Integrand size = 43, antiderivative size = 423 \[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {9}{2}}(c+d x) \, dx=-\frac {2 \left (3 a^4 B+30 a^2 b^2 B-5 b^4 B+20 a b^3 (A-C)+4 a^3 b (3 A+5 C)\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{5 d}+\frac {2 \left (28 a^3 b B+84 a b^3 B+7 b^4 (3 A+C)+42 a^2 b^2 (A+3 C)+a^4 (5 A+7 C)\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{21 d}-\frac {2 b^2 \left (98 a b B+b^2 (87 A-35 C)+5 a^2 (5 A+7 C)\right ) \sin (c+d x)}{105 d \sqrt {\sec (c+d x)}}+\frac {2 a \left (192 A b^3+63 a^3 B+413 a b^2 B+a^2 (202 A b+350 b C)\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{105 d}+\frac {2 \left (48 A b^2+77 a b B+5 a^2 (5 A+7 C)\right ) (a+b \cos (c+d x))^2 \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac {2 (8 A b+7 a B) (a+b \cos (c+d x))^3 \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac {2 A (a+b \cos (c+d x))^4 \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{7 d} \]
2/105*(48*A*b^2+77*B*a*b+5*a^2*(5*A+7*C))*(a+b*cos(d*x+c))^2*sec(d*x+c)^(3 /2)*sin(d*x+c)/d+2/35*(8*A*b+7*B*a)*(a+b*cos(d*x+c))^3*sec(d*x+c)^(5/2)*si n(d*x+c)/d+2/7*A*(a+b*cos(d*x+c))^4*sec(d*x+c)^(7/2)*sin(d*x+c)/d-2/105*b^ 2*(98*B*a*b+b^2*(87*A-35*C)+5*a^2*(5*A+7*C))*sin(d*x+c)/d/sec(d*x+c)^(1/2) +2/105*a*(192*A*b^3+63*B*a^3+413*B*a*b^2+a^2*(202*A*b+350*C*b))*sin(d*x+c) *sec(d*x+c)^(1/2)/d-2/5*(3*B*a^4+30*B*a^2*b^2-5*B*b^4+20*a*b^3*(A-C)+4*a^3 *b*(3*A+5*C))*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticE(si n(1/2*d*x+1/2*c),2^(1/2))*cos(d*x+c)^(1/2)*sec(d*x+c)^(1/2)/d+2/21*(28*B*a ^3*b+84*B*a*b^3+7*b^4*(3*A+C)+42*a^2*b^2*(A+3*C)+a^4*(5*A+7*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) )*cos(d*x+c)^(1/2)*sec(d*x+c)^(1/2)/d
Time = 7.06 (sec) , antiderivative size = 294, normalized size of antiderivative = 0.70 \[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {9}{2}}(c+d x) \, dx=\frac {\sqrt {\sec (c+d x)} \left (-42 \left (3 a^4 B+30 a^2 b^2 B-5 b^4 B+20 a b^3 (A-C)+4 a^3 b (3 A+5 C)\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+10 \left (28 a^3 b B+84 a b^3 B+7 b^4 (3 A+C)+42 a^2 b^2 (A+3 C)+a^4 (5 A+7 C)\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+42 a \left (20 A b^3+3 a^3 B+30 a b^2 B+4 a^2 b (3 A+5 C)\right ) \sin (c+d x)+35 b^4 C \sin (2 (c+d x))+10 a^2 \left (42 A b^2+28 a b B+a^2 (5 A+7 C)\right ) \tan (c+d x)+42 a^3 (4 A b+a B) \sec (c+d x) \tan (c+d x)+30 a^4 A \sec ^2(c+d x) \tan (c+d x)\right )}{105 d} \]
Integrate[(a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*S ec[c + d*x]^(9/2),x]
(Sqrt[Sec[c + d*x]]*(-42*(3*a^4*B + 30*a^2*b^2*B - 5*b^4*B + 20*a*b^3*(A - C) + 4*a^3*b*(3*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 10*(28*a^3*b*B + 84*a*b^3*B + 7*b^4*(3*A + C) + 42*a^2*b^2*(A + 3*C) + a^4 *(5*A + 7*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + 42*a*(20*A*b^ 3 + 3*a^3*B + 30*a*b^2*B + 4*a^2*b*(3*A + 5*C))*Sin[c + d*x] + 35*b^4*C*Si n[2*(c + d*x)] + 10*a^2*(42*A*b^2 + 28*a*b*B + a^2*(5*A + 7*C))*Tan[c + d* x] + 42*a^3*(4*A*b + a*B)*Sec[c + d*x]*Tan[c + d*x] + 30*a^4*A*Sec[c + d*x ]^2*Tan[c + d*x]))/(105*d)
Time = 2.75 (sec) , antiderivative size = 411, normalized size of antiderivative = 0.97, number of steps used = 22, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.512, Rules used = {3042, 4709, 3042, 3526, 27, 3042, 3526, 27, 3042, 3526, 27, 3042, 3510, 27, 3042, 3502, 27, 3042, 3227, 3042, 3119, 3120}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \sec ^{\frac {9}{2}}(c+d x) (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sec (c+d x)^{9/2} (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos (c+d x)^2\right )dx\) |
| \(\Big \downarrow \) 4709 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {(a+b \cos (c+d x))^4 \left (C \cos ^2(c+d x)+B \cos (c+d x)+A\right )}{\cos ^{\frac {9}{2}}(c+d x)}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^4 \left (C \sin \left (c+d x+\frac {\pi }{2}\right )^2+B \sin \left (c+d x+\frac {\pi }{2}\right )+A\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{9/2}}dx\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {2}{7} \int \frac {(a+b \cos (c+d x))^3 \left (-b (3 A-7 C) \cos ^2(c+d x)+(5 a A+7 b B+7 a C) \cos (c+d x)+8 A b+7 a B\right )}{2 \cos ^{\frac {7}{2}}(c+d x)}dx+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{7} \int \frac {(a+b \cos (c+d x))^3 \left (-b (3 A-7 C) \cos ^2(c+d x)+(5 a A+7 b B+7 a C) \cos (c+d x)+8 A b+7 a B\right )}{\cos ^{\frac {7}{2}}(c+d x)}dx+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{7} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^3 \left (-b (3 A-7 C) \sin \left (c+d x+\frac {\pi }{2}\right )^2+(5 a A+7 b B+7 a C) \sin \left (c+d x+\frac {\pi }{2}\right )+8 A b+7 a B\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{7/2}}dx+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{7} \left (\frac {2}{5} \int \frac {(a+b \cos (c+d x))^2 \left (5 (5 A+7 C) a^2+77 b B a+48 A b^2-b (39 A b-35 C b+21 a B) \cos ^2(c+d x)+\left (21 B a^2+34 A b a+70 b C a+35 b^2 B\right ) \cos (c+d x)\right )}{2 \cos ^{\frac {5}{2}}(c+d x)}dx+\frac {2 (7 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{7} \left (\frac {1}{5} \int \frac {(a+b \cos (c+d x))^2 \left (5 (5 A+7 C) a^2+77 b B a+48 A b^2-b (39 A b-35 C b+21 a B) \cos ^2(c+d x)+\left (21 B a^2+34 A b a+70 b C a+35 b^2 B\right ) \cos (c+d x)\right )}{\cos ^{\frac {5}{2}}(c+d x)}dx+\frac {2 (7 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{7} \left (\frac {1}{5} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^2 \left (5 (5 A+7 C) a^2+77 b B a+48 A b^2-b (39 A b-35 C b+21 a B) \sin \left (c+d x+\frac {\pi }{2}\right )^2+\left (21 B a^2+34 A b a+70 b C a+35 b^2 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{5/2}}dx+\frac {2 (7 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {2}{3} \int \frac {(a+b \cos (c+d x)) \left (63 B a^3+2 b (101 A+175 C) a^2+413 b^2 B a+192 A b^3-3 b \left (5 (5 A+7 C) a^2+98 b B a+b^2 (87 A-35 C)\right ) \cos ^2(c+d x)+\left (5 (5 A+7 C) a^3+77 b B a^2+3 b^2 (11 A+105 C) a+105 b^3 B\right ) \cos (c+d x)\right )}{2 \cos ^{\frac {3}{2}}(c+d x)}dx+\frac {2 \sin (c+d x) \left (5 a^2 (5 A+7 C)+77 a b B+48 A b^2\right ) (a+b \cos (c+d x))^2}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (7 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \int \frac {(a+b \cos (c+d x)) \left (63 B a^3+(202 A b+350 C b) a^2+413 b^2 B a+192 A b^3-3 b \left (5 (5 A+7 C) a^2+98 b B a+b^2 (87 A-35 C)\right ) \cos ^2(c+d x)+\left (5 (5 A+7 C) a^3+77 b B a^2+3 b^2 (11 A+105 C) a+105 b^3 B\right ) \cos (c+d x)\right )}{\cos ^{\frac {3}{2}}(c+d x)}dx+\frac {2 \sin (c+d x) \left (5 a^2 (5 A+7 C)+77 a b B+48 A b^2\right ) (a+b \cos (c+d x))^2}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (7 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right ) \left (63 B a^3+(202 A b+350 C b) a^2+413 b^2 B a+192 A b^3-3 b \left (5 (5 A+7 C) a^2+98 b B a+b^2 (87 A-35 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+\left (5 (5 A+7 C) a^3+77 b B a^2+3 b^2 (11 A+105 C) a+105 b^3 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2}}dx+\frac {2 \sin (c+d x) \left (5 a^2 (5 A+7 C)+77 a b B+48 A b^2\right ) (a+b \cos (c+d x))^2}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (7 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3510 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {2 a \sin (c+d x) \left (63 a^3 B+a^2 (202 A b+350 b C)+413 a b^2 B+192 A b^3\right )}{d \sqrt {\cos (c+d x)}}-2 \int -\frac {5 (5 A+7 C) a^4+140 b B a^3+5 b^2 (47 A+133 C) a^2+518 b^3 B a+192 A b^4-3 b^2 \left (5 (5 A+7 C) a^2+98 b B a+b^2 (87 A-35 C)\right ) \cos ^2(c+d x)-21 \left (3 B a^4+4 b (3 A+5 C) a^3+30 b^2 B a^2+20 b^3 (A-C) a-5 b^4 B\right ) \cos (c+d x)}{2 \sqrt {\cos (c+d x)}}dx\right )+\frac {2 \sin (c+d x) \left (5 a^2 (5 A+7 C)+77 a b B+48 A b^2\right ) (a+b \cos (c+d x))^2}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (7 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\int \frac {5 (5 A+7 C) a^4+140 b B a^3+5 b^2 (47 A+133 C) a^2+518 b^3 B a+192 A b^4-3 b^2 \left (5 (5 A+7 C) a^2+98 b B a+b^2 (87 A-35 C)\right ) \cos ^2(c+d x)-21 \left (3 B a^4+4 b (3 A+5 C) a^3+30 b^2 B a^2+20 b^3 (A-C) a-5 b^4 B\right ) \cos (c+d x)}{\sqrt {\cos (c+d x)}}dx+\frac {2 a \sin (c+d x) \left (63 a^3 B+a^2 (202 A b+350 b C)+413 a b^2 B+192 A b^3\right )}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 \sin (c+d x) \left (5 a^2 (5 A+7 C)+77 a b B+48 A b^2\right ) (a+b \cos (c+d x))^2}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (7 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\int \frac {5 (5 A+7 C) a^4+140 b B a^3+5 b^2 (47 A+133 C) a^2+518 b^3 B a+192 A b^4-3 b^2 \left (5 (5 A+7 C) a^2+98 b B a+b^2 (87 A-35 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2-21 \left (3 B a^4+4 b (3 A+5 C) a^3+30 b^2 B a^2+20 b^3 (A-C) a-5 b^4 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 a \sin (c+d x) \left (63 a^3 B+a^2 (202 A b+350 b C)+413 a b^2 B+192 A b^3\right )}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 \sin (c+d x) \left (5 a^2 (5 A+7 C)+77 a b B+48 A b^2\right ) (a+b \cos (c+d x))^2}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (7 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {2}{3} \int \frac {3 \left (5 \left ((5 A+7 C) a^4+28 b B a^3+42 b^2 (A+3 C) a^2+84 b^3 B a+7 b^4 (3 A+C)\right )-21 \left (3 B a^4+4 b (3 A+5 C) a^3+30 b^2 B a^2+20 b^3 (A-C) a-5 b^4 B\right ) \cos (c+d x)\right )}{2 \sqrt {\cos (c+d x)}}dx-\frac {2 b^2 \sin (c+d x) \sqrt {\cos (c+d x)} \left (5 a^2 (5 A+7 C)+98 a b B+b^2 (87 A-35 C)\right )}{d}+\frac {2 a \sin (c+d x) \left (63 a^3 B+a^2 (202 A b+350 b C)+413 a b^2 B+192 A b^3\right )}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 \sin (c+d x) \left (5 a^2 (5 A+7 C)+77 a b B+48 A b^2\right ) (a+b \cos (c+d x))^2}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (7 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\int \frac {5 \left ((5 A+7 C) a^4+28 b B a^3+42 b^2 (A+3 C) a^2+84 b^3 B a+7 b^4 (3 A+C)\right )-21 \left (3 B a^4+4 b (3 A+5 C) a^3+30 b^2 B a^2+20 b^3 (A-C) a-5 b^4 B\right ) \cos (c+d x)}{\sqrt {\cos (c+d x)}}dx-\frac {2 b^2 \sin (c+d x) \sqrt {\cos (c+d x)} \left (5 a^2 (5 A+7 C)+98 a b B+b^2 (87 A-35 C)\right )}{d}+\frac {2 a \sin (c+d x) \left (63 a^3 B+a^2 (202 A b+350 b C)+413 a b^2 B+192 A b^3\right )}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 \sin (c+d x) \left (5 a^2 (5 A+7 C)+77 a b B+48 A b^2\right ) (a+b \cos (c+d x))^2}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (7 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\int \frac {5 \left ((5 A+7 C) a^4+28 b B a^3+42 b^2 (A+3 C) a^2+84 b^3 B a+7 b^4 (3 A+C)\right )-21 \left (3 B a^4+4 b (3 A+5 C) a^3+30 b^2 B a^2+20 b^3 (A-C) a-5 b^4 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {2 b^2 \sin (c+d x) \sqrt {\cos (c+d x)} \left (5 a^2 (5 A+7 C)+98 a b B+b^2 (87 A-35 C)\right )}{d}+\frac {2 a \sin (c+d x) \left (63 a^3 B+a^2 (202 A b+350 b C)+413 a b^2 B+192 A b^3\right )}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 \sin (c+d x) \left (5 a^2 (5 A+7 C)+77 a b B+48 A b^2\right ) (a+b \cos (c+d x))^2}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (7 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (5 \left (a^4 (5 A+7 C)+28 a^3 b B+42 a^2 b^2 (A+3 C)+84 a b^3 B+7 b^4 (3 A+C)\right ) \int \frac {1}{\sqrt {\cos (c+d x)}}dx-21 \left (3 a^4 B+4 a^3 b (3 A+5 C)+30 a^2 b^2 B+20 a b^3 (A-C)-5 b^4 B\right ) \int \sqrt {\cos (c+d x)}dx-\frac {2 b^2 \sin (c+d x) \sqrt {\cos (c+d x)} \left (5 a^2 (5 A+7 C)+98 a b B+b^2 (87 A-35 C)\right )}{d}+\frac {2 a \sin (c+d x) \left (63 a^3 B+a^2 (202 A b+350 b C)+413 a b^2 B+192 A b^3\right )}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 \sin (c+d x) \left (5 a^2 (5 A+7 C)+77 a b B+48 A b^2\right ) (a+b \cos (c+d x))^2}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (7 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (5 \left (a^4 (5 A+7 C)+28 a^3 b B+42 a^2 b^2 (A+3 C)+84 a b^3 B+7 b^4 (3 A+C)\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx-21 \left (3 a^4 B+4 a^3 b (3 A+5 C)+30 a^2 b^2 B+20 a b^3 (A-C)-5 b^4 B\right ) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx-\frac {2 b^2 \sin (c+d x) \sqrt {\cos (c+d x)} \left (5 a^2 (5 A+7 C)+98 a b B+b^2 (87 A-35 C)\right )}{d}+\frac {2 a \sin (c+d x) \left (63 a^3 B+a^2 (202 A b+350 b C)+413 a b^2 B+192 A b^3\right )}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 \sin (c+d x) \left (5 a^2 (5 A+7 C)+77 a b B+48 A b^2\right ) (a+b \cos (c+d x))^2}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (7 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (5 \left (a^4 (5 A+7 C)+28 a^3 b B+42 a^2 b^2 (A+3 C)+84 a b^3 B+7 b^4 (3 A+C)\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {2 b^2 \sin (c+d x) \sqrt {\cos (c+d x)} \left (5 a^2 (5 A+7 C)+98 a b B+b^2 (87 A-35 C)\right )}{d}+\frac {2 a \sin (c+d x) \left (63 a^3 B+a^2 (202 A b+350 b C)+413 a b^2 B+192 A b^3\right )}{d \sqrt {\cos (c+d x)}}-\frac {42 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (3 a^4 B+4 a^3 b (3 A+5 C)+30 a^2 b^2 B+20 a b^3 (A-C)-5 b^4 B\right )}{d}\right )+\frac {2 \sin (c+d x) \left (5 a^2 (5 A+7 C)+77 a b B+48 A b^2\right ) (a+b \cos (c+d x))^2}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (7 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {2 \sin (c+d x) \left (5 a^2 (5 A+7 C)+77 a b B+48 A b^2\right ) (a+b \cos (c+d x))^2}{3 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {1}{3} \left (-\frac {2 b^2 \sin (c+d x) \sqrt {\cos (c+d x)} \left (5 a^2 (5 A+7 C)+98 a b B+b^2 (87 A-35 C)\right )}{d}+\frac {2 a \sin (c+d x) \left (63 a^3 B+a^2 (202 A b+350 b C)+413 a b^2 B+192 A b^3\right )}{d \sqrt {\cos (c+d x)}}+\frac {10 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (a^4 (5 A+7 C)+28 a^3 b B+42 a^2 b^2 (A+3 C)+84 a b^3 B+7 b^4 (3 A+C)\right )}{d}-\frac {42 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (3 a^4 B+4 a^3 b (3 A+5 C)+30 a^2 b^2 B+20 a b^3 (A-C)-5 b^4 B\right )}{d}\right )\right )+\frac {2 (7 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )\) |
Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((2*A*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)) + ((2*(8*A*b + 7*a*B)*(a + b*Cos[c + d*x]) ^3*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + ((2*(48*A*b^2 + 77*a*b*B + 5*a ^2*(5*A + 7*C))*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/ 2)) + ((-42*(3*a^4*B + 30*a^2*b^2*B - 5*b^4*B + 20*a*b^3*(A - C) + 4*a^3*b *(3*A + 5*C))*EllipticE[(c + d*x)/2, 2])/d + (10*(28*a^3*b*B + 84*a*b^3*B + 7*b^4*(3*A + C) + 42*a^2*b^2*(A + 3*C) + a^4*(5*A + 7*C))*EllipticF[(c + d*x)/2, 2])/d + (2*a*(192*A*b^3 + 63*a^3*B + 413*a*b^2*B + a^2*(202*A*b + 350*b*C))*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) - (2*b^2*(98*a*b*B + b^2*( 87*A - 35*C) + 5*a^2*(5*A + 7*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/d)/3)/5 )/7)
3.15.76.3.1 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[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[(-C)*Co s[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2))), x] + Simp[1/(b*(m + 2)) Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && !LtQ[m, -1]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f _.)*(x_)]^2), x_Symbol] :> Simp[(-(b*c - a*d))*(A*b^2 - a*b*B + a^2*C)*Cos[ e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b^2*f*(m + 1)*(a^2 - b^2))), x] - S imp[1/(b^2*(m + 1)*(a^2 - b^2)) Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[b*( m + 1)*((b*B - a*C)*(b*c - a*d) - A*b*(a*c - b*d)) + (b*B*(a^2*d + b^2*d*(m + 1) - a*b*c*(m + 2)) + (b*c - a*d)*(A*b^2*(m + 2) + C*(a^2 + b^2*(m + 1)) ))*Sin[e + f*x] - b*C*d*(m + 1)*(a^2 - b^2)*Sin[e + f*x]^2, x], x], x] /; F reeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && LtQ[m, -1]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(c^2*C - B*c*d + A*d^2))*Cos[e + f*x ]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 - d^2))), x] + Simp[1/(d*(n + 1)*(c^2 - d^2)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(b*d*m + a*c*(n + 1)) + (c*C - B* d)*(b*c*m + a*d*(n + 1)) - (d*(A*(a*d*(n + 2) - b*c*(n + 1)) + B*(b*d*(n + 1) - a*c*(n + 2))) - C*(b*c*d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x ] + b*(d*(B*c - A*d)*(m + n + 2) - C*(c^2*(m + 1) + d^2*(n + 1)))*Sin[e + f *x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d , 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 0] && LtQ[n, -1]
Int[(u_)*((c_.)*sec[(a_.) + (b_.)*(x_)])^(m_.), x_Symbol] :> Simp[(c*Sec[a + b*x])^m*(c*Cos[a + b*x])^m Int[ActivateTrig[u]/(c*Cos[a + b*x])^m, x], x] /; FreeQ[{a, b, c, m}, x] && !IntegerQ[m] && KnownSineIntegrandQ[u, x]
Timed out.
hanged
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.16 (sec) , antiderivative size = 478, normalized size of antiderivative = 1.13 \[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {9}{2}}(c+d x) \, dx=-\frac {5 \, \sqrt {2} {\left (i \, {\left (5 \, A + 7 \, C\right )} a^{4} + 28 i \, B a^{3} b + 42 i \, {\left (A + 3 \, C\right )} a^{2} b^{2} + 84 i \, B a b^{3} + 7 i \, {\left (3 \, A + C\right )} b^{4}\right )} \cos \left (d x + c\right )^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 5 \, \sqrt {2} {\left (-i \, {\left (5 \, A + 7 \, C\right )} a^{4} - 28 i \, B a^{3} b - 42 i \, {\left (A + 3 \, C\right )} a^{2} b^{2} - 84 i \, B a b^{3} - 7 i \, {\left (3 \, A + C\right )} b^{4}\right )} \cos \left (d x + c\right )^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 21 \, \sqrt {2} {\left (3 i \, B a^{4} + 4 i \, {\left (3 \, A + 5 \, C\right )} a^{3} b + 30 i \, B a^{2} b^{2} + 20 i \, {\left (A - C\right )} a b^{3} - 5 i \, B b^{4}\right )} \cos \left (d x + c\right )^{3} {\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 ) + 21 \, \sqrt {2} {\left (-3 i \, B a^{4} - 4 i \, {\left (3 \, A + 5 \, C\right )} a^{3} b - 30 i \, B a^{2} b^{2} - 20 i \, {\left (A - C\right )} a b^{3} + 5 i \, B b^{4}\right )} \cos \left (d x + c\right )^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - \frac {2 \, {\left (35 \, C b^{4} \cos \left (d x + c\right )^{4} + 15 \, A a^{4} + 21 \, {\left (3 \, B a^{4} + 4 \, {\left (3 \, A + 5 \, C\right )} a^{3} b + 30 \, B a^{2} b^{2} + 20 \, A a b^{3}\right )} \cos \left (d x + c\right )^{3} + 5 \, {\left ({\left (5 \, A + 7 \, C\right )} a^{4} + 28 \, B a^{3} b + 42 \, A a^{2} b^{2}\right )} \cos \left (d x + c\right )^{2} + 21 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{105 \, d \cos \left (d x + c\right )^{3}} \]
integrate((a+b*cos(d*x+c))^4*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^(9 /2),x, algorithm="fricas")
-1/105*(5*sqrt(2)*(I*(5*A + 7*C)*a^4 + 28*I*B*a^3*b + 42*I*(A + 3*C)*a^2*b ^2 + 84*I*B*a*b^3 + 7*I*(3*A + C)*b^4)*cos(d*x + c)^3*weierstrassPInverse( -4, 0, cos(d*x + c) + I*sin(d*x + c)) + 5*sqrt(2)*(-I*(5*A + 7*C)*a^4 - 28 *I*B*a^3*b - 42*I*(A + 3*C)*a^2*b^2 - 84*I*B*a*b^3 - 7*I*(3*A + C)*b^4)*co s(d*x + c)^3*weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) + 2 1*sqrt(2)*(3*I*B*a^4 + 4*I*(3*A + 5*C)*a^3*b + 30*I*B*a^2*b^2 + 20*I*(A - C)*a*b^3 - 5*I*B*b^4)*cos(d*x + c)^3*weierstrassZeta(-4, 0, weierstrassPIn verse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) + 21*sqrt(2)*(-3*I*B*a^4 - 4* I*(3*A + 5*C)*a^3*b - 30*I*B*a^2*b^2 - 20*I*(A - C)*a*b^3 + 5*I*B*b^4)*cos (d*x + c)^3*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c))) - 2*(35*C*b^4*cos(d*x + c)^4 + 15*A*a^4 + 21*(3*B*a^4 + 4*(3*A + 5*C)*a^3*b + 30*B*a^2*b^2 + 20*A*a*b^3)*cos(d*x + c)^3 + 5*((5* A + 7*C)*a^4 + 28*B*a^3*b + 42*A*a^2*b^2)*cos(d*x + c)^2 + 21*(B*a^4 + 4*A *a^3*b)*cos(d*x + c))*sin(d*x + c)/sqrt(cos(d*x + c)))/(d*cos(d*x + c)^3)
Timed out. \[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {9}{2}}(c+d x) \, dx=\text {Timed out} \]
\[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {9}{2}}(c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{4} \sec \left (d x + c\right )^{\frac {9}{2}} \,d x } \]
integrate((a+b*cos(d*x+c))^4*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^(9 /2),x, algorithm="maxima")
integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)*(b*cos(d*x + c) + a)^4*s ec(d*x + c)^(9/2), x)
\[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {9}{2}}(c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{4} \sec \left (d x + c\right )^{\frac {9}{2}} \,d x } \]
integrate((a+b*cos(d*x+c))^4*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^(9 /2),x, algorithm="giac")
integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)*(b*cos(d*x + c) + a)^4*s ec(d*x + c)^(9/2), x)
Timed out. \[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {9}{2}}(c+d x) \, dx=\int {\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{9/2}\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^4\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right ) \,d x \]