\(\int (a+b \cos (c+d x))^3 (A+C \cos ^2(c+d x)) \sqrt {\sec (c+d x)} \, dx\) [1380]

Optimal result

Integrand size = 35, antiderivative size = 285 \[ \int (a+b \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \sqrt {\sec (c+d x)} \, dx=\frac {2 b \left (9 a^2 (5 A+3 C)+b^2 (9 A+7 C)\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{15 d}+\frac {2 a \left (7 a^2 (3 A+C)+3 b^2 (7 A+5 C)\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{21 d}+\frac {2 b \left (24 a^2 C+7 b^2 (9 A+7 C)\right ) \sin (c+d x)}{315 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 a \left (63 A b^2+8 a^2 C+45 b^2 C\right ) \sin (c+d x)}{63 d \sqrt {\sec (c+d x)}}+\frac {4 a C (a+b \cos (c+d x))^2 \sin (c+d x)}{21 d \sqrt {\sec (c+d x)}}+\frac {2 C (a+b \cos (c+d x))^3 \sin (c+d x)}{9 d \sqrt {\sec (c+d x)}} \] Output:

2/15*b*(9*a^2*(5*A+3*C)+b^2*(9*A+7*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*a*(7*a^2*(3*A+C)+3*b^2*(7*A+5* 
C))*cos(d*x+c)^(1/2)*InverseJacobiAM(1/2*d*x+1/2*c,2^(1/2))*sec(d*x+c)^(1/ 
2)/d+2/315*b*(24*a^2*C+7*b^2*(9*A+7*C))*sin(d*x+c)/d/sec(d*x+c)^(3/2)+2/63 
*a*(63*A*b^2+8*C*a^2+45*C*b^2)*sin(d*x+c)/d/sec(d*x+c)^(1/2)+4/21*a*C*(a+b 
*cos(d*x+c))^2*sin(d*x+c)/d/sec(d*x+c)^(1/2)+2/9*C*(a+b*cos(d*x+c))^3*sin( 
d*x+c)/d/sec(d*x+c)^(1/2)
 

Mathematica [A] (verified)

Time = 4.81 (sec) , antiderivative size = 203, normalized size of antiderivative = 0.71 \[ \int (a+b \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \sqrt {\sec (c+d x)} \, dx=\frac {\sqrt {\sec (c+d x)} \left (168 b \left (9 a^2 (5 A+3 C)+b^2 (9 A+7 C)\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+120 a \left (7 a^2 (3 A+C)+3 b^2 (7 A+5 C)\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+\left (7 b \left (36 A b^2+108 a^2 C+43 b^2 C\right ) \cos (c+d x)+5 \left (252 a A b^2+84 a^3 C+234 a b^2 C+54 a b^2 C \cos (2 (c+d x))+7 b^3 C \cos (3 (c+d x))\right )\right ) \sin (2 (c+d x))\right )}{1260 d} \] Input:

Integrate[(a + b*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]] 
,x]
 

Output:

(Sqrt[Sec[c + d*x]]*(168*b*(9*a^2*(5*A + 3*C) + b^2*(9*A + 7*C))*Sqrt[Cos[ 
c + d*x]]*EllipticE[(c + d*x)/2, 2] + 120*a*(7*a^2*(3*A + C) + 3*b^2*(7*A 
+ 5*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + (7*b*(36*A*b^2 + 10 
8*a^2*C + 43*b^2*C)*Cos[c + d*x] + 5*(252*a*A*b^2 + 84*a^3*C + 234*a*b^2*C 
 + 54*a*b^2*C*Cos[2*(c + d*x)] + 7*b^3*C*Cos[3*(c + d*x)]))*Sin[2*(c + d*x 
)]))/(1260*d)
 

Rubi [A] (verified)

Time = 1.77 (sec) , antiderivative size = 278, normalized size of antiderivative = 0.98, number of steps used = 19, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.543, Rules used = {3042, 4709, 3042, 3529, 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.

\(\displaystyle \int \sqrt {\sec (c+d x)} (a+b \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \, dx\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4709

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3529

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {2}{9} \int \frac {(a+b \cos (c+d x))^2 \left (6 a C \cos ^2(c+d x)+b (9 A+7 C) \cos (c+d x)+a (9 A+C)\right )}{2 \sqrt {\cos (c+d x)}}dx+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \int \frac {(a+b \cos (c+d x))^2 \left (6 a C \cos ^2(c+d x)+b (9 A+7 C) \cos (c+d x)+a (9 A+C)\right )}{\sqrt {\cos (c+d x)}}dx+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^2 \left (6 a C \sin \left (c+d x+\frac {\pi }{2}\right )^2+b (9 A+7 C) \sin \left (c+d x+\frac {\pi }{2}\right )+a (9 A+C)\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}\right )\)

\(\Big \downarrow \) 3528

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {2}{7} \int \frac {(a+b \cos (c+d x)) \left ((63 A+13 C) a^2+2 b (63 A+43 C) \cos (c+d x) a+\left (24 C a^2+7 b^2 (9 A+7 C)\right ) \cos ^2(c+d x)\right )}{2 \sqrt {\cos (c+d x)}}dx+\frac {12 a C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {1}{7} \int \frac {(a+b \cos (c+d x)) \left ((63 A+13 C) a^2+2 b (63 A+43 C) \cos (c+d x) a+\left (24 C a^2+7 b^2 (9 A+7 C)\right ) \cos ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}}dx+\frac {12 a C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {1}{7} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right ) \left ((63 A+13 C) a^2+2 b (63 A+43 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a+\left (24 C a^2+7 b^2 (9 A+7 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {12 a C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}\right )\)

\(\Big \downarrow \) 3512

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {2}{5} \int \frac {5 (63 A+13 C) a^3+15 \left (8 C a^2+63 A b^2+45 b^2 C\right ) \cos ^2(c+d x) a+21 b \left (9 (5 A+3 C) a^2+b^2 (9 A+7 C)\right ) \cos (c+d x)}{2 \sqrt {\cos (c+d x)}}dx+\frac {2 b \left (24 a^2 C+7 b^2 (9 A+7 C)\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+\frac {12 a C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \int \frac {5 (63 A+13 C) a^3+15 \left (8 C a^2+63 A b^2+45 b^2 C\right ) \cos ^2(c+d x) a+21 b \left (9 (5 A+3 C) a^2+b^2 (9 A+7 C)\right ) \cos (c+d x)}{\sqrt {\cos (c+d x)}}dx+\frac {2 b \left (24 a^2 C+7 b^2 (9 A+7 C)\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+\frac {12 a C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \int \frac {5 (63 A+13 C) a^3+15 \left (8 C a^2+63 A b^2+45 b^2 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2 a+21 b \left (9 (5 A+3 C) a^2+b^2 (9 A+7 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 b \left (24 a^2 C+7 b^2 (9 A+7 C)\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+\frac {12 a C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}\right )\)

\(\Big \downarrow \) 3502

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {2}{3} \int \frac {9 \left (5 a \left (7 (3 A+C) a^2+3 b^2 (7 A+5 C)\right )+7 b \left (9 (5 A+3 C) a^2+b^2 (9 A+7 C)\right ) \cos (c+d x)\right )}{2 \sqrt {\cos (c+d x)}}dx+\frac {10 a \left (8 a^2 C+63 A b^2+45 b^2 C\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{d}\right )+\frac {2 b \left (24 a^2 C+7 b^2 (9 A+7 C)\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+\frac {12 a C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (3 \int \frac {5 a \left (7 (3 A+C) a^2+3 b^2 (7 A+5 C)\right )+7 b \left (9 (5 A+3 C) a^2+b^2 (9 A+7 C)\right ) \cos (c+d x)}{\sqrt {\cos (c+d x)}}dx+\frac {10 a \left (8 a^2 C+63 A b^2+45 b^2 C\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{d}\right )+\frac {2 b \left (24 a^2 C+7 b^2 (9 A+7 C)\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+\frac {12 a C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (3 \int \frac {5 a \left (7 (3 A+C) a^2+3 b^2 (7 A+5 C)\right )+7 b \left (9 (5 A+3 C) a^2+b^2 (9 A+7 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {10 a \left (8 a^2 C+63 A b^2+45 b^2 C\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{d}\right )+\frac {2 b \left (24 a^2 C+7 b^2 (9 A+7 C)\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+\frac {12 a C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}\right )\)

\(\Big \downarrow \) 3227

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (3 \left (5 a \left (7 a^2 (3 A+C)+3 b^2 (7 A+5 C)\right ) \int \frac {1}{\sqrt {\cos (c+d x)}}dx+7 b \left (9 a^2 (5 A+3 C)+b^2 (9 A+7 C)\right ) \int \sqrt {\cos (c+d x)}dx\right )+\frac {10 a \left (8 a^2 C+63 A b^2+45 b^2 C\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{d}\right )+\frac {2 b \left (24 a^2 C+7 b^2 (9 A+7 C)\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+\frac {12 a C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (3 \left (5 a \left (7 a^2 (3 A+C)+3 b^2 (7 A+5 C)\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+7 b \left (9 a^2 (5 A+3 C)+b^2 (9 A+7 C)\right ) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx\right )+\frac {10 a \left (8 a^2 C+63 A b^2+45 b^2 C\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{d}\right )+\frac {2 b \left (24 a^2 C+7 b^2 (9 A+7 C)\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+\frac {12 a C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}\right )\)

\(\Big \downarrow \) 3119

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (3 \left (5 a \left (7 a^2 (3 A+C)+3 b^2 (7 A+5 C)\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {14 b \left (9 a^2 (5 A+3 C)+b^2 (9 A+7 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )+\frac {10 a \left (8 a^2 C+63 A b^2+45 b^2 C\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{d}\right )+\frac {2 b \left (24 a^2 C+7 b^2 (9 A+7 C)\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+\frac {12 a C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}\right )\)

\(\Big \downarrow \) 3120

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {2 b \left (24 a^2 C+7 b^2 (9 A+7 C)\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}+\frac {1}{5} \left (\frac {10 a \left (8 a^2 C+63 A b^2+45 b^2 C\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{d}+3 \left (\frac {10 a \left (7 a^2 (3 A+C)+3 b^2 (7 A+5 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{d}+\frac {14 b \left (9 a^2 (5 A+3 C)+b^2 (9 A+7 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )\right )\right )+\frac {12 a C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}\right )\)

Input:

Int[(a + b*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]],x]
 

Output:

Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((2*C*Sqrt[Cos[c + d*x]]*(a + b*Cos[ 
c + d*x])^3*Sin[c + d*x])/(9*d) + ((12*a*C*Sqrt[Cos[c + d*x]]*(a + b*Cos[c 
 + d*x])^2*Sin[c + d*x])/(7*d) + ((2*b*(24*a^2*C + 7*b^2*(9*A + 7*C))*Cos[ 
c + d*x]^(3/2)*Sin[c + d*x])/(5*d) + (3*((14*b*(9*a^2*(5*A + 3*C) + b^2*(9 
*A + 7*C))*EllipticE[(c + d*x)/2, 2])/d + (10*a*(7*a^2*(3*A + C) + 3*b^2*( 
7*A + 5*C))*EllipticF[(c + d*x)/2, 2])/d) + (10*a*(63*A*b^2 + 8*a^2*C + 45 
*b^2*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/d)/5)/7)/9)
 

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 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 3529

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)*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)) + (A*b*d*(m + n + 2) - C*(a*c - b*d*(m + n + 1)))*Sin[e + f*x] + C* 
(a*d*m - b*c*(m + 1))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f 
, A, 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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(717\) vs. \(2(260)=520\).

Time = 29.15 (sec) , antiderivative size = 718, normalized size of antiderivative = 2.52

method	result	size
default	\(-\frac {2 \sqrt {\left (2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \left (-1120 C \,b^{3} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}+\left (2160 C a \,b^{2}+2240 C \,b^{3}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{8} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-504 A \,b^{3}-1512 a^{2} b C -3240 C a \,b^{2}-2072 C \,b^{3}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (1260 a A \,b^{2}+504 A \,b^{3}+420 a^{3} C +1512 a^{2} b C +2520 C a \,b^{2}+952 C \,b^{3}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-630 a A \,b^{2}-126 A \,b^{3}-210 a^{3} C -378 a^{2} b C -720 C a \,b^{2}-168 C \,b^{3}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+315 A \,a^{3} \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+315 a A \,b^{2} \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-945 A \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) a^{2} b -189 A \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) b^{3}+105 a^{3} C \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+225 C a \,b^{2} \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-567 C \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) a^{2} b -147 C \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) b^{3}\right )}{315 \sqrt {-2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, d}\)	\(718\)
parts	\(\text {Expression too large to display}\)	\(1098\)

Input:

int((a+b*cos(d*x+c))^3*(A+C*cos(d*x+c)^2)*sec(d*x+c)^(1/2),x,method=_RETUR 
NVERBOSE)
 

Output:

-2/315*((2*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(-1120*C*b^ 
3*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^10+(2160*C*a*b^2+2240*C*b^3)*sin(1 
/2*d*x+1/2*c)^8*cos(1/2*d*x+1/2*c)+(-504*A*b^3-1512*C*a^2*b-3240*C*a*b^2-2 
072*C*b^3)*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)+(1260*A*a*b^2+504*A*b^3 
+420*C*a^3+1512*C*a^2*b+2520*C*a*b^2+952*C*b^3)*sin(1/2*d*x+1/2*c)^4*cos(1 
/2*d*x+1/2*c)+(-630*A*a*b^2-126*A*b^3-210*C*a^3-378*C*a^2*b-720*C*a*b^2-16 
8*C*b^3)*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)+315*A*a^3*(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))+315*a*A*b^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))-945*A*(sin(1/2*d*x+1/2*c) 
^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^ 
(1/2))*a^2*b-189*A*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1) 
^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*b^3+105*a^3*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))+225*C*a*b^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))-567*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)*EllipticE(cos(1/2*d*x+1/2*c),2^ 
(1/2))*a^2*b-147*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)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*b^3)/(-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...
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.13 (sec) , antiderivative size = 307, normalized size of antiderivative = 1.08 \[ \int (a+b \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \sqrt {\sec (c+d x)} \, dx=-\frac {15 \, \sqrt {2} {\left (7 i \, {\left (3 \, A + C\right )} a^{3} + 3 i \, {\left (7 \, A + 5 \, C\right )} a b^{2}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 15 \, \sqrt {2} {\left (-7 i \, {\left (3 \, A + C\right )} a^{3} - 3 i \, {\left (7 \, A + 5 \, C\right )} a b^{2}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 21 \, \sqrt {2} {\left (-9 i \, {\left (5 \, A + 3 \, C\right )} a^{2} b - i \, {\left (9 \, A + 7 \, C\right )} b^{3}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 21 \, \sqrt {2} {\left (9 i \, {\left (5 \, A + 3 \, C\right )} a^{2} b + i \, {\left (9 \, A + 7 \, C\right )} b^{3}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - \frac {2 \, {\left (35 \, C b^{3} \cos \left (d x + c\right )^{4} + 135 \, C a b^{2} \cos \left (d x + c\right )^{3} + 7 \, {\left (27 \, C a^{2} b + {\left (9 \, A + 7 \, C\right )} b^{3}\right )} \cos \left (d x + c\right )^{2} + 15 \, {\left (7 \, C a^{3} + 3 \, {\left (7 \, A + 5 \, C\right )} a b^{2}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{315 \, d} \] Input:

integrate((a+b*cos(d*x+c))^3*(A+C*cos(d*x+c)^2)*sec(d*x+c)^(1/2),x, algori 
thm="fricas")
 

Output:

-1/315*(15*sqrt(2)*(7*I*(3*A + C)*a^3 + 3*I*(7*A + 5*C)*a*b^2)*weierstrass 
PInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c)) + 15*sqrt(2)*(-7*I*(3*A + C 
)*a^3 - 3*I*(7*A + 5*C)*a*b^2)*weierstrassPInverse(-4, 0, cos(d*x + c) - I 
*sin(d*x + c)) + 21*sqrt(2)*(-9*I*(5*A + 3*C)*a^2*b - I*(9*A + 7*C)*b^3)*w 
eierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x 
+ c))) + 21*sqrt(2)*(9*I*(5*A + 3*C)*a^2*b + I*(9*A + 7*C)*b^3)*weierstras 
sZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c))) - 
2*(35*C*b^3*cos(d*x + c)^4 + 135*C*a*b^2*cos(d*x + c)^3 + 7*(27*C*a^2*b + 
(9*A + 7*C)*b^3)*cos(d*x + c)^2 + 15*(7*C*a^3 + 3*(7*A + 5*C)*a*b^2)*cos(d 
*x + c))*sin(d*x + c)/sqrt(cos(d*x + c)))/d
 

Sympy [F(-1)]

Timed out. \[ \int (a+b \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \sqrt {\sec (c+d x)} \, dx=\text {Timed out} \] Input:

integrate((a+b*cos(d*x+c))**3*(A+C*cos(d*x+c)**2)*sec(d*x+c)**(1/2),x)
 

Output:

Timed out
 

Maxima [F]

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

integrate((a+b*cos(d*x+c))^3*(A+C*cos(d*x+c)^2)*sec(d*x+c)^(1/2),x, algori 
thm="maxima")
 

Output:

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

Giac [F]

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

integrate((a+b*cos(d*x+c))^3*(A+C*cos(d*x+c)^2)*sec(d*x+c)^(1/2),x, algori 
thm="giac")
 

Output:

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

Mupad [F(-1)]

Timed out. \[ \int (a+b \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \sqrt {\sec (c+d x)} \, dx=\int \left (C\,{\cos \left (c+d\,x\right )}^2+A\right )\,\sqrt {\frac {1}{\cos \left (c+d\,x\right )}}\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^3 \,d x \] Input:

int((A + C*cos(c + d*x)^2)*(1/cos(c + d*x))^(1/2)*(a + b*cos(c + d*x))^3,x 
)
 

Output:

int((A + C*cos(c + d*x)^2)*(1/cos(c + d*x))^(1/2)*(a + b*cos(c + d*x))^3, 
x)
 

Reduce [F]

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

int((a+b*cos(d*x+c))^3*(A+C*cos(d*x+c)^2)*sec(d*x+c)^(1/2),x)
 

Output:

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