\(\int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x)+C \cos ^2(c+d x)) \, dx\) [1129]

Optimal result

Integrand size = 45, antiderivative size = 834 \[ \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=-\frac {(a-b) \sqrt {a+b} \left (150 a^3 b B+2840 a b^3 B-45 a^4 C+256 b^4 (5 A+4 C)+12 a^2 b^2 (220 A+141 C)\right ) \cot (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{1920 a b^2 d}-\frac {\sqrt {a+b} \left (45 a^4 C-30 a^3 b (5 B+C)-16 b^4 (80 A+45 B+64 C)-8 a b^3 (260 A+355 B+193 C)-4 a^2 b^2 (660 A+295 B+423 C)\right ) \cot (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{1920 b^2 d}+\frac {\sqrt {a+b} \left (10 a^4 b B-240 a^2 b^3 B-96 b^5 B-3 a^5 C-40 a^3 b^2 (2 A+C)-80 a b^4 (4 A+3 C)\right ) \cot (c+d x) \operatorname {EllipticPi}\left (\frac {a+b}{b},\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{128 b^3 d}+\frac {\left (150 a^3 b B+2840 a b^3 B-45 a^4 C+256 b^4 (5 A+4 C)+12 a^2 b^2 (220 A+141 C)\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{1920 b^2 d \sqrt {\cos (c+d x)}}+\frac {\left (50 a^2 b B+120 b^3 B-15 a^3 C+4 a b^2 (60 A+43 C)\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{320 b d}+\frac {\left (80 A b^2+50 a b B-15 a^2 C+64 b^2 C\right ) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{240 b d}+\frac {(10 b B-3 a C) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{40 b d}+\frac {C \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{5 b d} \] Output:

-1/1920*(a-b)*(a+b)^(1/2)*(150*B*a^3*b+2840*B*a*b^3-45*a^4*C+256*b^4*(5*A+ 
4*C)+12*a^2*b^2*(220*A+141*C))*cot(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^2/d-1/1920*(a+b)^(1/2)*(45*a^4 
*C-30*a^3*b*(5*B+C)-16*b^4*(80*A+45*B+64*C)-8*a*b^3*(260*A+355*B+193*C)-4* 
a^2*b^2*(660*A+295*B+423*C))*cot(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+sec(d*x+c))/(a-b))^(1/2)/b^2/d+1/128*(a+b)^(1/2)*(10*B*a^4*b- 
240*B*a^2*b^3-96*B*b^5-3*C*a^5-40*a^3*b^2*(2*A+C)-80*a*b^4*(4*A+3*C))*cot( 
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^3/d+1/1920*(150*B*a^3*b+2840*B*a*b^3-45*a^4*C+256*b^4*(5*A 
+4*C)+12*a^2*b^2*(220*A+141*C))*(a+b*cos(d*x+c))^(1/2)*sin(d*x+c)/b^2/d/co 
s(d*x+c)^(1/2)+1/320*(50*B*a^2*b+120*B*b^3-15*a^3*C+4*a*b^2*(60*A+43*C))*c 
os(d*x+c)^(1/2)*(a+b*cos(d*x+c))^(1/2)*sin(d*x+c)/b/d+1/240*(80*A*b^2+50*B 
*a*b-15*C*a^2+64*C*b^2)*cos(d*x+c)^(1/2)*(a+b*cos(d*x+c))^(3/2)*sin(d*x+c) 
/b/d+1/40*(10*B*b-3*C*a)*cos(d*x+c)^(1/2)*(a+b*cos(d*x+c))^(5/2)*sin(d*x+c 
)/b/d+1/5*C*cos(d*x+c)^(1/2)*(a+b*cos(d*x+c))^(7/2)*sin(d*x+c)/b/d
                                                                                    
                                                                                    
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 8.22 (sec) , antiderivative size = 1410, normalized size of antiderivative = 1.69 \[ \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx =\text {Too large to display} \] Input:

Integrate[Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x 
] + C*Cos[c + d*x]^2),x]
 

Output:

-1/3840*((-4*a*(-4720*a^2*A*b^2 - 1280*A*b^4 - 1330*a^3*b*B - 3560*a*b^3*B 
 + 15*a^4*C - 3236*a^2*b^2*C - 1024*b^4*C)*Sqrt[((a + b)*Cot[(c + d*x)/2]^ 
2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a 
 + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sq 
rt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)] 
*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) 
 - 4*a*(-3840*a^3*A*b - 6080*a*A*b^3 - 6440*a^2*b^2*B - 1440*b^4*B - 2292* 
a^3*b*C - 4624*a*b^3*C)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt 
[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x] 
)*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c 
+ d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2] 
^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b) 
*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/ 
2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*E 
llipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a] 
/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt 
[a + b*Cos[c + d*x]])) + 2*(-2640*a^2*A*b^2 - 1280*A*b^4 - 150*a^3*b*B - 2 
840*a*b^3*B + 45*a^4*C - 1692*a^2*b^2*C - 1024*b^4*C)*((I*Cos[(c + d*x)/2] 
*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c 
+ d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec...
 

Rubi [A] (verified)

Time = 4.84 (sec) , antiderivative size = 841, normalized size of antiderivative = 1.01, number of steps used = 23, number of rules used = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.511, Rules used = {3042, 3528, 27, 3042, 3528, 27, 3042, 3528, 27, 3042, 3528, 27, 3042, 3540, 25, 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.

\(\displaystyle \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3528

\(\displaystyle \frac {\int \frac {(a+b \cos (c+d x))^{5/2} \left ((10 b B-3 a C) \cos ^2(c+d x)+2 b (5 A+4 C) \cos (c+d x)+a C\right )}{2 \sqrt {\cos (c+d x)}}dx}{5 b}+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{7/2}}{5 b d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {(a+b \cos (c+d x))^{5/2} \left ((10 b B-3 a C) \cos ^2(c+d x)+2 b (5 A+4 C) \cos (c+d x)+a C\right )}{\sqrt {\cos (c+d x)}}dx}{10 b}+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{7/2}}{5 b d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2} \left ((10 b B-3 a C) \sin \left (c+d x+\frac {\pi }{2}\right )^2+2 b (5 A+4 C) \sin \left (c+d x+\frac {\pi }{2}\right )+a C\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{10 b}+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{7/2}}{5 b d}\)

\(\Big \downarrow \) 3528

\(\displaystyle \frac {\frac {1}{4} \int \frac {(a+b \cos (c+d x))^{3/2} \left (\left (-15 C a^2+50 b B a+80 A b^2+64 b^2 C\right ) \cos ^2(c+d x)+2 b (40 a A+30 b B+27 a C) \cos (c+d x)+5 a (2 b B+a C)\right )}{2 \sqrt {\cos (c+d x)}}dx+\frac {(10 b B-3 a C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 d}}{10 b}+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{7/2}}{5 b d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{8} \int \frac {(a+b \cos (c+d x))^{3/2} \left (\left (-15 C a^2+50 b B a+80 A b^2+64 b^2 C\right ) \cos ^2(c+d x)+2 b (40 a A+30 b B+27 a C) \cos (c+d x)+5 a (2 b B+a C)\right )}{\sqrt {\cos (c+d x)}}dx+\frac {(10 b B-3 a C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 d}}{10 b}+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{7/2}}{5 b d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {1}{8} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (\left (-15 C a^2+50 b B a+80 A b^2+64 b^2 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+2 b (40 a A+30 b B+27 a C) \sin \left (c+d x+\frac {\pi }{2}\right )+5 a (2 b B+a C)\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {(10 b B-3 a C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 d}}{10 b}+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{7/2}}{5 b d}\)

\(\Big \downarrow \) 3528

\(\displaystyle \frac {\frac {1}{8} \left (\frac {1}{3} \int \frac {\sqrt {a+b \cos (c+d x)} \left (3 \left (-15 C a^3+50 b B a^2+4 b^2 (60 A+43 C) a+120 b^3 B\right ) \cos ^2(c+d x)+2 b \left (3 (80 A+49 C) a^2+310 b B a+32 b^2 (5 A+4 C)\right ) \cos (c+d x)+a \left (15 C a^2+110 b B a+80 A b^2+64 b^2 C\right )\right )}{2 \sqrt {\cos (c+d x)}}dx+\frac {\sin (c+d x) \sqrt {\cos (c+d x)} \left (-15 a^2 C+50 a b B+80 A b^2+64 b^2 C\right ) (a+b \cos (c+d x))^{3/2}}{3 d}\right )+\frac {(10 b B-3 a C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 d}}{10 b}+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{7/2}}{5 b d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{8} \left (\frac {1}{6} \int \frac {\sqrt {a+b \cos (c+d x)} \left (3 \left (-15 C a^3+50 b B a^2+4 b^2 (60 A+43 C) a+120 b^3 B\right ) \cos ^2(c+d x)+2 b \left (3 (80 A+49 C) a^2+310 b B a+32 b^2 (5 A+4 C)\right ) \cos (c+d x)+a \left (15 C a^2+110 b B a+80 A b^2+64 b^2 C\right )\right )}{\sqrt {\cos (c+d x)}}dx+\frac {\sin (c+d x) \sqrt {\cos (c+d x)} \left (-15 a^2 C+50 a b B+80 A b^2+64 b^2 C\right ) (a+b \cos (c+d x))^{3/2}}{3 d}\right )+\frac {(10 b B-3 a C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 d}}{10 b}+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{7/2}}{5 b d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {1}{8} \left (\frac {1}{6} \int \frac {\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )} \left (3 \left (-15 C a^3+50 b B a^2+4 b^2 (60 A+43 C) a+120 b^3 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+2 b \left (3 (80 A+49 C) a^2+310 b B a+32 b^2 (5 A+4 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+a \left (15 C a^2+110 b B a+80 A b^2+64 b^2 C\right )\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {\sin (c+d x) \sqrt {\cos (c+d x)} \left (-15 a^2 C+50 a b B+80 A b^2+64 b^2 C\right ) (a+b \cos (c+d x))^{3/2}}{3 d}\right )+\frac {(10 b B-3 a C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 d}}{10 b}+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{7/2}}{5 b d}\)

\(\Big \downarrow \) 3528

\(\displaystyle \frac {\frac {1}{8} \left (\frac {1}{6} \left (\frac {1}{2} \int \frac {\left (-45 C a^4+150 b B a^3+12 b^2 (220 A+141 C) a^2+2840 b^3 B a+256 b^4 (5 A+4 C)\right ) \cos ^2(c+d x)+2 b \left ((960 A+573 C) a^3+1610 b B a^2+4 b^2 (380 A+289 C) a+360 b^3 B\right ) \cos (c+d x)+a \left (15 C a^3+590 b B a^2+4 b^2 (260 A+193 C) a+360 b^3 B\right )}{2 \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}}dx+\frac {3 \sin (c+d x) \sqrt {\cos (c+d x)} \left (-15 a^3 C+50 a^2 b B+4 a b^2 (60 A+43 C)+120 b^3 B\right ) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {\sin (c+d x) \sqrt {\cos (c+d x)} \left (-15 a^2 C+50 a b B+80 A b^2+64 b^2 C\right ) (a+b \cos (c+d x))^{3/2}}{3 d}\right )+\frac {(10 b B-3 a C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 d}}{10 b}+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{7/2}}{5 b d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{8} \left (\frac {1}{6} \left (\frac {1}{4} \int \frac {\left (-45 C a^4+150 b B a^3+12 b^2 (220 A+141 C) a^2+2840 b^3 B a+256 b^4 (5 A+4 C)\right ) \cos ^2(c+d x)+2 b \left ((960 A+573 C) a^3+1610 b B a^2+4 b^2 (380 A+289 C) a+360 b^3 B\right ) \cos (c+d x)+a \left (15 C a^3+590 b B a^2+4 b^2 (260 A+193 C) a+360 b^3 B\right )}{\sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}}dx+\frac {3 \sin (c+d x) \sqrt {\cos (c+d x)} \left (-15 a^3 C+50 a^2 b B+4 a b^2 (60 A+43 C)+120 b^3 B\right ) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {\sin (c+d x) \sqrt {\cos (c+d x)} \left (-15 a^2 C+50 a b B+80 A b^2+64 b^2 C\right ) (a+b \cos (c+d x))^{3/2}}{3 d}\right )+\frac {(10 b B-3 a C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 d}}{10 b}+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{7/2}}{5 b d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {1}{8} \left (\frac {1}{6} \left (\frac {1}{4} \int \frac {\left (-45 C a^4+150 b B a^3+12 b^2 (220 A+141 C) a^2+2840 b^3 B a+256 b^4 (5 A+4 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+2 b \left ((960 A+573 C) a^3+1610 b B a^2+4 b^2 (380 A+289 C) a+360 b^3 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+a \left (15 C a^3+590 b B a^2+4 b^2 (260 A+193 C) a+360 b^3 B\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {3 \sin (c+d x) \sqrt {\cos (c+d x)} \left (-15 a^3 C+50 a^2 b B+4 a b^2 (60 A+43 C)+120 b^3 B\right ) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {\sin (c+d x) \sqrt {\cos (c+d x)} \left (-15 a^2 C+50 a b B+80 A b^2+64 b^2 C\right ) (a+b \cos (c+d x))^{3/2}}{3 d}\right )+\frac {(10 b B-3 a C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 d}}{10 b}+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{7/2}}{5 b d}\)

\(\Big \downarrow \) 3540

\(\displaystyle \frac {\frac {1}{8} \left (\frac {1}{6} \left (\frac {1}{4} \left (\frac {\int -\frac {15 \left (-3 C a^5+10 b B a^4-40 b^2 (2 A+C) a^3-240 b^3 B a^2-80 b^4 (4 A+3 C) a-96 b^5 B\right ) \cos ^2(c+d x)-2 a b \left (15 C a^3+590 b B a^2+4 b^2 (260 A+193 C) a+360 b^3 B\right ) \cos (c+d x)+a \left (-45 C a^4+150 b B a^3+12 b^2 (220 A+141 C) a^2+2840 b^3 B a+256 b^4 (5 A+4 C)\right )}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx}{2 b}+\frac {\sin (c+d x) \left (-45 a^4 C+150 a^3 b B+12 a^2 b^2 (220 A+141 C)+2840 a b^3 B+256 b^4 (5 A+4 C)\right ) \sqrt {a+b \cos (c+d x)}}{b d \sqrt {\cos (c+d x)}}\right )+\frac {3 \sin (c+d x) \sqrt {\cos (c+d x)} \left (-15 a^3 C+50 a^2 b B+4 a b^2 (60 A+43 C)+120 b^3 B\right ) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {\sin (c+d x) \sqrt {\cos (c+d x)} \left (-15 a^2 C+50 a b B+80 A b^2+64 b^2 C\right ) (a+b \cos (c+d x))^{3/2}}{3 d}\right )+\frac {(10 b B-3 a C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 d}}{10 b}+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{7/2}}{5 b d}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {1}{8} \left (\frac {1}{6} \left (\frac {1}{4} \left (\frac {\sin (c+d x) \left (-45 a^4 C+150 a^3 b B+12 a^2 b^2 (220 A+141 C)+2840 a b^3 B+256 b^4 (5 A+4 C)\right ) \sqrt {a+b \cos (c+d x)}}{b d \sqrt {\cos (c+d x)}}-\frac {\int \frac {15 \left (-3 C a^5+10 b B a^4-40 b^2 (2 A+C) a^3-240 b^3 B a^2-80 b^4 (4 A+3 C) a-96 b^5 B\right ) \cos ^2(c+d x)-2 a b \left (15 C a^3+590 b B a^2+4 b^2 (260 A+193 C) a+360 b^3 B\right ) \cos (c+d x)+a \left (-45 C a^4+150 b B a^3+12 b^2 (220 A+141 C) a^2+2840 b^3 B a+256 b^4 (5 A+4 C)\right )}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx}{2 b}\right )+\frac {3 \sin (c+d x) \sqrt {\cos (c+d x)} \left (-15 a^3 C+50 a^2 b B+4 a b^2 (60 A+43 C)+120 b^3 B\right ) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {\sin (c+d x) \sqrt {\cos (c+d x)} \left (-15 a^2 C+50 a b B+80 A b^2+64 b^2 C\right ) (a+b \cos (c+d x))^{3/2}}{3 d}\right )+\frac {(10 b B-3 a C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 d}}{10 b}+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{7/2}}{5 b d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {1}{8} \left (\frac {1}{6} \left (\frac {1}{4} \left (\frac {\sin (c+d x) \left (-45 a^4 C+150 a^3 b B+12 a^2 b^2 (220 A+141 C)+2840 a b^3 B+256 b^4 (5 A+4 C)\right ) \sqrt {a+b \cos (c+d x)}}{b d \sqrt {\cos (c+d x)}}-\frac {\int \frac {15 \left (-3 C a^5+10 b B a^4-40 b^2 (2 A+C) a^3-240 b^3 B a^2-80 b^4 (4 A+3 C) a-96 b^5 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2-2 a b \left (15 C a^3+590 b B a^2+4 b^2 (260 A+193 C) a+360 b^3 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+a \left (-45 C a^4+150 b B a^3+12 b^2 (220 A+141 C) a^2+2840 b^3 B a+256 b^4 (5 A+4 C)\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{2 b}\right )+\frac {3 \sin (c+d x) \sqrt {\cos (c+d x)} \left (-15 a^3 C+50 a^2 b B+4 a b^2 (60 A+43 C)+120 b^3 B\right ) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {\sin (c+d x) \sqrt {\cos (c+d x)} \left (-15 a^2 C+50 a b B+80 A b^2+64 b^2 C\right ) (a+b \cos (c+d x))^{3/2}}{3 d}\right )+\frac {(10 b B-3 a C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 d}}{10 b}+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{7/2}}{5 b d}\)

\(\Big \downarrow \) 3532

\(\displaystyle \frac {\frac {1}{8} \left (\frac {1}{6} \left (\frac {1}{4} \left (\frac {\sin (c+d x) \left (-45 a^4 C+150 a^3 b B+12 a^2 b^2 (220 A+141 C)+2840 a b^3 B+256 b^4 (5 A+4 C)\right ) \sqrt {a+b \cos (c+d x)}}{b d \sqrt {\cos (c+d x)}}-\frac {\int \frac {a \left (-45 C a^4+150 b B a^3+12 b^2 (220 A+141 C) a^2+2840 b^3 B a+256 b^4 (5 A+4 C)\right )-2 a b \left (15 C a^3+590 b B a^2+4 b^2 (260 A+193 C) a+360 b^3 B\right ) \cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx+15 \left (-3 a^5 C+10 a^4 b B-40 a^3 b^2 (2 A+C)-240 a^2 b^3 B-80 a b^4 (4 A+3 C)-96 b^5 B\right ) \int \frac {\sqrt {\cos (c+d x)}}{\sqrt {a+b \cos (c+d x)}}dx}{2 b}\right )+\frac {3 \sin (c+d x) \sqrt {\cos (c+d x)} \left (-15 a^3 C+50 a^2 b B+4 a b^2 (60 A+43 C)+120 b^3 B\right ) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {\sin (c+d x) \sqrt {\cos (c+d x)} \left (-15 a^2 C+50 a b B+80 A b^2+64 b^2 C\right ) (a+b \cos (c+d x))^{3/2}}{3 d}\right )+\frac {(10 b B-3 a C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 d}}{10 b}+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{7/2}}{5 b d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {1}{8} \left (\frac {1}{6} \left (\frac {1}{4} \left (\frac {\sin (c+d x) \left (-45 a^4 C+150 a^3 b B+12 a^2 b^2 (220 A+141 C)+2840 a b^3 B+256 b^4 (5 A+4 C)\right ) \sqrt {a+b \cos (c+d x)}}{b d \sqrt {\cos (c+d x)}}-\frac {\int \frac {a \left (-45 C a^4+150 b B a^3+12 b^2 (220 A+141 C) a^2+2840 b^3 B a+256 b^4 (5 A+4 C)\right )-2 a b \left (15 C a^3+590 b B a^2+4 b^2 (260 A+193 C) a+360 b^3 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+15 \left (-3 a^5 C+10 a^4 b B-40 a^3 b^2 (2 A+C)-240 a^2 b^3 B-80 a b^4 (4 A+3 C)-96 b^5 B\right ) \int \frac {\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{2 b}\right )+\frac {3 \sin (c+d x) \sqrt {\cos (c+d x)} \left (-15 a^3 C+50 a^2 b B+4 a b^2 (60 A+43 C)+120 b^3 B\right ) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {\sin (c+d x) \sqrt {\cos (c+d x)} \left (-15 a^2 C+50 a b B+80 A b^2+64 b^2 C\right ) (a+b \cos (c+d x))^{3/2}}{3 d}\right )+\frac {(10 b B-3 a C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 d}}{10 b}+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{7/2}}{5 b d}\)

\(\Big \downarrow \) 3288

\(\displaystyle \frac {\frac {1}{8} \left (\frac {1}{6} \left (\frac {1}{4} \left (\frac {\sin (c+d x) \left (-45 a^4 C+150 a^3 b B+12 a^2 b^2 (220 A+141 C)+2840 a b^3 B+256 b^4 (5 A+4 C)\right ) \sqrt {a+b \cos (c+d x)}}{b d \sqrt {\cos (c+d x)}}-\frac {\int \frac {a \left (-45 C a^4+150 b B a^3+12 b^2 (220 A+141 C) a^2+2840 b^3 B a+256 b^4 (5 A+4 C)\right )-2 a b \left (15 C a^3+590 b B a^2+4 b^2 (260 A+193 C) a+360 b^3 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {30 \sqrt {a+b} \cot (c+d x) \left (-3 a^5 C+10 a^4 b B-40 a^3 b^2 (2 A+C)-240 a^2 b^3 B-80 a b^4 (4 A+3 C)-96 b^5 B\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{b},\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{b d}}{2 b}\right )+\frac {3 \sin (c+d x) \sqrt {\cos (c+d x)} \left (-15 a^3 C+50 a^2 b B+4 a b^2 (60 A+43 C)+120 b^3 B\right ) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {\sin (c+d x) \sqrt {\cos (c+d x)} \left (-15 a^2 C+50 a b B+80 A b^2+64 b^2 C\right ) (a+b \cos (c+d x))^{3/2}}{3 d}\right )+\frac {(10 b B-3 a C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 d}}{10 b}+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{7/2}}{5 b d}\)

\(\Big \downarrow \) 3477

\(\displaystyle \frac {\frac {1}{8} \left (\frac {1}{6} \left (\frac {1}{4} \left (\frac {\sin (c+d x) \left (-45 a^4 C+150 a^3 b B+12 a^2 b^2 (220 A+141 C)+2840 a b^3 B+256 b^4 (5 A+4 C)\right ) \sqrt {a+b \cos (c+d x)}}{b d \sqrt {\cos (c+d x)}}-\frac {a \left (-45 a^4 C+150 a^3 b B+12 a^2 b^2 (220 A+141 C)+2840 a b^3 B+256 b^4 (5 A+4 C)\right ) \int \frac {\cos (c+d x)+1}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx+a \left (45 a^4 C-30 a^3 b (5 B+C)-4 a^2 b^2 (660 A+295 B+423 C)-8 a b^3 (260 A+355 B+193 C)-16 b^4 (80 A+45 B+64 C)\right ) \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}}dx-\frac {30 \sqrt {a+b} \cot (c+d x) \left (-3 a^5 C+10 a^4 b B-40 a^3 b^2 (2 A+C)-240 a^2 b^3 B-80 a b^4 (4 A+3 C)-96 b^5 B\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{b},\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{b d}}{2 b}\right )+\frac {3 \sin (c+d x) \sqrt {\cos (c+d x)} \left (-15 a^3 C+50 a^2 b B+4 a b^2 (60 A+43 C)+120 b^3 B\right ) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {\sin (c+d x) \sqrt {\cos (c+d x)} \left (-15 a^2 C+50 a b B+80 A b^2+64 b^2 C\right ) (a+b \cos (c+d x))^{3/2}}{3 d}\right )+\frac {(10 b B-3 a C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 d}}{10 b}+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{7/2}}{5 b d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {1}{8} \left (\frac {1}{6} \left (\frac {1}{4} \left (\frac {\sin (c+d x) \left (-45 a^4 C+150 a^3 b B+12 a^2 b^2 (220 A+141 C)+2840 a b^3 B+256 b^4 (5 A+4 C)\right ) \sqrt {a+b \cos (c+d x)}}{b d \sqrt {\cos (c+d x)}}-\frac {a \left (45 a^4 C-30 a^3 b (5 B+C)-4 a^2 b^2 (660 A+295 B+423 C)-8 a b^3 (260 A+355 B+193 C)-16 b^4 (80 A+45 B+64 C)\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+a \left (-45 a^4 C+150 a^3 b B+12 a^2 b^2 (220 A+141 C)+2840 a b^3 B+256 b^4 (5 A+4 C)\right ) \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )+1}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {30 \sqrt {a+b} \cot (c+d x) \left (-3 a^5 C+10 a^4 b B-40 a^3 b^2 (2 A+C)-240 a^2 b^3 B-80 a b^4 (4 A+3 C)-96 b^5 B\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{b},\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{b d}}{2 b}\right )+\frac {3 \sin (c+d x) \sqrt {\cos (c+d x)} \left (-15 a^3 C+50 a^2 b B+4 a b^2 (60 A+43 C)+120 b^3 B\right ) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {\sin (c+d x) \sqrt {\cos (c+d x)} \left (-15 a^2 C+50 a b B+80 A b^2+64 b^2 C\right ) (a+b \cos (c+d x))^{3/2}}{3 d}\right )+\frac {(10 b B-3 a C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 d}}{10 b}+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{7/2}}{5 b d}\)

\(\Big \downarrow \) 3295

\(\displaystyle \frac {\frac {1}{8} \left (\frac {1}{6} \left (\frac {1}{4} \left (\frac {\sin (c+d x) \left (-45 a^4 C+150 a^3 b B+12 a^2 b^2 (220 A+141 C)+2840 a b^3 B+256 b^4 (5 A+4 C)\right ) \sqrt {a+b \cos (c+d x)}}{b d \sqrt {\cos (c+d x)}}-\frac {a \left (-45 a^4 C+150 a^3 b B+12 a^2 b^2 (220 A+141 C)+2840 a b^3 B+256 b^4 (5 A+4 C)\right ) \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )+1}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sqrt {a+b} \cot (c+d x) \left (45 a^4 C-30 a^3 b (5 B+C)-4 a^2 b^2 (660 A+295 B+423 C)-8 a b^3 (260 A+355 B+193 C)-16 b^4 (80 A+45 B+64 C)\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{d}-\frac {30 \sqrt {a+b} \cot (c+d x) \left (-3 a^5 C+10 a^4 b B-40 a^3 b^2 (2 A+C)-240 a^2 b^3 B-80 a b^4 (4 A+3 C)-96 b^5 B\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{b},\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{b d}}{2 b}\right )+\frac {3 \sin (c+d x) \sqrt {\cos (c+d x)} \left (-15 a^3 C+50 a^2 b B+4 a b^2 (60 A+43 C)+120 b^3 B\right ) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {\sin (c+d x) \sqrt {\cos (c+d x)} \left (-15 a^2 C+50 a b B+80 A b^2+64 b^2 C\right ) (a+b \cos (c+d x))^{3/2}}{3 d}\right )+\frac {(10 b B-3 a C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 d}}{10 b}+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{7/2}}{5 b d}\)

\(\Big \downarrow \) 3473

\(\displaystyle \frac {C \sqrt {\cos (c+d x)} \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{5 b d}+\frac {\frac {(10 b B-3 a C) \sqrt {\cos (c+d x)} \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{4 d}+\frac {1}{8} \left (\frac {\left (-15 C a^2+50 b B a+80 A b^2+64 b^2 C\right ) \sqrt {\cos (c+d x)} \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{3 d}+\frac {1}{6} \left (\frac {3 \left (-15 C a^3+50 b B a^2+4 b^2 (60 A+43 C) a+120 b^3 B\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{2 d}+\frac {1}{4} \left (\frac {\left (-45 C a^4+150 b B a^3+12 b^2 (220 A+141 C) a^2+2840 b^3 B a+256 b^4 (5 A+4 C)\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{b d \sqrt {\cos (c+d x)}}-\frac {\frac {2 (a-b) \sqrt {a+b} \left (-45 C a^4+150 b B a^3+12 b^2 (220 A+141 C) a^2+2840 b^3 B a+256 b^4 (5 A+4 C)\right ) \cot (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right )}{a d}+\frac {2 \sqrt {a+b} \left (45 C a^4-30 b (5 B+C) a^3-4 b^2 (660 A+295 B+423 C) a^2-8 b^3 (260 A+355 B+193 C) a-16 b^4 (80 A+45 B+64 C)\right ) \cot (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}}}{d}-\frac {30 \sqrt {a+b} \left (-3 C a^5+10 b B a^4-40 b^2 (2 A+C) a^3-240 b^3 B a^2-80 b^4 (4 A+3 C) a-96 b^5 B\right ) \cot (c+d x) \operatorname {EllipticPi}\left (\frac {a+b}{b},\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}}}{b d}}{2 b}\right )\right )\right )}{10 b}\)

Input:

Int[Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C* 
Cos[c + d*x]^2),x]
 

Output:

(C*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(5*b*d) + ( 
((10*b*B - 3*a*C)*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d* 
x])/(4*d) + (((80*A*b^2 + 50*a*b*B - 15*a^2*C + 64*b^2*C)*Sqrt[Cos[c + d*x 
]]*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*d) + ((3*(50*a^2*b*B + 120* 
b^3*B - 15*a^3*C + 4*a*b^2*(60*A + 43*C))*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Co 
s[c + d*x]]*Sin[c + d*x])/(2*d) + (-1/2*((2*(a - b)*Sqrt[a + b]*(150*a^3*b 
*B + 2840*a*b^3*B - 45*a^4*C + 256*b^4*(5*A + 4*C) + 12*a^2*b^2*(220*A + 1 
41*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]*(45*a^4 
*C - 30*a^3*b*(5*B + C) - 16*b^4*(80*A + 45*B + 64*C) - 8*a*b^3*(260*A + 3 
55*B + 193*C) - 4*a^2*b^2*(660*A + 295*B + 423*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 - (30*Sqrt[a + b]*(10*a^4*b*B - 240*a^2*b^3*B - 96*b^5*B - 
 3*a^5*C - 40*a^3*b^2*(2*A + C) - 80*a*b^4*(4*A + 3*C))*Cot[c + d*x]*Ellip 
ticPi[(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))/b + ((150*a^3*b*B + 2840*a*b^3*B - 4 
5*a^4*C + 256*b^4*(5*A + 4*C) + 12*a^2*b^2*(220*A + 141*C))*Sqrt[a + b*...
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(4057\) vs. \(2(765)=1530\).

Time = 57.92 (sec) , antiderivative size = 4058, normalized size of antiderivative = 4.87

method	result	size
default	\(\text {Expression too large to display}\)	\(4058\)
parts	\(\text {Expression too large to display}\)	\(4102\)

Input:

int(cos(d*x+c)^(1/2)*(a+b*cos(d*x+c))^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2 
),x,method=_RETURNVERBOSE)
 

Output:

1/1920/d*((-2640*cos(d*x+c)^2-5280*cos(d*x+c)-2640)*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)*a^3*b^2*Elli 
pticE(cot(d*x+c)-csc(d*x+c),(-(a-b)/(a+b))^(1/2))+(-2640*cos(d*x+c)^2-5280 
*cos(d*x+c)-2640)*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)*a^2*b^3*EllipticE(cot(d*x+c)-csc(d*x+c),(-(a-b 
)/(a+b))^(1/2))+(-1280*cos(d*x+c)^2-2560*cos(d*x+c)-1280)*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)*a*b^4* 
EllipticE(cot(d*x+c)-csc(d*x+c),(-(a-b)/(a+b))^(1/2))+(-150*cos(d*x+c)^2-3 
00*cos(d*x+c)-150)*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)*a^4*b*EllipticE(cot(d*x+c)-csc(d*x+c),(-(a-b) 
/(a+b))^(1/2))+(-150*cos(d*x+c)^2-300*cos(d*x+c)-150)*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)*a^3*b^2*El 
lipticE(cot(d*x+c)-csc(d*x+c),(-(a-b)/(a+b))^(1/2))+(-2840*cos(d*x+c)^2-56 
80*cos(d*x+c)-2840)*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)*a^2*b^3*EllipticE(cot(d*x+c)-csc(d*x+c),(-(a 
-b)/(a+b))^(1/2))+(-2840*cos(d*x+c)^2-5680*cos(d*x+c)-2840)*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)*a*b^ 
4*EllipticE(cot(d*x+c)-csc(d*x+c),(-(a-b)/(a+b))^(1/2))+(45*cos(d*x+c)^2+9 
0*cos(d*x+c)+45)*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*b*EllipticE(cot(d*x+c)-csc(d*x+c),(-(a-b...
 

Fricas [F]

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

integrate(cos(d*x+c)^(1/2)*(a+b*cos(d*x+c))^(5/2)*(A+B*cos(d*x+c)+C*cos(d* 
x+c)^2),x, algorithm="fricas")
 

Output:

integral((C*b^2*cos(d*x + c)^4 + (2*C*a*b + B*b^2)*cos(d*x + c)^3 + A*a^2 
+ (C*a^2 + 2*B*a*b + A*b^2)*cos(d*x + c)^2 + (B*a^2 + 2*A*a*b)*cos(d*x + c 
))*sqrt(b*cos(d*x + c) + a)*sqrt(cos(d*x + c)), x)
 

Sympy [F(-1)]

Timed out. \[ \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\text {Timed out} \] Input:

integrate(cos(d*x+c)**(1/2)*(a+b*cos(d*x+c))**(5/2)*(A+B*cos(d*x+c)+C*cos( 
d*x+c)**2),x)
 

Output:

Timed out
 

Maxima [F]

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

integrate(cos(d*x+c)^(1/2)*(a+b*cos(d*x+c))^(5/2)*(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)^(5/ 
2)*sqrt(cos(d*x + c)), x)
 

Giac [F]

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

integrate(cos(d*x+c)^(1/2)*(a+b*cos(d*x+c))^(5/2)*(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)^(5/ 
2)*sqrt(cos(d*x + c)), x)
 

Mupad [F(-1)]

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

int(cos(c + d*x)^(1/2)*(a + b*cos(c + d*x))^(5/2)*(A + B*cos(c + d*x) + C* 
cos(c + d*x)^2),x)
 

Output:

int(cos(c + d*x)^(1/2)*(a + b*cos(c + d*x))^(5/2)*(A + B*cos(c + d*x) + C* 
cos(c + d*x)^2), x)
 

Reduce [F]

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

int(cos(d*x+c)^(1/2)*(a+b*cos(d*x+c))^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2 
),x)
 

Output:

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