Integrand size = 43, antiderivative size = 383 \[ \int \frac {(a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {9}{2}}(c+d x)} \, dx=-\frac {2 \left (3 a^4 B+30 a^2 b^2 B-5 b^4 B+20 a b^3 (A-C)+4 a^3 b (3 A+5 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 \left (28 a^3 b B+84 a b^3 B+7 b^4 (3 A+C)+42 a^2 b^2 (A+3 C)+a^4 (5 A+7 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 d}+\frac {2 a \left (192 A b^3+63 a^3 B+413 a b^2 B+a^2 (202 A b+350 b C)\right ) \sin (c+d x)}{105 d \sqrt {\cos (c+d x)}}-\frac {2 b^2 \left (98 a b B+b^2 (87 A-35 C)+5 a^2 (5 A+7 C)\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{105 d}+\frac {2 \left (48 A b^2+77 a b B+5 a^2 (5 A+7 C)\right ) (a+b \cos (c+d x))^2 \sin (c+d x)}{105 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 (8 A b+7 a B) (a+b \cos (c+d x))^3 \sin (c+d x)}{35 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 A (a+b \cos (c+d x))^4 \sin (c+d x)}{7 d \cos ^{\frac {7}{2}}(c+d x)} \] Output:
-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))*Ellip ticE(sin(1/2*d*x+1/2*c),2^(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))*InverseJacobiAM(1/2*d*x+1/2*c,2^(1/2) )/d+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)/d/cos(d*x+c)^(1/2)-2/105*b^2*(98*B*a*b+b^2*(87*A-35*C)+5*a^2*(5*A+7*C) )*cos(d*x+c)^(1/2)*sin(d*x+c)/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*sin(d*x+c)/d/cos(d*x+c)^(3/2)+2/35*(8*A*b+7*B*a)*(a+b*c os(d*x+c))^3*sin(d*x+c)/d/cos(d*x+c)^(5/2)+2/7*A*(a+b*cos(d*x+c))^4*sin(d* x+c)/d/cos(d*x+c)^(7/2)
Time = 10.37 (sec) , antiderivative size = 271, normalized size of antiderivative = 0.71 \[ \int \frac {(a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {9}{2}}(c+d x)} \, dx=\frac {-42 \left (3 a^4 B+30 a^2 b^2 B-5 b^4 B+20 a b^3 (A-C)+4 a^3 b (3 A+5 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+10 \left (28 a^3 b B+84 a b^3 B+7 b^4 (3 A+C)+42 a^2 b^2 (A+3 C)+a^4 (5 A+7 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+\frac {14 \left (3 a^3 (4 A b+a B)+3 a \left (20 A b^3+3 a^3 B+30 a b^2 B+4 a^2 b (3 A+5 C)\right ) \cos ^2(c+d x)+5 b^4 C \cos ^3(c+d x)\right ) \sin (c+d x)+5 \left (a^2 \left (42 A b^2+28 a b B+a^2 (5 A+7 C)\right ) \sin (2 (c+d x))+6 a^4 A \tan (c+d x)\right )}{\cos ^{\frac {5}{2}}(c+d x)}}{105 d} \] 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]^(9/2),x]
Output:
(-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] + 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] + (14*(3*a^3*(4*A*b + a*B) + 3*a*(20*A*b^3 + 3*a^3*B + 30*a*b^2*B + 4*a^2 *b*(3*A + 5*C))*Cos[c + d*x]^2 + 5*b^4*C*Cos[c + d*x]^3)*Sin[c + d*x] + 5* (a^2*(42*A*b^2 + 28*a*b*B + a^2*(5*A + 7*C))*Sin[2*(c + d*x)] + 6*a^4*A*Ta n[c + d*x]))/Cos[c + d*x]^(5/2))/(105*d)
Time = 2.65 (sec) , antiderivative size = 390, 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, 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.
| \(\displaystyle \int \frac {(a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {9}{2}}(c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^4 \left (A+B \sin \left (c+d x+\frac {\pi }{2}\right )+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{9/2}}dx\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle \frac {2}{7} \int \frac {(a+b \cos (c+d x))^3 \left (-b (3 A-7 C) \cos ^2(c+d x)+(5 a A+7 b B+7 a C) \cos (c+d x)+8 A b+7 a B\right )}{2 \cos ^{\frac {7}{2}}(c+d x)}dx+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} \int \frac {(a+b \cos (c+d x))^3 \left (-b (3 A-7 C) \cos ^2(c+d x)+(5 a A+7 b B+7 a C) \cos (c+d x)+8 A b+7 a B\right )}{\cos ^{\frac {7}{2}}(c+d x)}dx+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^3 \left (-b (3 A-7 C) \sin \left (c+d x+\frac {\pi }{2}\right )^2+(5 a A+7 b B+7 a C) \sin \left (c+d x+\frac {\pi }{2}\right )+8 A b+7 a B\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{7/2}}dx+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle \frac {1}{7} \left (\frac {2}{5} \int \frac {(a+b \cos (c+d x))^2 \left (5 (5 A+7 C) a^2+77 b B a+48 A b^2-b (39 A b-35 C b+21 a B) \cos ^2(c+d x)+\left (21 B a^2+34 A b a+70 b C a+35 b^2 B\right ) \cos (c+d x)\right )}{2 \cos ^{\frac {5}{2}}(c+d x)}dx+\frac {2 (7 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \int \frac {(a+b \cos (c+d x))^2 \left (5 (5 A+7 C) a^2+77 b B a+48 A b^2-b (39 A b-35 C b+21 a B) \cos ^2(c+d x)+\left (21 B a^2+34 A b a+70 b C a+35 b^2 B\right ) \cos (c+d x)\right )}{\cos ^{\frac {5}{2}}(c+d x)}dx+\frac {2 (7 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^2 \left (5 (5 A+7 C) a^2+77 b B a+48 A b^2-b (39 A b-35 C b+21 a B) \sin \left (c+d x+\frac {\pi }{2}\right )^2+\left (21 B a^2+34 A b a+70 b C a+35 b^2 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{5/2}}dx+\frac {2 (7 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {2}{3} \int \frac {(a+b \cos (c+d x)) \left (63 B a^3+2 b (101 A+175 C) a^2+413 b^2 B a+192 A b^3-3 b \left (5 (5 A+7 C) a^2+98 b B a+b^2 (87 A-35 C)\right ) \cos ^2(c+d x)+\left (5 (5 A+7 C) a^3+77 b B a^2+3 b^2 (11 A+105 C) a+105 b^3 B\right ) \cos (c+d x)\right )}{2 \cos ^{\frac {3}{2}}(c+d x)}dx+\frac {2 \sin (c+d x) \left (5 a^2 (5 A+7 C)+77 a b B+48 A b^2\right ) (a+b \cos (c+d x))^2}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (7 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \int \frac {(a+b \cos (c+d x)) \left (63 B a^3+(202 A b+350 C b) a^2+413 b^2 B a+192 A b^3-3 b \left (5 (5 A+7 C) a^2+98 b B a+b^2 (87 A-35 C)\right ) \cos ^2(c+d x)+\left (5 (5 A+7 C) a^3+77 b B a^2+3 b^2 (11 A+105 C) a+105 b^3 B\right ) \cos (c+d x)\right )}{\cos ^{\frac {3}{2}}(c+d x)}dx+\frac {2 \sin (c+d x) \left (5 a^2 (5 A+7 C)+77 a b B+48 A b^2\right ) (a+b \cos (c+d x))^2}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (7 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right ) \left (63 B a^3+(202 A b+350 C b) a^2+413 b^2 B a+192 A b^3-3 b \left (5 (5 A+7 C) a^2+98 b B a+b^2 (87 A-35 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+\left (5 (5 A+7 C) a^3+77 b B a^2+3 b^2 (11 A+105 C) a+105 b^3 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2}}dx+\frac {2 \sin (c+d x) \left (5 a^2 (5 A+7 C)+77 a b B+48 A b^2\right ) (a+b \cos (c+d x))^2}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (7 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3510 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {2 a \sin (c+d x) \left (63 a^3 B+a^2 (202 A b+350 b C)+413 a b^2 B+192 A b^3\right )}{d \sqrt {\cos (c+d x)}}-2 \int -\frac {5 (5 A+7 C) a^4+140 b B a^3+5 b^2 (47 A+133 C) a^2+518 b^3 B a+192 A b^4-3 b^2 \left (5 (5 A+7 C) a^2+98 b B a+b^2 (87 A-35 C)\right ) \cos ^2(c+d x)-21 \left (3 B a^4+4 b (3 A+5 C) a^3+30 b^2 B a^2+20 b^3 (A-C) a-5 b^4 B\right ) \cos (c+d x)}{2 \sqrt {\cos (c+d x)}}dx\right )+\frac {2 \sin (c+d x) \left (5 a^2 (5 A+7 C)+77 a b B+48 A b^2\right ) (a+b \cos (c+d x))^2}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (7 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\int \frac {5 (5 A+7 C) a^4+140 b B a^3+5 b^2 (47 A+133 C) a^2+518 b^3 B a+192 A b^4-3 b^2 \left (5 (5 A+7 C) a^2+98 b B a+b^2 (87 A-35 C)\right ) \cos ^2(c+d x)-21 \left (3 B a^4+4 b (3 A+5 C) a^3+30 b^2 B a^2+20 b^3 (A-C) a-5 b^4 B\right ) \cos (c+d x)}{\sqrt {\cos (c+d x)}}dx+\frac {2 a \sin (c+d x) \left (63 a^3 B+a^2 (202 A b+350 b C)+413 a b^2 B+192 A b^3\right )}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 \sin (c+d x) \left (5 a^2 (5 A+7 C)+77 a b B+48 A b^2\right ) (a+b \cos (c+d x))^2}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (7 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\int \frac {5 (5 A+7 C) a^4+140 b B a^3+5 b^2 (47 A+133 C) a^2+518 b^3 B a+192 A b^4-3 b^2 \left (5 (5 A+7 C) a^2+98 b B a+b^2 (87 A-35 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2-21 \left (3 B a^4+4 b (3 A+5 C) a^3+30 b^2 B a^2+20 b^3 (A-C) a-5 b^4 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 a \sin (c+d x) \left (63 a^3 B+a^2 (202 A b+350 b C)+413 a b^2 B+192 A b^3\right )}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 \sin (c+d x) \left (5 a^2 (5 A+7 C)+77 a b B+48 A b^2\right ) (a+b \cos (c+d x))^2}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (7 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {2}{3} \int \frac {3 \left (5 \left ((5 A+7 C) a^4+28 b B a^3+42 b^2 (A+3 C) a^2+84 b^3 B a+7 b^4 (3 A+C)\right )-21 \left (3 B a^4+4 b (3 A+5 C) a^3+30 b^2 B a^2+20 b^3 (A-C) a-5 b^4 B\right ) \cos (c+d x)\right )}{2 \sqrt {\cos (c+d x)}}dx-\frac {2 b^2 \sin (c+d x) \sqrt {\cos (c+d x)} \left (5 a^2 (5 A+7 C)+98 a b B+b^2 (87 A-35 C)\right )}{d}+\frac {2 a \sin (c+d x) \left (63 a^3 B+a^2 (202 A b+350 b C)+413 a b^2 B+192 A b^3\right )}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 \sin (c+d x) \left (5 a^2 (5 A+7 C)+77 a b B+48 A b^2\right ) (a+b \cos (c+d x))^2}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (7 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\int \frac {5 \left ((5 A+7 C) a^4+28 b B a^3+42 b^2 (A+3 C) a^2+84 b^3 B a+7 b^4 (3 A+C)\right )-21 \left (3 B a^4+4 b (3 A+5 C) a^3+30 b^2 B a^2+20 b^3 (A-C) a-5 b^4 B\right ) \cos (c+d x)}{\sqrt {\cos (c+d x)}}dx-\frac {2 b^2 \sin (c+d x) \sqrt {\cos (c+d x)} \left (5 a^2 (5 A+7 C)+98 a b B+b^2 (87 A-35 C)\right )}{d}+\frac {2 a \sin (c+d x) \left (63 a^3 B+a^2 (202 A b+350 b C)+413 a b^2 B+192 A b^3\right )}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 \sin (c+d x) \left (5 a^2 (5 A+7 C)+77 a b B+48 A b^2\right ) (a+b \cos (c+d x))^2}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (7 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\int \frac {5 \left ((5 A+7 C) a^4+28 b B a^3+42 b^2 (A+3 C) a^2+84 b^3 B a+7 b^4 (3 A+C)\right )-21 \left (3 B a^4+4 b (3 A+5 C) a^3+30 b^2 B a^2+20 b^3 (A-C) a-5 b^4 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {2 b^2 \sin (c+d x) \sqrt {\cos (c+d x)} \left (5 a^2 (5 A+7 C)+98 a b B+b^2 (87 A-35 C)\right )}{d}+\frac {2 a \sin (c+d x) \left (63 a^3 B+a^2 (202 A b+350 b C)+413 a b^2 B+192 A b^3\right )}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 \sin (c+d x) \left (5 a^2 (5 A+7 C)+77 a b B+48 A b^2\right ) (a+b \cos (c+d x))^2}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (7 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (5 \left (a^4 (5 A+7 C)+28 a^3 b B+42 a^2 b^2 (A+3 C)+84 a b^3 B+7 b^4 (3 A+C)\right ) \int \frac {1}{\sqrt {\cos (c+d x)}}dx-21 \left (3 a^4 B+4 a^3 b (3 A+5 C)+30 a^2 b^2 B+20 a b^3 (A-C)-5 b^4 B\right ) \int \sqrt {\cos (c+d x)}dx-\frac {2 b^2 \sin (c+d x) \sqrt {\cos (c+d x)} \left (5 a^2 (5 A+7 C)+98 a b B+b^2 (87 A-35 C)\right )}{d}+\frac {2 a \sin (c+d x) \left (63 a^3 B+a^2 (202 A b+350 b C)+413 a b^2 B+192 A b^3\right )}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 \sin (c+d x) \left (5 a^2 (5 A+7 C)+77 a b B+48 A b^2\right ) (a+b \cos (c+d x))^2}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (7 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (5 \left (a^4 (5 A+7 C)+28 a^3 b B+42 a^2 b^2 (A+3 C)+84 a b^3 B+7 b^4 (3 A+C)\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx-21 \left (3 a^4 B+4 a^3 b (3 A+5 C)+30 a^2 b^2 B+20 a b^3 (A-C)-5 b^4 B\right ) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx-\frac {2 b^2 \sin (c+d x) \sqrt {\cos (c+d x)} \left (5 a^2 (5 A+7 C)+98 a b B+b^2 (87 A-35 C)\right )}{d}+\frac {2 a \sin (c+d x) \left (63 a^3 B+a^2 (202 A b+350 b C)+413 a b^2 B+192 A b^3\right )}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 \sin (c+d x) \left (5 a^2 (5 A+7 C)+77 a b B+48 A b^2\right ) (a+b \cos (c+d x))^2}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (7 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (5 \left (a^4 (5 A+7 C)+28 a^3 b B+42 a^2 b^2 (A+3 C)+84 a b^3 B+7 b^4 (3 A+C)\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {2 b^2 \sin (c+d x) \sqrt {\cos (c+d x)} \left (5 a^2 (5 A+7 C)+98 a b B+b^2 (87 A-35 C)\right )}{d}+\frac {2 a \sin (c+d x) \left (63 a^3 B+a^2 (202 A b+350 b C)+413 a b^2 B+192 A b^3\right )}{d \sqrt {\cos (c+d x)}}-\frac {42 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (3 a^4 B+4 a^3 b (3 A+5 C)+30 a^2 b^2 B+20 a b^3 (A-C)-5 b^4 B\right )}{d}\right )+\frac {2 \sin (c+d x) \left (5 a^2 (5 A+7 C)+77 a b B+48 A b^2\right ) (a+b \cos (c+d x))^2}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (7 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {2 \sin (c+d x) \left (5 a^2 (5 A+7 C)+77 a b B+48 A b^2\right ) (a+b \cos (c+d x))^2}{3 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {1}{3} \left (-\frac {2 b^2 \sin (c+d x) \sqrt {\cos (c+d x)} \left (5 a^2 (5 A+7 C)+98 a b B+b^2 (87 A-35 C)\right )}{d}+\frac {2 a \sin (c+d x) \left (63 a^3 B+a^2 (202 A b+350 b C)+413 a b^2 B+192 A b^3\right )}{d \sqrt {\cos (c+d x)}}+\frac {10 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (a^4 (5 A+7 C)+28 a^3 b B+42 a^2 b^2 (A+3 C)+84 a b^3 B+7 b^4 (3 A+C)\right )}{d}-\frac {42 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (3 a^4 B+4 a^3 b (3 A+5 C)+30 a^2 b^2 B+20 a b^3 (A-C)-5 b^4 B\right )}{d}\right )\right )+\frac {2 (7 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\) |
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]^(9/2),x]
Output:
(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
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]
Leaf count of result is larger than twice the leaf count of optimal. \(1422\) vs. \(2(362)=724\).
Time = 14.62 (sec) , antiderivative size = 1423, normalized size of antiderivative = 3.72
| method | result | size |
| parts | \(\text {Expression too large to display}\) | \(1423\) |
| default | \(\text {Expression too large to display}\) | \(1593\) |
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)^(9/2),x, method=_RETURNVERBOSE)
Output:
-2/5*(4*A*a^3*b+B*a^4)*(-(-2*cos(1/2*d*x+1/2*c)^2+1)*sin(1/2*d*x+1/2*c)^2) ^(1/2)/(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)^3*(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(c os(1/2*d*x+1/2*c),2^(1/2))*sin(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)*EllipticE(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)/(-1+2*cos(1/2*d*x+1/2*c)^2)^(1/ 2)/d+2*(B*b^4+4*C*a*b^3)*((-1+2*cos(1/2*d*x+1/2*c)^2)*sin(1/2*d*x+1/2*c)^2 )^(1/2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)*Ell ipticE(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)/sin(1/2*d*x+1/2*c)/(-1+2*cos(1/2*d*x+1/2*c)^2)^(1/2)/d+2*(A* b^4+4*B*a*b^3+6*C*a^2*b^2)/d*InverseJacobiAM(1/2*d*x+1/2*c,2^(1/2))-2*(4*A *a*b^3+6*B*a^2*b^2+4*C*a^3*b)*(-2*(-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)*sin(1/2*d*x+1/2*c)^2+(sin(1/2*d*x+1/2*c)^2 )^(1/2)*(2*sin(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)*EllipticE(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)/sin(1/2*d*x+1/2*c)/(-1+2*cos(1/2*...
Result contains complex when optimal does not.
Time = 0.17 (sec) , antiderivative size = 478, normalized size of antiderivative = 1.25 \[ \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 {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 )^{4} {\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 )^{4} {\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 )^{4} {\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 )^{4} {\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 (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 )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )}{105 \, d \cos \left (d x + c\right )^{4}} \] 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)^(9 /2),x, algorithm="fricas")
Output:
-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)^4*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)^4*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)^4*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)^4*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))*sqrt(cos(d*x + c))*sin(d*x + c))/(d*cos(d*x + c)^4)
Timed out. \[ \int \frac {(a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {9}{2}}(c+d x)} \, dx=\text {Timed out} \] 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)* *(9/2),x)
Output:
Timed out
\[ \int \frac {(a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {9}{2}}(c+d x)} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{4}}{\cos \left (d x + c\right )^{\frac {9}{2}}} \,d x } \] 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)^(9 /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)^(9/2), x)
\[ \int \frac {(a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {9}{2}}(c+d x)} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{4}}{\cos \left (d x + c\right )^{\frac {9}{2}}} \,d x } \] 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)^(9 /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)^(9/2), x)
Time = 4.16 (sec) , antiderivative size = 559, normalized size of antiderivative = 1.46 \[ \int \frac {(a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {9}{2}}(c+d x)} \, dx =\text {Too large to display} \] 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)^(9/2),x)
Output:
(2*(C*b^4*ellipticF(c/2 + (d*x)/2, 2) + 12*C*a*b^3*ellipticE(c/2 + (d*x)/2 , 2) + C*b^4*cos(c + d*x)^(1/2)*sin(c + d*x) + 18*C*a^2*b^2*ellipticF(c/2 + (d*x)/2, 2)))/(3*d) + (2*A*b^4*ellipticF(c/2 + (d*x)/2, 2))/d + (2*B*b^4 *ellipticE(c/2 + (d*x)/2, 2))/d + (8*B*a*b^3*ellipticF(c/2 + (d*x)/2, 2))/ d + (2*A*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*B*a^4*sin(c + d*x)*hype rgeom([-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)) + (2*C*a^4*sin(c + d*x)*hypergeom([-3/4, 1/2], 1/4, cos(c + d*x)^2))/(3*d*cos(c + d*x)^(3/2)*(sin(c + d*x)^2)^(1/2)) + (8*A*a*b^3*si n(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*A*a^3*b*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)) + (8*B*a^3*b*sin(c + d*x)*hypergeom([-3/4, 1/2], 1/4, cos(c + d*x)^2))/(3* d*cos(c + d*x)^(3/2)*(sin(c + d*x)^2)^(1/2)) + (8*C*a^3*b*sin(c + d*x)*hyp ergeom([-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)) + (4*A*a^2*b^2*sin(c + d*x)*hypergeom([-3/4, 1/2], 1/4, cos( c + d*x)^2))/(d*cos(c + d*x)^(3/2)*(sin(c + d*x)^2)^(1/2)) + (12*B*a^2*b^2 *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))
\[ \int \frac {(a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {9}{2}}(c+d x)} \, dx=6 \left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )}d x \right ) a^{2} b^{2} c +5 \left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )}d x \right ) a \,b^{4}+\left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{5}}d x \right ) a^{5}+5 \left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{4}}d x \right ) a^{4} b +\left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{3}}d x \right ) a^{4} c +10 \left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{3}}d x \right ) a^{3} b^{2}+4 \left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{2}}d x \right ) a^{3} b c +10 \left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{2}}d x \right ) a^{2} b^{3}+4 \left (\int \sqrt {\cos \left (d x +c \right )}d x \right ) a \,b^{3} c +\left (\int \sqrt {\cos \left (d x +c \right )}d x \right ) b^{5}+\left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )d x \right ) b^{4} c \] 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)^(9/2),x)
Output:
6*int(sqrt(cos(c + d*x))/cos(c + d*x),x)*a**2*b**2*c + 5*int(sqrt(cos(c + d*x))/cos(c + d*x),x)*a*b**4 + int(sqrt(cos(c + d*x))/cos(c + d*x)**5,x)*a **5 + 5*int(sqrt(cos(c + d*x))/cos(c + d*x)**4,x)*a**4*b + int(sqrt(cos(c + d*x))/cos(c + d*x)**3,x)*a**4*c + 10*int(sqrt(cos(c + d*x))/cos(c + d*x) **3,x)*a**3*b**2 + 4*int(sqrt(cos(c + d*x))/cos(c + d*x)**2,x)*a**3*b*c + 10*int(sqrt(cos(c + d*x))/cos(c + d*x)**2,x)*a**2*b**3 + 4*int(sqrt(cos(c + d*x)),x)*a*b**3*c + int(sqrt(cos(c + d*x)),x)*b**5 + int(sqrt(cos(c + d* x))*cos(c + d*x),x)*b**4*c