Integrand size = 43, antiderivative size = 373 \[ \int \frac {(a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {5}{2}}(c+d x)} \, dx=-\frac {2 \left (5 a^4 B-30 a^2 b^2 B-3 b^4 B+20 a^3 b (A-C)-4 a b^3 (5 A+3 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 \left (84 a^3 b B+28 a b^3 B+42 a^2 b^2 (3 A+C)+7 a^4 (A+3 C)+b^4 (7 A+5 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 d}-\frac {2 b \left (42 a^3 B-28 a b^2 B+3 a^2 b (49 A-13 C)-b^3 (7 A+5 C)\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{21 d}-\frac {2 b^2 \left (350 a A b+105 a^2 B-21 b^2 B-54 a b C\right ) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{105 d}-\frac {2 b (21 A b+7 a B-b C) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2 \sin (c+d x)}{7 d}+\frac {2 (8 A b+3 a B) (a+b \cos (c+d x))^3 \sin (c+d x)}{3 d \sqrt {\cos (c+d x)}}+\frac {2 A (a+b \cos (c+d x))^4 \sin (c+d x)}{3 d \cos ^{\frac {3}{2}}(c+d x)} \] Output:
-2/5*(5*B*a^4-30*B*a^2*b^2-3*B*b^4+20*a^3*b*(A-C)-4*a*b^3*(5*A+3*C))*Ellip ticE(sin(1/2*d*x+1/2*c),2^(1/2))/d+2/21*(84*B*a^3*b+28*B*a*b^3+42*a^2*b^2* (3*A+C)+7*a^4*(A+3*C)+b^4*(7*A+5*C))*InverseJacobiAM(1/2*d*x+1/2*c,2^(1/2) )/d-2/21*b*(42*B*a^3-28*B*a*b^2+3*a^2*b*(49*A-13*C)-b^3*(7*A+5*C))*cos(d*x +c)^(1/2)*sin(d*x+c)/d-2/105*b^2*(350*A*a*b+105*B*a^2-21*B*b^2-54*C*a*b)*c os(d*x+c)^(3/2)*sin(d*x+c)/d-2/7*b*(21*A*b+7*B*a-C*b)*cos(d*x+c)^(1/2)*(a+ b*cos(d*x+c))^2*sin(d*x+c)/d+2/3*(8*A*b+3*B*a)*(a+b*cos(d*x+c))^3*sin(d*x+ c)/d/cos(d*x+c)^(1/2)+2/3*A*(a+b*cos(d*x+c))^4*sin(d*x+c)/d/cos(d*x+c)^(3/ 2)
Time = 7.39 (sec) , antiderivative size = 257, normalized size of antiderivative = 0.69 \[ \int \frac {(a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {5}{2}}(c+d x)} \, dx=\frac {-42 \left (5 a^4 B-30 a^2 b^2 B-3 b^4 B+20 a^3 b (A-C)-4 a b^3 (5 A+3 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+10 \left (84 a^3 b B+28 a b^3 B+42 a^2 b^2 (3 A+C)+7 a^4 (A+3 C)+b^4 (7 A+5 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+\frac {168 \left (5 a^3 (4 A b+a B)+b^3 (b B+4 a C) \cos ^2(c+d x)\right ) \sin (c+d x)+5 \left (b^2 \left (28 A b^2+112 a b B+168 a^2 C+23 b^2 C\right ) \sin (2 (c+d x))+6 b^4 C \cos (c+d x) \sin (3 (c+d x))+56 a^4 A \tan (c+d x)\right )}{4 \sqrt {\cos (c+d x)}}}{105 d} \] 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]^(5/2),x]
Output:
(-42*(5*a^4*B - 30*a^2*b^2*B - 3*b^4*B + 20*a^3*b*(A - C) - 4*a*b^3*(5*A + 3*C))*EllipticE[(c + d*x)/2, 2] + 10*(84*a^3*b*B + 28*a*b^3*B + 42*a^2*b^ 2*(3*A + C) + 7*a^4*(A + 3*C) + b^4*(7*A + 5*C))*EllipticF[(c + d*x)/2, 2] + (168*(5*a^3*(4*A*b + a*B) + b^3*(b*B + 4*a*C)*Cos[c + d*x]^2)*Sin[c + d *x] + 5*(b^2*(28*A*b^2 + 112*a*b*B + 168*a^2*C + 23*b^2*C)*Sin[2*(c + d*x) ] + 6*b^4*C*Cos[c + d*x]*Sin[3*(c + d*x)] + 56*a^4*A*Tan[c + d*x]))/(4*Sqr t[Cos[c + d*x]]))/(105*d)
Time = 2.53 (sec) , antiderivative size = 380, 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, 3528, 27, 3042, 3512, 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.
| \(\displaystyle \int \frac {(a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {5}{2}}(c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^4 \left (A+B \sin \left (c+d x+\frac {\pi }{2}\right )+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{5/2}}dx\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle \frac {2}{3} \int \frac {(a+b \cos (c+d x))^3 \left (-b (7 A-3 C) \cos ^2(c+d x)+(3 b B+a (A+3 C)) \cos (c+d x)+8 A b+3 a B\right )}{2 \cos ^{\frac {3}{2}}(c+d x)}dx+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{3 d \cos ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} \int \frac {(a+b \cos (c+d x))^3 \left (-b (7 A-3 C) \cos ^2(c+d x)+(3 b B+a (A+3 C)) \cos (c+d x)+8 A b+3 a B\right )}{\cos ^{\frac {3}{2}}(c+d x)}dx+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{3 d \cos ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^3 \left (-b (7 A-3 C) \sin \left (c+d x+\frac {\pi }{2}\right )^2+(3 b B+a (A+3 C)) \sin \left (c+d x+\frac {\pi }{2}\right )+8 A b+3 a B\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2}}dx+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{3 d \cos ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle \frac {1}{3} \left (2 \int \frac {(a+b \cos (c+d x))^2 \left ((A+3 C) a^2+21 b B a+48 A b^2-3 b (21 A b-C b+7 a B) \cos ^2(c+d x)-\left (3 B a^2+14 A b a-6 b C a-3 b^2 B\right ) \cos (c+d x)\right )}{2 \sqrt {\cos (c+d x)}}dx+\frac {2 (3 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{3 d \cos ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} \left (\int \frac {(a+b \cos (c+d x))^2 \left ((A+3 C) a^2+21 b B a+48 A b^2-3 b (21 A b-C b+7 a B) \cos ^2(c+d x)-\left (3 B a^2+14 A b a-6 b C a-3 b^2 B\right ) \cos (c+d x)\right )}{\sqrt {\cos (c+d x)}}dx+\frac {2 (3 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{3 d \cos ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \left (\int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^2 \left ((A+3 C) a^2+21 b B a+48 A b^2-3 b (21 A b-C b+7 a B) \sin \left (c+d x+\frac {\pi }{2}\right )^2+\left (-3 B a^2-14 A b a+6 b C a+3 b^2 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 (3 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{3 d \cos ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \frac {1}{3} \left (\frac {2}{7} \int \frac {(a+b \cos (c+d x)) \left (-b \left (105 B a^2+350 A b a-54 b C a-21 b^2 B\right ) \cos ^2(c+d x)-\left (21 B a^3+(91 A b-63 b C) a^2-63 b^2 B a-3 b^3 (7 A+5 C)\right ) \cos (c+d x)+a \left (7 (A+3 C) a^2+126 b B a+3 b^2 (91 A+C)\right )\right )}{2 \sqrt {\cos (c+d x)}}dx-\frac {6 b \sin (c+d x) \sqrt {\cos (c+d x)} (7 a B+21 A b-b C) (a+b \cos (c+d x))^2}{7 d}+\frac {2 (3 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{3 d \cos ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{7} \int \frac {(a+b \cos (c+d x)) \left (-b \left (105 B a^2+350 A b a-54 b C a-21 b^2 B\right ) \cos ^2(c+d x)-\left (21 B a^3+(91 A b-63 b C) a^2-63 b^2 B a-3 b^3 (7 A+5 C)\right ) \cos (c+d x)+a \left (7 (A+3 C) a^2+126 b B a+3 b^2 (91 A+C)\right )\right )}{\sqrt {\cos (c+d x)}}dx-\frac {6 b \sin (c+d x) \sqrt {\cos (c+d x)} (7 a B+21 A b-b C) (a+b \cos (c+d x))^2}{7 d}+\frac {2 (3 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{3 d \cos ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{7} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right ) \left (-b \left (105 B a^2+350 A b a-54 b C a-21 b^2 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+\left (-21 B a^3-(91 A b-63 b C) a^2+63 b^2 B a+3 b^3 (7 A+5 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+a \left (7 (A+3 C) a^2+126 b B a+3 b^2 (91 A+C)\right )\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {6 b \sin (c+d x) \sqrt {\cos (c+d x)} (7 a B+21 A b-b C) (a+b \cos (c+d x))^2}{7 d}+\frac {2 (3 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{3 d \cos ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3512 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{7} \left (\frac {2}{5} \int \frac {5 \left (7 (A+3 C) a^2+126 b B a+3 b^2 (91 A+C)\right ) a^2-15 b \left (42 B a^3+3 b (49 A-13 C) a^2-28 b^2 B a-b^3 (7 A+5 C)\right ) \cos ^2(c+d x)-21 \left (5 B a^4+20 b (A-C) a^3-30 b^2 B a^2-4 b^3 (5 A+3 C) a-3 b^4 B\right ) \cos (c+d x)}{2 \sqrt {\cos (c+d x)}}dx-\frac {2 b^2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (105 a^2 B+350 a A b-54 a b C-21 b^2 B\right )}{5 d}\right )-\frac {6 b \sin (c+d x) \sqrt {\cos (c+d x)} (7 a B+21 A b-b C) (a+b \cos (c+d x))^2}{7 d}+\frac {2 (3 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{3 d \cos ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{7} \left (\frac {1}{5} \int \frac {5 \left (7 (A+3 C) a^2+126 b B a+3 b^2 (91 A+C)\right ) a^2-15 b \left (42 B a^3+3 b (49 A-13 C) a^2-28 b^2 B a-b^3 (7 A+5 C)\right ) \cos ^2(c+d x)-21 \left (5 B a^4+20 b (A-C) a^3-30 b^2 B a^2-4 b^3 (5 A+3 C) a-3 b^4 B\right ) \cos (c+d x)}{\sqrt {\cos (c+d x)}}dx-\frac {2 b^2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (105 a^2 B+350 a A b-54 a b C-21 b^2 B\right )}{5 d}\right )-\frac {6 b \sin (c+d x) \sqrt {\cos (c+d x)} (7 a B+21 A b-b C) (a+b \cos (c+d x))^2}{7 d}+\frac {2 (3 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{3 d \cos ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{7} \left (\frac {1}{5} \int \frac {5 \left (7 (A+3 C) a^2+126 b B a+3 b^2 (91 A+C)\right ) a^2-15 b \left (42 B a^3+3 b (49 A-13 C) a^2-28 b^2 B a-b^3 (7 A+5 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2-21 \left (5 B a^4+20 b (A-C) a^3-30 b^2 B a^2-4 b^3 (5 A+3 C) a-3 b^4 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {2 b^2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (105 a^2 B+350 a A b-54 a b C-21 b^2 B\right )}{5 d}\right )-\frac {6 b \sin (c+d x) \sqrt {\cos (c+d x)} (7 a B+21 A b-b C) (a+b \cos (c+d x))^2}{7 d}+\frac {2 (3 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{3 d \cos ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {2}{3} \int \frac {3 \left (5 \left (7 (A+3 C) a^4+84 b B a^3+42 b^2 (3 A+C) a^2+28 b^3 B a+b^4 (7 A+5 C)\right )-21 \left (5 B a^4+20 b (A-C) a^3-30 b^2 B a^2-4 b^3 (5 A+3 C) a-3 b^4 B\right ) \cos (c+d x)\right )}{2 \sqrt {\cos (c+d x)}}dx-\frac {10 b \sin (c+d x) \sqrt {\cos (c+d x)} \left (42 a^3 B+3 a^2 b (49 A-13 C)-28 a b^2 B-b^3 (7 A+5 C)\right )}{d}\right )-\frac {2 b^2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (105 a^2 B+350 a A b-54 a b C-21 b^2 B\right )}{5 d}\right )-\frac {6 b \sin (c+d x) \sqrt {\cos (c+d x)} (7 a B+21 A b-b C) (a+b \cos (c+d x))^2}{7 d}+\frac {2 (3 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{3 d \cos ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{7} \left (\frac {1}{5} \left (\int \frac {5 \left (7 (A+3 C) a^4+84 b B a^3+42 b^2 (3 A+C) a^2+28 b^3 B a+b^4 (7 A+5 C)\right )-21 \left (5 B a^4+20 b (A-C) a^3-30 b^2 B a^2-4 b^3 (5 A+3 C) a-3 b^4 B\right ) \cos (c+d x)}{\sqrt {\cos (c+d x)}}dx-\frac {10 b \sin (c+d x) \sqrt {\cos (c+d x)} \left (42 a^3 B+3 a^2 b (49 A-13 C)-28 a b^2 B-b^3 (7 A+5 C)\right )}{d}\right )-\frac {2 b^2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (105 a^2 B+350 a A b-54 a b C-21 b^2 B\right )}{5 d}\right )-\frac {6 b \sin (c+d x) \sqrt {\cos (c+d x)} (7 a B+21 A b-b C) (a+b \cos (c+d x))^2}{7 d}+\frac {2 (3 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{3 d \cos ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{7} \left (\frac {1}{5} \left (\int \frac {5 \left (7 (A+3 C) a^4+84 b B a^3+42 b^2 (3 A+C) a^2+28 b^3 B a+b^4 (7 A+5 C)\right )-21 \left (5 B a^4+20 b (A-C) a^3-30 b^2 B a^2-4 b^3 (5 A+3 C) a-3 b^4 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {10 b \sin (c+d x) \sqrt {\cos (c+d x)} \left (42 a^3 B+3 a^2 b (49 A-13 C)-28 a b^2 B-b^3 (7 A+5 C)\right )}{d}\right )-\frac {2 b^2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (105 a^2 B+350 a A b-54 a b C-21 b^2 B\right )}{5 d}\right )-\frac {6 b \sin (c+d x) \sqrt {\cos (c+d x)} (7 a B+21 A b-b C) (a+b \cos (c+d x))^2}{7 d}+\frac {2 (3 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{3 d \cos ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{7} \left (\frac {1}{5} \left (5 \left (7 a^4 (A+3 C)+84 a^3 b B+42 a^2 b^2 (3 A+C)+28 a b^3 B+b^4 (7 A+5 C)\right ) \int \frac {1}{\sqrt {\cos (c+d x)}}dx-21 \left (5 a^4 B+20 a^3 b (A-C)-30 a^2 b^2 B-4 a b^3 (5 A+3 C)-3 b^4 B\right ) \int \sqrt {\cos (c+d x)}dx-\frac {10 b \sin (c+d x) \sqrt {\cos (c+d x)} \left (42 a^3 B+3 a^2 b (49 A-13 C)-28 a b^2 B-b^3 (7 A+5 C)\right )}{d}\right )-\frac {2 b^2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (105 a^2 B+350 a A b-54 a b C-21 b^2 B\right )}{5 d}\right )-\frac {6 b \sin (c+d x) \sqrt {\cos (c+d x)} (7 a B+21 A b-b C) (a+b \cos (c+d x))^2}{7 d}+\frac {2 (3 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{3 d \cos ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{7} \left (\frac {1}{5} \left (5 \left (7 a^4 (A+3 C)+84 a^3 b B+42 a^2 b^2 (3 A+C)+28 a b^3 B+b^4 (7 A+5 C)\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx-21 \left (5 a^4 B+20 a^3 b (A-C)-30 a^2 b^2 B-4 a b^3 (5 A+3 C)-3 b^4 B\right ) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx-\frac {10 b \sin (c+d x) \sqrt {\cos (c+d x)} \left (42 a^3 B+3 a^2 b (49 A-13 C)-28 a b^2 B-b^3 (7 A+5 C)\right )}{d}\right )-\frac {2 b^2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (105 a^2 B+350 a A b-54 a b C-21 b^2 B\right )}{5 d}\right )-\frac {6 b \sin (c+d x) \sqrt {\cos (c+d x)} (7 a B+21 A b-b C) (a+b \cos (c+d x))^2}{7 d}+\frac {2 (3 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{3 d \cos ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{7} \left (\frac {1}{5} \left (5 \left (7 a^4 (A+3 C)+84 a^3 b B+42 a^2 b^2 (3 A+C)+28 a b^3 B+b^4 (7 A+5 C)\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {10 b \sin (c+d x) \sqrt {\cos (c+d x)} \left (42 a^3 B+3 a^2 b (49 A-13 C)-28 a b^2 B-b^3 (7 A+5 C)\right )}{d}-\frac {42 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (5 a^4 B+20 a^3 b (A-C)-30 a^2 b^2 B-4 a b^3 (5 A+3 C)-3 b^4 B\right )}{d}\right )-\frac {2 b^2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (105 a^2 B+350 a A b-54 a b C-21 b^2 B\right )}{5 d}\right )-\frac {6 b \sin (c+d x) \sqrt {\cos (c+d x)} (7 a B+21 A b-b C) (a+b \cos (c+d x))^2}{7 d}+\frac {2 (3 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{3 d \cos ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{7} \left (\frac {1}{5} \left (-\frac {10 b \sin (c+d x) \sqrt {\cos (c+d x)} \left (42 a^3 B+3 a^2 b (49 A-13 C)-28 a b^2 B-b^3 (7 A+5 C)\right )}{d}+\frac {10 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (7 a^4 (A+3 C)+84 a^3 b B+42 a^2 b^2 (3 A+C)+28 a b^3 B+b^4 (7 A+5 C)\right )}{d}-\frac {42 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (5 a^4 B+20 a^3 b (A-C)-30 a^2 b^2 B-4 a b^3 (5 A+3 C)-3 b^4 B\right )}{d}\right )-\frac {2 b^2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (105 a^2 B+350 a A b-54 a b C-21 b^2 B\right )}{5 d}\right )-\frac {6 b \sin (c+d x) \sqrt {\cos (c+d x)} (7 a B+21 A b-b C) (a+b \cos (c+d x))^2}{7 d}+\frac {2 (3 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{3 d \cos ^{\frac {3}{2}}(c+d x)}\) |
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]^(5/2),x]
Output:
(2*A*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + ((-6* b*(21*A*b + 7*a*B - b*C)*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(7*d) + (2*(8*A*b + 3*a*B)*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(d* Sqrt[Cos[c + d*x]]) + ((-2*b^2*(350*a*A*b + 105*a^2*B - 21*b^2*B - 54*a*b* C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d) + ((-42*(5*a^4*B - 30*a^2*b^2*B - 3*b^4*B + 20*a^3*b*(A - C) - 4*a*b^3*(5*A + 3*C))*EllipticE[(c + d*x)/2, 2])/d + (10*(84*a^3*b*B + 28*a*b^3*B + 42*a^2*b^2*(3*A + C) + 7*a^4*(A + 3*C) + b^4*(7*A + 5*C))*EllipticF[(c + d*x)/2, 2])/d - (10*b*(42*a^3*B - 2 8*a*b^2*B + 3*a^2*b*(49*A - 13*C) - b^3*(7*A + 5*C))*Sqrt[Cos[c + d*x]]*Si n[c + d*x])/d)/5)/7)/3
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[(-C)*d*Cos[e + f*x]*Sin[e + f*x]*((a + b*Si n[e + f*x])^(m + 1)/(b*f*(m + 3))), x] + Simp[1/(b*(m + 3)) Int[(a + b*Si n[e + f*x])^m*Simp[a*C*d + A*b*c*(m + 3) + b*(B*c*(m + 3) + d*(C*(m + 2) + A*(m + 3)))*Sin[e + f*x] - (2*a*C*d - b*(c*C + B*d)*(m + 3))*Sin[e + f*x]^2 , x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m}, 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_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_ .) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Cos[e + f*x]*(a + b*Sin[e + f*x ])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n + 2))), x] + Simp[1/(d*(m + n + 2)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A* d*(m + n + 2) + C*(b*c*m + a*d*(n + 1)) + (d*(A*b + a*B)*(m + n + 2) - C*(a *c - b*d*(m + n + 1)))*Sin[e + f*x] + (C*(a*d*m - b*c*(m + 1)) + b*B*d*(m + n + 2))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n} , x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[ m, 0] && !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 0])))
Leaf count of result is larger than twice the leaf count of optimal. \(1222\) vs. \(2(352)=704\).
Time = 14.85 (sec) , antiderivative size = 1223, normalized size of antiderivative = 3.28
| method | result | size |
| parts | \(\text {Expression too large to display}\) | \(1223\) |
| default | \(\text {Expression too large to display}\) | \(2507\) |
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)^(5/2),x, method=_RETURNVERBOSE)
Output:
-2*(4*A*a^3*b+B*a^4)*(-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*d*x+1/2*c)^2 )^(1/2)/d-2/5*(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)*(-8*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^6+8*cos(1/2*d*x+1 /2*c)*sin(1/2*d*x+1/2*c)^4-2*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^2-3*(si n(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)/sin(1/2*d*x+1/2*c)/(-1+2*cos(1/2*d*x+1/2*c)^2)^(1/2)/d-2/3*(A*b^4+4*B*a *b^3+6*C*a^2*b^2)*((-1+2*cos(1/2*d*x+1/2*c)^2)*sin(1/2*d*x+1/2*c)^2)^(1/2) *(4*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^4-2*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)* EllipticF(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 *(4*A*a*b^3+6*B*a^2*b^2+4*C*a^3*b)*((-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)*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*d*x+1/2*c)^2)^...
Result contains complex when optimal does not.
Time = 0.16 (sec) , antiderivative size = 455, normalized size of antiderivative = 1.22 \[ \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 {5}{2}}(c+d x)} \, dx=-\frac {5 \, \sqrt {2} {\left (7 i \, {\left (A + 3 \, C\right )} a^{4} + 84 i \, B a^{3} b + 42 i \, {\left (3 \, A + C\right )} a^{2} b^{2} + 28 i \, B a b^{3} + i \, {\left (7 \, A + 5 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{2} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 5 \, \sqrt {2} {\left (-7 i \, {\left (A + 3 \, C\right )} a^{4} - 84 i \, B a^{3} b - 42 i \, {\left (3 \, A + C\right )} a^{2} b^{2} - 28 i \, B a b^{3} - i \, {\left (7 \, A + 5 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{2} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 21 \, \sqrt {2} {\left (5 i \, B a^{4} + 20 i \, {\left (A - C\right )} a^{3} b - 30 i \, B a^{2} b^{2} - 4 i \, {\left (5 \, A + 3 \, C\right )} a b^{3} - 3 i \, B b^{4}\right )} \cos \left (d x + c\right )^{2} {\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 (-5 i \, B a^{4} - 20 i \, {\left (A - C\right )} a^{3} b + 30 i \, B a^{2} b^{2} + 4 i \, {\left (5 \, A + 3 \, C\right )} a b^{3} + 3 i \, B b^{4}\right )} \cos \left (d x + c\right )^{2} {\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 (15 \, C b^{4} \cos \left (d x + c\right )^{4} + 35 \, A a^{4} + 21 \, {\left (4 \, C a b^{3} + B b^{4}\right )} \cos \left (d x + c\right )^{3} + 5 \, {\left (42 \, C a^{2} b^{2} + 28 \, B a b^{3} + {\left (7 \, A + 5 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{2} + 105 \, {\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 )^{2}} \] 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)^(5 /2),x, algorithm="fricas")
Output:
-1/105*(5*sqrt(2)*(7*I*(A + 3*C)*a^4 + 84*I*B*a^3*b + 42*I*(3*A + C)*a^2*b ^2 + 28*I*B*a*b^3 + I*(7*A + 5*C)*b^4)*cos(d*x + c)^2*weierstrassPInverse( -4, 0, cos(d*x + c) + I*sin(d*x + c)) + 5*sqrt(2)*(-7*I*(A + 3*C)*a^4 - 84 *I*B*a^3*b - 42*I*(3*A + C)*a^2*b^2 - 28*I*B*a*b^3 - I*(7*A + 5*C)*b^4)*co s(d*x + c)^2*weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) + 2 1*sqrt(2)*(5*I*B*a^4 + 20*I*(A - C)*a^3*b - 30*I*B*a^2*b^2 - 4*I*(5*A + 3* C)*a*b^3 - 3*I*B*b^4)*cos(d*x + c)^2*weierstrassZeta(-4, 0, weierstrassPIn verse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) + 21*sqrt(2)*(-5*I*B*a^4 - 20 *I*(A - C)*a^3*b + 30*I*B*a^2*b^2 + 4*I*(5*A + 3*C)*a*b^3 + 3*I*B*b^4)*cos (d*x + c)^2*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c))) - 2*(15*C*b^4*cos(d*x + c)^4 + 35*A*a^4 + 21*(4*C*a*b^ 3 + B*b^4)*cos(d*x + c)^3 + 5*(42*C*a^2*b^2 + 28*B*a*b^3 + (7*A + 5*C)*b^4 )*cos(d*x + c)^2 + 105*(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)^2)
Timed out. \[ \int \frac {(a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {5}{2}}(c+d x)} \, dx=\text {Timed out} \] 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)* *(5/2),x)
Output:
Timed out
\[ \int \frac {(a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {5}{2}}(c+d x)} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{4}}{\cos \left (d x + c\right )^{\frac {5}{2}}} \,d x } \] 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)^(5 /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)^(5/2), x)
\[ \int \frac {(a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {5}{2}}(c+d x)} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{4}}{\cos \left (d x + c\right )^{\frac {5}{2}}} \,d x } \] 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)^(5 /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)^(5/2), x)
Time = 2.15 (sec) , antiderivative size = 516, normalized size of antiderivative = 1.38 \[ \int \frac {(a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {5}{2}}(c+d x)} \, dx =\text {Too large to display} \] 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)^(5/2),x)
Output:
(2*(C*a^4*ellipticF(c/2 + (d*x)/2, 2) + 4*C*a^3*b*ellipticE(c/2 + (d*x)/2, 2) + 2*C*a^2*b^2*ellipticF(c/2 + (d*x)/2, 2) + 2*C*a^2*b^2*cos(c + d*x)^( 1/2)*sin(c + d*x)))/d + (2*(A*b^4*ellipticF(c/2 + (d*x)/2, 2) + 12*A*a*b^3 *ellipticE(c/2 + (d*x)/2, 2) + A*b^4*cos(c + d*x)^(1/2)*sin(c + d*x) + 18* A*a^2*b^2*ellipticF(c/2 + (d*x)/2, 2)))/(3*d) + (8*B*a^3*b*ellipticF(c/2 + (d*x)/2, 2))/d + (4*B*a*b^3*((2*cos(c + d*x)^(1/2)*sin(c + d*x))/3 + (2*e llipticF(c/2 + (d*x)/2, 2))/3))/d + (12*B*a^2*b^2*ellipticE(c/2 + (d*x)/2, 2))/d + (2*A*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)) + (2*B*a^4*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)) - (2*B*b^4*cos(c + d*x)^(7/2)*sin(c + d*x)*hypergeom([1/2 , 7/4], 11/4, cos(c + d*x)^2))/(7*d*(sin(c + d*x)^2)^(1/2)) - (2*C*b^4*cos (c + d*x)^(9/2)*sin(c + d*x)*hypergeom([1/2, 9/4], 13/4, cos(c + d*x)^2))/ (9*d*(sin(c + d*x)^2)^(1/2)) + (8*A*a^3*b*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*b^3*cos(c + d*x)^(7/2)*sin(c + d*x)*hypergeom([1/2, 7/4], 11/4, cos (c + d*x)^2))/(7*d*(sin(c + d*x)^2)^(1/2))
\[ \int \frac {(a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {5}{2}}(c+d x)} \, dx=\left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )}d x \right ) a^{4} c +10 \left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )}d x \right ) a^{3} b^{2}+\left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{3}}d x \right ) a^{5}+5 \left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{2}}d x \right ) a^{4} b +4 \left (\int \sqrt {\cos \left (d x +c \right )}d x \right ) a^{3} b c +10 \left (\int \sqrt {\cos \left (d x +c \right )}d x \right ) a^{2} b^{3}+6 \left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )d x \right ) a^{2} b^{2} c +5 \left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )d x \right ) a \,b^{4}+\left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{3}d x \right ) b^{4} c +4 \left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{2}d x \right ) a \,b^{3} c +\left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{2}d x \right ) b^{5} \] 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)^(5/2),x)
Output:
int(sqrt(cos(c + d*x))/cos(c + d*x),x)*a**4*c + 10*int(sqrt(cos(c + d*x))/ cos(c + d*x),x)*a**3*b**2 + int(sqrt(cos(c + d*x))/cos(c + d*x)**3,x)*a**5 + 5*int(sqrt(cos(c + d*x))/cos(c + d*x)**2,x)*a**4*b + 4*int(sqrt(cos(c + d*x)),x)*a**3*b*c + 10*int(sqrt(cos(c + d*x)),x)*a**2*b**3 + 6*int(sqrt(c os(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)**3,x)*b**4*c + 4*in t(sqrt(cos(c + d*x))*cos(c + d*x)**2,x)*a*b**3*c + int(sqrt(cos(c + d*x))* cos(c + d*x)**2,x)*b**5