\(\int \frac {\sqrt {a+b \cos (c+d x)} (A+B \cos (c+d x)+C \cos ^2(c+d x))}{\sec ^{\frac {3}{2}}(c+d x)} \, dx\) [1509]

Optimal result

Integrand size = 45, antiderivative size = 766 \[ \int \frac {\sqrt {a+b \cos (c+d x)} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sec ^{\frac {3}{2}}(c+d x)} \, dx=\frac {(a-b) \sqrt {a+b} \left (24 a^2 b B-128 b^3 B-15 a^3 C-4 a b^2 (12 A+7 C)\right ) \sqrt {\cos (c+d x)} \csc (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{192 a b^3 d \sqrt {\sec (c+d x)}}+\frac {\sqrt {a+b} \left (15 a^3 C-2 a^2 b (12 B+5 C)+4 a b^2 (12 A+4 B+7 C)+8 b^3 (12 A+16 B+9 C)\right ) \sqrt {\cos (c+d x)} \csc (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{192 b^3 d \sqrt {\sec (c+d x)}}-\frac {\sqrt {a+b} \left (8 a^3 b B+32 a b^3 B-5 a^4 C-8 a^2 b^2 (2 A+C)+16 b^4 (4 A+3 C)\right ) \sqrt {\cos (c+d x)} \csc (c+d x) \operatorname {EllipticPi}\left (\frac {a+b}{b},\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{64 b^4 d \sqrt {\sec (c+d x)}}+\frac {C (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{4 b d \sec ^{\frac {3}{2}}(c+d x)}+\frac {\left (16 A b^2-8 a b B+5 a^2 C+12 b^2 C\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{32 b^2 d \sqrt {\sec (c+d x)}}+\frac {(8 b B-5 a C) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{24 b^2 d \sqrt {\sec (c+d x)}}-\frac {\left (24 a^2 b B-128 b^3 B-15 a^3 C-4 a b^2 (12 A+7 C)\right ) \sqrt {a+b \cos (c+d x)} \sqrt {\sec (c+d x)} \sin (c+d x)}{192 b^3 d} \] Output:

1/192*(a-b)*(a+b)^(1/2)*(24*B*a^2*b-128*B*b^3-15*a^3*C-4*a*b^2*(12*A+7*C)) 
*cos(d*x+c)^(1/2)*csc(d*x+c)*EllipticE((a+b*cos(d*x+c))^(1/2)/(a+b)^(1/2)/ 
cos(d*x+c)^(1/2),(-(a+b)/(a-b))^(1/2))*(a*(1-sec(d*x+c))/(a+b))^(1/2)*(a*( 
1+sec(d*x+c))/(a-b))^(1/2)/a/b^3/d/sec(d*x+c)^(1/2)+1/192*(a+b)^(1/2)*(15* 
a^3*C-2*a^2*b*(12*B+5*C)+4*a*b^2*(12*A+4*B+7*C)+8*b^3*(12*A+16*B+9*C))*cos 
(d*x+c)^(1/2)*csc(d*x+c)*EllipticF((a+b*cos(d*x+c))^(1/2)/(a+b)^(1/2)/cos( 
d*x+c)^(1/2),(-(a+b)/(a-b))^(1/2))*(a*(1-sec(d*x+c))/(a+b))^(1/2)*(a*(1+se 
c(d*x+c))/(a-b))^(1/2)/b^3/d/sec(d*x+c)^(1/2)-1/64*(a+b)^(1/2)*(8*B*a^3*b+ 
32*B*a*b^3-5*a^4*C-8*a^2*b^2*(2*A+C)+16*b^4*(4*A+3*C))*cos(d*x+c)^(1/2)*cs 
c(d*x+c)*EllipticPi((a+b*cos(d*x+c))^(1/2)/(a+b)^(1/2)/cos(d*x+c)^(1/2),(a 
+b)/b,(-(a+b)/(a-b))^(1/2))*(a*(1-sec(d*x+c))/(a+b))^(1/2)*(a*(1+sec(d*x+c 
))/(a-b))^(1/2)/b^4/d/sec(d*x+c)^(1/2)+1/4*C*(a+b*cos(d*x+c))^(3/2)*sin(d* 
x+c)/b/d/sec(d*x+c)^(3/2)+1/32*(16*A*b^2-8*B*a*b+5*C*a^2+12*C*b^2)*(a+b*co 
s(d*x+c))^(1/2)*sin(d*x+c)/b^2/d/sec(d*x+c)^(1/2)+1/24*(8*B*b-5*C*a)*(a+b* 
cos(d*x+c))^(3/2)*sin(d*x+c)/b^2/d/sec(d*x+c)^(1/2)-1/192*(24*B*a^2*b-128* 
B*b^3-15*a^3*C-4*a*b^2*(12*A+7*C))*(a+b*cos(d*x+c))^(1/2)*sec(d*x+c)^(1/2) 
*sin(d*x+c)/b^3/d
                                                                                    
                                                                                    
 

Mathematica [A] (warning: unable to verify)

Time = 14.39 (sec) , antiderivative size = 704, normalized size of antiderivative = 0.92 \[ \int \frac {\sqrt {a+b \cos (c+d x)} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sec ^{\frac {3}{2}}(c+d x)} \, dx=\frac {\sqrt {a+b \cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {(8 b B+a C) \sin (c+d x)}{96 b}+\frac {\left (48 A b^2+8 a b B-5 a^2 C+48 b^2 C\right ) \sin (2 (c+d x))}{192 b^2}+\frac {(8 b B+a C) \sin (3 (c+d x))}{96 b}+\frac {1}{32} C \sin (4 (c+d x))\right )}{d}+\frac {\cos ^4\left (\frac {1}{2} (c+d x)\right ) \sqrt {\sec (c+d x)} \left (-2 b (-a+b) (a+b) \left (-24 a^2 b B+128 b^3 B+15 a^3 C+4 a b^2 (12 A+7 C)\right ) \sqrt {\frac {\cos (c+d x)}{1+\cos (c+d x)}} \sqrt {\frac {a+b \cos (c+d x)}{(a+b) (1+\cos (c+d x))}} E\left (\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right )|\frac {-a+b}{a+b}\right ) \sec ^2\left (\frac {1}{2} (c+d x)\right )+a (a-b) (a+b) \left (15 a^3 C-6 a^2 b (4 B+5 C)-8 b^3 (12 A+16 B+9 C)+4 a b^2 (12 A+12 B+11 C)\right ) \operatorname {EllipticF}\left (\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right ),\frac {-a+b}{a+b}\right ) \left (\cos (c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right )\right )^{3/2} \sqrt {\frac {(a+b \cos (c+d x)) \sec ^2\left (\frac {1}{2} (c+d x)\right )}{a+b}} \sec (c+d x)-3 (a-b) \left (-8 a^3 b B-32 a b^3 B+5 a^4 C+8 a^2 b^2 (2 A+C)-16 b^4 (4 A+3 C)\right ) \left ((a-b) \operatorname {EllipticF}\left (\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right ),\frac {-a+b}{a+b}\right )+2 b \operatorname {EllipticPi}\left (-1,\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right ),\frac {-a+b}{a+b}\right )\right ) \left (\cos (c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right )\right )^{3/2} \sqrt {\frac {(a+b \cos (c+d x)) \sec ^2\left (\frac {1}{2} (c+d x)\right )}{a+b}} \sec (c+d x)+(a-b) b \left (-24 a^2 b B+128 b^3 B+15 a^3 C+4 a b^2 (12 A+7 C)\right ) \cos (c+d x) (a+b \cos (c+d x)) \sec ^4\left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {1}{2} (c+d x)\right )\right )}{192 (a-b) b^4 d \sqrt {a+b \cos (c+d x)}} \] Input:

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

Output:

(Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*(((8*b*B + a*C)*Sin[c + d*x]) 
/(96*b) + ((48*A*b^2 + 8*a*b*B - 5*a^2*C + 48*b^2*C)*Sin[2*(c + d*x)])/(19 
2*b^2) + ((8*b*B + a*C)*Sin[3*(c + d*x)])/(96*b) + (C*Sin[4*(c + d*x)])/32 
))/d + (Cos[(c + d*x)/2]^4*Sqrt[Sec[c + d*x]]*(-2*b*(-a + b)*(a + b)*(-24* 
a^2*b*B + 128*b^3*B + 15*a^3*C + 4*a*b^2*(12*A + 7*C))*Sqrt[Cos[c + d*x]/( 
1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] 
*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2 
+ a*(a - b)*(a + b)*(15*a^3*C - 6*a^2*b*(4*B + 5*C) - 8*b^3*(12*A + 16*B + 
 9*C) + 4*a*b^2*(12*A + 12*B + 11*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], 
(-a + b)/(a + b)]*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sqrt[((a + b*Cos 
[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Sec[c + d*x] - 3*(a - b)*(-8*a^3*b 
*B - 32*a*b^3*B + 5*a^4*C + 8*a^2*b^2*(2*A + C) - 16*b^4*(4*A + 3*C))*((a 
- b)*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + 2*b*EllipticP 
i[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)])*(Cos[c + d*x]*Sec[(c + 
d*x)/2]^2)^(3/2)*Sqrt[((a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*S 
ec[c + d*x] + (a - b)*b*(-24*a^2*b*B + 128*b^3*B + 15*a^3*C + 4*a*b^2*(12* 
A + 7*C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^4*Tan[(c + d* 
x)/2]))/(192*(a - b)*b^4*d*Sqrt[a + b*Cos[c + d*x]])
 

Rubi [A] (verified)

Time = 4.01 (sec) , antiderivative size = 739, normalized size of antiderivative = 0.96, number of steps used = 21, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {3042, 4709, 3042, 3528, 27, 3042, 3528, 27, 3042, 3528, 27, 3042, 3540, 3042, 3532, 3042, 3288, 3477, 3042, 3295, 3473}

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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4709

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3528

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3528

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {\int \frac {\sqrt {a+b \cos (c+d x)} \left (3 \left (5 C a^2-8 b B a+16 A b^2+12 b^2 C\right ) \cos ^2(c+d x)+2 b (16 b B-a C) \cos (c+d x)+a (8 b B-5 a C)\right )}{2 \sqrt {\cos (c+d x)}}dx}{3 b}+\frac {(8 b B-5 a C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 b d}}{8 b}+\frac {C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}}{4 b d}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {\int \frac {\sqrt {a+b \cos (c+d x)} \left (3 \left (5 C a^2-8 b B a+16 A b^2+12 b^2 C\right ) \cos ^2(c+d x)+2 b (16 b B-a C) \cos (c+d x)+a (8 b B-5 a C)\right )}{\sqrt {\cos (c+d x)}}dx}{6 b}+\frac {(8 b B-5 a C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 b d}}{8 b}+\frac {C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}}{4 b d}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {\int \frac {\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )} \left (3 \left (5 C a^2-8 b B a+16 A b^2+12 b^2 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+2 b (16 b B-a C) \sin \left (c+d x+\frac {\pi }{2}\right )+a (8 b B-5 a C)\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{6 b}+\frac {(8 b B-5 a C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 b d}}{8 b}+\frac {C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}}{4 b d}\right )\)

\(\Big \downarrow \) 3528

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {\frac {1}{2} \int \frac {-\left (\left (-15 C a^3+24 b B a^2-4 b^2 (12 A+7 C) a-128 b^3 B\right ) \cos ^2(c+d x)\right )+2 b \left (C a^2+56 b B a+48 A b^2+36 b^2 C\right ) \cos (c+d x)+a \left (-5 C a^2+8 b B a+48 A b^2+36 b^2 C\right )}{2 \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}}dx+\frac {3 \sin (c+d x) \sqrt {\cos (c+d x)} \left (5 a^2 C-8 a b B+16 A b^2+12 b^2 C\right ) \sqrt {a+b \cos (c+d x)}}{2 d}}{6 b}+\frac {(8 b B-5 a C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 b d}}{8 b}+\frac {C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}}{4 b d}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {\frac {1}{4} \int \frac {-\left (\left (-15 C a^3+24 b B a^2-4 b^2 (12 A+7 C) a-128 b^3 B\right ) \cos ^2(c+d x)\right )+2 b \left (C a^2+56 b B a+48 A b^2+36 b^2 C\right ) \cos (c+d x)+a \left (-5 C a^2+8 b B a+48 A b^2+36 b^2 C\right )}{\sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}}dx+\frac {3 \sin (c+d x) \sqrt {\cos (c+d x)} \left (5 a^2 C-8 a b B+16 A b^2+12 b^2 C\right ) \sqrt {a+b \cos (c+d x)}}{2 d}}{6 b}+\frac {(8 b B-5 a C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 b d}}{8 b}+\frac {C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}}{4 b d}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {\frac {1}{4} \int \frac {\left (15 C a^3-24 b B a^2+4 b^2 (12 A+7 C) a+128 b^3 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+2 b \left (C a^2+56 b B a+48 A b^2+36 b^2 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+a \left (-5 C a^2+8 b B a+48 A b^2+36 b^2 C\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {3 \sin (c+d x) \sqrt {\cos (c+d x)} \left (5 a^2 C-8 a b B+16 A b^2+12 b^2 C\right ) \sqrt {a+b \cos (c+d x)}}{2 d}}{6 b}+\frac {(8 b B-5 a C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 b d}}{8 b}+\frac {C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}}{4 b d}\right )\)

\(\Big \downarrow \) 3540

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {\frac {1}{4} \left (\frac {\int \frac {3 \left (-5 C a^4+8 b B a^3-8 b^2 (2 A+C) a^2+32 b^3 B a+16 b^4 (4 A+3 C)\right ) \cos ^2(c+d x)+2 a b \left (-5 C a^2+8 b B a+48 A b^2+36 b^2 C\right ) \cos (c+d x)+a \left (-15 C a^3+24 b B a^2-4 b^2 (12 A+7 C) a-128 b^3 B\right )}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx}{2 b}-\frac {\sin (c+d x) \left (-15 a^3 C+24 a^2 b B-4 a b^2 (12 A+7 C)-128 b^3 B\right ) \sqrt {a+b \cos (c+d x)}}{b d \sqrt {\cos (c+d x)}}\right )+\frac {3 \sin (c+d x) \sqrt {\cos (c+d x)} \left (5 a^2 C-8 a b B+16 A b^2+12 b^2 C\right ) \sqrt {a+b \cos (c+d x)}}{2 d}}{6 b}+\frac {(8 b B-5 a C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 b d}}{8 b}+\frac {C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}}{4 b d}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {\frac {1}{4} \left (\frac {\int \frac {3 \left (-5 C a^4+8 b B a^3-8 b^2 (2 A+C) a^2+32 b^3 B a+16 b^4 (4 A+3 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+2 a b \left (-5 C a^2+8 b B a+48 A b^2+36 b^2 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+a \left (-15 C a^3+24 b B a^2-4 b^2 (12 A+7 C) a-128 b^3 B\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{2 b}-\frac {\sin (c+d x) \left (-15 a^3 C+24 a^2 b B-4 a b^2 (12 A+7 C)-128 b^3 B\right ) \sqrt {a+b \cos (c+d x)}}{b d \sqrt {\cos (c+d x)}}\right )+\frac {3 \sin (c+d x) \sqrt {\cos (c+d x)} \left (5 a^2 C-8 a b B+16 A b^2+12 b^2 C\right ) \sqrt {a+b \cos (c+d x)}}{2 d}}{6 b}+\frac {(8 b B-5 a C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 b d}}{8 b}+\frac {C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}}{4 b d}\right )\)

\(\Big \downarrow \) 3532

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {\frac {1}{4} \left (\frac {\int \frac {a \left (-15 C a^3+24 b B a^2-4 b^2 (12 A+7 C) a-128 b^3 B\right )+2 a b \left (-5 C a^2+8 b B a+48 A b^2+36 b^2 C\right ) \cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx+3 \left (-5 a^4 C+8 a^3 b B-8 a^2 b^2 (2 A+C)+32 a b^3 B+16 b^4 (4 A+3 C)\right ) \int \frac {\sqrt {\cos (c+d x)}}{\sqrt {a+b \cos (c+d x)}}dx}{2 b}-\frac {\sin (c+d x) \left (-15 a^3 C+24 a^2 b B-4 a b^2 (12 A+7 C)-128 b^3 B\right ) \sqrt {a+b \cos (c+d x)}}{b d \sqrt {\cos (c+d x)}}\right )+\frac {3 \sin (c+d x) \sqrt {\cos (c+d x)} \left (5 a^2 C-8 a b B+16 A b^2+12 b^2 C\right ) \sqrt {a+b \cos (c+d x)}}{2 d}}{6 b}+\frac {(8 b B-5 a C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 b d}}{8 b}+\frac {C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}}{4 b d}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {\frac {1}{4} \left (\frac {\int \frac {a \left (-15 C a^3+24 b B a^2-4 b^2 (12 A+7 C) a-128 b^3 B\right )+2 a b \left (-5 C a^2+8 b B a+48 A b^2+36 b^2 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+3 \left (-5 a^4 C+8 a^3 b B-8 a^2 b^2 (2 A+C)+32 a b^3 B+16 b^4 (4 A+3 C)\right ) \int \frac {\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{2 b}-\frac {\sin (c+d x) \left (-15 a^3 C+24 a^2 b B-4 a b^2 (12 A+7 C)-128 b^3 B\right ) \sqrt {a+b \cos (c+d x)}}{b d \sqrt {\cos (c+d x)}}\right )+\frac {3 \sin (c+d x) \sqrt {\cos (c+d x)} \left (5 a^2 C-8 a b B+16 A b^2+12 b^2 C\right ) \sqrt {a+b \cos (c+d x)}}{2 d}}{6 b}+\frac {(8 b B-5 a C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 b d}}{8 b}+\frac {C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}}{4 b d}\right )\)

\(\Big \downarrow \) 3288

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {\frac {1}{4} \left (\frac {\int \frac {a \left (-15 C a^3+24 b B a^2-4 b^2 (12 A+7 C) a-128 b^3 B\right )+2 a b \left (-5 C a^2+8 b B a+48 A b^2+36 b^2 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {6 \sqrt {a+b} \cot (c+d x) \left (-5 a^4 C+8 a^3 b B-8 a^2 b^2 (2 A+C)+32 a b^3 B+16 b^4 (4 A+3 C)\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{b},\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{b d}}{2 b}-\frac {\sin (c+d x) \left (-15 a^3 C+24 a^2 b B-4 a b^2 (12 A+7 C)-128 b^3 B\right ) \sqrt {a+b \cos (c+d x)}}{b d \sqrt {\cos (c+d x)}}\right )+\frac {3 \sin (c+d x) \sqrt {\cos (c+d x)} \left (5 a^2 C-8 a b B+16 A b^2+12 b^2 C\right ) \sqrt {a+b \cos (c+d x)}}{2 d}}{6 b}+\frac {(8 b B-5 a C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 b d}}{8 b}+\frac {C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}}{4 b d}\right )\)

\(\Big \downarrow \) 3477

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {\frac {1}{4} \left (\frac {a \left (-15 a^3 C+24 a^2 b B-4 a b^2 (12 A+7 C)-128 b^3 B\right ) \int \frac {\cos (c+d x)+1}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx+a \left (15 a^3 C-2 a^2 b (12 B+5 C)+4 a b^2 (12 A+4 B+7 C)+8 b^3 (12 A+16 B+9 C)\right ) \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}}dx-\frac {6 \sqrt {a+b} \cot (c+d x) \left (-5 a^4 C+8 a^3 b B-8 a^2 b^2 (2 A+C)+32 a b^3 B+16 b^4 (4 A+3 C)\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{b},\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{b d}}{2 b}-\frac {\sin (c+d x) \left (-15 a^3 C+24 a^2 b B-4 a b^2 (12 A+7 C)-128 b^3 B\right ) \sqrt {a+b \cos (c+d x)}}{b d \sqrt {\cos (c+d x)}}\right )+\frac {3 \sin (c+d x) \sqrt {\cos (c+d x)} \left (5 a^2 C-8 a b B+16 A b^2+12 b^2 C\right ) \sqrt {a+b \cos (c+d x)}}{2 d}}{6 b}+\frac {(8 b B-5 a C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 b d}}{8 b}+\frac {C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}}{4 b d}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {\frac {1}{4} \left (\frac {a \left (15 a^3 C-2 a^2 b (12 B+5 C)+4 a b^2 (12 A+4 B+7 C)+8 b^3 (12 A+16 B+9 C)\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+a \left (-15 a^3 C+24 a^2 b B-4 a b^2 (12 A+7 C)-128 b^3 B\right ) \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )+1}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {6 \sqrt {a+b} \cot (c+d x) \left (-5 a^4 C+8 a^3 b B-8 a^2 b^2 (2 A+C)+32 a b^3 B+16 b^4 (4 A+3 C)\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{b},\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{b d}}{2 b}-\frac {\sin (c+d x) \left (-15 a^3 C+24 a^2 b B-4 a b^2 (12 A+7 C)-128 b^3 B\right ) \sqrt {a+b \cos (c+d x)}}{b d \sqrt {\cos (c+d x)}}\right )+\frac {3 \sin (c+d x) \sqrt {\cos (c+d x)} \left (5 a^2 C-8 a b B+16 A b^2+12 b^2 C\right ) \sqrt {a+b \cos (c+d x)}}{2 d}}{6 b}+\frac {(8 b B-5 a C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 b d}}{8 b}+\frac {C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}}{4 b d}\right )\)

\(\Big \downarrow \) 3295

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {\frac {1}{4} \left (\frac {a \left (-15 a^3 C+24 a^2 b B-4 a b^2 (12 A+7 C)-128 b^3 B\right ) \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )+1}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sqrt {a+b} \cot (c+d x) \left (15 a^3 C-2 a^2 b (12 B+5 C)+4 a b^2 (12 A+4 B+7 C)+8 b^3 (12 A+16 B+9 C)\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{d}-\frac {6 \sqrt {a+b} \cot (c+d x) \left (-5 a^4 C+8 a^3 b B-8 a^2 b^2 (2 A+C)+32 a b^3 B+16 b^4 (4 A+3 C)\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{b},\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{b d}}{2 b}-\frac {\sin (c+d x) \left (-15 a^3 C+24 a^2 b B-4 a b^2 (12 A+7 C)-128 b^3 B\right ) \sqrt {a+b \cos (c+d x)}}{b d \sqrt {\cos (c+d x)}}\right )+\frac {3 \sin (c+d x) \sqrt {\cos (c+d x)} \left (5 a^2 C-8 a b B+16 A b^2+12 b^2 C\right ) \sqrt {a+b \cos (c+d x)}}{2 d}}{6 b}+\frac {(8 b B-5 a C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 b d}}{8 b}+\frac {C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}}{4 b d}\right )\)

\(\Big \downarrow \) 3473

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {\frac {3 \sin (c+d x) \sqrt {\cos (c+d x)} \left (5 a^2 C-8 a b B+16 A b^2+12 b^2 C\right ) \sqrt {a+b \cos (c+d x)}}{2 d}+\frac {1}{4} \left (\frac {\frac {2 \sqrt {a+b} \cot (c+d x) \left (15 a^3 C-2 a^2 b (12 B+5 C)+4 a b^2 (12 A+4 B+7 C)+8 b^3 (12 A+16 B+9 C)\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{d}+\frac {2 (a-b) \sqrt {a+b} \cot (c+d x) \left (-15 a^3 C+24 a^2 b B-4 a b^2 (12 A+7 C)-128 b^3 B\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right )}{a d}-\frac {6 \sqrt {a+b} \cot (c+d x) \left (-5 a^4 C+8 a^3 b B-8 a^2 b^2 (2 A+C)+32 a b^3 B+16 b^4 (4 A+3 C)\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{b},\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{b d}}{2 b}-\frac {\sin (c+d x) \left (-15 a^3 C+24 a^2 b B-4 a b^2 (12 A+7 C)-128 b^3 B\right ) \sqrt {a+b \cos (c+d x)}}{b d \sqrt {\cos (c+d x)}}\right )}{6 b}+\frac {(8 b B-5 a C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 b d}}{8 b}+\frac {C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}}{4 b d}\right )\)

Input:

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

Output:

Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((C*Cos[c + d*x]^(3/2)*(a + b*Cos[c 
+ d*x])^(3/2)*Sin[c + d*x])/(4*b*d) + (((8*b*B - 5*a*C)*Sqrt[Cos[c + d*x]] 
*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*b*d) + ((3*(16*A*b^2 - 8*a*b* 
B + 5*a^2*C + 12*b^2*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c 
+ d*x])/(2*d) + (((2*(a - b)*Sqrt[a + b]*(24*a^2*b*B - 128*b^3*B - 15*a^3* 
C - 4*a*b^2*(12*A + 7*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + 
 d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - 
 Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*d) + (2* 
Sqrt[a + b]*(15*a^3*C - 2*a^2*b*(12*B + 5*C) + 4*a*b^2*(12*A + 4*B + 7*C) 
+ 8*b^3*(12*A + 16*B + 9*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[ 
c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*( 
1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/d - (6*S 
qrt[a + b]*(8*a^3*b*B + 32*a*b^3*B - 5*a^4*C - 8*a^2*b^2*(2*A + C) + 16*b^ 
4*(4*A + 3*C))*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c 
+ d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 
- Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b*d))/(2* 
b) - ((24*a^2*b*B - 128*b^3*B - 15*a^3*C - 4*a*b^2*(12*A + 7*C))*Sqrt[a + 
b*Cos[c + d*x]]*Sin[c + d*x])/(b*d*Sqrt[Cos[c + d*x]]))/4)/(6*b))/(8*b))
 

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 3288

Int[Sqrt[(b_.)*sin[(e_.) + (f_.)*(x_)]]/Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.) 
*(x_)]], x_Symbol] :> Simp[2*b*(Tan[e + f*x]/(d*f))*Rt[(c + d)/b, 2]*Sqrt[c 
*((1 + Csc[e + f*x])/(c - d))]*Sqrt[c*((1 - Csc[e + f*x])/(c + d))]*Ellipti 
cPi[(c + d)/d, ArcSin[Sqrt[c + d*Sin[e + f*x]]/Sqrt[b*Sin[e + f*x]]/Rt[(c + 
 d)/b, 2]], -(c + d)/(c - d)], x] /; FreeQ[{b, c, d, e, f}, x] && NeQ[c^2 - 
 d^2, 0] && PosQ[(c + d)/b]
 

rule 3295

Int[1/(Sqrt[(d_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(a_) + (b_.)*sin[(e_.) + (f 
_.)*(x_)]]), x_Symbol] :> Simp[-2*(Tan[e + f*x]/(a*f))*Rt[(a + b)/d, 2]*Sqr 
t[a*((1 - Csc[e + f*x])/(a + b))]*Sqrt[a*((1 + Csc[e + f*x])/(a - b))]*Elli 
pticF[ArcSin[Sqrt[a + b*Sin[e + f*x]]/Sqrt[d*Sin[e + f*x]]/Rt[(a + b)/d, 2] 
], -(a + b)/(a - b)], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] 
&& PosQ[(a + b)/d]
 

rule 3473

Int[((A_) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((b_.)*sin[(e_.) + (f_.)*(x_)]) 
^(3/2)*Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[-2*A* 
(c - d)*(Tan[e + f*x]/(f*b*c^2))*Rt[(c + d)/b, 2]*Sqrt[c*((1 + Csc[e + f*x] 
)/(c - d))]*Sqrt[c*((1 - Csc[e + f*x])/(c + d))]*EllipticE[ArcSin[Sqrt[c + 
d*Sin[e + f*x]]/Sqrt[b*Sin[e + f*x]]/Rt[(c + d)/b, 2]], -(c + d)/(c - d)], 
x] /; FreeQ[{b, c, d, e, f, A, B}, x] && NeQ[c^2 - d^2, 0] && EqQ[A, B] && 
PosQ[(c + d)/b]
 

rule 3477

Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((a_.) + (b_.)*sin[(e_.) + (f_ 
.)*(x_)])^(3/2)*Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> S 
imp[(A - B)/(a - b)   Int[1/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f* 
x]]), x], x] - Simp[(A*b - a*B)/(a - b)   Int[(1 + Sin[e + f*x])/((a + b*Si 
n[e + f*x])^(3/2)*Sqrt[c + d*Sin[e + f*x]]), x], x] /; FreeQ[{a, b, c, d, e 
, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 
0] && NeQ[A, B]
 

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 3532

Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^ 
2)/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(3/2)*Sqrt[(c_.) + (d_.)*sin[(e 
_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[C/b^2   Int[Sqrt[a + b*Sin[e + f*x]] 
/Sqrt[c + d*Sin[e + f*x]], x], x] + Simp[1/b^2   Int[(A*b^2 - a^2*C + b*(b* 
B - 2*a*C)*Sin[e + f*x])/((a + b*Sin[e + f*x])^(3/2)*Sqrt[c + d*Sin[e + f*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]
 

rule 3540

Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^ 
2)/(Sqrt[(a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(c_) + (d_.)*sin[(e_.) 
 + (f_.)*(x_)]]), x_Symbol] :> Simp[(-C)*Cos[e + f*x]*(Sqrt[c + d*Sin[e + f 
*x]]/(d*f*Sqrt[a + b*Sin[e + f*x]])), x] + Simp[1/(2*d)   Int[(1/((a + b*Si 
n[e + f*x])^(3/2)*Sqrt[c + d*Sin[e + f*x]]))*Simp[2*a*A*d - C*(b*c - a*d) - 
 2*(a*c*C - d*(A*b + a*B))*Sin[e + f*x] + (2*b*B*d - C*(b*c + a*d))*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]
 

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. \(2960\) vs. \(2(691)=1382\).

Time = 28.14 (sec) , antiderivative size = 2961, normalized size of antiderivative = 3.87

method	result	size
default	\(\text {Expression too large to display}\)	\(2961\)
parts	\(\text {Expression too large to display}\)	\(3021\)

Input:

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

Output:

1/192/d/b^3*(a+b*cos(d*x+c))^(1/2)/(b*cos(d*x+c)^2+a*cos(d*x+c)+b*cos(d*x+ 
c)+a)/sec(d*x+c)^(3/2)*(A*(1/(a+b)*(a+b*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)* 
(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*b^4*EllipticPi(-csc(d*x+c)+cot(d*x+c),-1 
,(-(a-b)/(a+b))^(1/2))*(-384-768*sec(d*x+c)-384*sec(d*x+c)^2)+C*(1/(a+b)*( 
a+b*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*a^ 
4*EllipticPi(-csc(d*x+c)+cot(d*x+c),-1,(-(a-b)/(a+b))^(1/2))*(30+60*sec(d* 
x+c)+30*sec(d*x+c)^2)+C*(1/(a+b)*(a+b*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*(c 
os(d*x+c)/(1+cos(d*x+c)))^(1/2)*b^4*EllipticPi(-csc(d*x+c)+cot(d*x+c),-1,( 
-(a-b)/(a+b))^(1/2))*(-288-576*sec(d*x+c)-288*sec(d*x+c)^2)+A*(1/(a+b)*(a+ 
b*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*b^4* 
EllipticF(-csc(d*x+c)+cot(d*x+c),(-(a-b)/(a+b))^(1/2))*(192+384*sec(d*x+c) 
+192*sec(d*x+c)^2)+C*(1/(a+b)*(a+b*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*(cos( 
d*x+c)/(1+cos(d*x+c)))^(1/2)*b^4*EllipticF(-csc(d*x+c)+cot(d*x+c),(-(a-b)/ 
(a+b))^(1/2))*(144+288*sec(d*x+c)+144*sec(d*x+c)^2)+B*(1/(a+b)*(a+b*cos(d* 
x+c))/(1+cos(d*x+c)))^(1/2)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*b^4*Elliptic 
E(-csc(d*x+c)+cot(d*x+c),(-(a-b)/(a+b))^(1/2))*(-128-256*sec(d*x+c)-128*se 
c(d*x+c)^2)+C*(1/(a+b)*(a+b*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*(cos(d*x+c)/ 
(1+cos(d*x+c)))^(1/2)*a^4*EllipticE(-csc(d*x+c)+cot(d*x+c),(-(a-b)/(a+b))^ 
(1/2))*(-15-30*sec(d*x+c)-15*sec(d*x+c)^2)+15*a^4*C*tan(d*x+c)+a*A*b^3*(14 
4*sin(d*x+c)+96*tan(d*x+c))+B*a^2*b^2*(-8*sin(d*x+c)+16*tan(d*x+c))+a^3...
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

Integral(sqrt(a + b*cos(c + d*x))*(A + B*cos(c + d*x) + C*cos(c + d*x)**2) 
/sec(c + d*x)**(3/2), x)
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

int((sqrt(sec(c + d*x))*sqrt(cos(c + d*x)*b + a)*cos(c + d*x))/sec(c + d*x 
)**2,x)*b + int((sqrt(sec(c + d*x))*sqrt(cos(c + d*x)*b + a)*cos(c + d*x)* 
*2)/sec(c + d*x)**2,x)*c + int((sqrt(sec(c + d*x))*sqrt(cos(c + d*x)*b + a 
))/sec(c + d*x)**2,x)*a