\(\int (a+b \cos (c+d x))^4 (A+C \cos ^2(c+d x)) \sec ^{\frac {5}{2}}(c+d x) \, dx\) [1387]

Optimal result

Integrand size = 35, antiderivative size = 340 \[ \int (a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {5}{2}}(c+d x) \, dx=-\frac {8 a b \left (5 a^2 (A-C)-b^2 (5 A+3 C)\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{5 d}+\frac {2 \left (42 a^2 b^2 (3 A+C)+7 a^4 (A+3 C)+b^4 (7 A+5 C)\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{21 d}-\frac {4 a b^3 (175 A-27 C) \sin (c+d x)}{105 d \sec ^{\frac {3}{2}}(c+d x)}-\frac {2 b^2 \left (3 a^2 (49 A-13 C)-b^2 (7 A+5 C)\right ) \sin (c+d x)}{21 d \sqrt {\sec (c+d x)}}-\frac {2 b^2 (21 A-C) (a+b \cos (c+d x))^2 \sin (c+d x)}{7 d \sqrt {\sec (c+d x)}}+\frac {16 A b (a+b \cos (c+d x))^3 \sqrt {\sec (c+d x)} \sin (c+d x)}{3 d}+\frac {2 A (a+b \cos (c+d x))^4 \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{3 d} \] Output:

-8/5*a*b*(5*a^2*(A-C)-b^2*(5*A+3*C))*cos(d*x+c)^(1/2)*EllipticE(sin(1/2*d* 
x+1/2*c),2^(1/2))*sec(d*x+c)^(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(d*x+c)^(1/2)*InverseJacobiAM(1/2*d*x+1/2*c,2^(1/2))*s 
ec(d*x+c)^(1/2)/d-4/105*a*b^3*(175*A-27*C)*sin(d*x+c)/d/sec(d*x+c)^(3/2)-2 
/21*b^2*(3*a^2*(49*A-13*C)-b^2*(7*A+5*C))*sin(d*x+c)/d/sec(d*x+c)^(1/2)-2/ 
7*b^2*(21*A-C)*(a+b*cos(d*x+c))^2*sin(d*x+c)/d/sec(d*x+c)^(1/2)+16/3*A*b*( 
a+b*cos(d*x+c))^3*sec(d*x+c)^(1/2)*sin(d*x+c)/d+2/3*A*(a+b*cos(d*x+c))^4*s 
ec(d*x+c)^(3/2)*sin(d*x+c)/d
 

Mathematica [A] (verified)

Time = 6.21 (sec) , antiderivative size = 243, normalized size of antiderivative = 0.71 \[ \int (a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {5}{2}}(c+d x) \, dx=\frac {\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (-672 a b \left (5 a^2 (A-C)-b^2 (5 A+3 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+40 \left (42 a^2 b^2 (3 A+C)+7 a^4 (A+3 C)+b^4 (7 A+5 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+\frac {\left (280 a^4 A+140 A b^4+840 a^2 b^2 C+145 b^4 C+168 a b \left (20 a^2 A+3 b^2 C\right ) \cos (c+d x)+20 \left (7 A b^4+42 a^2 b^2 C+8 b^4 C\right ) \cos (2 (c+d x))+168 a b^3 C \cos (3 (c+d x))+15 b^4 C \cos (4 (c+d x))\right ) \sin (c+d x)}{\cos ^{\frac {3}{2}}(c+d x)}\right )}{420 d} \] Input:

Integrate[(a + b*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2) 
,x]
 

Output:

(Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*(-672*a*b*(5*a^2*(A - C) - b^2*(5*A 
 + 3*C))*EllipticE[(c + d*x)/2, 2] + 40*(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] + ((280*a^4*A + 140*A*b 
^4 + 840*a^2*b^2*C + 145*b^4*C + 168*a*b*(20*a^2*A + 3*b^2*C)*Cos[c + d*x] 
 + 20*(7*A*b^4 + 42*a^2*b^2*C + 8*b^4*C)*Cos[2*(c + d*x)] + 168*a*b^3*C*Co 
s[3*(c + d*x)] + 15*b^4*C*Cos[4*(c + d*x)])*Sin[c + d*x])/Cos[c + d*x]^(3/ 
2)))/(420*d)
 

Rubi [A] (verified)

Time = 2.40 (sec) , antiderivative size = 328, normalized size of antiderivative = 0.96, number of steps used = 22, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.629, Rules used = {3042, 4709, 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.

\(\displaystyle \int \sec ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \sec (c+d x)^{5/2} (a+b \cos (c+d x))^4 \left (A+C \cos (c+d x)^2\right )dx\)

\(\Big \downarrow \) 4709

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {(a+b \cos (c+d x))^4 \left (C \cos ^2(c+d x)+A\right )}{\cos ^{\frac {5}{2}}(c+d x)}dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^4 \left (C \sin \left (c+d x+\frac {\pi }{2}\right )^2+A\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{5/2}}dx\)

\(\Big \downarrow \) 3527

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {2}{3} \int \frac {(a+b \cos (c+d x))^3 \left (-b (7 A-3 C) \cos ^2(c+d x)+a (A+3 C) \cos (c+d x)+8 A b\right )}{2 \cos ^{\frac {3}{2}}(c+d x)}dx+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{3} \int \frac {(a+b \cos (c+d x))^3 \left (-b (7 A-3 C) \cos ^2(c+d x)+a (A+3 C) \cos (c+d x)+8 A b\right )}{\cos ^{\frac {3}{2}}(c+d x)}dx+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{3} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^3 \left (-b (7 A-3 C) \sin \left (c+d x+\frac {\pi }{2}\right )^2+a (A+3 C) \sin \left (c+d x+\frac {\pi }{2}\right )+8 A b\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2}}dx+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )\)

\(\Big \downarrow \) 3526

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{3} \left (2 \int \frac {(a+b \cos (c+d x))^2 \left ((A+3 C) a^2-2 b (7 A-3 C) \cos (c+d x) a+48 A b^2-3 b^2 (21 A-C) \cos ^2(c+d x)\right )}{2 \sqrt {\cos (c+d x)}}dx+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{3} \left (\int \frac {(a+b \cos (c+d x))^2 \left ((A+3 C) a^2-2 b (7 A-3 C) \cos (c+d x) a+48 A b^2-3 b^2 (21 A-C) \cos ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}}dx+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{3} \left (\int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^2 \left ((A+3 C) a^2-2 b (7 A-3 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a+48 A b^2-3 b^2 (21 A-C) \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )\)

\(\Big \downarrow \) 3528

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{3} \left (\frac {2}{7} \int \frac {(a+b \cos (c+d x)) \left (-2 a b^2 (175 A-27 C) \cos ^2(c+d x)-b \left (a^2 (91 A-63 C)-3 b^2 (7 A+5 C)\right ) \cos (c+d x)+a \left (7 (A+3 C) a^2+3 b^2 (91 A+C)\right )\right )}{2 \sqrt {\cos (c+d x)}}dx-\frac {6 b^2 (21 A-C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{3} \left (\frac {1}{7} \int \frac {(a+b \cos (c+d x)) \left (-2 a b^2 (175 A-27 C) \cos ^2(c+d x)-b \left (7 a^2 (13 A-9 C)-3 b^2 (7 A+5 C)\right ) \cos (c+d x)+a \left (7 (A+3 C) a^2+3 b^2 (91 A+C)\right )\right )}{\sqrt {\cos (c+d x)}}dx-\frac {6 b^2 (21 A-C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{3} \left (\frac {1}{7} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right ) \left (-2 a b^2 (175 A-27 C) \sin \left (c+d x+\frac {\pi }{2}\right )^2-b \left (7 a^2 (13 A-9 C)-3 b^2 (7 A+5 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+a \left (7 (A+3 C) a^2+3 b^2 (91 A+C)\right )\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {6 b^2 (21 A-C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )\)

\(\Big \downarrow \) 3512

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{3} \left (\frac {1}{7} \left (\frac {2}{5} \int \frac {5 \left (7 (A+3 C) a^2+3 b^2 (91 A+C)\right ) a^2-84 b \left (5 a^2 (A-C)-b^2 (5 A+3 C)\right ) \cos (c+d x) a-15 b^2 \left (3 a^2 (49 A-13 C)-b^2 (7 A+5 C)\right ) \cos ^2(c+d x)}{2 \sqrt {\cos (c+d x)}}dx-\frac {4 a b^3 (175 A-27 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )-\frac {6 b^2 (21 A-C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{3} \left (\frac {1}{7} \left (\frac {1}{5} \int \frac {5 \left (7 (A+3 C) a^2+3 b^2 (91 A+C)\right ) a^2-84 b \left (5 a^2 (A-C)-b^2 (5 A+3 C)\right ) \cos (c+d x) a-15 b^2 \left (3 a^2 (49 A-13 C)-b^2 (7 A+5 C)\right ) \cos ^2(c+d x)}{\sqrt {\cos (c+d x)}}dx-\frac {4 a b^3 (175 A-27 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )-\frac {6 b^2 (21 A-C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{3} \left (\frac {1}{7} \left (\frac {1}{5} \int \frac {5 \left (7 (A+3 C) a^2+3 b^2 (91 A+C)\right ) a^2-84 b \left (5 a^2 (A-C)-b^2 (5 A+3 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) a-15 b^2 \left (3 a^2 (49 A-13 C)-b^2 (7 A+5 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {4 a b^3 (175 A-27 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )-\frac {6 b^2 (21 A-C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )\)

\(\Big \downarrow \) 3502

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{3} \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {2}{3} \int \frac {3 \left (5 \left (7 (A+3 C) a^4+42 b^2 (3 A+C) a^2+b^4 (7 A+5 C)\right )-84 a b \left (5 a^2 (A-C)-b^2 (5 A+3 C)\right ) \cos (c+d x)\right )}{2 \sqrt {\cos (c+d x)}}dx-\frac {10 b^2 \left (3 a^2 (49 A-13 C)-b^2 (7 A+5 C)\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{d}\right )-\frac {4 a b^3 (175 A-27 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )-\frac {6 b^2 (21 A-C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{3} \left (\frac {1}{7} \left (\frac {1}{5} \left (\int \frac {5 \left (7 (A+3 C) a^4+42 b^2 (3 A+C) a^2+b^4 (7 A+5 C)\right )-84 a b \left (5 a^2 (A-C)-b^2 (5 A+3 C)\right ) \cos (c+d x)}{\sqrt {\cos (c+d x)}}dx-\frac {10 b^2 \left (3 a^2 (49 A-13 C)-b^2 (7 A+5 C)\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{d}\right )-\frac {4 a b^3 (175 A-27 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )-\frac {6 b^2 (21 A-C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{3} \left (\frac {1}{7} \left (\frac {1}{5} \left (\int \frac {5 \left (7 (A+3 C) a^4+42 b^2 (3 A+C) a^2+b^4 (7 A+5 C)\right )-84 a b \left (5 a^2 (A-C)-b^2 (5 A+3 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {10 b^2 \left (3 a^2 (49 A-13 C)-b^2 (7 A+5 C)\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{d}\right )-\frac {4 a b^3 (175 A-27 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )-\frac {6 b^2 (21 A-C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )\)

\(\Big \downarrow \) 3227

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{3} \left (\frac {1}{7} \left (\frac {1}{5} \left (-84 a b \left (5 a^2 (A-C)-b^2 (5 A+3 C)\right ) \int \sqrt {\cos (c+d x)}dx+5 \left (7 a^4 (A+3 C)+42 a^2 b^2 (3 A+C)+b^4 (7 A+5 C)\right ) \int \frac {1}{\sqrt {\cos (c+d x)}}dx-\frac {10 b^2 \left (3 a^2 (49 A-13 C)-b^2 (7 A+5 C)\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{d}\right )-\frac {4 a b^3 (175 A-27 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )-\frac {6 b^2 (21 A-C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{3} \left (\frac {1}{7} \left (\frac {1}{5} \left (-84 a b \left (5 a^2 (A-C)-b^2 (5 A+3 C)\right ) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx+5 \left (7 a^4 (A+3 C)+42 a^2 b^2 (3 A+C)+b^4 (7 A+5 C)\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {10 b^2 \left (3 a^2 (49 A-13 C)-b^2 (7 A+5 C)\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{d}\right )-\frac {4 a b^3 (175 A-27 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )-\frac {6 b^2 (21 A-C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )\)

\(\Big \downarrow \) 3119

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{3} \left (\frac {1}{7} \left (\frac {1}{5} \left (5 \left (7 a^4 (A+3 C)+42 a^2 b^2 (3 A+C)+b^4 (7 A+5 C)\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {168 a b \left (5 a^2 (A-C)-b^2 (5 A+3 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}-\frac {10 b^2 \left (3 a^2 (49 A-13 C)-b^2 (7 A+5 C)\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{d}\right )-\frac {4 a b^3 (175 A-27 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )-\frac {6 b^2 (21 A-C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )\)

\(\Big \downarrow \) 3120

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{3} \left (\frac {1}{7} \left (\frac {1}{5} \left (-\frac {168 a b \left (5 a^2 (A-C)-b^2 (5 A+3 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}-\frac {10 b^2 \left (3 a^2 (49 A-13 C)-b^2 (7 A+5 C)\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{d}+\frac {10 \left (7 a^4 (A+3 C)+42 a^2 b^2 (3 A+C)+b^4 (7 A+5 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{d}\right )-\frac {4 a b^3 (175 A-27 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )-\frac {6 b^2 (21 A-C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )\)

Input:

Int[(a + b*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2),x]
 

Output:

Sqrt[Cos[c + d*x]]*Sqrt[Sec[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)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3119

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]
 

rule 3120

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]
 

rule 3227

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]
 

rule 3502

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]
 

rule 3512

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]
 

rule 3526

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]
 

rule 3527

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]
 

rule 3528

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])))
 

rule 4709

Int[(u_)*((c_.)*sec[(a_.) + (b_.)*(x_)])^(m_.), x_Symbol] :> Simp[(c*Sec[a 
+ b*x])^m*(c*Cos[a + b*x])^m   Int[ActivateTrig[u]/(c*Cos[a + b*x])^m, x], 
x] /; FreeQ[{a, b, c, m}, x] &&  !IntegerQ[m] && KnownSineIntegrandQ[u, x]
 

Maple [F(-1)]

Timed out.

hanged

Input:

int((a+b*cos(d*x+c))^4*(A+C*cos(d*x+c)^2)*sec(d*x+c)^(5/2),x)
 

Output:

int((a+b*cos(d*x+c))^4*(A+C*cos(d*x+c)^2)*sec(d*x+c)^(5/2),x)
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.14 (sec) , antiderivative size = 354, normalized size of antiderivative = 1.04 \[ \int (a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {5}{2}}(c+d x) \, dx=-\frac {5 \, \sqrt {2} {\left (7 i \, {\left (A + 3 \, C\right )} a^{4} + 42 i \, {\left (3 \, A + C\right )} a^{2} b^{2} + i \, {\left (7 \, A + 5 \, C\right )} b^{4}\right )} \cos \left (d x + c\right ) {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 5 \, \sqrt {2} {\left (-7 i \, {\left (A + 3 \, C\right )} a^{4} - 42 i \, {\left (3 \, A + C\right )} a^{2} b^{2} - i \, {\left (7 \, A + 5 \, C\right )} b^{4}\right )} \cos \left (d x + c\right ) {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 84 \, \sqrt {2} {\left (5 i \, {\left (A - C\right )} a^{3} b - i \, {\left (5 \, A + 3 \, C\right )} a b^{3}\right )} \cos \left (d x + c\right ) {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 84 \, \sqrt {2} {\left (-5 i \, {\left (A - C\right )} a^{3} b + i \, {\left (5 \, A + 3 \, C\right )} a b^{3}\right )} \cos \left (d x + c\right ) {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - \frac {2 \, {\left (15 \, C b^{4} \cos \left (d x + c\right )^{4} + 84 \, C a b^{3} \cos \left (d x + c\right )^{3} + 420 \, A a^{3} b \cos \left (d x + c\right ) + 35 \, A a^{4} + 5 \, {\left (42 \, C a^{2} b^{2} + {\left (7 \, A + 5 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{105 \, d \cos \left (d x + c\right )} \] Input:

integrate((a+b*cos(d*x+c))^4*(A+C*cos(d*x+c)^2)*sec(d*x+c)^(5/2),x, algori 
thm="fricas")
 

Output:

-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)*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)*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) 
*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)*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)*sin(d*x + c)/sqrt(cos(d*x + c)))/(d*cos(d*x + c))
 

Sympy [F(-1)]

Timed out. \[ \int (a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {5}{2}}(c+d x) \, dx=\text {Timed out} \] Input:

integrate((a+b*cos(d*x+c))**4*(A+C*cos(d*x+c)**2)*sec(d*x+c)**(5/2),x)
 

Output:

Timed out
                                                                                    
                                                                                    
 

Maxima [F]

\[ \int (a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {5}{2}}(c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{4} \sec \left (d x + c\right )^{\frac {5}{2}} \,d x } \] Input:

integrate((a+b*cos(d*x+c))^4*(A+C*cos(d*x+c)^2)*sec(d*x+c)^(5/2),x, algori 
thm="maxima")
 

Output:

integrate((C*cos(d*x + c)^2 + A)*(b*cos(d*x + c) + a)^4*sec(d*x + c)^(5/2) 
, x)
 

Giac [F]

\[ \int (a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {5}{2}}(c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{4} \sec \left (d x + c\right )^{\frac {5}{2}} \,d x } \] Input:

integrate((a+b*cos(d*x+c))^4*(A+C*cos(d*x+c)^2)*sec(d*x+c)^(5/2),x, algori 
thm="giac")
 

Output:

integrate((C*cos(d*x + c)^2 + A)*(b*cos(d*x + c) + a)^4*sec(d*x + c)^(5/2) 
, x)
 

Mupad [F(-1)]

Timed out. \[ \int (a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {5}{2}}(c+d x) \, dx=\int \left (C\,{\cos \left (c+d\,x\right )}^2+A\right )\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{5/2}\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^4 \,d x \] Input:

int((A + C*cos(c + d*x)^2)*(1/cos(c + d*x))^(5/2)*(a + b*cos(c + d*x))^4,x 
)
 

Output:

int((A + C*cos(c + d*x)^2)*(1/cos(c + d*x))^(5/2)*(a + b*cos(c + d*x))^4, 
x)
 

Reduce [F]

\[ \int (a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {5}{2}}(c+d x) \, dx=4 \left (\int \sqrt {\sec \left (d x +c \right )}\, \cos \left (d x +c \right ) \sec \left (d x +c \right )^{2}d x \right ) a^{4} b +\left (\int \sqrt {\sec \left (d x +c \right )}\, \cos \left (d x +c \right )^{6} \sec \left (d x +c \right )^{2}d x \right ) b^{4} c +4 \left (\int \sqrt {\sec \left (d x +c \right )}\, \cos \left (d x +c \right )^{5} \sec \left (d x +c \right )^{2}d x \right ) a \,b^{3} c +6 \left (\int \sqrt {\sec \left (d x +c \right )}\, \cos \left (d x +c \right )^{4} \sec \left (d x +c \right )^{2}d x \right ) a^{2} b^{2} c +\left (\int \sqrt {\sec \left (d x +c \right )}\, \cos \left (d x +c \right )^{4} \sec \left (d x +c \right )^{2}d x \right ) a \,b^{4}+4 \left (\int \sqrt {\sec \left (d x +c \right )}\, \cos \left (d x +c \right )^{3} \sec \left (d x +c \right )^{2}d x \right ) a^{3} b c +4 \left (\int \sqrt {\sec \left (d x +c \right )}\, \cos \left (d x +c \right )^{3} \sec \left (d x +c \right )^{2}d x \right ) a^{2} b^{3}+\left (\int \sqrt {\sec \left (d x +c \right )}\, \cos \left (d x +c \right )^{2} \sec \left (d x +c \right )^{2}d x \right ) a^{4} c +6 \left (\int \sqrt {\sec \left (d x +c \right )}\, \cos \left (d x +c \right )^{2} \sec \left (d x +c \right )^{2}d x \right ) a^{3} b^{2}+\left (\int \sqrt {\sec \left (d x +c \right )}\, \sec \left (d x +c \right )^{2}d x \right ) a^{5} \] Input:

int((a+b*cos(d*x+c))^4*(A+C*cos(d*x+c)^2)*sec(d*x+c)^(5/2),x)
 

Output:

4*int(sqrt(sec(c + d*x))*cos(c + d*x)*sec(c + d*x)**2,x)*a**4*b + int(sqrt 
(sec(c + d*x))*cos(c + d*x)**6*sec(c + d*x)**2,x)*b**4*c + 4*int(sqrt(sec( 
c + d*x))*cos(c + d*x)**5*sec(c + d*x)**2,x)*a*b**3*c + 6*int(sqrt(sec(c + 
 d*x))*cos(c + d*x)**4*sec(c + d*x)**2,x)*a**2*b**2*c + int(sqrt(sec(c + d 
*x))*cos(c + d*x)**4*sec(c + d*x)**2,x)*a*b**4 + 4*int(sqrt(sec(c + d*x))* 
cos(c + d*x)**3*sec(c + d*x)**2,x)*a**3*b*c + 4*int(sqrt(sec(c + d*x))*cos 
(c + d*x)**3*sec(c + d*x)**2,x)*a**2*b**3 + int(sqrt(sec(c + d*x))*cos(c + 
 d*x)**2*sec(c + d*x)**2,x)*a**4*c + 6*int(sqrt(sec(c + d*x))*cos(c + d*x) 
**2*sec(c + d*x)**2,x)*a**3*b**2 + int(sqrt(sec(c + d*x))*sec(c + d*x)**2, 
x)*a**5