Integrand size = 35, antiderivative size = 316 \[ \int \frac {(a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {9}{2}}(c+d x)} \, dx=-\frac {8 a b \left (5 b^2 (A-C)+a^2 (3 A+5 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 \left (7 b^4 (3 A+C)+42 a^2 b^2 (A+3 C)+a^4 (5 A+7 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 d}+\frac {4 a b \left (96 A b^2+a^2 (101 A+175 C)\right ) \sin (c+d x)}{105 d \sqrt {\cos (c+d x)}}-\frac {2 b^2 \left (b^2 (87 A-35 C)+5 a^2 (5 A+7 C)\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{105 d}+\frac {2 \left (48 A b^2+5 a^2 (5 A+7 C)\right ) (a+b \cos (c+d x))^2 \sin (c+d x)}{105 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {16 A b (a+b \cos (c+d x))^3 \sin (c+d x)}{35 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 A (a+b \cos (c+d x))^4 \sin (c+d x)}{7 d \cos ^{\frac {7}{2}}(c+d x)} \] Output:
-8/5*a*b*(5*b^2*(A-C)+a^2*(3*A+5*C))*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2)) /d+2/21*(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+4/105*a*b*(96*A*b^2+a^2*(101*A+175*C))*sin(d*x+c)/ d/cos(d*x+c)^(1/2)-2/105*b^2*(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+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)+16/35*A*b*(a+b*cos(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 = 6.45 (sec) , antiderivative size = 276, normalized size of antiderivative = 0.87 \[ \int \frac {(a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {9}{2}}(c+d x)} \, dx=\frac {-168 a b \left (5 b^2 (A-C)+a^2 (3 A+5 C)\right ) \cos ^{\frac {5}{2}}(c+d x) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+10 \left (7 b^4 (3 A+C)+42 a^2 b^2 (A+3 C)+a^4 (5 A+7 C)\right ) \cos ^{\frac {5}{2}}(c+d x) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+168 a^3 A b \sin (c+d x)+504 a^3 A b \cos ^2(c+d x) \sin (c+d x)+840 a A b^3 \cos ^2(c+d x) \sin (c+d x)+840 a^3 b C \cos ^2(c+d x) \sin (c+d x)+70 b^4 C \cos ^3(c+d x) \sin (c+d x)+25 a^4 A \sin (2 (c+d x))+210 a^2 A b^2 \sin (2 (c+d x))+35 a^4 C \sin (2 (c+d x))+30 a^4 A \tan (c+d x)}{105 d \cos ^{\frac {5}{2}}(c+d x)} \] Input:
Integrate[((a + b*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/ 2),x]
Output:
(-168*a*b*(5*b^2*(A - C) + a^2*(3*A + 5*C))*Cos[c + d*x]^(5/2)*EllipticE[( c + d*x)/2, 2] + 10*(7*b^4*(3*A + C) + 42*a^2*b^2*(A + 3*C) + a^4*(5*A + 7 *C))*Cos[c + d*x]^(5/2)*EllipticF[(c + d*x)/2, 2] + 168*a^3*A*b*Sin[c + d* x] + 504*a^3*A*b*Cos[c + d*x]^2*Sin[c + d*x] + 840*a*A*b^3*Cos[c + d*x]^2* Sin[c + d*x] + 840*a^3*b*C*Cos[c + d*x]^2*Sin[c + d*x] + 70*b^4*C*Cos[c + d*x]^3*Sin[c + d*x] + 25*a^4*A*Sin[2*(c + d*x)] + 210*a^2*A*b^2*Sin[2*(c + d*x)] + 35*a^4*C*Sin[2*(c + d*x)] + 30*a^4*A*Tan[c + d*x])/(105*d*Cos[c + d*x]^(5/2))
Time = 2.26 (sec) , antiderivative size = 323, 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, 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.
| \(\displaystyle \int \frac {(a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {9}{2}}(c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^4 \left (A+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{9/2}}dx\) |
| \(\Big \downarrow \) 3527 |
| \(\displaystyle \frac {2}{7} \int \frac {(a+b \cos (c+d x))^3 \left (-b (3 A-7 C) \cos ^2(c+d x)+a (5 A+7 C) \cos (c+d x)+8 A b\right )}{2 \cos ^{\frac {7}{2}}(c+d x)}dx+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} \int \frac {(a+b \cos (c+d x))^3 \left (-b (3 A-7 C) \cos ^2(c+d x)+a (5 A+7 C) \cos (c+d x)+8 A b\right )}{\cos ^{\frac {7}{2}}(c+d x)}dx+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^3 \left (-b (3 A-7 C) \sin \left (c+d x+\frac {\pi }{2}\right )^2+a (5 A+7 C) \sin \left (c+d x+\frac {\pi }{2}\right )+8 A b\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{7/2}}dx+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle \frac {1}{7} \left (\frac {2}{5} \int \frac {(a+b \cos (c+d x))^2 \left (5 (5 A+7 C) a^2+2 b (17 A+35 C) \cos (c+d x) a+48 A b^2-b^2 (39 A-35 C) \cos ^2(c+d x)\right )}{2 \cos ^{\frac {5}{2}}(c+d x)}dx+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \int \frac {(a+b \cos (c+d x))^2 \left (5 (5 A+7 C) a^2+2 b (17 A+35 C) \cos (c+d x) a+48 A b^2-b^2 (39 A-35 C) \cos ^2(c+d x)\right )}{\cos ^{\frac {5}{2}}(c+d x)}dx+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^2 \left (5 (5 A+7 C) a^2+2 b (17 A+35 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a+48 A b^2-b^2 (39 A-35 C) \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{5/2}}dx+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {2}{3} \int \frac {(a+b \cos (c+d x)) \left (-3 b \left (5 (5 A+7 C) a^2+b^2 (87 A-35 C)\right ) \cos ^2(c+d x)+a \left (5 (5 A+7 C) a^2+3 b^2 (11 A+105 C)\right ) \cos (c+d x)+2 b \left ((101 A+175 C) a^2+96 A b^2\right )\right )}{2 \cos ^{\frac {3}{2}}(c+d x)}dx+\frac {2 \left (5 a^2 (5 A+7 C)+48 A b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^2}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \int \frac {(a+b \cos (c+d x)) \left (-3 b \left (5 (5 A+7 C) a^2+b^2 (87 A-35 C)\right ) \cos ^2(c+d x)+a \left (5 (5 A+7 C) a^2+3 b^2 (11 A+105 C)\right ) \cos (c+d x)+2 b \left ((101 A+175 C) a^2+96 A b^2\right )\right )}{\cos ^{\frac {3}{2}}(c+d x)}dx+\frac {2 \left (5 a^2 (5 A+7 C)+48 A b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^2}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right ) \left (-3 b \left (5 (5 A+7 C) a^2+b^2 (87 A-35 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+a \left (5 (5 A+7 C) a^2+3 b^2 (11 A+105 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+2 b \left ((101 A+175 C) a^2+96 A b^2\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2}}dx+\frac {2 \left (5 a^2 (5 A+7 C)+48 A b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^2}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3510 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {4 a b \left (a^2 (101 A+175 C)+96 A b^2\right ) \sin (c+d x)}{d \sqrt {\cos (c+d x)}}-2 \int -\frac {5 (5 A+7 C) a^4+5 b^2 (47 A+133 C) a^2-84 b \left ((3 A+5 C) a^2+5 b^2 (A-C)\right ) \cos (c+d x) a+192 A b^4-3 b^2 \left (5 (5 A+7 C) a^2+b^2 (87 A-35 C)\right ) \cos ^2(c+d x)}{2 \sqrt {\cos (c+d x)}}dx\right )+\frac {2 \left (5 a^2 (5 A+7 C)+48 A b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^2}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\int \frac {5 (5 A+7 C) a^4+5 b^2 (47 A+133 C) a^2-84 b \left ((3 A+5 C) a^2+5 b^2 (A-C)\right ) \cos (c+d x) a+192 A b^4-3 b^2 \left (5 (5 A+7 C) a^2+b^2 (87 A-35 C)\right ) \cos ^2(c+d x)}{\sqrt {\cos (c+d x)}}dx+\frac {4 a b \left (a^2 (101 A+175 C)+96 A b^2\right ) \sin (c+d x)}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 \left (5 a^2 (5 A+7 C)+48 A b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^2}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\int \frac {5 (5 A+7 C) a^4+5 b^2 (47 A+133 C) a^2-84 b \left ((3 A+5 C) a^2+5 b^2 (A-C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) a+192 A b^4-3 b^2 \left (5 (5 A+7 C) a^2+b^2 (87 A-35 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {4 a b \left (a^2 (101 A+175 C)+96 A b^2\right ) \sin (c+d x)}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 \left (5 a^2 (5 A+7 C)+48 A b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^2}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {2}{3} \int \frac {3 \left (5 \left ((5 A+7 C) a^4+42 b^2 (A+3 C) a^2+7 b^4 (3 A+C)\right )-84 a b \left ((3 A+5 C) a^2+5 b^2 (A-C)\right ) \cos (c+d x)\right )}{2 \sqrt {\cos (c+d x)}}dx-\frac {2 b^2 \left (5 a^2 (5 A+7 C)+b^2 (87 A-35 C)\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{d}+\frac {4 a b \left (a^2 (101 A+175 C)+96 A b^2\right ) \sin (c+d x)}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 \left (5 a^2 (5 A+7 C)+48 A b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^2}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\int \frac {5 \left ((5 A+7 C) a^4+42 b^2 (A+3 C) a^2+7 b^4 (3 A+C)\right )-84 a b \left ((3 A+5 C) a^2+5 b^2 (A-C)\right ) \cos (c+d x)}{\sqrt {\cos (c+d x)}}dx-\frac {2 b^2 \left (5 a^2 (5 A+7 C)+b^2 (87 A-35 C)\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{d}+\frac {4 a b \left (a^2 (101 A+175 C)+96 A b^2\right ) \sin (c+d x)}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 \left (5 a^2 (5 A+7 C)+48 A b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^2}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\int \frac {5 \left ((5 A+7 C) a^4+42 b^2 (A+3 C) a^2+7 b^4 (3 A+C)\right )-84 a b \left ((3 A+5 C) a^2+5 b^2 (A-C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {2 b^2 \left (5 a^2 (5 A+7 C)+b^2 (87 A-35 C)\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{d}+\frac {4 a b \left (a^2 (101 A+175 C)+96 A b^2\right ) \sin (c+d x)}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 \left (5 a^2 (5 A+7 C)+48 A b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^2}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (-84 a b \left (a^2 (3 A+5 C)+5 b^2 (A-C)\right ) \int \sqrt {\cos (c+d x)}dx+5 \left (a^4 (5 A+7 C)+42 a^2 b^2 (A+3 C)+7 b^4 (3 A+C)\right ) \int \frac {1}{\sqrt {\cos (c+d x)}}dx-\frac {2 b^2 \left (5 a^2 (5 A+7 C)+b^2 (87 A-35 C)\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{d}+\frac {4 a b \left (a^2 (101 A+175 C)+96 A b^2\right ) \sin (c+d x)}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 \left (5 a^2 (5 A+7 C)+48 A b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^2}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (-84 a b \left (a^2 (3 A+5 C)+5 b^2 (A-C)\right ) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx+5 \left (a^4 (5 A+7 C)+42 a^2 b^2 (A+3 C)+7 b^4 (3 A+C)\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {2 b^2 \left (5 a^2 (5 A+7 C)+b^2 (87 A-35 C)\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{d}+\frac {4 a b \left (a^2 (101 A+175 C)+96 A b^2\right ) \sin (c+d x)}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 \left (5 a^2 (5 A+7 C)+48 A b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^2}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (5 \left (a^4 (5 A+7 C)+42 a^2 b^2 (A+3 C)+7 b^4 (3 A+C)\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {168 a b \left (a^2 (3 A+5 C)+5 b^2 (A-C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}-\frac {2 b^2 \left (5 a^2 (5 A+7 C)+b^2 (87 A-35 C)\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{d}+\frac {4 a b \left (a^2 (101 A+175 C)+96 A b^2\right ) \sin (c+d x)}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 \left (5 a^2 (5 A+7 C)+48 A b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^2}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {2 \left (5 a^2 (5 A+7 C)+48 A b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^2}{3 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {1}{3} \left (-\frac {168 a b \left (a^2 (3 A+5 C)+5 b^2 (A-C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}-\frac {2 b^2 \left (5 a^2 (5 A+7 C)+b^2 (87 A-35 C)\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{d}+\frac {4 a b \left (a^2 (101 A+175 C)+96 A b^2\right ) \sin (c+d x)}{d \sqrt {\cos (c+d x)}}+\frac {10 \left (a^4 (5 A+7 C)+42 a^2 b^2 (A+3 C)+7 b^4 (3 A+C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{d}\right )\right )+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\) |
Input:
Int[((a + b*Cos[c + d*x])^4*(A + 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)) + ((16* A*b*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + ((2*(4 8*A*b^2 + 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)) + ((-168*a*b*(5*b^2*(A - C) + a^2*(3*A + 5*C))*EllipticE[ (c + d*x)/2, 2])/d + (10*(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 + (4*a*b*(96*A*b^2 + a^2*(101*A + 1 75*C))*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) - (2*b^2*(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]
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]
Leaf count of result is larger than twice the leaf count of optimal. \(1383\) vs. \(2(295)=590\).
Time = 38.27 (sec) , antiderivative size = 1384, normalized size of antiderivative = 4.38
| method | result | size |
| parts | \(\text {Expression too large to display}\) | \(1384\) |
| default | \(\text {Expression too large to display}\) | \(1504\) |
Input:
int((a+b*cos(d*x+c))^4*(A+C*cos(d*x+c)^2)/cos(d*x+c)^(9/2),x,method=_RETUR NVERBOSE)
Output:
2*(A*b^4+6*C*a^2*b^2)/d*InverseJacobiAM(1/2*d*x+1/2*c,2^(1/2))-2*(4*A*a*b^ 3+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)/(2*cos(1/2*d*x+1/2*c)^2-1)^(1/2)/d -2/3*(6*A*a^2*b^2+C*a^4)*(-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*c )^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/(2*cos(1/2*d*x+1/2*c)^2-1)^(3/2)/sin(1/2*d *x+1/2*c)/d-A*a^4*(-(1-2*cos(1/2*d*x+1/2*c)^2)*sin(1/2*d*x+1/2*c)^2)^(1/2) *(-1/28*cos(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^ (1/2)/(cos(1/2*d*x+1/2*c)^2-1/2)^4-5/21*cos(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x +1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/(cos(1/2*d*x+1/2*c)^2-1/2)^2+10/21*( sin(1/2*d*x+1/2*c)^2)^(1/2)*(1-2*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)*EllipticF(cos(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*C*b^4*((2*c os(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...
Result contains complex when optimal does not.
Time = 0.15 (sec) , antiderivative size = 377, normalized size of antiderivative = 1.19 \[ \int \frac {(a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {9}{2}}(c+d x)} \, dx=-\frac {5 \, \sqrt {2} {\left (i \, {\left (5 \, A + 7 \, C\right )} a^{4} + 42 i \, {\left (A + 3 \, C\right )} a^{2} b^{2} + 7 i \, {\left (3 \, A + C\right )} b^{4}\right )} \cos \left (d x + c\right )^{4} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 5 \, \sqrt {2} {\left (-i \, {\left (5 \, A + 7 \, C\right )} a^{4} - 42 i \, {\left (A + 3 \, C\right )} a^{2} b^{2} - 7 i \, {\left (3 \, A + C\right )} b^{4}\right )} \cos \left (d x + c\right )^{4} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 84 \, \sqrt {2} {\left (i \, {\left (3 \, A + 5 \, C\right )} a^{3} b + 5 i \, {\left (A - C\right )} a b^{3}\right )} \cos \left (d x + c\right )^{4} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 84 \, \sqrt {2} {\left (-i \, {\left (3 \, A + 5 \, C\right )} a^{3} b - 5 i \, {\left (A - C\right )} a b^{3}\right )} \cos \left (d x + c\right )^{4} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - 2 \, {\left (35 \, C b^{4} \cos \left (d x + c\right )^{4} + 84 \, A a^{3} b \cos \left (d x + c\right ) + 15 \, A a^{4} + 84 \, {\left ({\left (3 \, A + 5 \, C\right )} a^{3} b + 5 \, A a b^{3}\right )} \cos \left (d x + c\right )^{3} + 5 \, {\left ({\left (5 \, A + 7 \, C\right )} a^{4} + 42 \, A a^{2} b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )}{105 \, d \cos \left (d x + c\right )^{4}} \] Input:
integrate((a+b*cos(d*x+c))^4*(A+C*cos(d*x+c)^2)/cos(d*x+c)^(9/2),x, algori thm="fricas")
Output:
-1/105*(5*sqrt(2)*(I*(5*A + 7*C)*a^4 + 42*I*(A + 3*C)*a^2*b^2 + 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 - 42*I*(A + 3*C)*a^2*b^2 - 7*I*(3* A + C)*b^4)*cos(d*x + c)^4*weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin (d*x + c)) + 84*sqrt(2)*(I*(3*A + 5*C)*a^3*b + 5*I*(A - C)*a*b^3)*cos(d*x + c)^4*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I* sin(d*x + c))) + 84*sqrt(2)*(-I*(3*A + 5*C)*a^3*b - 5*I*(A - C)*a*b^3)*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 + 84*A*a^3*b*cos(d*x + c) + 15*A*a^4 + 84*((3*A + 5*C)*a^3*b + 5*A*a*b^3)*cos(d*x + c)^3 + 5*((5*A + 7*C)*a^4 + 42*A*a^2*b^2)*cos(d*x + c)^2)*sqrt(cos(d*x + c))*sin(d*x + c) )/(d*cos(d*x + c)^4)
Timed out. \[ \int \frac {(a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {9}{2}}(c+d x)} \, dx=\text {Timed out} \] Input:
integrate((a+b*cos(d*x+c))**4*(A+C*cos(d*x+c)**2)/cos(d*x+c)**(9/2),x)
Output:
Timed out
\[ \int \frac {(a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {9}{2}}(c+d x)} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{4}}{\cos \left (d x + c\right )^{\frac {9}{2}}} \,d x } \] Input:
integrate((a+b*cos(d*x+c))^4*(A+C*cos(d*x+c)^2)/cos(d*x+c)^(9/2),x, algori thm="maxima")
Output:
integrate((C*cos(d*x + c)^2 + A)*(b*cos(d*x + c) + a)^4/cos(d*x + c)^(9/2) , x)
\[ \int \frac {(a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {9}{2}}(c+d x)} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{4}}{\cos \left (d x + c\right )^{\frac {9}{2}}} \,d x } \] Input:
integrate((a+b*cos(d*x+c))^4*(A+C*cos(d*x+c)^2)/cos(d*x+c)^(9/2),x, algori thm="giac")
Output:
integrate((C*cos(d*x + c)^2 + A)*(b*cos(d*x + c) + a)^4/cos(d*x + c)^(9/2) , x)
Time = 2.71 (sec) , antiderivative size = 378, normalized size of antiderivative = 1.20 \[ \int \frac {(a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {9}{2}}(c+d x)} \, dx=\frac {2\,\left (C\,b^4\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )+12\,C\,a\,b^3\,\mathrm {E}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )+C\,b^4\,\sqrt {\cos \left (c+d\,x\right )}\,\sin \left (c+d\,x\right )+18\,C\,a^2\,b^2\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )\right )}{3\,d}+\frac {2\,A\,b^4\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{d}+\frac {2\,A\,a^4\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {7}{4},\frac {1}{2};\ -\frac {3}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{7\,d\,{\cos \left (c+d\,x\right )}^{7/2}\,\sqrt {{\sin \left (c+d\,x\right )}^2}}+\frac {2\,C\,a^4\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{4},\frac {1}{2};\ \frac {1}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{3\,d\,{\cos \left (c+d\,x\right )}^{3/2}\,\sqrt {{\sin \left (c+d\,x\right )}^2}}+\frac {8\,A\,a\,b^3\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{4},\frac {1}{2};\ \frac {3}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{d\,\sqrt {\cos \left (c+d\,x\right )}\,\sqrt {{\sin \left (c+d\,x\right )}^2}}+\frac {8\,A\,a^3\,b\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {5}{4},\frac {1}{2};\ -\frac {1}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{5\,d\,{\cos \left (c+d\,x\right )}^{5/2}\,\sqrt {{\sin \left (c+d\,x\right )}^2}}+\frac {8\,C\,a^3\,b\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{4},\frac {1}{2};\ \frac {3}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{d\,\sqrt {\cos \left (c+d\,x\right )}\,\sqrt {{\sin \left (c+d\,x\right )}^2}}+\frac {4\,A\,a^2\,b^2\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{4},\frac {1}{2};\ \frac {1}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{d\,{\cos \left (c+d\,x\right )}^{3/2}\,\sqrt {{\sin \left (c+d\,x\right )}^2}} \] Input:
int(((A + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^4)/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*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*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*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*A*a^3*b*sin(c + d*x)*hy pergeom([-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*C*a^3*b*sin(c + d*x)*hypergeom([-1/4, 1/2], 3/4, co s(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))
\[ \int \frac {(a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {9}{2}}(c+d x)} \, dx=6 \left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )}d x \right ) a^{2} b^{2} c +\left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )}d x \right ) a \,b^{4}+\left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{5}}d x \right ) a^{5}+4 \left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{4}}d x \right ) a^{4} b +\left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{3}}d x \right ) a^{4} c +6 \left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{3}}d x \right ) a^{3} b^{2}+4 \left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{2}}d x \right ) a^{3} b c +4 \left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{2}}d x \right ) a^{2} b^{3}+4 \left (\int \sqrt {\cos \left (d x +c \right )}d x \right ) a \,b^{3} c +\left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )d x \right ) b^{4} c \] Input:
int((a+b*cos(d*x+c))^4*(A+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 + 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 + 4*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 + 6*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 + 4*i nt(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))*cos(c + d*x),x)*b**4*c