\(\int \frac {A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^3} \, dx\) [1115]

Optimal result

Integrand size = 43, antiderivative size = 609 \[ \int \frac {A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^3} \, dx=\frac {\left (35 A b^5-8 a^5 B+29 a^3 b^2 B-15 a b^4 B+3 a^4 b (8 A-3 C)-a^2 b^3 (65 A-3 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{4 a^4 \left (a^2-b^2\right )^2 d}+\frac {\left (35 A b^4+33 a^3 b B-15 a b^3 B+a^4 (8 A-21 C)-a^2 b^2 (61 A-3 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{12 a^3 \left (a^2-b^2\right )^2 d}+\frac {\left (35 A b^6-35 a^5 b B+38 a^3 b^3 B-15 a b^5 B-a^2 b^4 (86 A-3 C)+3 a^4 b^2 (21 A-2 C)+15 a^6 C\right ) \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right )}{4 a^4 (a-b)^2 (a+b)^3 d}+\frac {\left (35 A b^4+33 a^3 b B-15 a b^3 B+a^4 (8 A-21 C)-a^2 b^2 (61 A-3 C)\right ) \sin (c+d x)}{12 a^3 \left (a^2-b^2\right )^2 d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\left (35 A b^5-8 a^5 B+29 a^3 b^2 B-15 a b^4 B+3 a^4 b (8 A-3 C)-a^2 b^3 (65 A-3 C)\right ) \sin (c+d x)}{4 a^4 \left (a^2-b^2\right )^2 d \sqrt {\cos (c+d x)}}+\frac {\left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}-\frac {\left (7 A b^4+9 a^3 b B-3 a b^3 B-5 a^4 C-a^2 b^2 (13 A+C)\right ) \sin (c+d x)}{4 a^2 \left (a^2-b^2\right )^2 d \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))} \] Output:

1/4*(35*A*b^5-8*a^5*B+29*a^3*b^2*B-15*a*b^4*B+3*a^4*b*(8*A-3*C)-a^2*b^3*(6 
5*A-3*C))*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2))/a^4/(a^2-b^2)^2/d+1/12*(35 
*A*b^4+33*B*a^3*b-15*B*a*b^3+a^4*(8*A-21*C)-a^2*b^2*(61*A-3*C))*InverseJac 
obiAM(1/2*d*x+1/2*c,2^(1/2))/a^3/(a^2-b^2)^2/d+1/4*(35*A*b^6-35*B*a^5*b+38 
*B*a^3*b^3-15*B*a*b^5-a^2*b^4*(86*A-3*C)+3*a^4*b^2*(21*A-2*C)+15*a^6*C)*El 
lipticPi(sin(1/2*d*x+1/2*c),2*b/(a+b),2^(1/2))/a^4/(a-b)^2/(a+b)^3/d+1/12* 
(35*A*b^4+33*B*a^3*b-15*B*a*b^3+a^4*(8*A-21*C)-a^2*b^2*(61*A-3*C))*sin(d*x 
+c)/a^3/(a^2-b^2)^2/d/cos(d*x+c)^(3/2)-1/4*(35*A*b^5-8*a^5*B+29*a^3*b^2*B- 
15*a*b^4*B+3*a^4*b*(8*A-3*C)-a^2*b^3*(65*A-3*C))*sin(d*x+c)/a^4/(a^2-b^2)^ 
2/d/cos(d*x+c)^(1/2)+1/2*(A*b^2-a*(B*b-C*a))*sin(d*x+c)/a/(a^2-b^2)/d/cos( 
d*x+c)^(3/2)/(a+b*cos(d*x+c))^2-1/4*(7*A*b^4+9*B*a^3*b-3*B*a*b^3-5*a^4*C-a 
^2*b^2*(13*A+C))*sin(d*x+c)/a^2/(a^2-b^2)^2/d/cos(d*x+c)^(3/2)/(a+b*cos(d* 
x+c))
                                                                                    
                                                                                    
 

Mathematica [A] (warning: unable to verify)

Time = 8.38 (sec) , antiderivative size = 672, normalized size of antiderivative = 1.10 \[ \int \frac {A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^3} \, dx=\frac {\frac {2 \left (16 a^6 A+328 a^4 A b^2-641 a^2 A b^4+315 A b^6-168 a^5 b B+285 a^3 b^3 B-135 a b^5 B+48 a^6 C-57 a^4 b^2 C+27 a^2 b^4 C\right ) \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right )}{a+b}+\frac {\left (160 a^5 A b-512 a^3 A b^3+280 a A b^5-48 a^6 B+240 a^4 b^2 B-120 a^2 b^4 B-96 a^5 b C+24 a^3 b^3 C\right ) \left (2 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )-\frac {2 a \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right )}{a+b}\right )}{b}+\frac {2 \left (72 a^4 A b^2-195 a^2 A b^4+105 A b^6-24 a^5 b B+87 a^3 b^3 B-45 a b^5 B-27 a^4 b^2 C+9 a^2 b^4 C\right ) \cos (2 (c+d x)) \left (-2 a b E\left (\left .\arcsin \left (\sqrt {\cos (c+d x)}\right )\right |-1\right )+2 a (a+b) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\cos (c+d x)}\right ),-1\right )+\left (-2 a^2+b^2\right ) \operatorname {EllipticPi}\left (-\frac {b}{a},\arcsin \left (\sqrt {\cos (c+d x)}\right ),-1\right )\right ) \sin (c+d x)}{a b^2 \sqrt {1-\cos ^2(c+d x)} \left (-1+2 \cos ^2(c+d x)\right )}}{48 a^4 (a-b)^2 (a+b)^2 d}+\frac {\sqrt {\cos (c+d x)} \left (\frac {2 \sec (c+d x) (-3 A b \sin (c+d x)+a B \sin (c+d x))}{a^4}+\frac {A b^4 \sin (c+d x)-a b^3 B \sin (c+d x)+a^2 b^2 C \sin (c+d x)}{2 a^3 \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}+\frac {17 a^2 A b^4 \sin (c+d x)-11 A b^6 \sin (c+d x)-13 a^3 b^3 B \sin (c+d x)+7 a b^5 B \sin (c+d x)+9 a^4 b^2 C \sin (c+d x)-3 a^2 b^4 C \sin (c+d x)}{4 a^4 \left (a^2-b^2\right )^2 (a+b \cos (c+d x))}+\frac {2 A \sec (c+d x) \tan (c+d x)}{3 a^3}\right )}{d} \] Input:

Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + 
 b*Cos[c + d*x])^3),x]
 

Output:

((2*(16*a^6*A + 328*a^4*A*b^2 - 641*a^2*A*b^4 + 315*A*b^6 - 168*a^5*b*B + 
285*a^3*b^3*B - 135*a*b^5*B + 48*a^6*C - 57*a^4*b^2*C + 27*a^2*b^4*C)*Elli 
pticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b) + ((160*a^5*A*b - 512*a^3*A 
*b^3 + 280*a*A*b^5 - 48*a^6*B + 240*a^4*b^2*B - 120*a^2*b^4*B - 96*a^5*b*C 
 + 24*a^3*b^3*C)*(2*EllipticF[(c + d*x)/2, 2] - (2*a*EllipticPi[(2*b)/(a + 
 b), (c + d*x)/2, 2])/(a + b)))/b + (2*(72*a^4*A*b^2 - 195*a^2*A*b^4 + 105 
*A*b^6 - 24*a^5*b*B + 87*a^3*b^3*B - 45*a*b^5*B - 27*a^4*b^2*C + 9*a^2*b^4 
*C)*Cos[2*(c + d*x)]*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2 
*a*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (-2*a^2 + b^2)*Elli 
pticPi[-(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)))/(48*a^4*(a - b)^2*(a + b)^2* 
d) + (Sqrt[Cos[c + d*x]]*((2*Sec[c + d*x]*(-3*A*b*Sin[c + d*x] + a*B*Sin[c 
 + d*x]))/a^4 + (A*b^4*Sin[c + d*x] - a*b^3*B*Sin[c + d*x] + a^2*b^2*C*Sin 
[c + d*x])/(2*a^3*(a^2 - b^2)*(a + b*Cos[c + d*x])^2) + (17*a^2*A*b^4*Sin[ 
c + d*x] - 11*A*b^6*Sin[c + d*x] - 13*a^3*b^3*B*Sin[c + d*x] + 7*a*b^5*B*S 
in[c + d*x] + 9*a^4*b^2*C*Sin[c + d*x] - 3*a^2*b^4*C*Sin[c + d*x])/(4*a^4* 
(a^2 - b^2)^2*(a + b*Cos[c + d*x])) + (2*A*Sec[c + d*x]*Tan[c + d*x])/(3*a 
^3)))/d
 

Rubi [A] (verified)

Time = 4.86 (sec) , antiderivative size = 598, normalized size of antiderivative = 0.98, number of steps used = 21, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.488, Rules used = {3042, 3534, 27, 3042, 3534, 27, 3042, 3534, 27, 3042, 3534, 27, 3042, 3538, 25, 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.

\(\displaystyle \int \frac {A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^3} \, dx\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3534

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3534

\(\displaystyle \frac {\sin (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}-\frac {\frac {\int -\frac {(8 A-21 C) a^4+33 b B a^3-b^2 (61 A-3 C) a^2-15 b^3 B a+4 \left (2 B a^3-b (4 A+3 C) a^2+b^2 B a+A b^3\right ) \cos (c+d x) a+35 A b^4-3 \left (-5 C a^4+9 b B a^3-b^2 (13 A+C) a^2-3 b^3 B a+7 A b^4\right ) \cos ^2(c+d x)}{2 \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))}dx}{a \left (a^2-b^2\right )}+\frac {\sin (c+d x) \left (-5 a^4 C+9 a^3 b B-a^2 b^2 (13 A+C)-3 a b^3 B+7 A b^4\right )}{a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))}}{4 a \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sin (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}-\frac {\frac {\sin (c+d x) \left (-5 a^4 C+9 a^3 b B-a^2 b^2 (13 A+C)-3 a b^3 B+7 A b^4\right )}{a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))}-\frac {\int \frac {(8 A-21 C) a^4+33 b B a^3-b^2 (61 A-3 C) a^2-15 b^3 B a+4 \left (2 B a^3-b (4 A+3 C) a^2+b^2 B a+A b^3\right ) \cos (c+d x) a+35 A b^4-3 \left (-5 C a^4+9 b B a^3-b^2 (13 A+C) a^2-3 b^3 B a+7 A b^4\right ) \cos ^2(c+d x)}{\cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))}dx}{2 a \left (a^2-b^2\right )}}{4 a \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\sin (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}-\frac {\frac {\sin (c+d x) \left (-5 a^4 C+9 a^3 b B-a^2 b^2 (13 A+C)-3 a b^3 B+7 A b^4\right )}{a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))}-\frac {\int \frac {(8 A-21 C) a^4+33 b B a^3-b^2 (61 A-3 C) a^2-15 b^3 B a+4 \left (2 B a^3-b (4 A+3 C) a^2+b^2 B a+A b^3\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) a+35 A b^4-3 \left (-5 C a^4+9 b B a^3-b^2 (13 A+C) a^2-3 b^3 B a+7 A b^4\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2}{\sin \left (c+d x+\frac {\pi }{2}\right )^{5/2} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{2 a \left (a^2-b^2\right )}}{4 a \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3534

\(\displaystyle \frac {\sin (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}-\frac {\frac {\sin (c+d x) \left (-5 a^4 C+9 a^3 b B-a^2 b^2 (13 A+C)-3 a b^3 B+7 A b^4\right )}{a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))}-\frac {\frac {2 \int -\frac {-b \left ((8 A-21 C) a^4+33 b B a^3-b^2 (61 A-3 C) a^2-15 b^3 B a+35 A b^4\right ) \cos ^2(c+d x)+4 a \left (-2 (A+3 C) a^4+12 b B a^3-b^2 (14 A+3 C) a^2-3 b^3 B a+7 A b^4\right ) \cos (c+d x)+3 \left (-8 B a^5+(24 A b-9 b C) a^4+29 b^2 B a^3-b^3 (65 A-3 C) a^2-15 b^4 B a+35 A b^5\right )}{2 \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))}dx}{3 a}+\frac {2 \sin (c+d x) \left (a^4 (8 A-21 C)+33 a^3 b B-a^2 b^2 (61 A-3 C)-15 a b^3 B+35 A b^4\right )}{3 a d \cos ^{\frac {3}{2}}(c+d x)}}{2 a \left (a^2-b^2\right )}}{4 a \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sin (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}-\frac {\frac {\sin (c+d x) \left (-5 a^4 C+9 a^3 b B-a^2 b^2 (13 A+C)-3 a b^3 B+7 A b^4\right )}{a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))}-\frac {\frac {2 \sin (c+d x) \left (a^4 (8 A-21 C)+33 a^3 b B-a^2 b^2 (61 A-3 C)-15 a b^3 B+35 A b^4\right )}{3 a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\int \frac {-b \left ((8 A-21 C) a^4+33 b B a^3-b^2 (61 A-3 C) a^2-15 b^3 B a+35 A b^4\right ) \cos ^2(c+d x)+4 a \left (-2 (A+3 C) a^4+12 b B a^3-b^2 (14 A+3 C) a^2-3 b^3 B a+7 A b^4\right ) \cos (c+d x)+3 \left (-8 B a^5+3 b (8 A-3 C) a^4+29 b^2 B a^3-b^3 (65 A-3 C) a^2-15 b^4 B a+35 A b^5\right )}{\cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))}dx}{3 a}}{2 a \left (a^2-b^2\right )}}{4 a \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\sin (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}-\frac {\frac {\sin (c+d x) \left (-5 a^4 C+9 a^3 b B-a^2 b^2 (13 A+C)-3 a b^3 B+7 A b^4\right )}{a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))}-\frac {\frac {2 \sin (c+d x) \left (a^4 (8 A-21 C)+33 a^3 b B-a^2 b^2 (61 A-3 C)-15 a b^3 B+35 A b^4\right )}{3 a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\int \frac {-b \left ((8 A-21 C) a^4+33 b B a^3-b^2 (61 A-3 C) a^2-15 b^3 B a+35 A b^4\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+4 a \left (-2 (A+3 C) a^4+12 b B a^3-b^2 (14 A+3 C) a^2-3 b^3 B a+7 A b^4\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+3 \left (-8 B a^5+3 b (8 A-3 C) a^4+29 b^2 B a^3-b^3 (65 A-3 C) a^2-15 b^4 B a+35 A b^5\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{3 a}}{2 a \left (a^2-b^2\right )}}{4 a \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3534

\(\displaystyle \frac {\sin (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}-\frac {\frac {\sin (c+d x) \left (-5 a^4 C+9 a^3 b B-a^2 b^2 (13 A+C)-3 a b^3 B+7 A b^4\right )}{a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))}-\frac {\frac {2 \sin (c+d x) \left (a^4 (8 A-21 C)+33 a^3 b B-a^2 b^2 (61 A-3 C)-15 a b^3 B+35 A b^4\right )}{3 a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \int -\frac {8 (A+3 C) a^6-72 b B a^5+b^2 (128 A-15 C) a^4+99 b^3 B a^3-b^4 (223 A-9 C) a^2-45 b^5 B a+4 \left (-6 B a^5+4 b (5 A-3 C) a^4+30 b^2 B a^3-b^3 (64 A-3 C) a^2-15 b^4 B a+35 A b^5\right ) \cos (c+d x) a+105 A b^6+3 b \left (-8 B a^5+3 b (8 A-3 C) a^4+29 b^2 B a^3-b^3 (65 A-3 C) a^2-15 b^4 B a+35 A b^5\right ) \cos ^2(c+d x)}{2 \sqrt {\cos (c+d x)} (a+b \cos (c+d x))}dx}{a}+\frac {6 \sin (c+d x) \left (-8 a^5 B+3 a^4 b (8 A-3 C)+29 a^3 b^2 B-a^2 b^3 (65 A-3 C)-15 a b^4 B+35 A b^5\right )}{a d \sqrt {\cos (c+d x)}}}{3 a}}{2 a \left (a^2-b^2\right )}}{4 a \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sin (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}-\frac {\frac {\sin (c+d x) \left (-5 a^4 C+9 a^3 b B-a^2 b^2 (13 A+C)-3 a b^3 B+7 A b^4\right )}{a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))}-\frac {\frac {2 \sin (c+d x) \left (a^4 (8 A-21 C)+33 a^3 b B-a^2 b^2 (61 A-3 C)-15 a b^3 B+35 A b^4\right )}{3 a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\frac {6 \sin (c+d x) \left (-8 a^5 B+3 a^4 b (8 A-3 C)+29 a^3 b^2 B-a^2 b^3 (65 A-3 C)-15 a b^4 B+35 A b^5\right )}{a d \sqrt {\cos (c+d x)}}-\frac {\int \frac {8 (A+3 C) a^6-72 b B a^5+b^2 (128 A-15 C) a^4+99 b^3 B a^3-b^4 (223 A-9 C) a^2-45 b^5 B a+4 \left (-6 B a^5+4 b (5 A-3 C) a^4+30 b^2 B a^3-b^3 (64 A-3 C) a^2-15 b^4 B a+35 A b^5\right ) \cos (c+d x) a+105 A b^6+3 b \left (-8 B a^5+3 b (8 A-3 C) a^4+29 b^2 B a^3-b^3 (65 A-3 C) a^2-15 b^4 B a+35 A b^5\right ) \cos ^2(c+d x)}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))}dx}{a}}{3 a}}{2 a \left (a^2-b^2\right )}}{4 a \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\sin (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}-\frac {\frac {\sin (c+d x) \left (-5 a^4 C+9 a^3 b B-a^2 b^2 (13 A+C)-3 a b^3 B+7 A b^4\right )}{a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))}-\frac {\frac {2 \sin (c+d x) \left (a^4 (8 A-21 C)+33 a^3 b B-a^2 b^2 (61 A-3 C)-15 a b^3 B+35 A b^4\right )}{3 a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\frac {6 \sin (c+d x) \left (-8 a^5 B+3 a^4 b (8 A-3 C)+29 a^3 b^2 B-a^2 b^3 (65 A-3 C)-15 a b^4 B+35 A b^5\right )}{a d \sqrt {\cos (c+d x)}}-\frac {\int \frac {8 (A+3 C) a^6-72 b B a^5+b^2 (128 A-15 C) a^4+99 b^3 B a^3-b^4 (223 A-9 C) a^2-45 b^5 B a+4 \left (-6 B a^5+4 b (5 A-3 C) a^4+30 b^2 B a^3-b^3 (64 A-3 C) a^2-15 b^4 B a+35 A b^5\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) a+105 A b^6+3 b \left (-8 B a^5+3 b (8 A-3 C) a^4+29 b^2 B a^3-b^3 (65 A-3 C) a^2-15 b^4 B a+35 A b^5\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a}}{3 a}}{2 a \left (a^2-b^2\right )}}{4 a \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3538

\(\displaystyle \frac {\sin (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}-\frac {\frac {\sin (c+d x) \left (-5 a^4 C+9 a^3 b B-a^2 b^2 (13 A+C)-3 a b^3 B+7 A b^4\right )}{a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))}-\frac {\frac {2 \sin (c+d x) \left (a^4 (8 A-21 C)+33 a^3 b B-a^2 b^2 (61 A-3 C)-15 a b^3 B+35 A b^4\right )}{3 a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\frac {6 \sin (c+d x) \left (-8 a^5 B+3 a^4 b (8 A-3 C)+29 a^3 b^2 B-a^2 b^3 (65 A-3 C)-15 a b^4 B+35 A b^5\right )}{a d \sqrt {\cos (c+d x)}}-\frac {3 \left (-8 a^5 B+3 a^4 b (8 A-3 C)+29 a^3 b^2 B-a^2 b^3 (65 A-3 C)-15 a b^4 B+35 A b^5\right ) \int \sqrt {\cos (c+d x)}dx-\frac {\int -\frac {a \left ((8 A-21 C) a^4+33 b B a^3-b^2 (61 A-3 C) a^2-15 b^3 B a+35 A b^4\right ) \cos (c+d x) b^2+\left (8 (A+3 C) a^6-72 b B a^5+b^2 (128 A-15 C) a^4+99 b^3 B a^3-b^4 (223 A-9 C) a^2-45 b^5 B a+105 A b^6\right ) b}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))}dx}{b}}{a}}{3 a}}{2 a \left (a^2-b^2\right )}}{4 a \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\sin (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}-\frac {\frac {\sin (c+d x) \left (-5 a^4 C+9 a^3 b B-a^2 b^2 (13 A+C)-3 a b^3 B+7 A b^4\right )}{a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))}-\frac {\frac {2 \sin (c+d x) \left (a^4 (8 A-21 C)+33 a^3 b B-a^2 b^2 (61 A-3 C)-15 a b^3 B+35 A b^4\right )}{3 a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\frac {6 \sin (c+d x) \left (-8 a^5 B+3 a^4 b (8 A-3 C)+29 a^3 b^2 B-a^2 b^3 (65 A-3 C)-15 a b^4 B+35 A b^5\right )}{a d \sqrt {\cos (c+d x)}}-\frac {3 \left (-8 a^5 B+3 a^4 b (8 A-3 C)+29 a^3 b^2 B-a^2 b^3 (65 A-3 C)-15 a b^4 B+35 A b^5\right ) \int \sqrt {\cos (c+d x)}dx+\frac {\int \frac {a \left ((8 A-21 C) a^4+33 b B a^3-b^2 (61 A-3 C) a^2-15 b^3 B a+35 A b^4\right ) \cos (c+d x) b^2+\left (8 (A+3 C) a^6-72 b B a^5+b^2 (128 A-15 C) a^4+99 b^3 B a^3-b^4 (223 A-9 C) a^2-45 b^5 B a+105 A b^6\right ) b}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))}dx}{b}}{a}}{3 a}}{2 a \left (a^2-b^2\right )}}{4 a \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\sin (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}-\frac {\frac {\sin (c+d x) \left (-5 a^4 C+9 a^3 b B-a^2 b^2 (13 A+C)-3 a b^3 B+7 A b^4\right )}{a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))}-\frac {\frac {2 \sin (c+d x) \left (a^4 (8 A-21 C)+33 a^3 b B-a^2 b^2 (61 A-3 C)-15 a b^3 B+35 A b^4\right )}{3 a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\frac {6 \sin (c+d x) \left (-8 a^5 B+3 a^4 b (8 A-3 C)+29 a^3 b^2 B-a^2 b^3 (65 A-3 C)-15 a b^4 B+35 A b^5\right )}{a d \sqrt {\cos (c+d x)}}-\frac {3 \left (-8 a^5 B+3 a^4 b (8 A-3 C)+29 a^3 b^2 B-a^2 b^3 (65 A-3 C)-15 a b^4 B+35 A b^5\right ) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {\int \frac {a \left ((8 A-21 C) a^4+33 b B a^3-b^2 (61 A-3 C) a^2-15 b^3 B a+35 A b^4\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) b^2+\left (8 (A+3 C) a^6-72 b B a^5+b^2 (128 A-15 C) a^4+99 b^3 B a^3-b^4 (223 A-9 C) a^2-45 b^5 B a+105 A b^6\right ) b}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{b}}{a}}{3 a}}{2 a \left (a^2-b^2\right )}}{4 a \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3119

\(\displaystyle \frac {\sin (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}-\frac {\frac {\sin (c+d x) \left (-5 a^4 C+9 a^3 b B-a^2 b^2 (13 A+C)-3 a b^3 B+7 A b^4\right )}{a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))}-\frac {\frac {2 \sin (c+d x) \left (a^4 (8 A-21 C)+33 a^3 b B-a^2 b^2 (61 A-3 C)-15 a b^3 B+35 A b^4\right )}{3 a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\frac {6 \sin (c+d x) \left (-8 a^5 B+3 a^4 b (8 A-3 C)+29 a^3 b^2 B-a^2 b^3 (65 A-3 C)-15 a b^4 B+35 A b^5\right )}{a d \sqrt {\cos (c+d x)}}-\frac {\frac {\int \frac {a \left ((8 A-21 C) a^4+33 b B a^3-b^2 (61 A-3 C) a^2-15 b^3 B a+35 A b^4\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) b^2+\left (8 (A+3 C) a^6-72 b B a^5+b^2 (128 A-15 C) a^4+99 b^3 B a^3-b^4 (223 A-9 C) a^2-45 b^5 B a+105 A b^6\right ) b}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{b}+\frac {6 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (-8 a^5 B+3 a^4 b (8 A-3 C)+29 a^3 b^2 B-a^2 b^3 (65 A-3 C)-15 a b^4 B+35 A b^5\right )}{d}}{a}}{3 a}}{2 a \left (a^2-b^2\right )}}{4 a \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3481

\(\displaystyle \frac {\sin (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}-\frac {\frac {\sin (c+d x) \left (-5 a^4 C+9 a^3 b B-a^2 b^2 (13 A+C)-3 a b^3 B+7 A b^4\right )}{a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))}-\frac {\frac {2 \sin (c+d x) \left (a^4 (8 A-21 C)+33 a^3 b B-a^2 b^2 (61 A-3 C)-15 a b^3 B+35 A b^4\right )}{3 a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\frac {6 \sin (c+d x) \left (-8 a^5 B+3 a^4 b (8 A-3 C)+29 a^3 b^2 B-a^2 b^3 (65 A-3 C)-15 a b^4 B+35 A b^5\right )}{a d \sqrt {\cos (c+d x)}}-\frac {\frac {a b \left (a^4 (8 A-21 C)+33 a^3 b B-a^2 b^2 (61 A-3 C)-15 a b^3 B+35 A b^4\right ) \int \frac {1}{\sqrt {\cos (c+d x)}}dx+3 b \left (15 a^6 C-35 a^5 b B+3 a^4 b^2 (21 A-2 C)+38 a^3 b^3 B-a^2 b^4 (86 A-3 C)-15 a b^5 B+35 A b^6\right ) \int \frac {1}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))}dx}{b}+\frac {6 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (-8 a^5 B+3 a^4 b (8 A-3 C)+29 a^3 b^2 B-a^2 b^3 (65 A-3 C)-15 a b^4 B+35 A b^5\right )}{d}}{a}}{3 a}}{2 a \left (a^2-b^2\right )}}{4 a \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\sin (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}-\frac {\frac {\sin (c+d x) \left (-5 a^4 C+9 a^3 b B-a^2 b^2 (13 A+C)-3 a b^3 B+7 A b^4\right )}{a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))}-\frac {\frac {2 \sin (c+d x) \left (a^4 (8 A-21 C)+33 a^3 b B-a^2 b^2 (61 A-3 C)-15 a b^3 B+35 A b^4\right )}{3 a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\frac {6 \sin (c+d x) \left (-8 a^5 B+3 a^4 b (8 A-3 C)+29 a^3 b^2 B-a^2 b^3 (65 A-3 C)-15 a b^4 B+35 A b^5\right )}{a d \sqrt {\cos (c+d x)}}-\frac {\frac {a b \left (a^4 (8 A-21 C)+33 a^3 b B-a^2 b^2 (61 A-3 C)-15 a b^3 B+35 A b^4\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+3 b \left (15 a^6 C-35 a^5 b B+3 a^4 b^2 (21 A-2 C)+38 a^3 b^3 B-a^2 b^4 (86 A-3 C)-15 a b^5 B+35 A b^6\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{b}+\frac {6 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (-8 a^5 B+3 a^4 b (8 A-3 C)+29 a^3 b^2 B-a^2 b^3 (65 A-3 C)-15 a b^4 B+35 A b^5\right )}{d}}{a}}{3 a}}{2 a \left (a^2-b^2\right )}}{4 a \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3120

\(\displaystyle \frac {\sin (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}-\frac {\frac {\sin (c+d x) \left (-5 a^4 C+9 a^3 b B-a^2 b^2 (13 A+C)-3 a b^3 B+7 A b^4\right )}{a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))}-\frac {\frac {2 \sin (c+d x) \left (a^4 (8 A-21 C)+33 a^3 b B-a^2 b^2 (61 A-3 C)-15 a b^3 B+35 A b^4\right )}{3 a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\frac {6 \sin (c+d x) \left (-8 a^5 B+3 a^4 b (8 A-3 C)+29 a^3 b^2 B-a^2 b^3 (65 A-3 C)-15 a b^4 B+35 A b^5\right )}{a d \sqrt {\cos (c+d x)}}-\frac {\frac {3 b \left (15 a^6 C-35 a^5 b B+3 a^4 b^2 (21 A-2 C)+38 a^3 b^3 B-a^2 b^4 (86 A-3 C)-15 a b^5 B+35 A b^6\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx+\frac {2 a b \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (a^4 (8 A-21 C)+33 a^3 b B-a^2 b^2 (61 A-3 C)-15 a b^3 B+35 A b^4\right )}{d}}{b}+\frac {6 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (-8 a^5 B+3 a^4 b (8 A-3 C)+29 a^3 b^2 B-a^2 b^3 (65 A-3 C)-15 a b^4 B+35 A b^5\right )}{d}}{a}}{3 a}}{2 a \left (a^2-b^2\right )}}{4 a \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3284

\(\displaystyle \frac {\sin (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}-\frac {\frac {\sin (c+d x) \left (-5 a^4 C+9 a^3 b B-a^2 b^2 (13 A+C)-3 a b^3 B+7 A b^4\right )}{a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))}-\frac {\frac {2 \sin (c+d x) \left (a^4 (8 A-21 C)+33 a^3 b B-a^2 b^2 (61 A-3 C)-15 a b^3 B+35 A b^4\right )}{3 a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\frac {6 \sin (c+d x) \left (-8 a^5 B+3 a^4 b (8 A-3 C)+29 a^3 b^2 B-a^2 b^3 (65 A-3 C)-15 a b^4 B+35 A b^5\right )}{a d \sqrt {\cos (c+d x)}}-\frac {\frac {6 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (-8 a^5 B+3 a^4 b (8 A-3 C)+29 a^3 b^2 B-a^2 b^3 (65 A-3 C)-15 a b^4 B+35 A b^5\right )}{d}+\frac {\frac {2 a b \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (a^4 (8 A-21 C)+33 a^3 b B-a^2 b^2 (61 A-3 C)-15 a b^3 B+35 A b^4\right )}{d}+\frac {6 b \left (15 a^6 C-35 a^5 b B+3 a^4 b^2 (21 A-2 C)+38 a^3 b^3 B-a^2 b^4 (86 A-3 C)-15 a b^5 B+35 A b^6\right ) \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right )}{d (a+b)}}{b}}{a}}{3 a}}{2 a \left (a^2-b^2\right )}}{4 a \left (a^2-b^2\right )}\)

Input:

Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + b*Cos 
[c + d*x])^3),x]
 

Output:

((A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*Cos[c + d*x]^(3/ 
2)*(a + b*Cos[c + d*x])^2) - (((7*A*b^4 + 9*a^3*b*B - 3*a*b^3*B - 5*a^4*C 
- a^2*b^2*(13*A + C))*Sin[c + d*x])/(a*(a^2 - b^2)*d*Cos[c + d*x]^(3/2)*(a 
 + b*Cos[c + d*x])) - ((2*(35*A*b^4 + 33*a^3*b*B - 15*a*b^3*B + a^4*(8*A - 
 21*C) - a^2*b^2*(61*A - 3*C))*Sin[c + d*x])/(3*a*d*Cos[c + d*x]^(3/2)) - 
(-(((6*(35*A*b^5 - 8*a^5*B + 29*a^3*b^2*B - 15*a*b^4*B + 3*a^4*b*(8*A - 3* 
C) - a^2*b^3*(65*A - 3*C))*EllipticE[(c + d*x)/2, 2])/d + ((2*a*b*(35*A*b^ 
4 + 33*a^3*b*B - 15*a*b^3*B + a^4*(8*A - 21*C) - a^2*b^2*(61*A - 3*C))*Ell 
ipticF[(c + d*x)/2, 2])/d + (6*b*(35*A*b^6 - 35*a^5*b*B + 38*a^3*b^3*B - 1 
5*a*b^5*B - a^2*b^4*(86*A - 3*C) + 3*a^4*b^2*(21*A - 2*C) + 15*a^6*C)*Elli 
pticPi[(2*b)/(a + b), (c + d*x)/2, 2])/((a + b)*d))/b)/a) + (6*(35*A*b^5 - 
 8*a^5*B + 29*a^3*b^2*B - 15*a*b^4*B + 3*a^4*b*(8*A - 3*C) - a^2*b^3*(65*A 
 - 3*C))*Sin[c + d*x])/(a*d*Sqrt[Cos[c + d*x]]))/(3*a))/(2*a*(a^2 - b^2))) 
/(4*a*(a^2 - b^2))
 

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 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 3534

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[(-(A*b^2 - a*b*B + a^2*C))*Cos[e + f*x 
]*(a + b*Sin[e + f*x])^(m + 1)*((c + d*Sin[e + f*x])^(n + 1)/(f*(m + 1)*(b* 
c - a*d)*(a^2 - b^2))), x] + Simp[1/((m + 1)*(b*c - a*d)*(a^2 - b^2))   Int 
[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[(m + 1)*(b*c - a* 
d)*(a*A - b*B + a*C) + d*(A*b^2 - a*b*B + a^2*C)*(m + n + 2) - (c*(A*b^2 - 
a*b*B + a^2*C) + (m + 1)*(b*c - a*d)*(A*b - a*B + b*C))*Sin[e + f*x] - d*(A 
*b^2 - a*b*B + a^2*C)*(m + n + 3)*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] && LtQ[m, -1] && ((EqQ[a, 0] && IntegerQ[m] &&  !IntegerQ 
[n]) ||  !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] &&  !IntegerQ[m]) | 
| EqQ[a, 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. \(2138\) vs. \(2(592)=1184\).

Time = 11.58 (sec) , antiderivative size = 2139, normalized size of antiderivative = 3.51

method	result	size
default	\(\text {Expression too large to display}\)	\(2139\)

Input:

int((A+B*cos(d*x+c)+C*cos(d*x+c)^2)/cos(d*x+c)^(5/2)/(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)*(2*A/a^3*(-1/6* 
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)/(c 
os(1/2*d*x+1/2*c)^2-1/2)^2+1/3*(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)*E 
llipticF(cos(1/2*d*x+1/2*c),2^(1/2)))-2*(3*A*b-B*a)/a^4/sin(1/2*d*x+1/2*c) 
^2/(2*sin(1/2*d*x+1/2*c)^2-1)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^ 
2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)-(sin(1/2*d*x+1/2*c)^2) 
^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/ 
2)))+2*(A*b^2-B*a*b+C*a^2)/a^2*(-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+sin(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(c 
os(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...
 

Fricas [F(-1)]

Timed out. \[ \int \frac {A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^3} \, dx=\text {Timed out} \] Input:

integrate((A+B*cos(d*x+c)+C*cos(d*x+c)^2)/cos(d*x+c)^(5/2)/(a+b*cos(d*x+c) 
)^3,x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^3} \, dx=\text {Timed out} \] Input:

integrate((A+B*cos(d*x+c)+C*cos(d*x+c)**2)/cos(d*x+c)**(5/2)/(a+b*cos(d*x+ 
c))**3,x)
 

Output:

Timed out
 

Maxima [F(-1)]

Timed out. \[ \int \frac {A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^3} \, dx=\text {Timed out} \] Input:

integrate((A+B*cos(d*x+c)+C*cos(d*x+c)^2)/cos(d*x+c)^(5/2)/(a+b*cos(d*x+c) 
)^3,x, algorithm="maxima")
 

Output:

Timed out
 

Giac [F]

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

integrate((A+B*cos(d*x+c)+C*cos(d*x+c)^2)/cos(d*x+c)^(5/2)/(a+b*cos(d*x+c) 
)^3,x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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