Integrand size = 35, antiderivative size = 300 \[ \int \frac {(a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {5}{2}}(c+d x)} \, dx=-\frac {8 a b \left (5 a^2 (A-C)-b^2 (5 A+3 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 \left (42 a^2 b^2 (3 A+C)+7 a^4 (A+3 C)+b^4 (7 A+5 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 d}-\frac {2 b^2 \left (3 a^2 (49 A-13 C)-b^2 (7 A+5 C)\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{21 d}-\frac {4 a b^3 (175 A-27 C) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{105 d}-\frac {2 b^2 (21 A-C) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2 \sin (c+d x)}{7 d}+\frac {16 A b (a+b \cos (c+d x))^3 \sin (c+d x)}{3 d \sqrt {\cos (c+d x)}}+\frac {2 A (a+b \cos (c+d x))^4 \sin (c+d x)}{3 d \cos ^{\frac {3}{2}}(c+d x)} \]
-8/5*a*b*(5*a^2*(A-C)-b^2*(5*A+3*C))*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2* d*x+1/2*c)*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2))/d+2/21*(42*a^2*b^2*(3*A+C )+7*a^4*(A+3*C)+b^4*(7*A+5*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))/d-4/105*a*b^3*(175*A-27*C)*cos( d*x+c)^(3/2)*sin(d*x+c)/d+2/3*A*(a+b*cos(d*x+c))^4*sin(d*x+c)/d/cos(d*x+c) ^(3/2)+16/3*A*b*(a+b*cos(d*x+c))^3*sin(d*x+c)/d/cos(d*x+c)^(1/2)-2/21*b^2* (3*a^2*(49*A-13*C)-b^2*(7*A+5*C))*sin(d*x+c)*cos(d*x+c)^(1/2)/d-2/7*b^2*(2 1*A-C)*(a+b*cos(d*x+c))^2*sin(d*x+c)*cos(d*x+c)^(1/2)/d
Time = 5.83 (sec) , antiderivative size = 206, normalized size of antiderivative = 0.69 \[ \int \frac {(a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {5}{2}}(c+d x)} \, dx=\frac {-168 \left (5 a^3 b (A-C)-a b^3 (5 A+3 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+10 \left (42 a^2 b^2 (3 A+C)+7 a^4 (A+3 C)+b^4 (7 A+5 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+105 \sqrt {\cos (c+d x)} \left (\left (\frac {2 A b^4}{3}+4 a^2 b^2 C+\frac {23 b^4 C}{42}\right ) \sin (c+d x)+\frac {4}{5} a b^3 C \sin (2 (c+d x))+\frac {1}{14} b^4 C \sin (3 (c+d x))+8 a^3 A b \tan (c+d x)+\frac {2}{3} a^4 A \sec (c+d x) \tan (c+d x)\right )}{105 d} \]
(-168*(5*a^3*b*(A - C) - a*b^3*(5*A + 3*C))*EllipticE[(c + d*x)/2, 2] + 10 *(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] + 105*Sqrt[Cos[c + d*x]]*(((2*A*b^4)/3 + 4*a^2*b^2*C + (23*b^4 *C)/42)*Sin[c + d*x] + (4*a*b^3*C*Sin[2*(c + d*x)])/5 + (b^4*C*Sin[3*(c + d*x)])/14 + 8*a^3*A*b*Tan[c + d*x] + (2*a^4*A*Sec[c + d*x]*Tan[c + d*x])/3 ))/(105*d)
Time = 2.14 (sec) , antiderivative size = 307, normalized size of antiderivative = 1.02, number of steps used = 20, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {3042, 3527, 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.
| \(\displaystyle \int \frac {(a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {5}{2}}(c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^4 \left (A+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{5/2}}dx\) |
| \(\Big \downarrow \) 3527 |
| \(\displaystyle \frac {2}{3} \int \frac {(a+b \cos (c+d x))^3 \left (-b (7 A-3 C) \cos ^2(c+d x)+a (A+3 C) \cos (c+d x)+8 A b\right )}{2 \cos ^{\frac {3}{2}}(c+d x)}dx+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{3 d \cos ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} \int \frac {(a+b \cos (c+d x))^3 \left (-b (7 A-3 C) \cos ^2(c+d x)+a (A+3 C) \cos (c+d x)+8 A b\right )}{\cos ^{\frac {3}{2}}(c+d x)}dx+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{3 d \cos ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^3 \left (-b (7 A-3 C) \sin \left (c+d x+\frac {\pi }{2}\right )^2+a (A+3 C) \sin \left (c+d x+\frac {\pi }{2}\right )+8 A b\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2}}dx+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{3 d \cos ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle \frac {1}{3} \left (2 \int \frac {(a+b \cos (c+d x))^2 \left ((A+3 C) a^2-2 b (7 A-3 C) \cos (c+d x) a+48 A b^2-3 b^2 (21 A-C) \cos ^2(c+d x)\right )}{2 \sqrt {\cos (c+d x)}}dx+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{3 d \cos ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} \left (\int \frac {(a+b \cos (c+d x))^2 \left ((A+3 C) a^2-2 b (7 A-3 C) \cos (c+d x) a+48 A b^2-3 b^2 (21 A-C) \cos ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}}dx+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{3 d \cos ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \left (\int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^2 \left ((A+3 C) a^2-2 b (7 A-3 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a+48 A b^2-3 b^2 (21 A-C) \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{3 d \cos ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \frac {1}{3} \left (\frac {2}{7} \int \frac {(a+b \cos (c+d x)) \left (-2 a b^2 (175 A-27 C) \cos ^2(c+d x)-b \left (a^2 (91 A-63 C)-3 b^2 (7 A+5 C)\right ) \cos (c+d x)+a \left (7 (A+3 C) a^2+3 b^2 (91 A+C)\right )\right )}{2 \sqrt {\cos (c+d x)}}dx-\frac {6 b^2 (21 A-C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{3 d \cos ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{7} \int \frac {(a+b \cos (c+d x)) \left (-2 a b^2 (175 A-27 C) \cos ^2(c+d x)-b \left (7 a^2 (13 A-9 C)-3 b^2 (7 A+5 C)\right ) \cos (c+d x)+a \left (7 (A+3 C) a^2+3 b^2 (91 A+C)\right )\right )}{\sqrt {\cos (c+d x)}}dx-\frac {6 b^2 (21 A-C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{3 d \cos ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{7} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right ) \left (-2 a b^2 (175 A-27 C) \sin \left (c+d x+\frac {\pi }{2}\right )^2-b \left (7 a^2 (13 A-9 C)-3 b^2 (7 A+5 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+a \left (7 (A+3 C) a^2+3 b^2 (91 A+C)\right )\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {6 b^2 (21 A-C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{3 d \cos ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3512 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{7} \left (\frac {2}{5} \int \frac {5 \left (7 (A+3 C) a^2+3 b^2 (91 A+C)\right ) a^2-84 b \left (5 a^2 (A-C)-b^2 (5 A+3 C)\right ) \cos (c+d x) a-15 b^2 \left (3 a^2 (49 A-13 C)-b^2 (7 A+5 C)\right ) \cos ^2(c+d x)}{2 \sqrt {\cos (c+d x)}}dx-\frac {4 a b^3 (175 A-27 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )-\frac {6 b^2 (21 A-C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{3 d \cos ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{7} \left (\frac {1}{5} \int \frac {5 \left (7 (A+3 C) a^2+3 b^2 (91 A+C)\right ) a^2-84 b \left (5 a^2 (A-C)-b^2 (5 A+3 C)\right ) \cos (c+d x) a-15 b^2 \left (3 a^2 (49 A-13 C)-b^2 (7 A+5 C)\right ) \cos ^2(c+d x)}{\sqrt {\cos (c+d x)}}dx-\frac {4 a b^3 (175 A-27 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )-\frac {6 b^2 (21 A-C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{3 d \cos ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{7} \left (\frac {1}{5} \int \frac {5 \left (7 (A+3 C) a^2+3 b^2 (91 A+C)\right ) a^2-84 b \left (5 a^2 (A-C)-b^2 (5 A+3 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) a-15 b^2 \left (3 a^2 (49 A-13 C)-b^2 (7 A+5 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {4 a b^3 (175 A-27 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )-\frac {6 b^2 (21 A-C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{3 d \cos ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {2}{3} \int \frac {3 \left (5 \left (7 (A+3 C) a^4+42 b^2 (3 A+C) a^2+b^4 (7 A+5 C)\right )-84 a b \left (5 a^2 (A-C)-b^2 (5 A+3 C)\right ) \cos (c+d x)\right )}{2 \sqrt {\cos (c+d x)}}dx-\frac {10 b^2 \left (3 a^2 (49 A-13 C)-b^2 (7 A+5 C)\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{d}\right )-\frac {4 a b^3 (175 A-27 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )-\frac {6 b^2 (21 A-C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{3 d \cos ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{7} \left (\frac {1}{5} \left (\int \frac {5 \left (7 (A+3 C) a^4+42 b^2 (3 A+C) a^2+b^4 (7 A+5 C)\right )-84 a b \left (5 a^2 (A-C)-b^2 (5 A+3 C)\right ) \cos (c+d x)}{\sqrt {\cos (c+d x)}}dx-\frac {10 b^2 \left (3 a^2 (49 A-13 C)-b^2 (7 A+5 C)\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{d}\right )-\frac {4 a b^3 (175 A-27 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )-\frac {6 b^2 (21 A-C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{3 d \cos ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{7} \left (\frac {1}{5} \left (\int \frac {5 \left (7 (A+3 C) a^4+42 b^2 (3 A+C) a^2+b^4 (7 A+5 C)\right )-84 a b \left (5 a^2 (A-C)-b^2 (5 A+3 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {10 b^2 \left (3 a^2 (49 A-13 C)-b^2 (7 A+5 C)\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{d}\right )-\frac {4 a b^3 (175 A-27 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )-\frac {6 b^2 (21 A-C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{3 d \cos ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{7} \left (\frac {1}{5} \left (-84 a b \left (5 a^2 (A-C)-b^2 (5 A+3 C)\right ) \int \sqrt {\cos (c+d x)}dx+5 \left (7 a^4 (A+3 C)+42 a^2 b^2 (3 A+C)+b^4 (7 A+5 C)\right ) \int \frac {1}{\sqrt {\cos (c+d x)}}dx-\frac {10 b^2 \left (3 a^2 (49 A-13 C)-b^2 (7 A+5 C)\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{d}\right )-\frac {4 a b^3 (175 A-27 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )-\frac {6 b^2 (21 A-C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{3 d \cos ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{7} \left (\frac {1}{5} \left (-84 a b \left (5 a^2 (A-C)-b^2 (5 A+3 C)\right ) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx+5 \left (7 a^4 (A+3 C)+42 a^2 b^2 (3 A+C)+b^4 (7 A+5 C)\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {10 b^2 \left (3 a^2 (49 A-13 C)-b^2 (7 A+5 C)\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{d}\right )-\frac {4 a b^3 (175 A-27 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )-\frac {6 b^2 (21 A-C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{3 d \cos ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{7} \left (\frac {1}{5} \left (5 \left (7 a^4 (A+3 C)+42 a^2 b^2 (3 A+C)+b^4 (7 A+5 C)\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {168 a b \left (5 a^2 (A-C)-b^2 (5 A+3 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}-\frac {10 b^2 \left (3 a^2 (49 A-13 C)-b^2 (7 A+5 C)\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{d}\right )-\frac {4 a b^3 (175 A-27 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )-\frac {6 b^2 (21 A-C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{3 d \cos ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{7} \left (\frac {1}{5} \left (-\frac {168 a b \left (5 a^2 (A-C)-b^2 (5 A+3 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}-\frac {10 b^2 \left (3 a^2 (49 A-13 C)-b^2 (7 A+5 C)\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{d}+\frac {10 \left (7 a^4 (A+3 C)+42 a^2 b^2 (3 A+C)+b^4 (7 A+5 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{d}\right )-\frac {4 a b^3 (175 A-27 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )-\frac {6 b^2 (21 A-C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{3 d \cos ^{\frac {3}{2}}(c+d x)}\) |
(2*A*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + ((-6* b^2*(21*A - C)*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(7* d) + (16*A*b*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) + ((-4*a*b^3*(175*A - 27*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d) + ((-168 *a*b*(5*a^2*(A - C) - b^2*(5*A + 3*C))*EllipticE[(c + d*x)/2, 2])/d + (10* (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^2*(3*a^2*(49*A - 13*C) - b^2*(7*A + 5*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/d)/5)/7)/3
3.7.99.3.1 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[(-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_.) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(c^2*C + A*d^2))*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 - d^2))), x] + Simp[1/(d*(n + 1)*(c^2 - d^ 2)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp[A* d*(b*d*m + a*c*(n + 1)) + c*C*(b*c*m + a*d*(n + 1)) - (A*d*(a*d*(n + 2) - b *c*(n + 1)) - C*(b*c*d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x] - b*( A*d^2*(m + n + 2) + C*(c^2*(m + 1) + d^2*(n + 1)))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 0] && LtQ[n, -1]
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])))
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 30.52 (sec) , antiderivative size = 1184, normalized size of antiderivative = 3.95
| method | result | size |
| parts | \(\text {Expression too large to display}\) | \(1184\) |
| default | \(\text {Expression too large to display}\) | \(1715\) |
-2/3*(A*b^4+6*C*a^2*b^2)*((2*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2) ^(1/2)*(4*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)-2*sin(1/2*d*x+1/2*c)^2*c os(1/2*d*x+1/2*c)+(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)/(2*cos(1/2*d*x+1/2*c)^2-1)^(1/2 )/d+2*(4*A*a*b^3+4*C*a^3*b)*((2*cos(1/2*d*x+1/2*c)^2-1)*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)*E llipticE(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)/(2*cos(1/2*d*x+1/2*c)^2-1)^(1/2)/d+2*(6 *A*a^2*b^2+C*a^4)/d*InverseJacobiAM(1/2*d*x+1/2*c,2^(1/2))-2/21*C*b^4*((2* cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(48*cos(1/2*d*x+1/2*c) ^9-120*cos(1/2*d*x+1/2*c)^7+128*cos(1/2*d*x+1/2*c)^5-72*cos(1/2*d*x+1/2*c) ^3+5*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)*Ellipt icF(cos(1/2*d*x+1/2*c),2^(1/2))+16*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)/sin(1/2*d*x+1/2*c)/(2*cos(1/2*d*x+1/2*c) ^2-1)^(1/2)/d-2/3*a^4*A*(-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))*sin(1/2*d*x+1/2*c)^2 -2*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)+(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* cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)/(-2*sin(1/2*d*x+1/2...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.15 (sec) , antiderivative size = 362, normalized size of antiderivative = 1.21 \[ \int \frac {(a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {5}{2}}(c+d x)} \, dx=-\frac {5 \, \sqrt {2} {\left (7 i \, {\left (A + 3 \, C\right )} a^{4} + 42 i \, {\left (3 \, A + C\right )} a^{2} b^{2} + i \, {\left (7 \, A + 5 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{2} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 5 \, \sqrt {2} {\left (-7 i \, {\left (A + 3 \, C\right )} a^{4} - 42 i \, {\left (3 \, A + C\right )} a^{2} b^{2} - i \, {\left (7 \, A + 5 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{2} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 84 \, \sqrt {2} {\left (5 i \, {\left (A - C\right )} a^{3} b - i \, {\left (5 \, A + 3 \, C\right )} a b^{3}\right )} \cos \left (d x + c\right )^{2} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 84 \, \sqrt {2} {\left (-5 i \, {\left (A - C\right )} a^{3} b + i \, {\left (5 \, A + 3 \, C\right )} a b^{3}\right )} \cos \left (d x + c\right )^{2} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - 2 \, {\left (15 \, C b^{4} \cos \left (d x + c\right )^{4} + 84 \, C a b^{3} \cos \left (d x + c\right )^{3} + 420 \, A a^{3} b \cos \left (d x + c\right ) + 35 \, A a^{4} + 5 \, {\left (42 \, C a^{2} b^{2} + {\left (7 \, A + 5 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{2}\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )}{105 \, d \cos \left (d x + c\right )^{2}} \]
-1/105*(5*sqrt(2)*(7*I*(A + 3*C)*a^4 + 42*I*(3*A + C)*a^2*b^2 + 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 - 42*I*(3*A + C)*a^2*b^2 - I*(7*A + 5*C)*b^4)*cos(d*x + c)^2*weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin (d*x + c)) + 84*sqrt(2)*(5*I*(A - C)*a^3*b - I*(5*A + 3*C)*a*b^3)*cos(d*x + c)^2*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I* sin(d*x + c))) + 84*sqrt(2)*(-5*I*(A - C)*a^3*b + I*(5*A + 3*C)*a*b^3)*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 + 84*C*a*b^3*cos(d*x + c) ^3 + 420*A*a^3*b*cos(d*x + c) + 35*A*a^4 + 5*(42*C*a^2*b^2 + (7*A + 5*C)*b ^4)*cos(d*x + c)^2)*sqrt(cos(d*x + c))*sin(d*x + c))/(d*cos(d*x + c)^2)
Timed out. \[ \int \frac {(a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {5}{2}}(c+d x)} \, dx=\text {Timed out} \]
\[ \int \frac {(a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {5}{2}}(c+d x)} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{4}}{\cos \left (d x + c\right )^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {(a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {5}{2}}(c+d x)} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{4}}{\cos \left (d x + c\right )^{\frac {5}{2}}} \,d x } \]
Time = 3.34 (sec) , antiderivative size = 343, normalized size of antiderivative = 1.14 \[ \int \frac {(a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {5}{2}}(c+d x)} \, dx=\frac {2\,\left (C\,a^4\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )+4\,C\,a^3\,b\,\mathrm {E}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )+2\,C\,a^2\,b^2\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )+2\,C\,a^2\,b^2\,\sqrt {\cos \left (c+d\,x\right )}\,\sin \left (c+d\,x\right )\right )}{d}+\frac {2\,\left (A\,b^4\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )+12\,A\,a\,b^3\,\mathrm {E}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )+A\,b^4\,\sqrt {\cos \left (c+d\,x\right )}\,\sin \left (c+d\,x\right )+18\,A\,a^2\,b^2\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )\right )}{3\,d}+\frac {2\,A\,a^4\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{4},\frac {1}{2};\ \frac {1}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{3\,d\,{\cos \left (c+d\,x\right )}^{3/2}\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {2\,C\,b^4\,{\cos \left (c+d\,x\right )}^{9/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {9}{4};\ \frac {13}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{9\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}+\frac {8\,A\,a^3\,b\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{4},\frac {1}{2};\ \frac {3}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{d\,\sqrt {\cos \left (c+d\,x\right )}\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {8\,C\,a\,b^3\,{\cos \left (c+d\,x\right )}^{7/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {7}{4};\ \frac {11}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{7\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}} \]
(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) + (2*A*a^4*sin(c + d*x)*hype rgeom([-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*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)*hy pergeom([1/2, 7/4], 11/4, cos(c + d*x)^2))/(7*d*(sin(c + d*x)^2)^(1/2))