^3 (A+3 C)+21 b^4 B\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {14 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (a^4 (7 A+9 C)+36 a^3 b B+18 a^2 b^2 (3 A+5 C)+60 a b^3 B+15 b^4 (A-C)\right )}{d}\right )+\frac {2 a \sin (c+d x) \left (75 a^3 B+a^2 (202 A b+294 b C)+261 a b^2 B+64 A b^3\right )}{d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 \sin (c+d x) \left (21 a^4 (7 A+9 C)+756 a^3 b B+7 a^2 b^2 (155 A+261 C)+1098 a b^3 B+192 A b^4\right )}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 \sin (c+d x) \left (7 a^2 (7 A+9 C)+117 a b B+48 A b^2\right ) (a+b \cos (c+d x))^2}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (9 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {2 \sin (c+d x) \left (7 a^2 (7 A+9 C)+117 a b B+48 A b^2\right ) (a+b \cos (c+d x))^2}{5 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {1}{5} \left (\frac {2 a \sin (c+d x) \left (75 a^3 B+a^2 (202 A b+294 b C)+261 a b^2 B+64 A b^3\right )}{d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 \sin (c+d x) \left (21 a^4 (7 A+9 C)+756 a^3 b B+7 a^2 b^2 (155 A+261 C)+1098 a b^3 B+192 A b^4\right )}{d \sqrt {\cos (c+d x)}}+3 \left (\frac {10 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (5 a^4 B+4 a^3 b (5 A+7 C)+42 a^2 b^2 B+28 a b^3 (A+3 C)+21 b^4 B\right )}{d}-\frac {14 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (a^4 (7 A+9 C)+36 a^3 b B+18 a^2 b^2 (3 A+5 C)+60 a b^3 B+15 b^4 (A-C)\right )}{d}\right )\right )\right )+\frac {2 (9 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
Input:
Int[((a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(11/2),x]
Output:
(2*A*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(9*d*Cos[c + d*x]^(9/2)) + ((2*( 8*A*b + 9*a*B)*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2 )) + ((2*(48*A*b^2 + 117*a*b*B + 7*a^2*(7*A + 9*C))*(a + b*Cos[c + d*x])^2 *Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (3*((-14*(36*a^3*b*B + 60*a*b^3* B + 15*b^4*(A - C) + 18*a^2*b^2*(3*A + 5*C) + a^4*(7*A + 9*C))*EllipticE[( c + d*x)/2, 2])/d + (10*(5*a^4*B + 42*a^2*b^2*B + 21*b^4*B + 28*a*b^3*(A + 3*C) + 4*a^3*b*(5*A + 7*C))*EllipticF[(c + d*x)/2, 2])/d) + (2*a*(64*A*b^ 3 + 75*a^3*B + 261*a*b^2*B + a^2*(202*A*b + 294*b*C))*Sin[c + d*x])/(d*Cos [c + d*x]^(3/2)) + (2*(192*A*b^4 + 756*a^3*b*B + 1098*a*b^3*B + 21*a^4*(7* A + 9*C) + 7*a^2*b^2*(155*A + 261*C))*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) )/5)/7)/9
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)* (c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(1/2 )*(c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x _)]), x_Symbol] :> Simp[c Int[(b*Sin[e + f*x])^m, x], x] + Simp[d/b Int [(b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*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. \(2(380)=760\).
Time = 16.41 (sec) , antiderivative size = 1523, normalized size of antiderivative = 3.80
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1523\) |
| parts | \(\text {Expression too large to display}\) | \(1627\) |
Input:
int((a+b*cos(d*x+c))^4*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/cos(d*x+c)^(11/2),x ,method=_RETURNVERBOSE)
Output:
-(-(-2*cos(1/2*d*x+1/2*c)^2+1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*B*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*(-1/144*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*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)^ 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))-7/15*(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))-EllipticE(cos(1/2*d *x+1/2*c),2^(1/2))))+2/5*a^2*(6*A*b^2+4*B*a*b+C*a^2)/(8*sin(1/2*d*x+1/2*c) ^6-12*sin(1/2*d*x+1/2*c)^4+6*sin(1/2*d*x+1/2*c)^2-1)/sin(1/2*d*x+1/2*c)^2* (24*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^6-12*(2*sin(1/2*d*x+1/2*c)^2-1)^ (1/2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*s in(1/2*d*x+1/2*c)^4-24*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^4+12*(2*sin(1 /2*d*x+1/2*c)^2-1)^(1/2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticE(cos(1/2*d* x+1/2*c),2^(1/2))*sin(1/2*d*x+1/2*c)^2+8*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/ 2*c)^2-3*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*...
Result contains complex when optimal does not.
Time = 0.16 (sec) , antiderivative size = 525, normalized size of antiderivative = 1.31 \[ \int \frac {(a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {11}{2}}(c+d x)} \, dx =\text {Too large to display} \] Input:
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 1/2),x, algorithm="fricas")
Output:
-1/315*(15*sqrt(2)*(5*I*B*a^4 + 4*I*(5*A + 7*C)*a^3*b + 42*I*B*a^2*b^2 + 2 8*I*(A + 3*C)*a*b^3 + 21*I*B*b^4)*cos(d*x + c)^5*weierstrassPInverse(-4, 0 , cos(d*x + c) + I*sin(d*x + c)) + 15*sqrt(2)*(-5*I*B*a^4 - 4*I*(5*A + 7*C )*a^3*b - 42*I*B*a^2*b^2 - 28*I*(A + 3*C)*a*b^3 - 21*I*B*b^4)*cos(d*x + c) ^5*weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) + 21*sqrt(2)* (I*(7*A + 9*C)*a^4 + 36*I*B*a^3*b + 18*I*(3*A + 5*C)*a^2*b^2 + 60*I*B*a*b^ 3 + 15*I*(A - C)*b^4)*cos(d*x + c)^5*weierstrassZeta(-4, 0, weierstrassPIn verse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) + 21*sqrt(2)*(-I*(7*A + 9*C)* a^4 - 36*I*B*a^3*b - 18*I*(3*A + 5*C)*a^2*b^2 - 60*I*B*a*b^3 - 15*I*(A - C )*b^4)*cos(d*x + c)^5*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, co s(d*x + c) - I*sin(d*x + c))) - 2*(35*A*a^4 + 21*((7*A + 9*C)*a^4 + 36*B*a ^3*b + 18*(3*A + 5*C)*a^2*b^2 + 60*B*a*b^3 + 15*A*b^4)*cos(d*x + c)^4 + 15 *(5*B*a^4 + 4*(5*A + 7*C)*a^3*b + 42*B*a^2*b^2 + 28*A*a*b^3)*cos(d*x + c)^ 3 + 7*((7*A + 9*C)*a^4 + 36*B*a^3*b + 54*A*a^2*b^2)*cos(d*x + c)^2 + 45*(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)^5)
Timed out. \[ \int \frac {(a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {11}{2}}(c+d x)} \, dx=\text {Timed out} \] Input:
integrate((a+b*cos(d*x+c))**4*(A+B*cos(d*x+c)+C*cos(d*x+c)**2)/cos(d*x+c)* *(11/2),x)
Output:
Timed out
\[ \int \frac {(a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {11}{2}}(c+d x)} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{4}}{\cos \left (d x + c\right )^{\frac {11}{2}}} \,d x } \] Input:
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 1/2),x, algorithm="maxima")
Output:
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)^(11/2), x)
\[ \int \frac {(a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {11}{2}}(c+d x)} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{4}}{\cos \left (d x + c\right )^{\frac {11}{2}}} \,d x } \] Input:
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 1/2),x, algorithm="giac")
Output:
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)^(11/2), x)
Time = 5.46 (sec) , antiderivative size = 866, normalized size of antiderivative = 2.16 \[ \int \frac {(a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {11}{2}}(c+d x)} \, dx=\text {Too large to display} \] Input:
int(((a + b*cos(c + d*x))^4*(A + B*cos(c + d*x) + C*cos(c + d*x)^2))/cos(c + d*x)^(11/2),x)
Output:
(8*hypergeom([-3/4, 1/2], 1/4, cos(c + d*x)^2)*((7*A*a*b^3*sin(c + d*x))/( cos(c + d*x)^(3/2)*(sin(c + d*x)^2)^(1/2)) + (4*A*a^3*b*sin(c + d*x))/(cos (c + d*x)^(3/2)*(sin(c + d*x)^2)^(1/2)) + (3*A*a^3*b*sin(c + d*x))/(cos(c + d*x)^(7/2)*(sin(c + d*x)^2)^(1/2))))/(21*d) - (8*hypergeom([-1/4, 1/2], 7/4, cos(c + d*x)^2)*((7*A*a^4*sin(c + d*x))/(cos(c + d*x)^(1/2)*(sin(c + d*x)^2)^(1/2)) + (5*A*a^4*sin(c + d*x))/(cos(c + d*x)^(5/2)*(sin(c + d*x)^ 2)^(1/2)) + (54*A*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([-1/4, 1/2], 3/4, cos(c + d*x)^2)*((28* A*a^4*sin(c + d*x))/(cos(c + d*x)^(1/2)*(sin(c + d*x)^2)^(1/2)) + (12*A*a^ 4*sin(c + d*x))/(cos(c + d*x)^(5/2)*(sin(c + d*x)^2)^(1/2)) + (5*A*a^4*sin (c + d*x))/(cos(c + d*x)^(9/2)*(sin(c + d*x)^2)^(1/2)) + (45*A*b^4*sin(c + d*x))/(cos(c + d*x)^(1/2)*(sin(c + d*x)^2)^(1/2)) + (216*A*a^2*b^2*sin(c + d*x))/(cos(c + d*x)^(1/2)*(sin(c + d*x)^2)^(1/2)) + (54*A*a^2*b^2*sin(c + d*x))/(cos(c + d*x)^(5/2)*(sin(c + d*x)^2)^(1/2))))/(45*d) + (2*B*b^4*el lipticF(c/2 + (d*x)/2, 2))/d + (2*C*b^4*ellipticE(c/2 + (d*x)/2, 2))/d + ( 8*C*a*b^3*ellipticF(c/2 + (d*x)/2, 2))/d + (2*B*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)) + (2*C*a^4*sin(c + d*x)*hypergeom([-5/4, 1/2], -1/4, cos(c + d* x)^2))/(5*d*cos(c + d*x)^(5/2)*(sin(c + d*x)^2)^(1/2)) + (32*A*a^3*b*sin(c + d*x)*hypergeom([-3/4, 1/2], 5/4, cos(c + d*x)^2))/(21*d*cos(c + d*x)...
\[ \int \frac {(a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {11}{2}}(c+d x)} \, dx=4 \left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )}d x \right ) a \,b^{3} c +\left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )}d x \right ) b^{5}+\left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{6}}d x \right ) a^{5}+5 \left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{5}}d x \right ) a^{4} b +\left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{4}}d x \right ) a^{4} c +10 \left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{4}}d x \right ) a^{3} b^{2}+4 \left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{3}}d x \right ) a^{3} b c +10 \left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{3}}d x \right ) a^{2} b^{3}+6 \left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{2}}d x \right ) a^{2} b^{2} c +5 \left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{2}}d x \right ) a \,b^{4}+\left (\int \sqrt {\cos \left (d x +c \right )}d x \right ) b^{4} c \] Input:
int((a+b*cos(d*x+c))^4*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/cos(d*x+c)^(11/2),x )
Output:
4*int(sqrt(cos(c + d*x))/cos(c + d*x),x)*a*b**3*c + int(sqrt(cos(c + d*x)) /cos(c + d*x),x)*b**5 + int(sqrt(cos(c + d*x))/cos(c + d*x)**6,x)*a**5 + 5 *int(sqrt(cos(c + d*x))/cos(c + d*x)**5,x)*a**4*b + int(sqrt(cos(c + d*x)) /cos(c + d*x)**4,x)*a**4*c + 10*int(sqrt(cos(c + d*x))/cos(c + d*x)**4,x)* a**3*b**2 + 4*int(sqrt(cos(c + d*x))/cos(c + d*x)**3,x)*a**3*b*c + 10*int( sqrt(cos(c + d*x))/cos(c + d*x)**3,x)*a**2*b**3 + 6*int(sqrt(cos(c + d*x)) /cos(c + d*x)**2,x)*a**2*b**2*c + 5*int(sqrt(cos(c + d*x))/cos(c + d*x)**2 ,x)*a*b**4 + int(sqrt(cos(c + d*x)),x)*b**4*c