\(\int \frac {\cos ^{\frac {7}{2}}(c+d x) (A+B \cos (c+d x)+C \cos ^2(c+d x))}{(a+b \cos (c+d x))^3} \, dx\) [1109]

Optimal result

Integrand size = 43, antiderivative size = 654 \[ \int \frac {\cos ^{\frac {7}{2}}(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^3} \, dx=-\frac {\left (175 a^5 b B-325 a^3 b^3 B+120 a b^5 B+a^2 b^4 (145 A-192 C)-3 a^4 b^2 (25 A-187 C)-315 a^6 C-8 b^6 (5 A+3 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{20 b^5 \left (a^2-b^2\right )^2 d}+\frac {\left (105 a^6 b B-223 a^4 b^3 B+128 a^2 b^5 B+8 b^7 B+3 a^3 b^4 (33 A-64 C)-9 a^5 b^2 (5 A-43 C)-189 a^7 C-24 a b^6 (3 A+C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{12 b^6 \left (a^2-b^2\right )^2 d}+\frac {a^2 \left (35 A b^6-35 a^5 b B+86 a^3 b^3 B-63 a b^5 B-a^2 b^4 (38 A-99 C)+15 a^4 b^2 (A-10 C)+63 a^6 C\right ) \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right )}{4 (a-b)^2 b^6 (a+b)^3 d}+\frac {\left (35 a^4 b B-61 a^2 b^3 B+8 b^5 B+3 a b^4 (11 A-8 C)-15 a^3 b^2 (A-7 C)-63 a^5 C\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{12 b^4 \left (a^2-b^2\right )^2 d}-\frac {\left (35 a^3 b B-65 a b^3 B-a^2 b^2 (15 A-101 C)+b^4 (45 A-8 C)-63 a^4 C\right ) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{20 b^3 \left (a^2-b^2\right )^2 d}-\frac {\left (A b^2-a (b B-a C)\right ) \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^2}+\frac {\left (7 A b^4+5 a^3 b B-11 a b^3 B-a^2 b^2 (A-15 C)-9 a^4 C\right ) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{4 b^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))} \] Output:

-1/20*(175*B*a^5*b-325*B*a^3*b^3+120*B*a*b^5+a^2*b^4*(145*A-192*C)-3*a^4*b 
^2*(25*A-187*C)-315*a^6*C-8*b^6*(5*A+3*C))*EllipticE(sin(1/2*d*x+1/2*c),2^ 
(1/2))/b^5/(a^2-b^2)^2/d+1/12*(105*a^6*b*B-223*a^4*b^3*B+128*a^2*b^5*B+8*b 
^7*B+3*a^3*b^4*(33*A-64*C)-9*a^5*b^2*(5*A-43*C)-189*a^7*C-24*a*b^6*(3*A+C) 
)*InverseJacobiAM(1/2*d*x+1/2*c,2^(1/2))/b^6/(a^2-b^2)^2/d+1/4*a^2*(35*A*b 
^6-35*B*a^5*b+86*B*a^3*b^3-63*B*a*b^5-a^2*b^4*(38*A-99*C)+15*a^4*b^2*(A-10 
*C)+63*a^6*C)*EllipticPi(sin(1/2*d*x+1/2*c),2*b/(a+b),2^(1/2))/(a-b)^2/b^6 
/(a+b)^3/d+1/12*(35*B*a^4*b-61*B*a^2*b^3+8*B*b^5+3*a*b^4*(11*A-8*C)-15*a^3 
*b^2*(A-7*C)-63*C*a^5)*cos(d*x+c)^(1/2)*sin(d*x+c)/b^4/(a^2-b^2)^2/d-1/20* 
(35*B*a^3*b-65*B*a*b^3-a^2*b^2*(15*A-101*C)+b^4*(45*A-8*C)-63*a^4*C)*cos(d 
*x+c)^(3/2)*sin(d*x+c)/b^3/(a^2-b^2)^2/d-1/2*(A*b^2-a*(B*b-C*a))*cos(d*x+c 
)^(7/2)*sin(d*x+c)/b/(a^2-b^2)/d/(a+b*cos(d*x+c))^2+1/4*(7*A*b^4+5*B*a^3*b 
-11*B*a*b^3-a^2*b^2*(A-15*C)-9*a^4*C)*cos(d*x+c)^(5/2)*sin(d*x+c)/b^2/(a^2 
-b^2)^2/d/(a+b*cos(d*x+c))
 

Mathematica [A] (warning: unable to verify)

Time = 9.47 (sec) , antiderivative size = 670, normalized size of antiderivative = 1.02 \[ \int \frac {\cos ^{\frac {7}{2}}(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^3} \, dx=\frac {\frac {2 \left (75 a^4 A b^2-105 a^2 A b^4+120 A b^6-175 a^5 b B+365 a^3 b^3 B-280 a b^5 B+315 a^6 C-633 a^4 b^2 C+336 a^2 b^4 C+72 b^6 C\right ) \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right )}{a+b}+\frac {\left (120 a^3 A b^3-480 a A b^5-280 a^4 b^2 B+560 a^2 b^4 B+80 b^6 B+504 a^5 b C-768 a^3 b^3 C-96 a b^5 C\right ) \left (2 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )-\frac {2 a \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right )}{a+b}\right )}{b}+\frac {2 \left (225 a^4 A b^2-435 a^2 A b^4+120 A b^6-525 a^5 b B+975 a^3 b^3 B-360 a b^5 B+945 a^6 C-1683 a^4 b^2 C+576 a^2 b^4 C+72 b^6 C\right ) \cos (2 (c+d x)) \left (-2 a b E\left (\left .\arcsin \left (\sqrt {\cos (c+d x)}\right )\right |-1\right )+2 a (a+b) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\cos (c+d x)}\right ),-1\right )+\left (-2 a^2+b^2\right ) \operatorname {EllipticPi}\left (-\frac {b}{a},\arcsin \left (\sqrt {\cos (c+d x)}\right ),-1\right )\right ) \sin (c+d x)}{a b^2 \sqrt {1-\cos ^2(c+d x)} \left (-1+2 \cos ^2(c+d x)\right )}}{240 (a-b)^2 b^4 (a+b)^2 d}+\frac {\sqrt {\cos (c+d x)} \left (\frac {2 (b B-3 a C) \sin (c+d x)}{3 b^4}-\frac {a^3 A b^2 \sin (c+d x)-a^4 b B \sin (c+d x)+a^5 C \sin (c+d x)}{2 b^4 \left (-a^2+b^2\right ) (a+b \cos (c+d x))^2}+\frac {-7 a^4 A b^2 \sin (c+d x)+13 a^2 A b^4 \sin (c+d x)+11 a^5 b B \sin (c+d x)-17 a^3 b^3 B \sin (c+d x)-15 a^6 C \sin (c+d x)+21 a^4 b^2 C \sin (c+d x)}{4 b^4 \left (-a^2+b^2\right )^2 (a+b \cos (c+d x))}+\frac {C \sin (2 (c+d x))}{5 b^3}\right )}{d} \] Input:

Integrate[(Cos[c + d*x]^(7/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a 
+ b*Cos[c + d*x])^3,x]
 

Output:

((2*(75*a^4*A*b^2 - 105*a^2*A*b^4 + 120*A*b^6 - 175*a^5*b*B + 365*a^3*b^3* 
B - 280*a*b^5*B + 315*a^6*C - 633*a^4*b^2*C + 336*a^2*b^4*C + 72*b^6*C)*El 
lipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b) + ((120*a^3*A*b^3 - 480*a 
*A*b^5 - 280*a^4*b^2*B + 560*a^2*b^4*B + 80*b^6*B + 504*a^5*b*C - 768*a^3* 
b^3*C - 96*a*b^5*C)*(2*EllipticF[(c + d*x)/2, 2] - (2*a*EllipticPi[(2*b)/( 
a + b), (c + d*x)/2, 2])/(a + b)))/b + (2*(225*a^4*A*b^2 - 435*a^2*A*b^4 + 
 120*A*b^6 - 525*a^5*b*B + 975*a^3*b^3*B - 360*a*b^5*B + 945*a^6*C - 1683* 
a^4*b^2*C + 576*a^2*b^4*C + 72*b^6*C)*Cos[2*(c + d*x)]*(-2*a*b*EllipticE[A 
rcSin[Sqrt[Cos[c + d*x]]], -1] + 2*a*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + 
 d*x]]], -1] + (-2*a^2 + b^2)*EllipticPi[-(b/a), ArcSin[Sqrt[Cos[c + d*x]] 
], -1])*Sin[c + d*x])/(a*b^2*Sqrt[1 - Cos[c + d*x]^2]*(-1 + 2*Cos[c + d*x] 
^2)))/(240*(a - b)^2*b^4*(a + b)^2*d) + (Sqrt[Cos[c + d*x]]*((2*(b*B - 3*a 
*C)*Sin[c + d*x])/(3*b^4) - (a^3*A*b^2*Sin[c + d*x] - a^4*b*B*Sin[c + d*x] 
 + a^5*C*Sin[c + d*x])/(2*b^4*(-a^2 + b^2)*(a + b*Cos[c + d*x])^2) + (-7*a 
^4*A*b^2*Sin[c + d*x] + 13*a^2*A*b^4*Sin[c + d*x] + 11*a^5*b*B*Sin[c + d*x 
] - 17*a^3*b^3*B*Sin[c + d*x] - 15*a^6*C*Sin[c + d*x] + 21*a^4*b^2*C*Sin[c 
 + d*x])/(4*b^4*(-a^2 + b^2)^2*(a + b*Cos[c + d*x])) + (C*Sin[2*(c + d*x)] 
)/(5*b^3)))/d
 

Rubi [A] (verified)

Time = 4.63 (sec) , antiderivative size = 655, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.488, Rules used = {3042, 3526, 27, 3042, 3526, 27, 3042, 3528, 27, 3042, 3528, 27, 3042, 3538, 27, 3042, 3119, 3481, 3042, 3120, 3284}

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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3526

\(\displaystyle -\frac {\int \frac {\cos ^{\frac {5}{2}}(c+d x) \left (-\left (\left (9 C a^2-5 b B a+5 A b^2-4 b^2 C\right ) \cos ^2(c+d x)\right )+4 b (b B-a (A+C)) \cos (c+d x)+7 \left (A b^2-a (b B-a C)\right )\right )}{2 (a+b \cos (c+d x))^2}dx}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^{\frac {7}{2}}(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\int \frac {\cos ^{\frac {5}{2}}(c+d x) \left (-\left (\left (9 C a^2-5 b B a+5 A b^2-4 b^2 C\right ) \cos ^2(c+d x)\right )+4 b (b B-a (A+C)) \cos (c+d x)+7 \left (A b^2-a (b B-a C)\right )\right )}{(a+b \cos (c+d x))^2}dx}{4 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^{\frac {7}{2}}(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^{5/2} \left (\left (-9 C a^2+5 b B a-5 A b^2+4 b^2 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+4 b (b B-a (A+C)) \sin \left (c+d x+\frac {\pi }{2}\right )+7 \left (A b^2-a (b B-a C)\right )\right )}{\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^2}dx}{4 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^{\frac {7}{2}}(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\)

\(\Big \downarrow \) 3526

\(\displaystyle -\frac {-\frac {\int \frac {\cos ^{\frac {3}{2}}(c+d x) \left (-\left (\left (-63 C a^4+35 b B a^3-b^2 (15 A-101 C) a^2-65 b^3 B a+b^4 (45 A-8 C)\right ) \cos ^2(c+d x)\right )+4 b \left (C a^3+b B a^2-b^2 (3 A+4 C) a+2 b^3 B\right ) \cos (c+d x)+5 \left (-9 C a^4+5 b B a^3-b^2 (A-15 C) a^2-11 b^3 B a+7 A b^4\right )\right )}{2 (a+b \cos (c+d x))}dx}{b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (-9 a^4 C+5 a^3 b B-a^2 b^2 (A-15 C)-11 a b^3 B+7 A b^4\right )}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{4 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^{\frac {7}{2}}(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {\int \frac {\cos ^{\frac {3}{2}}(c+d x) \left (-\left (\left (-63 C a^4+35 b B a^3-b^2 (15 A-101 C) a^2-65 b^3 B a+b^4 (45 A-8 C)\right ) \cos ^2(c+d x)\right )+4 b \left (C a^3+b B a^2-b^2 (3 A+4 C) a+2 b^3 B\right ) \cos (c+d x)+5 \left (-9 C a^4+5 b B a^3-b^2 (A-15 C) a^2-11 b^3 B a+7 A b^4\right )\right )}{a+b \cos (c+d x)}dx}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (-9 a^4 C+5 a^3 b B-a^2 b^2 (A-15 C)-11 a b^3 B+7 A b^4\right )}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{4 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^{\frac {7}{2}}(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {-\frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (\left (63 C a^4-35 b B a^3+b^2 (15 A-101 C) a^2+65 b^3 B a-b^4 (45 A-8 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+4 b \left (C a^3+b B a^2-b^2 (3 A+4 C) a+2 b^3 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+5 \left (-9 C a^4+5 b B a^3-b^2 (A-15 C) a^2-11 b^3 B a+7 A b^4\right )\right )}{a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (-9 a^4 C+5 a^3 b B-a^2 b^2 (A-15 C)-11 a b^3 B+7 A b^4\right )}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{4 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^{\frac {7}{2}}(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\)

\(\Big \downarrow \) 3528

\(\displaystyle -\frac {-\frac {\frac {2 \int -\frac {\sqrt {\cos (c+d x)} \left (-5 \left (-63 C a^5+35 b B a^4-15 b^2 (A-7 C) a^3-61 b^3 B a^2+3 b^4 (11 A-8 C) a+8 b^5 B\right ) \cos ^2(c+d x)-4 b \left (-9 C a^4+5 b B a^3+b^2 (5 A+18 C) a^2-20 b^3 B a+2 b^4 (5 A+3 C)\right ) \cos (c+d x)+3 a \left (-63 C a^4+35 b B a^3-b^2 (15 A-101 C) a^2-65 b^3 B a+b^4 (45 A-8 C)\right )\right )}{2 (a+b \cos (c+d x))}dx}{5 b}-\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (-63 a^4 C+35 a^3 b B-a^2 b^2 (15 A-101 C)-65 a b^3 B+b^4 (45 A-8 C)\right )}{5 b d}}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (-9 a^4 C+5 a^3 b B-a^2 b^2 (A-15 C)-11 a b^3 B+7 A b^4\right )}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{4 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^{\frac {7}{2}}(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {-\frac {\int \frac {\sqrt {\cos (c+d x)} \left (-5 \left (-63 C a^5+35 b B a^4-15 b^2 (A-7 C) a^3-61 b^3 B a^2+3 b^4 (11 A-8 C) a+8 b^5 B\right ) \cos ^2(c+d x)-4 b \left (-9 C a^4+5 b B a^3+b^2 (5 A+18 C) a^2-20 b^3 B a+2 b^4 (5 A+3 C)\right ) \cos (c+d x)+3 a \left (-63 C a^4+35 b B a^3-b^2 (15 A-101 C) a^2-65 b^3 B a+b^4 (45 A-8 C)\right )\right )}{a+b \cos (c+d x)}dx}{5 b}-\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (-63 a^4 C+35 a^3 b B-a^2 b^2 (15 A-101 C)-65 a b^3 B+b^4 (45 A-8 C)\right )}{5 b d}}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (-9 a^4 C+5 a^3 b B-a^2 b^2 (A-15 C)-11 a b^3 B+7 A b^4\right )}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{4 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^{\frac {7}{2}}(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {-\frac {-\frac {\int \frac {\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (-5 \left (-63 C a^5+35 b B a^4-15 b^2 (A-7 C) a^3-61 b^3 B a^2+3 b^4 (11 A-8 C) a+8 b^5 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2-4 b \left (-9 C a^4+5 b B a^3+b^2 (5 A+18 C) a^2-20 b^3 B a+2 b^4 (5 A+3 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+3 a \left (-63 C a^4+35 b B a^3-b^2 (15 A-101 C) a^2-65 b^3 B a+b^4 (45 A-8 C)\right )\right )}{a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{5 b}-\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (-63 a^4 C+35 a^3 b B-a^2 b^2 (15 A-101 C)-65 a b^3 B+b^4 (45 A-8 C)\right )}{5 b d}}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (-9 a^4 C+5 a^3 b B-a^2 b^2 (A-15 C)-11 a b^3 B+7 A b^4\right )}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{4 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^{\frac {7}{2}}(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\)

\(\Big \downarrow \) 3528

\(\displaystyle -\frac {-\frac {-\frac {\frac {2 \int -\frac {-3 \left (-315 C a^6+175 b B a^5-3 b^2 (25 A-187 C) a^4-325 b^3 B a^3+b^4 (145 A-192 C) a^2+120 b^5 B a-8 b^6 (5 A+3 C)\right ) \cos ^2(c+d x)-4 b \left (-63 C a^5+35 b B a^4-3 b^2 (5 A-32 C) a^3-70 b^3 B a^2+12 b^4 (5 A+C) a-10 b^5 B\right ) \cos (c+d x)+5 a \left (-63 C a^5+35 b B a^4-15 b^2 (A-7 C) a^3-61 b^3 B a^2+3 b^4 (11 A-8 C) a+8 b^5 B\right )}{2 \sqrt {\cos (c+d x)} (a+b \cos (c+d x))}dx}{3 b}-\frac {10 \sin (c+d x) \sqrt {\cos (c+d x)} \left (-63 a^5 C+35 a^4 b B-15 a^3 b^2 (A-7 C)-61 a^2 b^3 B+3 a b^4 (11 A-8 C)+8 b^5 B\right )}{3 b d}}{5 b}-\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (-63 a^4 C+35 a^3 b B-a^2 b^2 (15 A-101 C)-65 a b^3 B+b^4 (45 A-8 C)\right )}{5 b d}}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (-9 a^4 C+5 a^3 b B-a^2 b^2 (A-15 C)-11 a b^3 B+7 A b^4\right )}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{4 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^{\frac {7}{2}}(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {-\frac {-\frac {\int \frac {-3 \left (-315 C a^6+175 b B a^5-3 b^2 (25 A-187 C) a^4-325 b^3 B a^3+b^4 (145 A-192 C) a^2+120 b^5 B a-8 b^6 (5 A+3 C)\right ) \cos ^2(c+d x)-4 b \left (-63 C a^5+35 b B a^4-3 b^2 (5 A-32 C) a^3-70 b^3 B a^2+12 b^4 (5 A+C) a-10 b^5 B\right ) \cos (c+d x)+5 a \left (-63 C a^5+35 b B a^4-15 b^2 (A-7 C) a^3-61 b^3 B a^2+3 b^4 (11 A-8 C) a+8 b^5 B\right )}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))}dx}{3 b}-\frac {10 \sin (c+d x) \sqrt {\cos (c+d x)} \left (-63 a^5 C+35 a^4 b B-15 a^3 b^2 (A-7 C)-61 a^2 b^3 B+3 a b^4 (11 A-8 C)+8 b^5 B\right )}{3 b d}}{5 b}-\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (-63 a^4 C+35 a^3 b B-a^2 b^2 (15 A-101 C)-65 a b^3 B+b^4 (45 A-8 C)\right )}{5 b d}}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (-9 a^4 C+5 a^3 b B-a^2 b^2 (A-15 C)-11 a b^3 B+7 A b^4\right )}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{4 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^{\frac {7}{2}}(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {-\frac {-\frac {-\frac {\int \frac {-3 \left (-315 C a^6+175 b B a^5-3 b^2 (25 A-187 C) a^4-325 b^3 B a^3+b^4 (145 A-192 C) a^2+120 b^5 B a-8 b^6 (5 A+3 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2-4 b \left (-63 C a^5+35 b B a^4-3 b^2 (5 A-32 C) a^3-70 b^3 B a^2+12 b^4 (5 A+C) a-10 b^5 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+5 a \left (-63 C a^5+35 b B a^4-15 b^2 (A-7 C) a^3-61 b^3 B a^2+3 b^4 (11 A-8 C) a+8 b^5 B\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{3 b}-\frac {10 \sin (c+d x) \sqrt {\cos (c+d x)} \left (-63 a^5 C+35 a^4 b B-15 a^3 b^2 (A-7 C)-61 a^2 b^3 B+3 a b^4 (11 A-8 C)+8 b^5 B\right )}{3 b d}}{5 b}-\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (-63 a^4 C+35 a^3 b B-a^2 b^2 (15 A-101 C)-65 a b^3 B+b^4 (45 A-8 C)\right )}{5 b d}}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (-9 a^4 C+5 a^3 b B-a^2 b^2 (A-15 C)-11 a b^3 B+7 A b^4\right )}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{4 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^{\frac {7}{2}}(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\)

\(\Big \downarrow \) 3538

\(\displaystyle -\frac {-\frac {-\frac {-\frac {-\frac {3 \left (-315 a^6 C+175 a^5 b B-3 a^4 b^2 (25 A-187 C)-325 a^3 b^3 B+a^2 b^4 (145 A-192 C)+120 a b^5 B-8 b^6 (5 A+3 C)\right ) \int \sqrt {\cos (c+d x)}dx}{b}-\frac {\int -\frac {5 \left (a b \left (-63 C a^5+35 b B a^4-15 b^2 (A-7 C) a^3-61 b^3 B a^2+3 b^4 (11 A-8 C) a+8 b^5 B\right )+\left (-189 C a^7+105 b B a^6-9 b^2 (5 A-43 C) a^5-223 b^3 B a^4+3 b^4 (33 A-64 C) a^3+128 b^5 B a^2-24 b^6 (3 A+C) a+8 b^7 B\right ) \cos (c+d x)\right )}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))}dx}{b}}{3 b}-\frac {10 \sin (c+d x) \sqrt {\cos (c+d x)} \left (-63 a^5 C+35 a^4 b B-15 a^3 b^2 (A-7 C)-61 a^2 b^3 B+3 a b^4 (11 A-8 C)+8 b^5 B\right )}{3 b d}}{5 b}-\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (-63 a^4 C+35 a^3 b B-a^2 b^2 (15 A-101 C)-65 a b^3 B+b^4 (45 A-8 C)\right )}{5 b d}}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (-9 a^4 C+5 a^3 b B-a^2 b^2 (A-15 C)-11 a b^3 B+7 A b^4\right )}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{4 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^{\frac {7}{2}}(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {-\frac {-\frac {\frac {5 \int \frac {a b \left (-63 C a^5+35 b B a^4-15 b^2 (A-7 C) a^3-61 b^3 B a^2+3 b^4 (11 A-8 C) a+8 b^5 B\right )+\left (-189 C a^7+105 b B a^6-9 b^2 (5 A-43 C) a^5-223 b^3 B a^4+3 b^4 (33 A-64 C) a^3+128 b^5 B a^2-24 b^6 (3 A+C) a+8 b^7 B\right ) \cos (c+d x)}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))}dx}{b}-\frac {3 \left (-315 a^6 C+175 a^5 b B-3 a^4 b^2 (25 A-187 C)-325 a^3 b^3 B+a^2 b^4 (145 A-192 C)+120 a b^5 B-8 b^6 (5 A+3 C)\right ) \int \sqrt {\cos (c+d x)}dx}{b}}{3 b}-\frac {10 \sin (c+d x) \sqrt {\cos (c+d x)} \left (-63 a^5 C+35 a^4 b B-15 a^3 b^2 (A-7 C)-61 a^2 b^3 B+3 a b^4 (11 A-8 C)+8 b^5 B\right )}{3 b d}}{5 b}-\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (-63 a^4 C+35 a^3 b B-a^2 b^2 (15 A-101 C)-65 a b^3 B+b^4 (45 A-8 C)\right )}{5 b d}}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (-9 a^4 C+5 a^3 b B-a^2 b^2 (A-15 C)-11 a b^3 B+7 A b^4\right )}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{4 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^{\frac {7}{2}}(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {-\frac {-\frac {-\frac {\frac {5 \int \frac {a b \left (-63 C a^5+35 b B a^4-15 b^2 (A-7 C) a^3-61 b^3 B a^2+3 b^4 (11 A-8 C) a+8 b^5 B\right )+\left (-189 C a^7+105 b B a^6-9 b^2 (5 A-43 C) a^5-223 b^3 B a^4+3 b^4 (33 A-64 C) a^3+128 b^5 B a^2-24 b^6 (3 A+C) a+8 b^7 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{b}-\frac {3 \left (-315 a^6 C+175 a^5 b B-3 a^4 b^2 (25 A-187 C)-325 a^3 b^3 B+a^2 b^4 (145 A-192 C)+120 a b^5 B-8 b^6 (5 A+3 C)\right ) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx}{b}}{3 b}-\frac {10 \sin (c+d x) \sqrt {\cos (c+d x)} \left (-63 a^5 C+35 a^4 b B-15 a^3 b^2 (A-7 C)-61 a^2 b^3 B+3 a b^4 (11 A-8 C)+8 b^5 B\right )}{3 b d}}{5 b}-\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (-63 a^4 C+35 a^3 b B-a^2 b^2 (15 A-101 C)-65 a b^3 B+b^4 (45 A-8 C)\right )}{5 b d}}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (-9 a^4 C+5 a^3 b B-a^2 b^2 (A-15 C)-11 a b^3 B+7 A b^4\right )}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{4 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^{\frac {7}{2}}(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\)

\(\Big \downarrow \) 3119

\(\displaystyle -\frac {-\frac {-\frac {-\frac {\frac {5 \int \frac {a b \left (-63 C a^5+35 b B a^4-15 b^2 (A-7 C) a^3-61 b^3 B a^2+3 b^4 (11 A-8 C) a+8 b^5 B\right )+\left (-189 C a^7+105 b B a^6-9 b^2 (5 A-43 C) a^5-223 b^3 B a^4+3 b^4 (33 A-64 C) a^3+128 b^5 B a^2-24 b^6 (3 A+C) a+8 b^7 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{b}-\frac {6 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (-315 a^6 C+175 a^5 b B-3 a^4 b^2 (25 A-187 C)-325 a^3 b^3 B+a^2 b^4 (145 A-192 C)+120 a b^5 B-8 b^6 (5 A+3 C)\right )}{b d}}{3 b}-\frac {10 \sin (c+d x) \sqrt {\cos (c+d x)} \left (-63 a^5 C+35 a^4 b B-15 a^3 b^2 (A-7 C)-61 a^2 b^3 B+3 a b^4 (11 A-8 C)+8 b^5 B\right )}{3 b d}}{5 b}-\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (-63 a^4 C+35 a^3 b B-a^2 b^2 (15 A-101 C)-65 a b^3 B+b^4 (45 A-8 C)\right )}{5 b d}}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (-9 a^4 C+5 a^3 b B-a^2 b^2 (A-15 C)-11 a b^3 B+7 A b^4\right )}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{4 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^{\frac {7}{2}}(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\)

\(\Big \downarrow \) 3481

\(\displaystyle -\frac {-\frac {-\frac {-\frac {\frac {5 \left (\frac {3 a^2 \left (63 a^6 C-35 a^5 b B+15 a^4 b^2 (A-10 C)+86 a^3 b^3 B-a^2 b^4 (38 A-99 C)-63 a b^5 B+35 A b^6\right ) \int \frac {1}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))}dx}{b}+\frac {\left (-189 a^7 C+105 a^6 b B-9 a^5 b^2 (5 A-43 C)-223 a^4 b^3 B+3 a^3 b^4 (33 A-64 C)+128 a^2 b^5 B-24 a b^6 (3 A+C)+8 b^7 B\right ) \int \frac {1}{\sqrt {\cos (c+d x)}}dx}{b}\right )}{b}-\frac {6 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (-315 a^6 C+175 a^5 b B-3 a^4 b^2 (25 A-187 C)-325 a^3 b^3 B+a^2 b^4 (145 A-192 C)+120 a b^5 B-8 b^6 (5 A+3 C)\right )}{b d}}{3 b}-\frac {10 \sin (c+d x) \sqrt {\cos (c+d x)} \left (-63 a^5 C+35 a^4 b B-15 a^3 b^2 (A-7 C)-61 a^2 b^3 B+3 a b^4 (11 A-8 C)+8 b^5 B\right )}{3 b d}}{5 b}-\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (-63 a^4 C+35 a^3 b B-a^2 b^2 (15 A-101 C)-65 a b^3 B+b^4 (45 A-8 C)\right )}{5 b d}}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (-9 a^4 C+5 a^3 b B-a^2 b^2 (A-15 C)-11 a b^3 B+7 A b^4\right )}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{4 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^{\frac {7}{2}}(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {-\frac {-\frac {-\frac {\frac {5 \left (\frac {3 a^2 \left (63 a^6 C-35 a^5 b B+15 a^4 b^2 (A-10 C)+86 a^3 b^3 B-a^2 b^4 (38 A-99 C)-63 a b^5 B+35 A b^6\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{b}+\frac {\left (-189 a^7 C+105 a^6 b B-9 a^5 b^2 (5 A-43 C)-223 a^4 b^3 B+3 a^3 b^4 (33 A-64 C)+128 a^2 b^5 B-24 a b^6 (3 A+C)+8 b^7 B\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}\right )}{b}-\frac {6 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (-315 a^6 C+175 a^5 b B-3 a^4 b^2 (25 A-187 C)-325 a^3 b^3 B+a^2 b^4 (145 A-192 C)+120 a b^5 B-8 b^6 (5 A+3 C)\right )}{b d}}{3 b}-\frac {10 \sin (c+d x) \sqrt {\cos (c+d x)} \left (-63 a^5 C+35 a^4 b B-15 a^3 b^2 (A-7 C)-61 a^2 b^3 B+3 a b^4 (11 A-8 C)+8 b^5 B\right )}{3 b d}}{5 b}-\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (-63 a^4 C+35 a^3 b B-a^2 b^2 (15 A-101 C)-65 a b^3 B+b^4 (45 A-8 C)\right )}{5 b d}}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (-9 a^4 C+5 a^3 b B-a^2 b^2 (A-15 C)-11 a b^3 B+7 A b^4\right )}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{4 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^{\frac {7}{2}}(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\)

\(\Big \downarrow \) 3120

\(\displaystyle -\frac {-\frac {-\frac {-\frac {\frac {5 \left (\frac {3 a^2 \left (63 a^6 C-35 a^5 b B+15 a^4 b^2 (A-10 C)+86 a^3 b^3 B-a^2 b^4 (38 A-99 C)-63 a b^5 B+35 A b^6\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{b}+\frac {2 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (-189 a^7 C+105 a^6 b B-9 a^5 b^2 (5 A-43 C)-223 a^4 b^3 B+3 a^3 b^4 (33 A-64 C)+128 a^2 b^5 B-24 a b^6 (3 A+C)+8 b^7 B\right )}{b d}\right )}{b}-\frac {6 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (-315 a^6 C+175 a^5 b B-3 a^4 b^2 (25 A-187 C)-325 a^3 b^3 B+a^2 b^4 (145 A-192 C)+120 a b^5 B-8 b^6 (5 A+3 C)\right )}{b d}}{3 b}-\frac {10 \sin (c+d x) \sqrt {\cos (c+d x)} \left (-63 a^5 C+35 a^4 b B-15 a^3 b^2 (A-7 C)-61 a^2 b^3 B+3 a b^4 (11 A-8 C)+8 b^5 B\right )}{3 b d}}{5 b}-\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (-63 a^4 C+35 a^3 b B-a^2 b^2 (15 A-101 C)-65 a b^3 B+b^4 (45 A-8 C)\right )}{5 b d}}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (-9 a^4 C+5 a^3 b B-a^2 b^2 (A-15 C)-11 a b^3 B+7 A b^4\right )}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{4 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^{\frac {7}{2}}(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\)

\(\Big \downarrow \) 3284

\(\displaystyle -\frac {\sin (c+d x) \cos ^{\frac {7}{2}}(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {-\frac {\sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (-9 a^4 C+5 a^3 b B-a^2 b^2 (A-15 C)-11 a b^3 B+7 A b^4\right )}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {-\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (-63 a^4 C+35 a^3 b B-a^2 b^2 (15 A-101 C)-65 a b^3 B+b^4 (45 A-8 C)\right )}{5 b d}-\frac {-\frac {10 \sin (c+d x) \sqrt {\cos (c+d x)} \left (-63 a^5 C+35 a^4 b B-15 a^3 b^2 (A-7 C)-61 a^2 b^3 B+3 a b^4 (11 A-8 C)+8 b^5 B\right )}{3 b d}-\frac {\frac {5 \left (\frac {6 a^2 \left (63 a^6 C-35 a^5 b B+15 a^4 b^2 (A-10 C)+86 a^3 b^3 B-a^2 b^4 (38 A-99 C)-63 a b^5 B+35 A b^6\right ) \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right )}{b d (a+b)}+\frac {2 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (-189 a^7 C+105 a^6 b B-9 a^5 b^2 (5 A-43 C)-223 a^4 b^3 B+3 a^3 b^4 (33 A-64 C)+128 a^2 b^5 B-24 a b^6 (3 A+C)+8 b^7 B\right )}{b d}\right )}{b}-\frac {6 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (-315 a^6 C+175 a^5 b B-3 a^4 b^2 (25 A-187 C)-325 a^3 b^3 B+a^2 b^4 (145 A-192 C)+120 a b^5 B-8 b^6 (5 A+3 C)\right )}{b d}}{3 b}}{5 b}}{2 b \left (a^2-b^2\right )}}{4 b \left (a^2-b^2\right )}\)

Input:

Int[(Cos[c + d*x]^(7/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Co 
s[c + d*x])^3,x]
 

Output:

-1/2*((A*b^2 - a*(b*B - a*C))*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(b*(a^2 - b 
^2)*d*(a + b*Cos[c + d*x])^2) - (-(((7*A*b^4 + 5*a^3*b*B - 11*a*b^3*B - a^ 
2*b^2*(A - 15*C) - 9*a^4*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(b*(a^2 - b^2 
)*d*(a + b*Cos[c + d*x]))) - ((-2*(35*a^3*b*B - 65*a*b^3*B - a^2*b^2*(15*A 
 - 101*C) + b^4*(45*A - 8*C) - 63*a^4*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/ 
(5*b*d) - (-1/3*((-6*(175*a^5*b*B - 325*a^3*b^3*B + 120*a*b^5*B + a^2*b^4* 
(145*A - 192*C) - 3*a^4*b^2*(25*A - 187*C) - 315*a^6*C - 8*b^6*(5*A + 3*C) 
)*EllipticE[(c + d*x)/2, 2])/(b*d) + (5*((2*(105*a^6*b*B - 223*a^4*b^3*B + 
 128*a^2*b^5*B + 8*b^7*B + 3*a^3*b^4*(33*A - 64*C) - 9*a^5*b^2*(5*A - 43*C 
) - 189*a^7*C - 24*a*b^6*(3*A + C))*EllipticF[(c + d*x)/2, 2])/(b*d) + (6* 
a^2*(35*A*b^6 - 35*a^5*b*B + 86*a^3*b^3*B - 63*a*b^5*B - a^2*b^4*(38*A - 9 
9*C) + 15*a^4*b^2*(A - 10*C) + 63*a^6*C)*EllipticPi[(2*b)/(a + b), (c + d* 
x)/2, 2])/(b*(a + b)*d)))/b)/b - (10*(35*a^4*b*B - 61*a^2*b^3*B + 8*b^5*B 
+ 3*a*b^4*(11*A - 8*C) - 15*a^3*b^2*(A - 7*C) - 63*a^5*C)*Sqrt[Cos[c + d*x 
]]*Sin[c + d*x])/(3*b*d))/(5*b))/(2*b*(a^2 - b^2)))/(4*b*(a^2 - b^2))
 

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 3284

Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) 
 + (f_.)*(x_)]]), x_Symbol] :> Simp[(2/(f*(a + b)*Sqrt[c + d]))*EllipticPi[ 
2*(b/(a + b)), (1/2)*(e - Pi/2 + f*x), 2*(d/(c + d))], x] /; FreeQ[{a, b, c 
, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 
0] && GtQ[c + d, 0]
 

rule 3481

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

rule 3526

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^2*C - B*c*d + A*d^2))*Cos[e + f*x 
]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 - 
d^2))), x] + Simp[1/(d*(n + 1)*(c^2 - d^2))   Int[(a + b*Sin[e + f*x])^(m - 
 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(b*d*m + a*c*(n + 1)) + (c*C - B* 
d)*(b*c*m + a*d*(n + 1)) - (d*(A*(a*d*(n + 2) - b*c*(n + 1)) + B*(b*d*(n + 
1) - a*c*(n + 2))) - C*(b*c*d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x 
] + b*(d*(B*c - A*d)*(m + n + 2) - C*(c^2*(m + 1) + d^2*(n + 1)))*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] && GtQ[m, 0] && LtQ[n, -1]
 

rule 3528

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) 
+ (f_.)*(x_)])^(n_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_ 
.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Cos[e + f*x]*(a + b*Sin[e + f*x 
])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n + 2))), x] + Simp[1/(d*(m + 
n + 2))   Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A* 
d*(m + n + 2) + C*(b*c*m + a*d*(n + 1)) + (d*(A*b + a*B)*(m + n + 2) - C*(a 
*c - b*d*(m + n + 1)))*Sin[e + f*x] + (C*(a*d*m - b*c*(m + 1)) + b*B*d*(m + 
 n + 2))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n} 
, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[ 
m, 0] &&  !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 0])))
 

rule 3538

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

Leaf count of result is larger than twice the leaf count of optimal. \(2519\) vs. \(2(637)=1274\).

Time = 109.48 (sec) , antiderivative size = 2520, normalized size of antiderivative = 3.85

method	result	size
default	\(\text {Expression too large to display}\)	\(2520\)

Input:

int(cos(d*x+c)^(7/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+b*cos(d*x+c))^3,x, 
method=_RETURNVERBOSE)
 

Output:

-(-(-2*cos(1/2*d*x+1/2*c)^2+1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(-4*a^2/b^5*(6* 
A*b^2-10*B*a*b+15*C*a^2)/(-2*a*b+2*b^2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*c 
os(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2 
)^(1/2)*EllipticPi(cos(1/2*d*x+1/2*c),-2*b/(a-b),2^(1/2))+2*a^4*(A*b^2-B*a 
*b+C*a^2)/b^6*(-1/2/a*b^2/(a^2-b^2)*cos(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/2 
*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/(2*cos(1/2*d*x+1/2*c)^2*b+a-b)^2-3/4*b^2 
*(3*a^2-b^2)/a^2/(a^2-b^2)^2*cos(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/2*c)^4+s 
in(1/2*d*x+1/2*c)^2)^(1/2)/(2*cos(1/2*d*x+1/2*c)^2*b+a-b)-7/8/(a+b)/(a^2-b 
^2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin 
(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c) 
,2^(1/2))+1/4/(a+b)/(a^2-b^2)/a*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d 
*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)* 
EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))*b+3/8/(a+b)/(a^2-b^2)/a^2*(sin(1/2*d 
*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c 
)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))*b^2- 
9/8*b/(a^2-b^2)^2*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1) 
^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos( 
1/2*d*x+1/2*c),2^(1/2))+3/8*b^3/a^2/(a^2-b^2)^2*(sin(1/2*d*x+1/2*c)^2)^(1/ 
2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+ 
1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))+9/8*b/(a^2-b^2)^2...
 

Fricas [F]

\[ \int \frac {\cos ^{\frac {7}{2}}(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^3} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \cos \left (d x + c\right )^{\frac {7}{2}}}{{\left (b \cos \left (d x + c\right ) + a\right )}^{3}} \,d x } \] Input:

integrate(cos(d*x+c)^(7/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+b*cos(d*x+c) 
)^3,x, algorithm="fricas")
 

Output:

integral((C*cos(d*x + c)^5 + B*cos(d*x + c)^4 + A*cos(d*x + c)^3)*sqrt(cos 
(d*x + c))/(b^3*cos(d*x + c)^3 + 3*a*b^2*cos(d*x + c)^2 + 3*a^2*b*cos(d*x 
+ c) + a^3), x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\cos ^{\frac {7}{2}}(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^3} \, dx=\text {Timed out} \] Input:

integrate(cos(d*x+c)**(7/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)**2)/(a+b*cos(d*x+ 
c))**3,x)
 

Output:

Timed out
 

Maxima [F(-1)]

Timed out. \[ \int \frac {\cos ^{\frac {7}{2}}(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^3} \, dx=\text {Timed out} \] Input:

integrate(cos(d*x+c)^(7/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+b*cos(d*x+c) 
)^3,x, algorithm="maxima")
 

Output:

Timed out
 

Giac [F]

\[ \int \frac {\cos ^{\frac {7}{2}}(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^3} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \cos \left (d x + c\right )^{\frac {7}{2}}}{{\left (b \cos \left (d x + c\right ) + a\right )}^{3}} \,d x } \] Input:

integrate(cos(d*x+c)^(7/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+b*cos(d*x+c) 
)^3,x, algorithm="giac")
 

Output:

integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)*cos(d*x + c)^(7/2)/(b*co 
s(d*x + c) + a)^3, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\cos ^{\frac {7}{2}}(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^3} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^{7/2}\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right )}{{\left (a+b\,\cos \left (c+d\,x\right )\right )}^3} \,d x \] Input:

int((cos(c + d*x)^(7/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2))/(a + b*co 
s(c + d*x))^3,x)
 

Output:

int((cos(c + d*x)^(7/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2))/(a + b*co 
s(c + d*x))^3, x)
 

Reduce [F]

\[ \int \frac {\cos ^{\frac {7}{2}}(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^3} \, dx=\left (\int \frac {\sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{5}}{\cos \left (d x +c \right )^{3} b^{3}+3 \cos \left (d x +c \right )^{2} a \,b^{2}+3 \cos \left (d x +c \right ) a^{2} b +a^{3}}d x \right ) c +\left (\int \frac {\sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{4}}{\cos \left (d x +c \right )^{3} b^{3}+3 \cos \left (d x +c \right )^{2} a \,b^{2}+3 \cos \left (d x +c \right ) a^{2} b +a^{3}}d x \right ) b +\left (\int \frac {\sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{3}}{\cos \left (d x +c \right )^{3} b^{3}+3 \cos \left (d x +c \right )^{2} a \,b^{2}+3 \cos \left (d x +c \right ) a^{2} b +a^{3}}d x \right ) a \] Input:

int(cos(d*x+c)^(7/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+b*cos(d*x+c))^3,x)
 

Output:

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