\(\int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3 (A+B \cos (c+d x)+C \cos ^2(c+d x)) \, dx\) [1080]

Optimal result

Integrand size = 43, antiderivative size = 361 \[ \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {2 \left (27 a^2 b B+7 b^3 B+3 a^3 (5 A+3 C)+3 a b^2 (9 A+7 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {2 \left (77 a^3 B+165 a b^2 B+33 a^2 b (7 A+5 C)+5 b^3 (11 A+9 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{231 d}+\frac {2 \left (77 a^3 B+165 a b^2 B+33 a^2 b (7 A+5 C)+5 b^3 (11 A+9 C)\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{231 d}+\frac {2 \left (242 a^2 b B+77 b^3 B+24 a^3 C+33 a b^2 (9 A+7 C)\right ) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{495 d}+\frac {2 b \left (99 A b^2+143 a b B+24 a^2 C+81 b^2 C\right ) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{693 d}+\frac {2 (11 b B+6 a C) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{99 d}+\frac {2 C \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3 \sin (c+d x)}{11 d} \] Output:

2/15*(27*B*a^2*b+7*B*b^3+3*a^3*(5*A+3*C)+3*a*b^2*(9*A+7*C))*EllipticE(sin( 
1/2*d*x+1/2*c),2^(1/2))/d+2/231*(77*B*a^3+165*B*a*b^2+33*a^2*b*(7*A+5*C)+5 
*b^3*(11*A+9*C))*InverseJacobiAM(1/2*d*x+1/2*c,2^(1/2))/d+2/231*(77*B*a^3+ 
165*B*a*b^2+33*a^2*b*(7*A+5*C)+5*b^3*(11*A+9*C))*cos(d*x+c)^(1/2)*sin(d*x+ 
c)/d+2/495*(242*B*a^2*b+77*B*b^3+24*a^3*C+33*a*b^2*(9*A+7*C))*cos(d*x+c)^( 
3/2)*sin(d*x+c)/d+2/693*b*(99*A*b^2+143*B*a*b+24*C*a^2+81*C*b^2)*cos(d*x+c 
)^(5/2)*sin(d*x+c)/d+2/99*(11*B*b+6*C*a)*cos(d*x+c)^(3/2)*(a+b*cos(d*x+c)) 
^2*sin(d*x+c)/d+2/11*C*cos(d*x+c)^(3/2)*(a+b*cos(d*x+c))^3*sin(d*x+c)/d
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 3.65 (sec) , antiderivative size = 285, normalized size of antiderivative = 0.79 \[ \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {154 \left (27 a^2 b B+7 b^3 B+3 a^3 (5 A+3 C)+3 a b^2 (9 A+7 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+10 \left (77 a^3 B+165 a b^2 B+33 a^2 b (7 A+5 C)+5 b^3 (11 A+9 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+\frac {1}{12} \sqrt {\cos (c+d x)} \left (154 \left (108 a^2 b B+43 b^3 B+36 a^3 C+3 a b^2 (36 A+43 C)\right ) \cos (c+d x)+5 \left (1848 a^3 B+5148 a b^2 B+396 a^2 b (14 A+13 C)+3 b^3 (572 A+531 C)+36 b \left (11 A b^2+33 a b B+33 a^2 C+16 b^2 C\right ) \cos (2 (c+d x))+154 b^2 (b B+3 a C) \cos (3 (c+d x))+63 b^3 C \cos (4 (c+d x))\right )\right ) \sin (c+d x)}{1155 d} \] Input:

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

Output:

(154*(27*a^2*b*B + 7*b^3*B + 3*a^3*(5*A + 3*C) + 3*a*b^2*(9*A + 7*C))*Elli 
pticE[(c + d*x)/2, 2] + 10*(77*a^3*B + 165*a*b^2*B + 33*a^2*b*(7*A + 5*C) 
+ 5*b^3*(11*A + 9*C))*EllipticF[(c + d*x)/2, 2] + (Sqrt[Cos[c + d*x]]*(154 
*(108*a^2*b*B + 43*b^3*B + 36*a^3*C + 3*a*b^2*(36*A + 43*C))*Cos[c + d*x] 
+ 5*(1848*a^3*B + 5148*a*b^2*B + 396*a^2*b*(14*A + 13*C) + 3*b^3*(572*A + 
531*C) + 36*b*(11*A*b^2 + 33*a*b*B + 33*a^2*C + 16*b^2*C)*Cos[2*(c + d*x)] 
 + 154*b^2*(b*B + 3*a*C)*Cos[3*(c + d*x)] + 63*b^3*C*Cos[4*(c + d*x)]))*Si 
n[c + d*x])/12)/(1155*d)
 

Rubi [A] (verified)

Time = 1.99 (sec) , antiderivative size = 343, normalized size of antiderivative = 0.95, number of steps used = 19, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.442, Rules used = {3042, 3528, 27, 3042, 3528, 27, 3042, 3512, 27, 3042, 3502, 27, 3042, 3227, 3042, 3115, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3528

\(\displaystyle \frac {2}{11} \int \frac {1}{2} \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2 \left ((11 b B+6 a C) \cos ^2(c+d x)+(11 A b+9 C b+11 a B) \cos (c+d x)+a (11 A+3 C)\right )dx+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3}{11 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{11} \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2 \left ((11 b B+6 a C) \cos ^2(c+d x)+(11 A b+9 C b+11 a B) \cos (c+d x)+a (11 A+3 C)\right )dx+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3}{11 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{11} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^2 \left ((11 b B+6 a C) \sin \left (c+d x+\frac {\pi }{2}\right )^2+(11 A b+9 C b+11 a B) \sin \left (c+d x+\frac {\pi }{2}\right )+a (11 A+3 C)\right )dx+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3}{11 d}\)

\(\Big \downarrow \) 3528

\(\displaystyle \frac {1}{11} \left (\frac {2}{9} \int \frac {1}{2} \sqrt {\cos (c+d x)} (a+b \cos (c+d x)) \left (\left (24 C a^2+143 b B a+99 A b^2+81 b^2 C\right ) \cos ^2(c+d x)+\left (99 B a^2+198 A b a+150 b C a+77 b^2 B\right ) \cos (c+d x)+3 a (33 a A+11 b B+15 a C)\right )dx+\frac {2 (6 a C+11 b B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}{9 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3}{11 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{11} \left (\frac {1}{9} \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x)) \left (\left (24 C a^2+143 b B a+99 A b^2+81 b^2 C\right ) \cos ^2(c+d x)+\left (99 B a^2+198 A b a+150 b C a+77 b^2 B\right ) \cos (c+d x)+3 a (33 a A+11 b B+15 a C)\right )dx+\frac {2 (6 a C+11 b B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}{9 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3}{11 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{11} \left (\frac {1}{9} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right ) \left (\left (24 C a^2+143 b B a+99 A b^2+81 b^2 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+\left (99 B a^2+198 A b a+150 b C a+77 b^2 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+3 a (33 a A+11 b B+15 a C)\right )dx+\frac {2 (6 a C+11 b B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}{9 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3}{11 d}\)

\(\Big \downarrow \) 3512

\(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {2}{7} \int \frac {1}{2} \sqrt {\cos (c+d x)} \left (21 (33 a A+11 b B+15 a C) a^2+7 \left (24 C a^3+242 b B a^2+33 b^2 (9 A+7 C) a+77 b^3 B\right ) \cos ^2(c+d x)+9 \left (77 B a^3+33 b (7 A+5 C) a^2+165 b^2 B a+5 b^3 (11 A+9 C)\right ) \cos (c+d x)\right )dx+\frac {2 b \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (24 a^2 C+143 a b B+99 A b^2+81 b^2 C\right )}{7 d}\right )+\frac {2 (6 a C+11 b B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}{9 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3}{11 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \int \sqrt {\cos (c+d x)} \left (21 (33 a A+11 b B+15 a C) a^2+7 \left (24 C a^3+242 b B a^2+33 b^2 (9 A+7 C) a+77 b^3 B\right ) \cos ^2(c+d x)+9 \left (77 B a^3+33 b (7 A+5 C) a^2+165 b^2 B a+5 b^3 (11 A+9 C)\right ) \cos (c+d x)\right )dx+\frac {2 b \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (24 a^2 C+143 a b B+99 A b^2+81 b^2 C\right )}{7 d}\right )+\frac {2 (6 a C+11 b B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}{9 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3}{11 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (21 (33 a A+11 b B+15 a C) a^2+7 \left (24 C a^3+242 b B a^2+33 b^2 (9 A+7 C) a+77 b^3 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+9 \left (77 B a^3+33 b (7 A+5 C) a^2+165 b^2 B a+5 b^3 (11 A+9 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx+\frac {2 b \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (24 a^2 C+143 a b B+99 A b^2+81 b^2 C\right )}{7 d}\right )+\frac {2 (6 a C+11 b B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}{9 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3}{11 d}\)

\(\Big \downarrow \) 3502

\(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {2}{5} \int \frac {3}{2} \sqrt {\cos (c+d x)} \left (77 \left (3 (5 A+3 C) a^3+27 b B a^2+3 b^2 (9 A+7 C) a+7 b^3 B\right )+15 \left (77 B a^3+33 b (7 A+5 C) a^2+165 b^2 B a+5 b^3 (11 A+9 C)\right ) \cos (c+d x)\right )dx+\frac {14 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (24 a^3 C+242 a^2 b B+33 a b^2 (9 A+7 C)+77 b^3 B\right )}{5 d}\right )+\frac {2 b \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (24 a^2 C+143 a b B+99 A b^2+81 b^2 C\right )}{7 d}\right )+\frac {2 (6 a C+11 b B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}{9 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3}{11 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {3}{5} \int \sqrt {\cos (c+d x)} \left (77 \left (3 (5 A+3 C) a^3+27 b B a^2+3 b^2 (9 A+7 C) a+7 b^3 B\right )+15 \left (77 B a^3+33 b (7 A+5 C) a^2+165 b^2 B a+5 b^3 (11 A+9 C)\right ) \cos (c+d x)\right )dx+\frac {14 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (24 a^3 C+242 a^2 b B+33 a b^2 (9 A+7 C)+77 b^3 B\right )}{5 d}\right )+\frac {2 b \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (24 a^2 C+143 a b B+99 A b^2+81 b^2 C\right )}{7 d}\right )+\frac {2 (6 a C+11 b B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}{9 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3}{11 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {3}{5} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (77 \left (3 (5 A+3 C) a^3+27 b B a^2+3 b^2 (9 A+7 C) a+7 b^3 B\right )+15 \left (77 B a^3+33 b (7 A+5 C) a^2+165 b^2 B a+5 b^3 (11 A+9 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx+\frac {14 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (24 a^3 C+242 a^2 b B+33 a b^2 (9 A+7 C)+77 b^3 B\right )}{5 d}\right )+\frac {2 b \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (24 a^2 C+143 a b B+99 A b^2+81 b^2 C\right )}{7 d}\right )+\frac {2 (6 a C+11 b B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}{9 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3}{11 d}\)

\(\Big \downarrow \) 3227

\(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {3}{5} \left (15 \left (77 a^3 B+33 a^2 b (7 A+5 C)+165 a b^2 B+5 b^3 (11 A+9 C)\right ) \int \cos ^{\frac {3}{2}}(c+d x)dx+77 \left (3 a^3 (5 A+3 C)+27 a^2 b B+3 a b^2 (9 A+7 C)+7 b^3 B\right ) \int \sqrt {\cos (c+d x)}dx\right )+\frac {14 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (24 a^3 C+242 a^2 b B+33 a b^2 (9 A+7 C)+77 b^3 B\right )}{5 d}\right )+\frac {2 b \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (24 a^2 C+143 a b B+99 A b^2+81 b^2 C\right )}{7 d}\right )+\frac {2 (6 a C+11 b B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}{9 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3}{11 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {3}{5} \left (77 \left (3 a^3 (5 A+3 C)+27 a^2 b B+3 a b^2 (9 A+7 C)+7 b^3 B\right ) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx+15 \left (77 a^3 B+33 a^2 b (7 A+5 C)+165 a b^2 B+5 b^3 (11 A+9 C)\right ) \int \sin \left (c+d x+\frac {\pi }{2}\right )^{3/2}dx\right )+\frac {14 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (24 a^3 C+242 a^2 b B+33 a b^2 (9 A+7 C)+77 b^3 B\right )}{5 d}\right )+\frac {2 b \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (24 a^2 C+143 a b B+99 A b^2+81 b^2 C\right )}{7 d}\right )+\frac {2 (6 a C+11 b B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}{9 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3}{11 d}\)

\(\Big \downarrow \) 3115

\(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {3}{5} \left (77 \left (3 a^3 (5 A+3 C)+27 a^2 b B+3 a b^2 (9 A+7 C)+7 b^3 B\right ) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx+15 \left (77 a^3 B+33 a^2 b (7 A+5 C)+165 a b^2 B+5 b^3 (11 A+9 C)\right ) \left (\frac {1}{3} \int \frac {1}{\sqrt {\cos (c+d x)}}dx+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )\right )+\frac {14 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (24 a^3 C+242 a^2 b B+33 a b^2 (9 A+7 C)+77 b^3 B\right )}{5 d}\right )+\frac {2 b \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (24 a^2 C+143 a b B+99 A b^2+81 b^2 C\right )}{7 d}\right )+\frac {2 (6 a C+11 b B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}{9 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3}{11 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {3}{5} \left (77 \left (3 a^3 (5 A+3 C)+27 a^2 b B+3 a b^2 (9 A+7 C)+7 b^3 B\right ) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx+15 \left (77 a^3 B+33 a^2 b (7 A+5 C)+165 a b^2 B+5 b^3 (11 A+9 C)\right ) \left (\frac {1}{3} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )\right )+\frac {14 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (24 a^3 C+242 a^2 b B+33 a b^2 (9 A+7 C)+77 b^3 B\right )}{5 d}\right )+\frac {2 b \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (24 a^2 C+143 a b B+99 A b^2+81 b^2 C\right )}{7 d}\right )+\frac {2 (6 a C+11 b B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}{9 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3}{11 d}\)

\(\Big \downarrow \) 3119

\(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {3}{5} \left (15 \left (77 a^3 B+33 a^2 b (7 A+5 C)+165 a b^2 B+5 b^3 (11 A+9 C)\right ) \left (\frac {1}{3} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )+\frac {154 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (3 a^3 (5 A+3 C)+27 a^2 b B+3 a b^2 (9 A+7 C)+7 b^3 B\right )}{d}\right )+\frac {14 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (24 a^3 C+242 a^2 b B+33 a b^2 (9 A+7 C)+77 b^3 B\right )}{5 d}\right )+\frac {2 b \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (24 a^2 C+143 a b B+99 A b^2+81 b^2 C\right )}{7 d}\right )+\frac {2 (6 a C+11 b B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}{9 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3}{11 d}\)

\(\Big \downarrow \) 3120

\(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {2 b \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (24 a^2 C+143 a b B+99 A b^2+81 b^2 C\right )}{7 d}+\frac {1}{7} \left (\frac {14 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (24 a^3 C+242 a^2 b B+33 a b^2 (9 A+7 C)+77 b^3 B\right )}{5 d}+\frac {3}{5} \left (\frac {154 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (3 a^3 (5 A+3 C)+27 a^2 b B+3 a b^2 (9 A+7 C)+7 b^3 B\right )}{d}+15 \left (77 a^3 B+33 a^2 b (7 A+5 C)+165 a b^2 B+5 b^3 (11 A+9 C)\right ) \left (\frac {2 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d}+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )\right )\right )\right )+\frac {2 (6 a C+11 b B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}{9 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3}{11 d}\)

Input:

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

Output:

(2*C*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(11*d) + ((2* 
(11*b*B + 6*a*C)*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/( 
9*d) + ((2*b*(99*A*b^2 + 143*a*b*B + 24*a^2*C + 81*b^2*C)*Cos[c + d*x]^(5/ 
2)*Sin[c + d*x])/(7*d) + ((14*(242*a^2*b*B + 77*b^3*B + 24*a^3*C + 33*a*b^ 
2*(9*A + 7*C))*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d) + (3*((154*(27*a^2*b 
*B + 7*b^3*B + 3*a^3*(5*A + 3*C) + 3*a*b^2*(9*A + 7*C))*EllipticE[(c + d*x 
)/2, 2])/d + 15*(77*a^3*B + 165*a*b^2*B + 33*a^2*b*(7*A + 5*C) + 5*b^3*(11 
*A + 9*C))*((2*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*Sqrt[Cos[c + d*x]]*Si 
n[c + d*x])/(3*d))))/5)/7)/9)/11
 

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 3115

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* 
x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n)   Int[(b*Sin 
[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 
2*n]
 

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])))
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1081\) vs. \(2(340)=680\).

Time = 9.37 (sec) , antiderivative size = 1082, normalized size of antiderivative = 3.00

method	result	size
default	\(\text {Expression too large to display}\)	\(1082\)
parts	\(\text {Expression too large to display}\)	\(1220\)

Input:

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

Output:

-2/3465*((-1+2*cos(1/2*d*x+1/2*c)^2)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(20160*C* 
b^3*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^12+(-12320*B*b^3-36960*C*a*b^2-5 
0400*C*b^3)*sin(1/2*d*x+1/2*c)^10*cos(1/2*d*x+1/2*c)+(7920*A*b^3+23760*B*a 
*b^2+24640*B*b^3+23760*C*a^2*b+73920*C*a*b^2+56880*C*b^3)*sin(1/2*d*x+1/2* 
c)^8*cos(1/2*d*x+1/2*c)+(-16632*A*a*b^2-11880*A*b^3-16632*B*a^2*b-35640*B* 
a*b^2-22792*B*b^3-5544*C*a^3-35640*C*a^2*b-68376*C*a*b^2-34920*C*b^3)*sin( 
1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)+(13860*A*a^2*b+16632*A*a*b^2+9240*A*b^ 
3+4620*B*a^3+16632*B*a^2*b+27720*B*a*b^2+10472*B*b^3+5544*C*a^3+27720*C*a^ 
2*b+31416*C*a*b^2+13860*C*b^3)*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+(-6 
930*A*a^2*b-4158*A*a*b^2-2640*A*b^3-2310*B*a^3-4158*B*a^2*b-7920*B*a*b^2-1 
848*B*b^3-1386*C*a^3-7920*C*a^2*b-5544*C*a*b^2-2790*C*b^3)*sin(1/2*d*x+1/2 
*c)^2*cos(1/2*d*x+1/2*c)+3465*A*a^2*b*(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))+825*A*b^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(co 
s(1/2*d*x+1/2*c),2^(1/2))-3465*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^3-6237*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*b^2+1155*B*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))+2475*B*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)*Ellip...
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.13 (sec) , antiderivative size = 429, normalized size of antiderivative = 1.19 \[ \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {2 \, {\left (315 \, C b^{3} \cos \left (d x + c\right )^{4} + 1155 \, B a^{3} + 495 \, {\left (7 \, A + 5 \, C\right )} a^{2} b + 2475 \, B a b^{2} + 75 \, {\left (11 \, A + 9 \, C\right )} b^{3} + 385 \, {\left (3 \, C a b^{2} + B b^{3}\right )} \cos \left (d x + c\right )^{3} + 45 \, {\left (33 \, C a^{2} b + 33 \, B a b^{2} + {\left (11 \, A + 9 \, C\right )} b^{3}\right )} \cos \left (d x + c\right )^{2} + 77 \, {\left (9 \, C a^{3} + 27 \, B a^{2} b + 3 \, {\left (9 \, A + 7 \, C\right )} a b^{2} + 7 \, B b^{3}\right )} \cos \left (d x + c\right )\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) - 15 \, \sqrt {2} {\left (77 i \, B a^{3} + 33 i \, {\left (7 \, A + 5 \, C\right )} a^{2} b + 165 i \, B a b^{2} + 5 i \, {\left (11 \, A + 9 \, C\right )} b^{3}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 15 \, \sqrt {2} {\left (-77 i \, B a^{3} - 33 i \, {\left (7 \, A + 5 \, C\right )} a^{2} b - 165 i \, B a b^{2} - 5 i \, {\left (11 \, A + 9 \, C\right )} b^{3}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 231 \, \sqrt {2} {\left (-3 i \, {\left (5 \, A + 3 \, C\right )} a^{3} - 27 i \, B a^{2} b - 3 i \, {\left (9 \, A + 7 \, C\right )} a b^{2} - 7 i \, B b^{3}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - 231 \, \sqrt {2} {\left (3 i \, {\left (5 \, A + 3 \, C\right )} a^{3} + 27 i \, B a^{2} b + 3 i \, {\left (9 \, A + 7 \, C\right )} a b^{2} + 7 i \, B b^{3}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right )}{3465 \, d} \] Input:

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

Output:

1/3465*(2*(315*C*b^3*cos(d*x + c)^4 + 1155*B*a^3 + 495*(7*A + 5*C)*a^2*b + 
 2475*B*a*b^2 + 75*(11*A + 9*C)*b^3 + 385*(3*C*a*b^2 + B*b^3)*cos(d*x + c) 
^3 + 45*(33*C*a^2*b + 33*B*a*b^2 + (11*A + 9*C)*b^3)*cos(d*x + c)^2 + 77*( 
9*C*a^3 + 27*B*a^2*b + 3*(9*A + 7*C)*a*b^2 + 7*B*b^3)*cos(d*x + c))*sqrt(c 
os(d*x + c))*sin(d*x + c) - 15*sqrt(2)*(77*I*B*a^3 + 33*I*(7*A + 5*C)*a^2* 
b + 165*I*B*a*b^2 + 5*I*(11*A + 9*C)*b^3)*weierstrassPInverse(-4, 0, cos(d 
*x + c) + I*sin(d*x + c)) - 15*sqrt(2)*(-77*I*B*a^3 - 33*I*(7*A + 5*C)*a^2 
*b - 165*I*B*a*b^2 - 5*I*(11*A + 9*C)*b^3)*weierstrassPInverse(-4, 0, cos( 
d*x + c) - I*sin(d*x + c)) - 231*sqrt(2)*(-3*I*(5*A + 3*C)*a^3 - 27*I*B*a^ 
2*b - 3*I*(9*A + 7*C)*a*b^2 - 7*I*B*b^3)*weierstrassZeta(-4, 0, weierstras 
sPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) - 231*sqrt(2)*(3*I*(5*A + 
 3*C)*a^3 + 27*I*B*a^2*b + 3*I*(9*A + 7*C)*a*b^2 + 7*I*B*b^3)*weierstrassZ 
eta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c))))/d
 

Sympy [F(-1)]

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

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

Output:

Timed out
                                                                                    
                                                                                    
 

Maxima [F]

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

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

Output:

integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)*(b*cos(d*x + c) + a)^3*s 
qrt(cos(d*x + c)), x)
 

Giac [F]

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

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

Output:

integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)*(b*cos(d*x + c) + a)^3*s 
qrt(cos(d*x + c)), x)
 

Mupad [B] (verification not implemented)

Time = 1.83 (sec) , antiderivative size = 514, normalized size of antiderivative = 1.42 \[ \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx =\text {Too large to display} \] Input:

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

Output:

(2*(A*a^3*ellipticE(c/2 + (d*x)/2, 2) + A*a^2*b*ellipticF(c/2 + (d*x)/2, 2 
) + A*a^2*b*cos(c + d*x)^(1/2)*sin(c + d*x)))/d + (B*a^3*((2*cos(c + d*x)^ 
(1/2)*sin(c + d*x))/3 + (2*ellipticF(c/2 + (d*x)/2, 2))/3))/d - (2*A*b^3*c 
os(c + d*x)^(9/2)*sin(c + d*x)*hypergeom([1/2, 9/4], 13/4, cos(c + d*x)^2) 
)/(9*d*(sin(c + d*x)^2)^(1/2)) - (2*C*a^3*cos(c + d*x)^(7/2)*sin(c + d*x)* 
hypergeom([1/2, 7/4], 11/4, cos(c + d*x)^2))/(7*d*(sin(c + d*x)^2)^(1/2)) 
- (2*B*b^3*cos(c + d*x)^(11/2)*sin(c + d*x)*hypergeom([1/2, 11/4], 15/4, c 
os(c + d*x)^2))/(11*d*(sin(c + d*x)^2)^(1/2)) - (2*C*b^3*cos(c + d*x)^(13/ 
2)*sin(c + d*x)*hypergeom([1/2, 13/4], 17/4, cos(c + d*x)^2))/(13*d*(sin(c 
 + d*x)^2)^(1/2)) - (6*A*a*b^2*cos(c + d*x)^(7/2)*sin(c + d*x)*hypergeom([ 
1/2, 7/4], 11/4, cos(c + d*x)^2))/(7*d*(sin(c + d*x)^2)^(1/2)) - (6*B*a^2* 
b*cos(c + d*x)^(7/2)*sin(c + d*x)*hypergeom([1/2, 7/4], 11/4, cos(c + d*x) 
^2))/(7*d*(sin(c + d*x)^2)^(1/2)) - (2*B*a*b^2*cos(c + d*x)^(9/2)*sin(c + 
d*x)*hypergeom([1/2, 9/4], 13/4, cos(c + d*x)^2))/(3*d*(sin(c + d*x)^2)^(1 
/2)) - (2*C*a^2*b*cos(c + d*x)^(9/2)*sin(c + d*x)*hypergeom([1/2, 9/4], 13 
/4, cos(c + d*x)^2))/(3*d*(sin(c + d*x)^2)^(1/2)) - (6*C*a*b^2*cos(c + d*x 
)^(11/2)*sin(c + d*x)*hypergeom([1/2, 11/4], 15/4, cos(c + d*x)^2))/(11*d* 
(sin(c + d*x)^2)^(1/2))
 

Reduce [F]

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

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

Output:

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