\(\int (a+b \cos (c+d x))^4 (A+C \cos ^2(c+d x)) \sec ^{\frac {9}{2}}(c+d x) \, dx\) [1385]

Optimal result

Integrand size = 35, antiderivative size = 356 \[ \int (a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {9}{2}}(c+d x) \, dx=-\frac {8 a b \left (5 b^2 (A-C)+a^2 (3 A+5 C)\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{5 d}+\frac {2 \left (7 b^4 (3 A+C)+42 a^2 b^2 (A+3 C)+a^4 (5 A+7 C)\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{21 d}-\frac {2 b^2 \left (b^2 (87 A-35 C)+5 a^2 (5 A+7 C)\right ) \sin (c+d x)}{105 d \sqrt {\sec (c+d x)}}+\frac {4 a b \left (96 A b^2+a^2 (101 A+175 C)\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{105 d}+\frac {2 \left (48 A b^2+5 a^2 (5 A+7 C)\right ) (a+b \cos (c+d x))^2 \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac {16 A b (a+b \cos (c+d x))^3 \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac {2 A (a+b \cos (c+d x))^4 \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{7 d} \] Output:

-8/5*a*b*(5*b^2*(A-C)+a^2*(3*A+5*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*(7*b^4*(3*A+C)+42*a^2*b^2*(A+3*C 
)+a^4*(5*A+7*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-2/105*b^2*(b^2*(87*A-35*C)+5*a^2*(5*A+7*C))*sin(d*x+c)/d 
/sec(d*x+c)^(1/2)+4/105*a*b*(96*A*b^2+a^2*(101*A+175*C))*sec(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*sec(d*x+c 
)^(3/2)*sin(d*x+c)/d+16/35*A*b*(a+b*cos(d*x+c))^3*sec(d*x+c)^(5/2)*sin(d*x 
+c)/d+2/7*A*(a+b*cos(d*x+c))^4*sec(d*x+c)^(7/2)*sin(d*x+c)/d
 

Mathematica [A] (verified)

Time = 7.49 (sec) , antiderivative size = 296, normalized size of antiderivative = 0.83 \[ \int (a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {9}{2}}(c+d x) \, dx=\frac {2 \sec ^{\frac {7}{2}}(c+d x) \left (-84 a b \left (5 b^2 (A-C)+a^2 (3 A+5 C)\right ) \cos ^{\frac {7}{2}}(c+d x) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+5 \left (7 b^4 (3 A+C)+42 a^2 b^2 (A+3 C)+a^4 (5 A+7 C)\right ) \cos ^{\frac {7}{2}}(c+d x) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+15 a^4 A \sin (c+d x)+25 a^4 A \cos ^2(c+d x) \sin (c+d x)+210 a^2 A b^2 \cos ^2(c+d x) \sin (c+d x)+35 a^4 C \cos ^2(c+d x) \sin (c+d x)+252 a^3 A b \cos ^3(c+d x) \sin (c+d x)+420 a A b^3 \cos ^3(c+d x) \sin (c+d x)+420 a^3 b C \cos ^3(c+d x) \sin (c+d x)+35 b^4 C \cos ^4(c+d x) \sin (c+d x)+42 a^3 A b \sin (2 (c+d x))\right )}{105 d} \] Input:

Integrate[(a + b*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2) 
,x]
 

Output:

(2*Sec[c + d*x]^(7/2)*(-84*a*b*(5*b^2*(A - C) + a^2*(3*A + 5*C))*Cos[c + d 
*x]^(7/2)*EllipticE[(c + d*x)/2, 2] + 5*(7*b^4*(3*A + C) + 42*a^2*b^2*(A + 
 3*C) + a^4*(5*A + 7*C))*Cos[c + d*x]^(7/2)*EllipticF[(c + d*x)/2, 2] + 15 
*a^4*A*Sin[c + d*x] + 25*a^4*A*Cos[c + d*x]^2*Sin[c + d*x] + 210*a^2*A*b^2 
*Cos[c + d*x]^2*Sin[c + d*x] + 35*a^4*C*Cos[c + d*x]^2*Sin[c + d*x] + 252* 
a^3*A*b*Cos[c + d*x]^3*Sin[c + d*x] + 420*a*A*b^3*Cos[c + d*x]^3*Sin[c + d 
*x] + 420*a^3*b*C*Cos[c + d*x]^3*Sin[c + d*x] + 35*b^4*C*Cos[c + d*x]^4*Si 
n[c + d*x] + 42*a^3*A*b*Sin[2*(c + d*x)]))/(105*d)
 

Rubi [A] (verified)

Time = 2.48 (sec) , antiderivative size = 344, normalized size of antiderivative = 0.97, 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, 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.

\(\displaystyle \int \sec ^{\frac {9}{2}}(c+d x) (a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \sec (c+d x)^{9/2} (a+b \cos (c+d x))^4 \left (A+C \cos (c+d x)^2\right )dx\)

\(\Big \downarrow \) 4709

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {(a+b \cos (c+d x))^4 \left (C \cos ^2(c+d x)+A\right )}{\cos ^{\frac {9}{2}}(c+d x)}dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^4 \left (C \sin \left (c+d x+\frac {\pi }{2}\right )^2+A\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{9/2}}dx\)

\(\Big \downarrow \) 3527

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {2}{7} \int \frac {(a+b \cos (c+d x))^3 \left (-b (3 A-7 C) \cos ^2(c+d x)+a (5 A+7 C) \cos (c+d x)+8 A b\right )}{2 \cos ^{\frac {7}{2}}(c+d x)}dx+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{7} \int \frac {(a+b \cos (c+d x))^3 \left (-b (3 A-7 C) \cos ^2(c+d x)+a (5 A+7 C) \cos (c+d x)+8 A b\right )}{\cos ^{\frac {7}{2}}(c+d x)}dx+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{7} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^3 \left (-b (3 A-7 C) \sin \left (c+d x+\frac {\pi }{2}\right )^2+a (5 A+7 C) \sin \left (c+d x+\frac {\pi }{2}\right )+8 A b\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{7/2}}dx+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )\)

\(\Big \downarrow \) 3526

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{7} \left (\frac {2}{5} \int \frac {(a+b \cos (c+d x))^2 \left (5 (5 A+7 C) a^2+2 b (17 A+35 C) \cos (c+d x) a+48 A b^2-b^2 (39 A-35 C) \cos ^2(c+d x)\right )}{2 \cos ^{\frac {5}{2}}(c+d x)}dx+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{7} \left (\frac {1}{5} \int \frac {(a+b \cos (c+d x))^2 \left (5 (5 A+7 C) a^2+2 b (17 A+35 C) \cos (c+d x) a+48 A b^2-b^2 (39 A-35 C) \cos ^2(c+d x)\right )}{\cos ^{\frac {5}{2}}(c+d x)}dx+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{7} \left (\frac {1}{5} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^2 \left (5 (5 A+7 C) a^2+2 b (17 A+35 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a+48 A b^2-b^2 (39 A-35 C) \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{5/2}}dx+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )\)

\(\Big \downarrow \) 3526

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {2}{3} \int \frac {(a+b \cos (c+d x)) \left (-3 b \left (5 (5 A+7 C) a^2+b^2 (87 A-35 C)\right ) \cos ^2(c+d x)+a \left (5 (5 A+7 C) a^2+3 b^2 (11 A+105 C)\right ) \cos (c+d x)+2 b \left ((101 A+175 C) a^2+96 A b^2\right )\right )}{2 \cos ^{\frac {3}{2}}(c+d x)}dx+\frac {2 \left (5 a^2 (5 A+7 C)+48 A b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^2}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \int \frac {(a+b \cos (c+d x)) \left (-3 b \left (5 (5 A+7 C) a^2+b^2 (87 A-35 C)\right ) \cos ^2(c+d x)+a \left (5 (5 A+7 C) a^2+3 b^2 (11 A+105 C)\right ) \cos (c+d x)+2 b \left ((101 A+175 C) a^2+96 A b^2\right )\right )}{\cos ^{\frac {3}{2}}(c+d x)}dx+\frac {2 \left (5 a^2 (5 A+7 C)+48 A b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^2}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right ) \left (-3 b \left (5 (5 A+7 C) a^2+b^2 (87 A-35 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+a \left (5 (5 A+7 C) a^2+3 b^2 (11 A+105 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+2 b \left ((101 A+175 C) a^2+96 A b^2\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2}}dx+\frac {2 \left (5 a^2 (5 A+7 C)+48 A b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^2}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )\)

\(\Big \downarrow \) 3510

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {4 a b \left (a^2 (101 A+175 C)+96 A b^2\right ) \sin (c+d x)}{d \sqrt {\cos (c+d x)}}-2 \int -\frac {5 (5 A+7 C) a^4+5 b^2 (47 A+133 C) a^2-84 b \left ((3 A+5 C) a^2+5 b^2 (A-C)\right ) \cos (c+d x) a+192 A b^4-3 b^2 \left (5 (5 A+7 C) a^2+b^2 (87 A-35 C)\right ) \cos ^2(c+d x)}{2 \sqrt {\cos (c+d x)}}dx\right )+\frac {2 \left (5 a^2 (5 A+7 C)+48 A b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^2}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\int \frac {5 (5 A+7 C) a^4+5 b^2 (47 A+133 C) a^2-84 b \left ((3 A+5 C) a^2+5 b^2 (A-C)\right ) \cos (c+d x) a+192 A b^4-3 b^2 \left (5 (5 A+7 C) a^2+b^2 (87 A-35 C)\right ) \cos ^2(c+d x)}{\sqrt {\cos (c+d x)}}dx+\frac {4 a b \left (a^2 (101 A+175 C)+96 A b^2\right ) \sin (c+d x)}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 \left (5 a^2 (5 A+7 C)+48 A b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^2}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\int \frac {5 (5 A+7 C) a^4+5 b^2 (47 A+133 C) a^2-84 b \left ((3 A+5 C) a^2+5 b^2 (A-C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) a+192 A b^4-3 b^2 \left (5 (5 A+7 C) a^2+b^2 (87 A-35 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {4 a b \left (a^2 (101 A+175 C)+96 A b^2\right ) \sin (c+d x)}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 \left (5 a^2 (5 A+7 C)+48 A b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^2}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )\)

\(\Big \downarrow \) 3502

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {2}{3} \int \frac {3 \left (5 \left ((5 A+7 C) a^4+42 b^2 (A+3 C) a^2+7 b^4 (3 A+C)\right )-84 a b \left ((3 A+5 C) a^2+5 b^2 (A-C)\right ) \cos (c+d x)\right )}{2 \sqrt {\cos (c+d x)}}dx-\frac {2 b^2 \left (5 a^2 (5 A+7 C)+b^2 (87 A-35 C)\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{d}+\frac {4 a b \left (a^2 (101 A+175 C)+96 A b^2\right ) \sin (c+d x)}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 \left (5 a^2 (5 A+7 C)+48 A b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^2}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\int \frac {5 \left ((5 A+7 C) a^4+42 b^2 (A+3 C) a^2+7 b^4 (3 A+C)\right )-84 a b \left ((3 A+5 C) a^2+5 b^2 (A-C)\right ) \cos (c+d x)}{\sqrt {\cos (c+d x)}}dx-\frac {2 b^2 \left (5 a^2 (5 A+7 C)+b^2 (87 A-35 C)\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{d}+\frac {4 a b \left (a^2 (101 A+175 C)+96 A b^2\right ) \sin (c+d x)}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 \left (5 a^2 (5 A+7 C)+48 A b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^2}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\int \frac {5 \left ((5 A+7 C) a^4+42 b^2 (A+3 C) a^2+7 b^4 (3 A+C)\right )-84 a b \left ((3 A+5 C) a^2+5 b^2 (A-C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {2 b^2 \left (5 a^2 (5 A+7 C)+b^2 (87 A-35 C)\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{d}+\frac {4 a b \left (a^2 (101 A+175 C)+96 A b^2\right ) \sin (c+d x)}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 \left (5 a^2 (5 A+7 C)+48 A b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^2}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )\)

\(\Big \downarrow \) 3227

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (-84 a b \left (a^2 (3 A+5 C)+5 b^2 (A-C)\right ) \int \sqrt {\cos (c+d x)}dx+5 \left (a^4 (5 A+7 C)+42 a^2 b^2 (A+3 C)+7 b^4 (3 A+C)\right ) \int \frac {1}{\sqrt {\cos (c+d x)}}dx-\frac {2 b^2 \left (5 a^2 (5 A+7 C)+b^2 (87 A-35 C)\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{d}+\frac {4 a b \left (a^2 (101 A+175 C)+96 A b^2\right ) \sin (c+d x)}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 \left (5 a^2 (5 A+7 C)+48 A b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^2}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (-84 a b \left (a^2 (3 A+5 C)+5 b^2 (A-C)\right ) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx+5 \left (a^4 (5 A+7 C)+42 a^2 b^2 (A+3 C)+7 b^4 (3 A+C)\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {2 b^2 \left (5 a^2 (5 A+7 C)+b^2 (87 A-35 C)\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{d}+\frac {4 a b \left (a^2 (101 A+175 C)+96 A b^2\right ) \sin (c+d x)}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 \left (5 a^2 (5 A+7 C)+48 A b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^2}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )\)

\(\Big \downarrow \) 3119

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (5 \left (a^4 (5 A+7 C)+42 a^2 b^2 (A+3 C)+7 b^4 (3 A+C)\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {168 a b \left (a^2 (3 A+5 C)+5 b^2 (A-C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}-\frac {2 b^2 \left (5 a^2 (5 A+7 C)+b^2 (87 A-35 C)\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{d}+\frac {4 a b \left (a^2 (101 A+175 C)+96 A b^2\right ) \sin (c+d x)}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 \left (5 a^2 (5 A+7 C)+48 A b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^2}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )\)

\(\Big \downarrow \) 3120

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {2 \left (5 a^2 (5 A+7 C)+48 A b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^2}{3 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {1}{3} \left (-\frac {168 a b \left (a^2 (3 A+5 C)+5 b^2 (A-C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}-\frac {2 b^2 \left (5 a^2 (5 A+7 C)+b^2 (87 A-35 C)\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{d}+\frac {4 a b \left (a^2 (101 A+175 C)+96 A b^2\right ) \sin (c+d x)}{d \sqrt {\cos (c+d x)}}+\frac {10 \left (a^4 (5 A+7 C)+42 a^2 b^2 (A+3 C)+7 b^4 (3 A+C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{d}\right )\right )+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )\)

Input:

Int[(a + b*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2),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])/(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*(48*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 + 175*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)
 

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 3510

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]
 

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 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)^(9/2),x)
 

Output:

int((a+b*cos(d*x+c))^4*(A+C*cos(d*x+c)^2)*sec(d*x+c)^(9/2),x)
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.14 (sec) , antiderivative size = 377, normalized size of antiderivative = 1.06 \[ \int (a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {9}{2}}(c+d x) \, dx=-\frac {5 \, \sqrt {2} {\left (i \, {\left (5 \, A + 7 \, C\right )} a^{4} + 42 i \, {\left (A + 3 \, C\right )} a^{2} b^{2} + 7 i \, {\left (3 \, A + C\right )} b^{4}\right )} \cos \left (d x + c\right )^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 5 \, \sqrt {2} {\left (-i \, {\left (5 \, A + 7 \, C\right )} a^{4} - 42 i \, {\left (A + 3 \, C\right )} a^{2} b^{2} - 7 i \, {\left (3 \, A + C\right )} b^{4}\right )} \cos \left (d x + c\right )^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 84 \, \sqrt {2} {\left (i \, {\left (3 \, A + 5 \, C\right )} a^{3} b + 5 i \, {\left (A - C\right )} a b^{3}\right )} \cos \left (d x + c\right )^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 84 \, \sqrt {2} {\left (-i \, {\left (3 \, A + 5 \, C\right )} a^{3} b - 5 i \, {\left (A - C\right )} a b^{3}\right )} \cos \left (d x + c\right )^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - \frac {2 \, {\left (35 \, C b^{4} \cos \left (d x + c\right )^{4} + 84 \, A a^{3} b \cos \left (d x + c\right ) + 15 \, A a^{4} + 84 \, {\left ({\left (3 \, A + 5 \, C\right )} a^{3} b + 5 \, A a b^{3}\right )} \cos \left (d x + c\right )^{3} + 5 \, {\left ({\left (5 \, A + 7 \, C\right )} a^{4} + 42 \, A a^{2} b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{105 \, d \cos \left (d x + c\right )^{3}} \] Input:

integrate((a+b*cos(d*x+c))^4*(A+C*cos(d*x+c)^2)*sec(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)^3*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)^3*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)^3*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)^3*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)*sin(d*x + c)/sqrt(cos(d*x + c)) 
)/(d*cos(d*x + c)^3)
 

Sympy [F(-1)]

Timed out. \[ \int (a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {9}{2}}(c+d x) \, dx=\text {Timed out} \] Input:

integrate((a+b*cos(d*x+c))**4*(A+C*cos(d*x+c)**2)*sec(d*x+c)**(9/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int (a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {9}{2}}(c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{4} \sec \left (d x + c\right )^{\frac {9}{2}} \,d x } \] Input:

integrate((a+b*cos(d*x+c))^4*(A+C*cos(d*x+c)^2)*sec(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*sec(d*x + c)^(9/2) 
, x)
 

Giac [F]

\[ \int (a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {9}{2}}(c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{4} \sec \left (d x + c\right )^{\frac {9}{2}} \,d x } \] Input:

integrate((a+b*cos(d*x+c))^4*(A+C*cos(d*x+c)^2)*sec(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*sec(d*x + c)^(9/2) 
, x)
 

Mupad [F(-1)]

Timed out. \[ \int (a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {9}{2}}(c+d x) \, dx=\int \left (C\,{\cos \left (c+d\,x\right )}^2+A\right )\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{9/2}\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^4 \,d x \] Input:

int((A + C*cos(c + d*x)^2)*(1/cos(c + d*x))^(9/2)*(a + b*cos(c + d*x))^4,x 
)
 

Output:

int((A + C*cos(c + d*x)^2)*(1/cos(c + d*x))^(9/2)*(a + b*cos(c + d*x))^4, 
x)
 

Reduce [F]

\[ \int (a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {9}{2}}(c+d x) \, dx=4 \left (\int \sqrt {\sec \left (d x +c \right )}\, \cos \left (d x +c \right ) \sec \left (d x +c \right )^{4}d x \right ) a^{4} b +\left (\int \sqrt {\sec \left (d x +c \right )}\, \cos \left (d x +c \right )^{6} \sec \left (d x +c \right )^{4}d x \right ) b^{4} c +4 \left (\int \sqrt {\sec \left (d x +c \right )}\, \cos \left (d x +c \right )^{5} \sec \left (d x +c \right )^{4}d x \right ) a \,b^{3} c +6 \left (\int \sqrt {\sec \left (d x +c \right )}\, \cos \left (d x +c \right )^{4} \sec \left (d x +c \right )^{4}d x \right ) a^{2} b^{2} c +\left (\int \sqrt {\sec \left (d x +c \right )}\, \cos \left (d x +c \right )^{4} \sec \left (d x +c \right )^{4}d x \right ) a \,b^{4}+4 \left (\int \sqrt {\sec \left (d x +c \right )}\, \cos \left (d x +c \right )^{3} \sec \left (d x +c \right )^{4}d x \right ) a^{3} b c +4 \left (\int \sqrt {\sec \left (d x +c \right )}\, \cos \left (d x +c \right )^{3} \sec \left (d x +c \right )^{4}d x \right ) a^{2} b^{3}+\left (\int \sqrt {\sec \left (d x +c \right )}\, \cos \left (d x +c \right )^{2} \sec \left (d x +c \right )^{4}d x \right ) a^{4} c +6 \left (\int \sqrt {\sec \left (d x +c \right )}\, \cos \left (d x +c \right )^{2} \sec \left (d x +c \right )^{4}d x \right ) a^{3} b^{2}+\left (\int \sqrt {\sec \left (d x +c \right )}\, \sec \left (d x +c \right )^{4}d x \right ) a^{5} \] Input:

int((a+b*cos(d*x+c))^4*(A+C*cos(d*x+c)^2)*sec(d*x+c)^(9/2),x)
 

Output:

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