\(\int \frac {(a+b \cos (c+d x))^3 (A+B \cos (c+d x)+C \cos ^2(c+d x))}{\sqrt {\cos (c+d x)}} \, dx\) [1081]

Optimal result

Integrand size = 43, antiderivative size = 296 \[ \int \frac {(a+b \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx=\frac {2 \left (15 a^3 B+27 a b^2 B+9 a^2 b (5 A+3 C)+b^3 (9 A+7 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {2 \left (21 a^2 b B+5 b^3 B+7 a^3 (3 A+C)+3 a b^2 (7 A+5 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 d}+\frac {2 \left (54 a^2 b B+15 b^3 B+8 a^3 C+9 a b^2 (7 A+5 C)\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{63 d}+\frac {2 b \left (63 A b^2+99 a b B+24 a^2 C+49 b^2 C\right ) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{315 d}+\frac {2 (3 b B+2 a C) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2 \sin (c+d x)}{21 d}+\frac {2 C \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3 \sin (c+d x)}{9 d} \] Output:

2/15*(15*B*a^3+27*B*a*b^2+9*a^2*b*(5*A+3*C)+b^3*(9*A+7*C))*EllipticE(sin(1 
/2*d*x+1/2*c),2^(1/2))/d+2/21*(21*B*a^2*b+5*B*b^3+7*a^3*(3*A+C)+3*a*b^2*(7 
*A+5*C))*InverseJacobiAM(1/2*d*x+1/2*c,2^(1/2))/d+2/63*(54*B*a^2*b+15*B*b^ 
3+8*a^3*C+9*a*b^2*(7*A+5*C))*cos(d*x+c)^(1/2)*sin(d*x+c)/d+2/315*b*(63*A*b 
^2+99*B*a*b+24*C*a^2+49*C*b^2)*cos(d*x+c)^(3/2)*sin(d*x+c)/d+2/21*(3*B*b+2 
*C*a)*cos(d*x+c)^(1/2)*(a+b*cos(d*x+c))^2*sin(d*x+c)/d+2/9*C*cos(d*x+c)^(1 
/2)*(a+b*cos(d*x+c))^3*sin(d*x+c)/d
 

Mathematica [A] (verified)

Time = 4.61 (sec) , antiderivative size = 230, normalized size of antiderivative = 0.78 \[ \int \frac {(a+b \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx=\frac {84 \left (15 a^3 B+27 a b^2 B+9 a^2 b (5 A+3 C)+b^3 (9 A+7 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+60 \left (21 a^2 b B+5 b^3 B+7 a^3 (3 A+C)+3 a b^2 (7 A+5 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+\sqrt {\cos (c+d x)} \left (7 b \left (36 A b^2+108 a b B+108 a^2 C+43 b^2 C\right ) \cos (c+d x)+5 \left (252 a^2 b B+78 b^3 B+84 a^3 C+18 a b^2 (14 A+13 C)+18 b^2 (b B+3 a C) \cos (2 (c+d x))+7 b^3 C \cos (3 (c+d x))\right )\right ) \sin (c+d x)}{630 d} \] Input:

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

Output:

(84*(15*a^3*B + 27*a*b^2*B + 9*a^2*b*(5*A + 3*C) + b^3*(9*A + 7*C))*Ellipt 
icE[(c + d*x)/2, 2] + 60*(21*a^2*b*B + 5*b^3*B + 7*a^3*(3*A + C) + 3*a*b^2 
*(7*A + 5*C))*EllipticF[(c + d*x)/2, 2] + Sqrt[Cos[c + d*x]]*(7*b*(36*A*b^ 
2 + 108*a*b*B + 108*a^2*C + 43*b^2*C)*Cos[c + d*x] + 5*(252*a^2*b*B + 78*b 
^3*B + 84*a^3*C + 18*a*b^2*(14*A + 13*C) + 18*b^2*(b*B + 3*a*C)*Cos[2*(c + 
 d*x)] + 7*b^3*C*Cos[3*(c + d*x)]))*Sin[c + d*x])/(630*d)
 

Rubi [A] (verified)

Time = 1.81 (sec) , antiderivative size = 308, normalized size of antiderivative = 1.04, number of steps used = 17, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.395, Rules used = {3042, 3528, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3528

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3528

\(\displaystyle \frac {1}{9} \left (\frac {2}{7} \int \frac {(a+b \cos (c+d x)) \left (\left (24 C a^2+99 b B a+63 A b^2+49 b^2 C\right ) \cos ^2(c+d x)+\left (63 B a^2+126 A b a+86 b C a+45 b^2 B\right ) \cos (c+d x)+a (63 a A+9 b B+13 a C)\right )}{2 \sqrt {\cos (c+d x)}}dx+\frac {6 (2 a C+3 b B) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \int \frac {(a+b \cos (c+d x)) \left (\left (24 C a^2+99 b B a+63 A b^2+49 b^2 C\right ) \cos ^2(c+d x)+\left (63 B a^2+126 A b a+86 b C a+45 b^2 B\right ) \cos (c+d x)+a (63 a A+9 b B+13 a C)\right )}{\sqrt {\cos (c+d x)}}dx+\frac {6 (2 a C+3 b B) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right ) \left (\left (24 C a^2+99 b B a+63 A b^2+49 b^2 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+\left (63 B a^2+126 A b a+86 b C a+45 b^2 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+a (63 a A+9 b B+13 a C)\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {6 (2 a C+3 b B) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}\)

\(\Big \downarrow \) 3512

\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {2}{5} \int \frac {5 (63 a A+9 b B+13 a C) a^2+15 \left (8 C a^3+54 b B a^2+9 b^2 (7 A+5 C) a+15 b^3 B\right ) \cos ^2(c+d x)+21 \left (15 B a^3+9 b (5 A+3 C) a^2+27 b^2 B a+b^3 (9 A+7 C)\right ) \cos (c+d x)}{2 \sqrt {\cos (c+d x)}}dx+\frac {2 b \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (24 a^2 C+99 a b B+63 A b^2+49 b^2 C\right )}{5 d}\right )+\frac {6 (2 a C+3 b B) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \int \frac {5 (63 a A+9 b B+13 a C) a^2+15 \left (8 C a^3+54 b B a^2+9 b^2 (7 A+5 C) a+15 b^3 B\right ) \cos ^2(c+d x)+21 \left (15 B a^3+9 b (5 A+3 C) a^2+27 b^2 B a+b^3 (9 A+7 C)\right ) \cos (c+d x)}{\sqrt {\cos (c+d x)}}dx+\frac {2 b \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (24 a^2 C+99 a b B+63 A b^2+49 b^2 C\right )}{5 d}\right )+\frac {6 (2 a C+3 b B) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \int \frac {5 (63 a A+9 b B+13 a C) a^2+15 \left (8 C a^3+54 b B a^2+9 b^2 (7 A+5 C) a+15 b^3 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+21 \left (15 B a^3+9 b (5 A+3 C) a^2+27 b^2 B a+b^3 (9 A+7 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 b \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (24 a^2 C+99 a b B+63 A b^2+49 b^2 C\right )}{5 d}\right )+\frac {6 (2 a C+3 b B) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}\)

\(\Big \downarrow \) 3502

\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {2}{3} \int \frac {9 \left (5 \left (7 (3 A+C) a^3+21 b B a^2+3 b^2 (7 A+5 C) a+5 b^3 B\right )+7 \left (15 B a^3+9 b (5 A+3 C) a^2+27 b^2 B a+b^3 (9 A+7 C)\right ) \cos (c+d x)\right )}{2 \sqrt {\cos (c+d x)}}dx+\frac {10 \sin (c+d x) \sqrt {\cos (c+d x)} \left (8 a^3 C+54 a^2 b B+9 a b^2 (7 A+5 C)+15 b^3 B\right )}{d}\right )+\frac {2 b \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (24 a^2 C+99 a b B+63 A b^2+49 b^2 C\right )}{5 d}\right )+\frac {6 (2 a C+3 b B) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (3 \int \frac {5 \left (7 (3 A+C) a^3+21 b B a^2+3 b^2 (7 A+5 C) a+5 b^3 B\right )+7 \left (15 B a^3+9 b (5 A+3 C) a^2+27 b^2 B a+b^3 (9 A+7 C)\right ) \cos (c+d x)}{\sqrt {\cos (c+d x)}}dx+\frac {10 \sin (c+d x) \sqrt {\cos (c+d x)} \left (8 a^3 C+54 a^2 b B+9 a b^2 (7 A+5 C)+15 b^3 B\right )}{d}\right )+\frac {2 b \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (24 a^2 C+99 a b B+63 A b^2+49 b^2 C\right )}{5 d}\right )+\frac {6 (2 a C+3 b B) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (3 \int \frac {5 \left (7 (3 A+C) a^3+21 b B a^2+3 b^2 (7 A+5 C) a+5 b^3 B\right )+7 \left (15 B a^3+9 b (5 A+3 C) a^2+27 b^2 B a+b^3 (9 A+7 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {10 \sin (c+d x) \sqrt {\cos (c+d x)} \left (8 a^3 C+54 a^2 b B+9 a b^2 (7 A+5 C)+15 b^3 B\right )}{d}\right )+\frac {2 b \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (24 a^2 C+99 a b B+63 A b^2+49 b^2 C\right )}{5 d}\right )+\frac {6 (2 a C+3 b B) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}\)

\(\Big \downarrow \) 3227

\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (3 \left (5 \left (7 a^3 (3 A+C)+21 a^2 b B+3 a b^2 (7 A+5 C)+5 b^3 B\right ) \int \frac {1}{\sqrt {\cos (c+d x)}}dx+7 \left (15 a^3 B+9 a^2 b (5 A+3 C)+27 a b^2 B+b^3 (9 A+7 C)\right ) \int \sqrt {\cos (c+d x)}dx\right )+\frac {10 \sin (c+d x) \sqrt {\cos (c+d x)} \left (8 a^3 C+54 a^2 b B+9 a b^2 (7 A+5 C)+15 b^3 B\right )}{d}\right )+\frac {2 b \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (24 a^2 C+99 a b B+63 A b^2+49 b^2 C\right )}{5 d}\right )+\frac {6 (2 a C+3 b B) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (3 \left (5 \left (7 a^3 (3 A+C)+21 a^2 b B+3 a b^2 (7 A+5 C)+5 b^3 B\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+7 \left (15 a^3 B+9 a^2 b (5 A+3 C)+27 a b^2 B+b^3 (9 A+7 C)\right ) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx\right )+\frac {10 \sin (c+d x) \sqrt {\cos (c+d x)} \left (8 a^3 C+54 a^2 b B+9 a b^2 (7 A+5 C)+15 b^3 B\right )}{d}\right )+\frac {2 b \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (24 a^2 C+99 a b B+63 A b^2+49 b^2 C\right )}{5 d}\right )+\frac {6 (2 a C+3 b B) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}\)

\(\Big \downarrow \) 3119

\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (3 \left (5 \left (7 a^3 (3 A+C)+21 a^2 b B+3 a b^2 (7 A+5 C)+5 b^3 B\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {14 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (15 a^3 B+9 a^2 b (5 A+3 C)+27 a b^2 B+b^3 (9 A+7 C)\right )}{d}\right )+\frac {10 \sin (c+d x) \sqrt {\cos (c+d x)} \left (8 a^3 C+54 a^2 b B+9 a b^2 (7 A+5 C)+15 b^3 B\right )}{d}\right )+\frac {2 b \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (24 a^2 C+99 a b B+63 A b^2+49 b^2 C\right )}{5 d}\right )+\frac {6 (2 a C+3 b B) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}\)

\(\Big \downarrow \) 3120

\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {2 b \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (24 a^2 C+99 a b B+63 A b^2+49 b^2 C\right )}{5 d}+\frac {1}{5} \left (\frac {10 \sin (c+d x) \sqrt {\cos (c+d x)} \left (8 a^3 C+54 a^2 b B+9 a b^2 (7 A+5 C)+15 b^3 B\right )}{d}+3 \left (\frac {10 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (7 a^3 (3 A+C)+21 a^2 b B+3 a b^2 (7 A+5 C)+5 b^3 B\right )}{d}+\frac {14 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (15 a^3 B+9 a^2 b (5 A+3 C)+27 a b^2 B+b^3 (9 A+7 C)\right )}{d}\right )\right )\right )+\frac {6 (2 a C+3 b B) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}\)

Input:

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

Output:

(2*C*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(9*d) + ((6*( 
3*b*B + 2*a*C)*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(7* 
d) + ((2*b*(63*A*b^2 + 99*a*b*B + 24*a^2*C + 49*b^2*C)*Cos[c + d*x]^(3/2)* 
Sin[c + d*x])/(5*d) + (3*((14*(15*a^3*B + 27*a*b^2*B + 9*a^2*b*(5*A + 3*C) 
 + b^3*(9*A + 7*C))*EllipticE[(c + d*x)/2, 2])/d + (10*(21*a^2*b*B + 5*b^3 
*B + 7*a^3*(3*A + C) + 3*a*b^2*(7*A + 5*C))*EllipticF[(c + d*x)/2, 2])/d) 
+ (10*(54*a^2*b*B + 15*b^3*B + 8*a^3*C + 9*a*b^2*(7*A + 5*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])))
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(974\) vs. \(2(279)=558\).

Time = 9.00 (sec) , antiderivative size = 975, normalized size of antiderivative = 3.29

method	result	size
default	\(\text {Expression too large to display}\)	\(975\)
parts	\(\text {Expression too large to display}\)	\(1013\)

Input:

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

Output:

-2/315*((-1+2*cos(1/2*d*x+1/2*c)^2)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(-1120*C*s 
in(1/2*d*x+1/2*c)^10*cos(1/2*d*x+1/2*c)*b^3+(720*B*b^3+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*B*a*b^2-108 
0*B*b^3-1512*C*a^2*b-3240*C*a*b^2-2072*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+1260*B*a^2*b+1512*B*a*b^2+840*B*b^3+42 
0*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^3-630*B*a^2*b-378*B*a*b^2-240*B*b^3-210*C 
*a^3-378*C*a^2*b-720*C*a*b^2-168*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+315*B*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))+75*B*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(cos(1/2*d*x+1/2*c),2^(1 
/2))-315*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)*E 
llipticE(cos(1/2*d*x+1/2*c),2^(1/2))*a^3-567*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)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2)...
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.14 (sec) , antiderivative size = 376, normalized size of antiderivative = 1.27 \[ \int \frac {(a+b \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx=\frac {2 \, {\left (35 \, C b^{3} \cos \left (d x + c\right )^{3} + 105 \, C a^{3} + 315 \, B a^{2} b + 45 \, {\left (7 \, A + 5 \, C\right )} a b^{2} + 75 \, B b^{3} + 45 \, {\left (3 \, C a b^{2} + B b^{3}\right )} \cos \left (d x + c\right )^{2} + 7 \, {\left (27 \, C a^{2} b + 27 \, B a b^{2} + {\left (9 \, A + 7 \, C\right )} b^{3}\right )} \cos \left (d x + c\right )\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) - 15 \, \sqrt {2} {\left (7 i \, {\left (3 \, A + C\right )} a^{3} + 21 i \, B a^{2} b + 3 i \, {\left (7 \, A + 5 \, C\right )} a b^{2} + 5 i \, B b^{3}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 15 \, \sqrt {2} {\left (-7 i \, {\left (3 \, A + C\right )} a^{3} - 21 i \, B a^{2} b - 3 i \, {\left (7 \, A + 5 \, C\right )} a b^{2} - 5 i \, B b^{3}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 21 \, \sqrt {2} {\left (-15 i \, B a^{3} - 9 i \, {\left (5 \, A + 3 \, C\right )} a^{2} b - 27 i \, B a b^{2} - i \, {\left (9 \, A + 7 \, C\right )} b^{3}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - 21 \, \sqrt {2} {\left (15 i \, B a^{3} + 9 i \, {\left (5 \, A + 3 \, C\right )} a^{2} b + 27 i \, B a b^{2} + i \, {\left (9 \, A + 7 \, C\right )} b^{3}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right )}{315 \, d} \] Input:

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

Output:

1/315*(2*(35*C*b^3*cos(d*x + c)^3 + 105*C*a^3 + 315*B*a^2*b + 45*(7*A + 5* 
C)*a*b^2 + 75*B*b^3 + 45*(3*C*a*b^2 + B*b^3)*cos(d*x + c)^2 + 7*(27*C*a^2* 
b + 27*B*a*b^2 + (9*A + 7*C)*b^3)*cos(d*x + c))*sqrt(cos(d*x + c))*sin(d*x 
 + c) - 15*sqrt(2)*(7*I*(3*A + C)*a^3 + 21*I*B*a^2*b + 3*I*(7*A + 5*C)*a*b 
^2 + 5*I*B*b^3)*weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c)) 
- 15*sqrt(2)*(-7*I*(3*A + C)*a^3 - 21*I*B*a^2*b - 3*I*(7*A + 5*C)*a*b^2 - 
5*I*B*b^3)*weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) - 21* 
sqrt(2)*(-15*I*B*a^3 - 9*I*(5*A + 3*C)*a^2*b - 27*I*B*a*b^2 - I*(9*A + 7*C 
)*b^3)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I* 
sin(d*x + c))) - 21*sqrt(2)*(15*I*B*a^3 + 9*I*(5*A + 3*C)*a^2*b + 27*I*B*a 
*b^2 + I*(9*A + 7*C)*b^3)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0 
, cos(d*x + c) - I*sin(d*x + c))))/d
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

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

integrate((a+b*cos(d*x+c))^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/cos(d*x+c)^(1 
/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 = 1.66 (sec) , antiderivative size = 452, normalized size of antiderivative = 1.53 \[ \int \frac {(a+b \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx =\text {Too large to display} \] Input:

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

Output:

(2*(B*a^3*ellipticE(c/2 + (d*x)/2, 2) + B*a^2*b*ellipticF(c/2 + (d*x)/2, 2 
) + B*a^2*b*cos(c + d*x)^(1/2)*sin(c + d*x)))/d + (C*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*a^3*e 
llipticF(c/2 + (d*x)/2, 2))/d + (6*A*a^2*b*ellipticE(c/2 + (d*x)/2, 2))/d 
+ (3*A*a*b^2*((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*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*co 
s(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*b^3*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) 
) - (6*B*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*C*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*C*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))
 

Reduce [F]

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

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

Output:

int(sqrt(cos(c + d*x))/cos(c + d*x),x)*a**4 + 4*int(sqrt(cos(c + d*x)),x)* 
a**3*b + int(sqrt(cos(c + d*x))*cos(c + d*x),x)*a**3*c + 6*int(sqrt(cos(c 
+ d*x))*cos(c + d*x),x)*a**2*b**2 + int(sqrt(cos(c + d*x))*cos(c + d*x)**4 
,x)*b**3*c + 3*int(sqrt(cos(c + d*x))*cos(c + d*x)**3,x)*a*b**2*c + int(sq 
rt(cos(c + d*x))*cos(c + d*x)**3,x)*b**4 + 3*int(sqrt(cos(c + d*x))*cos(c 
+ d*x)**2,x)*a**2*b*c + 4*int(sqrt(cos(c + d*x))*cos(c + d*x)**2,x)*a*b**3