\(\int \frac {\cos ^{\frac {5}{2}}(c+d x) (A+C \cos ^2(c+d x))}{a+b \cos (c+d x)} \, dx\) [705]

Optimal result

Integrand size = 35, antiderivative size = 239 \[ \int \frac {\cos ^{\frac {5}{2}}(c+d x) \left (A+C \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx=-\frac {2 a \left (5 A b^2+5 a^2 C+3 b^2 C\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 b^4 d}+\frac {2 \left (21 a^4 C+7 a^2 b^2 (3 A+C)+b^4 (7 A+5 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 b^5 d}-\frac {2 a^3 \left (A b^2+a^2 C\right ) \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right )}{b^5 (a+b) d}+\frac {2 \left (7 a^2 C+b^2 (7 A+5 C)\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{21 b^3 d}-\frac {2 a C \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{5 b^2 d}+\frac {2 C \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{7 b d} \] Output:

-2/5*a*(5*A*b^2+5*C*a^2+3*C*b^2)*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2))/b^4 
/d+2/21*(21*a^4*C+7*a^2*b^2*(3*A+C)+b^4*(7*A+5*C))*InverseJacobiAM(1/2*d*x 
+1/2*c,2^(1/2))/b^5/d-2*a^3*(A*b^2+C*a^2)*EllipticPi(sin(1/2*d*x+1/2*c),2* 
b/(a+b),2^(1/2))/b^5/(a+b)/d+2/21*(7*a^2*C+b^2*(7*A+5*C))*cos(d*x+c)^(1/2) 
*sin(d*x+c)/b^3/d-2/5*a*C*cos(d*x+c)^(3/2)*sin(d*x+c)/b^2/d+2/7*C*cos(d*x+ 
c)^(5/2)*sin(d*x+c)/b/d
 

Mathematica [A] (warning: unable to verify)

Time = 3.85 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.22 \[ \int \frac {\cos ^{\frac {5}{2}}(c+d x) \left (A+C \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx=\frac {-\frac {2 a \left (35 A b^2+35 a^2 C+13 b^2 C\right ) \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right )}{a+b}+\frac {4 \left (35 A b^2-28 a^2 C+25 b^2 C\right ) \left ((a+b) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )-a \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right )\right )}{a+b}+2 \sqrt {\cos (c+d x)} \left (70 A b^2+70 a^2 C+65 b^2 C-42 a b C \cos (c+d x)+15 b^2 C \cos (2 (c+d x))\right ) \sin (c+d x)-\frac {42 \left (5 A b^2+5 a^2 C+3 b^2 C\right ) \left (-2 a b E\left (\left .\arcsin \left (\sqrt {\cos (c+d x)}\right )\right |-1\right )+2 a (a+b) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\cos (c+d x)}\right ),-1\right )+\left (-2 a^2+b^2\right ) \operatorname {EllipticPi}\left (-\frac {b}{a},\arcsin \left (\sqrt {\cos (c+d x)}\right ),-1\right )\right ) \sin (c+d x)}{b^2 \sqrt {\sin ^2(c+d x)}}}{210 b^3 d} \] Input:

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

Output:

((-2*a*(35*A*b^2 + 35*a^2*C + 13*b^2*C)*EllipticPi[(2*b)/(a + b), (c + d*x 
)/2, 2])/(a + b) + (4*(35*A*b^2 - 28*a^2*C + 25*b^2*C)*((a + b)*EllipticF[ 
(c + d*x)/2, 2] - a*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]))/(a + b) + 
2*Sqrt[Cos[c + d*x]]*(70*A*b^2 + 70*a^2*C + 65*b^2*C - 42*a*b*C*Cos[c + d* 
x] + 15*b^2*C*Cos[2*(c + d*x)])*Sin[c + d*x] - (42*(5*A*b^2 + 5*a^2*C + 3* 
b^2*C)*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*a*(a + b)*Ell 
ipticF[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])/(b^2*Sqrt[Sin[c + d*x]^2]) 
)/(210*b^3*d)
 

Rubi [A] (verified)

Time = 2.04 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.11, number of steps used = 18, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.514, Rules used = {3042, 3529, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3529

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3528

\(\displaystyle \frac {\frac {2 \int -\frac {\sqrt {\cos (c+d x)} \left (21 C a^2-4 b C \cos (c+d x) a-5 \left (7 C a^2+b^2 (7 A+5 C)\right ) \cos ^2(c+d x)\right )}{2 (a+b \cos (c+d x))}dx}{5 b}-\frac {14 a C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 b d}}{7 b}+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 b d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {\int \frac {\sqrt {\cos (c+d x)} \left (21 C a^2-4 b C \cos (c+d x) a-5 \left (7 C a^2+b^2 (7 A+5 C)\right ) \cos ^2(c+d x)\right )}{a+b \cos (c+d x)}dx}{5 b}-\frac {14 a C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 b d}}{7 b}+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 b d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {\int \frac {\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (21 C a^2-4 b C \sin \left (c+d x+\frac {\pi }{2}\right ) a-5 \left (7 C a^2+b^2 (7 A+5 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{5 b}-\frac {14 a C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 b d}}{7 b}+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 b d}\)

\(\Big \downarrow \) 3528

\(\displaystyle \frac {-\frac {\frac {2 \int -\frac {-21 a \left (5 C a^2+5 A b^2+3 b^2 C\right ) \cos ^2(c+d x)+b \left (-28 C a^2+35 A b^2+25 b^2 C\right ) \cos (c+d x)+5 a \left (7 C a^2+b^2 (7 A+5 C)\right )}{2 \sqrt {\cos (c+d x)} (a+b \cos (c+d x))}dx}{3 b}-\frac {10 \left (7 a^2 C+b^2 (7 A+5 C)\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{3 b d}}{5 b}-\frac {14 a C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 b d}}{7 b}+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 b d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {-\frac {\int \frac {-21 a \left (5 C a^2+5 A b^2+3 b^2 C\right ) \cos ^2(c+d x)+b \left (-28 C a^2+35 A b^2+25 b^2 C\right ) \cos (c+d x)+5 a \left (7 C a^2+b^2 (7 A+5 C)\right )}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))}dx}{3 b}-\frac {10 \left (7 a^2 C+b^2 (7 A+5 C)\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{3 b d}}{5 b}-\frac {14 a C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 b d}}{7 b}+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 b d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {-\frac {\int \frac {-21 a \left (5 C a^2+5 A b^2+3 b^2 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+b \left (-28 C a^2+35 A b^2+25 b^2 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+5 a \left (7 C a^2+b^2 (7 A+5 C)\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{3 b}-\frac {10 \left (7 a^2 C+b^2 (7 A+5 C)\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{3 b d}}{5 b}-\frac {14 a C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 b d}}{7 b}+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 b d}\)

\(\Big \downarrow \) 3538

\(\displaystyle \frac {-\frac {-\frac {-\frac {21 a \left (5 a^2 C+5 A b^2+3 b^2 C\right ) \int \sqrt {\cos (c+d x)}dx}{b}-\frac {\int -\frac {5 \left (a b \left (7 C a^2+b^2 (7 A+5 C)\right )+\left (21 C a^4+7 b^2 (3 A+C) a^2+b^4 (7 A+5 C)\right ) \cos (c+d x)\right )}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))}dx}{b}}{3 b}-\frac {10 \left (7 a^2 C+b^2 (7 A+5 C)\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{3 b d}}{5 b}-\frac {14 a C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 b d}}{7 b}+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 b d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {-\frac {\frac {5 \int \frac {a b \left (7 C a^2+b^2 (7 A+5 C)\right )+\left (21 C a^4+7 b^2 (3 A+C) a^2+b^4 (7 A+5 C)\right ) \cos (c+d x)}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))}dx}{b}-\frac {21 a \left (5 a^2 C+5 A b^2+3 b^2 C\right ) \int \sqrt {\cos (c+d x)}dx}{b}}{3 b}-\frac {10 \left (7 a^2 C+b^2 (7 A+5 C)\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{3 b d}}{5 b}-\frac {14 a C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 b d}}{7 b}+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 b d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {-\frac {\frac {5 \int \frac {a b \left (7 C a^2+b^2 (7 A+5 C)\right )+\left (21 C a^4+7 b^2 (3 A+C) a^2+b^4 (7 A+5 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{b}-\frac {21 a \left (5 a^2 C+5 A b^2+3 b^2 C\right ) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx}{b}}{3 b}-\frac {10 \left (7 a^2 C+b^2 (7 A+5 C)\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{3 b d}}{5 b}-\frac {14 a C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 b d}}{7 b}+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 b d}\)

\(\Big \downarrow \) 3119

\(\displaystyle \frac {-\frac {-\frac {\frac {5 \int \frac {a b \left (7 C a^2+b^2 (7 A+5 C)\right )+\left (21 C a^4+7 b^2 (3 A+C) a^2+b^4 (7 A+5 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{b}-\frac {42 a \left (5 a^2 C+5 A b^2+3 b^2 C\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{b d}}{3 b}-\frac {10 \left (7 a^2 C+b^2 (7 A+5 C)\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{3 b d}}{5 b}-\frac {14 a C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 b d}}{7 b}+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 b d}\)

\(\Big \downarrow \) 3481

\(\displaystyle \frac {-\frac {-\frac {\frac {5 \left (\frac {\left (21 a^4 C+7 a^2 b^2 (3 A+C)+b^4 (7 A+5 C)\right ) \int \frac {1}{\sqrt {\cos (c+d x)}}dx}{b}-\frac {21 a^3 \left (a^2 C+A b^2\right ) \int \frac {1}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))}dx}{b}\right )}{b}-\frac {42 a \left (5 a^2 C+5 A b^2+3 b^2 C\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{b d}}{3 b}-\frac {10 \left (7 a^2 C+b^2 (7 A+5 C)\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{3 b d}}{5 b}-\frac {14 a C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 b d}}{7 b}+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 b d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {-\frac {\frac {5 \left (\frac {\left (21 a^4 C+7 a^2 b^2 (3 A+C)+b^4 (7 A+5 C)\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {21 a^3 \left (a^2 C+A b^2\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{b}\right )}{b}-\frac {42 a \left (5 a^2 C+5 A b^2+3 b^2 C\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{b d}}{3 b}-\frac {10 \left (7 a^2 C+b^2 (7 A+5 C)\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{3 b d}}{5 b}-\frac {14 a C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 b d}}{7 b}+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 b d}\)

\(\Big \downarrow \) 3120

\(\displaystyle \frac {-\frac {-\frac {\frac {5 \left (\frac {2 \left (21 a^4 C+7 a^2 b^2 (3 A+C)+b^4 (7 A+5 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{b d}-\frac {21 a^3 \left (a^2 C+A b^2\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{b}\right )}{b}-\frac {42 a \left (5 a^2 C+5 A b^2+3 b^2 C\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{b d}}{3 b}-\frac {10 \left (7 a^2 C+b^2 (7 A+5 C)\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{3 b d}}{5 b}-\frac {14 a C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 b d}}{7 b}+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 b d}\)

\(\Big \downarrow \) 3284

\(\displaystyle \frac {-\frac {-\frac {10 \left (7 a^2 C+b^2 (7 A+5 C)\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{3 b d}-\frac {\frac {5 \left (\frac {2 \left (21 a^4 C+7 a^2 b^2 (3 A+C)+b^4 (7 A+5 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{b d}-\frac {42 a^3 \left (a^2 C+A b^2\right ) \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right )}{b d (a+b)}\right )}{b}-\frac {42 a \left (5 a^2 C+5 A b^2+3 b^2 C\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{b d}}{3 b}}{5 b}-\frac {14 a C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 b d}}{7 b}+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 b d}\)

Input:

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

Output:

(2*C*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*b*d) + ((-14*a*C*Cos[c + d*x]^(3/ 
2)*Sin[c + d*x])/(5*b*d) - (-1/3*((-42*a*(5*A*b^2 + 5*a^2*C + 3*b^2*C)*Ell 
ipticE[(c + d*x)/2, 2])/(b*d) + (5*((2*(21*a^4*C + 7*a^2*b^2*(3*A + C) + b 
^4*(7*A + 5*C))*EllipticF[(c + d*x)/2, 2])/(b*d) - (42*a^3*(A*b^2 + a^2*C) 
*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(b*(a + b)*d)))/b)/b - (10*(7* 
a^2*C + b^2*(7*A + 5*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*b*d))/(5*b))/ 
(7*b)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 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 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 3529

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) 
+ (f_.)*(x_)])^(n_.)*((A_.) + (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)) + (A*b*d*(m + n + 2) - C*(a*c - b*d*(m + n + 1)))*Sin[e + f*x] + C* 
(a*d*m - b*c*(m + 1))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f 
, A, 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] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1243\) vs. \(2(228)=456\).

Time = 34.45 (sec) , antiderivative size = 1244, normalized size of antiderivative = 5.21

method	result	size
default	\(\text {Expression too large to display}\)	\(1244\)

Input:

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

Output:

-2/105*((2*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*((240*C*a*b 
^4-240*C*b^5)*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^8+(168*C*a^2*b^3-528*C 
*a*b^4+360*C*b^5)*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)+(140*A*a*b^4-140 
*A*b^5+140*C*a^3*b^2-308*C*a^2*b^3+448*C*a*b^4-280*C*b^5)*sin(1/2*d*x+1/2* 
c)^4*cos(1/2*d*x+1/2*c)+(-70*A*a*b^4+70*A*b^5-70*C*a^3*b^2+112*C*a^2*b^3-1 
22*C*a*b^4+80*C*b^5)*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)+105*A*(sin(1/ 
2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF(cos(1/2*d 
*x+1/2*c),2^(1/2))*a^3*b^2-105*A*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d 
*x+1/2*c)^2-1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))*a^2*b^3+35*A*(s 
in(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF(cos( 
1/2*d*x+1/2*c),2^(1/2))*a*b^4-35*A*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2 
*d*x+1/2*c)^2-1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))*b^5+105*A*(si 
n(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1 
/2*d*x+1/2*c),2^(1/2))*a^2*b^3-105*A*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1 
/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*a*b^4-105*A 
*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticPi( 
cos(1/2*d*x+1/2*c),-2*b/(a-b),2^(1/2))*a^3*b^2+105*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)*EllipticF(cos(1/2*d*x+1/2*c),2^(1 
/2))*a^5-105*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)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))*a^4*b+35*C*(sin(1/2*d*x+1/2*c)...
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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