3.7.37 \(\int (a+b \cos (e+f x))^3 (A+B \cos (e+f x)) (c \sec (e+f x))^m \, dx\) [637]

3.7.37.1 Optimal result

Integrand size = 33, antiderivative size = 455 \[ \int (a+b \cos (e+f x))^3 (A+B \cos (e+f x)) (c \sec (e+f x))^m \, dx=-\frac {c^5 \left (a^3 A \left (8-6 m+m^2\right )+3 a A b^2 \left (4-5 m+m^2\right )+3 a^2 b B \left (4-5 m+m^2\right )+b^3 B \left (3-4 m+m^2\right )\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {5-m}{2},\frac {7-m}{2},\cos ^2(e+f x)\right ) (c \sec (e+f x))^{-5+m} \sin (e+f x)}{f (1-m) (3-m) (5-m) \sqrt {\sin ^2(e+f x)}}-\frac {c^4 \left (A b^3 (2-m)+3 a b^2 B (2-m)+3 a^2 A b (3-m)+a^3 B (3-m)\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {4-m}{2},\frac {6-m}{2},\cos ^2(e+f x)\right ) (c \sec (e+f x))^{-4+m} \sin (e+f x)}{f (2-m) (4-m) \sqrt {\sin ^2(e+f x)}}-\frac {a c^4 \left (3 a b B (1-m)+a^2 A (2-m)-2 A b^2 m\right ) (c \sec (e+f x))^{-4+m} \tan (e+f x)}{f (1-m) (3-m)}-\frac {a^2 c^4 (a B (1-m)-A b (1+m)) \sec (e+f x) (c \sec (e+f x))^{-4+m} \tan (e+f x)}{f (1-m) (2-m)}-\frac {a A c^4 (c \sec (e+f x))^{-4+m} (b+a \sec (e+f x))^2 \tan (e+f x)}{f (1-m)} \]

output

-c^5*(a^3*A*(m^2-6*m+8)+3*a*A*b^2*(m^2-5*m+4)+3*a^2*b*B*(m^2-5*m+4)+b^3*B* 
(m^2-4*m+3))*hypergeom([1/2, 5/2-1/2*m],[7/2-1/2*m],cos(f*x+e)^2)*(c*sec(f 
*x+e))^(-5+m)*sin(f*x+e)/f/(1-m)/(m^2-8*m+15)/(sin(f*x+e)^2)^(1/2)-c^4*(A* 
b^3*(2-m)+3*a*b^2*B*(2-m)+3*a^2*A*b*(3-m)+a^3*B*(3-m))*hypergeom([1/2, 2-1 
/2*m],[3-1/2*m],cos(f*x+e)^2)*(c*sec(f*x+e))^(-4+m)*sin(f*x+e)/f/(m^2-6*m+ 
8)/(sin(f*x+e)^2)^(1/2)-a*c^4*(3*a*b*B*(1-m)+a^2*A*(2-m)-2*A*b^2*m)*(c*sec 
(f*x+e))^(-4+m)*tan(f*x+e)/f/(m^2-4*m+3)-a^2*c^4*(a*B*(1-m)-A*b*(1+m))*sec 
(f*x+e)*(c*sec(f*x+e))^(-4+m)*tan(f*x+e)/f/(m^2-3*m+2)-a*A*c^4*(c*sec(f*x+ 
e))^(-4+m)*(b+a*sec(f*x+e))^2*tan(f*x+e)/f/(1-m)
 

3.7.37.2 Mathematica [A] (verified)

Time = 1.42 (sec) , antiderivative size = 259, normalized size of antiderivative = 0.57 \[ \int (a+b \cos (e+f x))^3 (A+B \cos (e+f x)) (c \sec (e+f x))^m \, dx=\frac {\cot (e+f x) \left (\frac {b^3 B \cos ^4(e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (-4+m),\frac {1}{2} (-2+m),\sec ^2(e+f x)\right )}{-4+m}+\frac {b^2 (A b+3 a B) \cos ^3(e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (-3+m),\frac {1}{2} (-1+m),\sec ^2(e+f x)\right )}{-3+m}+a \left (\frac {3 b (A b+a B) \cos ^2(e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (-2+m),\frac {m}{2},\sec ^2(e+f x)\right )}{-2+m}+a \left (\frac {(3 A b+a B) \cos (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (-1+m),\frac {1+m}{2},\sec ^2(e+f x)\right )}{-1+m}+\frac {a A \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m}{2},\frac {2+m}{2},\sec ^2(e+f x)\right )}{m}\right )\right )\right ) (c \sec (e+f x))^m \sqrt {-\tan ^2(e+f x)}}{f} \]

input

Integrate[(a + b*Cos[e + f*x])^3*(A + B*Cos[e + f*x])*(c*Sec[e + f*x])^m,x 
]
 

output

(Cot[e + f*x]*((b^3*B*Cos[e + f*x]^4*Hypergeometric2F1[1/2, (-4 + m)/2, (- 
2 + m)/2, Sec[e + f*x]^2])/(-4 + m) + (b^2*(A*b + 3*a*B)*Cos[e + f*x]^3*Hy 
pergeometric2F1[1/2, (-3 + m)/2, (-1 + m)/2, Sec[e + f*x]^2])/(-3 + m) + a 
*((3*b*(A*b + a*B)*Cos[e + f*x]^2*Hypergeometric2F1[1/2, (-2 + m)/2, m/2, 
Sec[e + f*x]^2])/(-2 + m) + a*(((3*A*b + a*B)*Cos[e + f*x]*Hypergeometric2 
F1[1/2, (-1 + m)/2, (1 + m)/2, Sec[e + f*x]^2])/(-1 + m) + (a*A*Hypergeome 
tric2F1[1/2, m/2, (2 + m)/2, Sec[e + f*x]^2])/m)))*(c*Sec[e + f*x])^m*Sqrt 
[-Tan[e + f*x]^2])/f
 

3.7.37.3 Rubi [A] (verified)

Time = 2.83 (sec) , antiderivative size = 450, normalized size of antiderivative = 0.99, number of steps used = 19, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.576, Rules used = {3042, 3439, 3042, 4514, 25, 3042, 4564, 25, 3042, 4535, 3042, 4259, 3042, 3122, 4534, 3042, 4259, 3042, 3122}

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.

\(\displaystyle \int (a+b \cos (e+f x))^3 (A+B \cos (e+f x)) (c \sec (e+f x))^m \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \left (a+b \sin \left (e+f x+\frac {\pi }{2}\right )\right )^3 \left (A+B \sin \left (e+f x+\frac {\pi }{2}\right )\right ) \left (c \csc \left (e+f x+\frac {\pi }{2}\right )\right )^mdx\)

\(\Big \downarrow \) 3439

\(\displaystyle c^4 \int (c \sec (e+f x))^{m-4} (b+a \sec (e+f x))^3 (B+A \sec (e+f x))dx\)

\(\Big \downarrow \) 3042

\(\displaystyle c^4 \int \left (c \csc \left (e+f x+\frac {\pi }{2}\right )\right )^{m-4} \left (b+a \csc \left (e+f x+\frac {\pi }{2}\right )\right )^3 \left (B+A \csc \left (e+f x+\frac {\pi }{2}\right )\right )dx\)

\(\Big \downarrow \) 4514

\(\displaystyle c^4 \left (-\frac {\int -(c \sec (e+f x))^{m-4} (b+a \sec (e+f x)) \left (a (a B (1-m)-A b (m+1)) \sec ^2(e+f x)+\left (A (2-m) a^2+b (A b+2 a B) (1-m)\right ) \sec (e+f x)+b (b B (1-m)+a A (4-m))\right )dx}{1-m}-\frac {a A \tan (e+f x) (a \sec (e+f x)+b)^2 (c \sec (e+f x))^{m-4}}{f (1-m)}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle c^4 \left (\frac {\int (c \sec (e+f x))^{m-4} (b+a \sec (e+f x)) \left (a (a B (1-m)-A b (m+1)) \sec ^2(e+f x)+\left (A (2-m) a^2+b (A b+2 a B) (1-m)\right ) \sec (e+f x)+b (b B (1-m)+a A (4-m))\right )dx}{1-m}-\frac {a A \tan (e+f x) (a \sec (e+f x)+b)^2 (c \sec (e+f x))^{m-4}}{f (1-m)}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle c^4 \left (\frac {\int \left (c \csc \left (e+f x+\frac {\pi }{2}\right )\right )^{m-4} \left (b+a \csc \left (e+f x+\frac {\pi }{2}\right )\right ) \left (a (a B (1-m)-A b (m+1)) \csc \left (e+f x+\frac {\pi }{2}\right )^2+\left (A (2-m) a^2+b (A b+2 a B) (1-m)\right ) \csc \left (e+f x+\frac {\pi }{2}\right )+b (b B (1-m)+a A (4-m))\right )dx}{1-m}-\frac {a A \tan (e+f x) (a \sec (e+f x)+b)^2 (c \sec (e+f x))^{m-4}}{f (1-m)}\right )\)

\(\Big \downarrow \) 4564

\(\displaystyle c^4 \left (\frac {-\frac {\int -(c \sec (e+f x))^{m-4} \left ((b B (1-m)+a A (4-m)) (2-m) b^2+a (2-m) \left (A (2-m) a^2+3 b B (1-m) a-2 A b^2 m\right ) \sec ^2(e+f x)+\left (B (3-m) a^3+3 A b (3-m) a^2+3 b^2 B (2-m) a+A b^3 (2-m)\right ) (1-m) \sec (e+f x)\right )dx}{2-m}-\frac {a^2 \tan (e+f x) \sec (e+f x) (a B (1-m)-A b (m+1)) (c \sec (e+f x))^{m-4}}{f (2-m)}}{1-m}-\frac {a A \tan (e+f x) (a \sec (e+f x)+b)^2 (c \sec (e+f x))^{m-4}}{f (1-m)}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle c^4 \left (\frac {\frac {\int (c \sec (e+f x))^{m-4} \left ((b B (1-m)+a A (4-m)) (2-m) b^2+a (2-m) \left (A (2-m) a^2+3 b B (1-m) a-2 A b^2 m\right ) \sec ^2(e+f x)+\left (B (3-m) a^3+3 A b (3-m) a^2+3 b^2 B (2-m) a+A b^3 (2-m)\right ) (1-m) \sec (e+f x)\right )dx}{2-m}-\frac {a^2 \tan (e+f x) \sec (e+f x) (a B (1-m)-A b (m+1)) (c \sec (e+f x))^{m-4}}{f (2-m)}}{1-m}-\frac {a A \tan (e+f x) (a \sec (e+f x)+b)^2 (c \sec (e+f x))^{m-4}}{f (1-m)}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle c^4 \left (\frac {\frac {\int \left (c \csc \left (e+f x+\frac {\pi }{2}\right )\right )^{m-4} \left ((b B (1-m)+a A (4-m)) (2-m) b^2+a (2-m) \left (A (2-m) a^2+3 b B (1-m) a-2 A b^2 m\right ) \csc \left (e+f x+\frac {\pi }{2}\right )^2+\left (B (3-m) a^3+3 A b (3-m) a^2+3 b^2 B (2-m) a+A b^3 (2-m)\right ) (1-m) \csc \left (e+f x+\frac {\pi }{2}\right )\right )dx}{2-m}-\frac {a^2 \tan (e+f x) \sec (e+f x) (a B (1-m)-A b (m+1)) (c \sec (e+f x))^{m-4}}{f (2-m)}}{1-m}-\frac {a A \tan (e+f x) (a \sec (e+f x)+b)^2 (c \sec (e+f x))^{m-4}}{f (1-m)}\right )\)

\(\Big \downarrow \) 4535

\(\displaystyle c^4 \left (\frac {\frac {\int (c \sec (e+f x))^{m-4} \left ((b B (1-m)+a A (4-m)) (2-m) b^2+a (2-m) \left (A (2-m) a^2+3 b B (1-m) a-2 A b^2 m\right ) \sec ^2(e+f x)\right )dx+\frac {(1-m) \left (a^3 B (3-m)+3 a^2 A b (3-m)+3 a b^2 B (2-m)+A b^3 (2-m)\right ) \int (c \sec (e+f x))^{m-3}dx}{c}}{2-m}-\frac {a^2 \tan (e+f x) \sec (e+f x) (a B (1-m)-A b (m+1)) (c \sec (e+f x))^{m-4}}{f (2-m)}}{1-m}-\frac {a A \tan (e+f x) (a \sec (e+f x)+b)^2 (c \sec (e+f x))^{m-4}}{f (1-m)}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle c^4 \left (\frac {\frac {\int \left (c \csc \left (e+f x+\frac {\pi }{2}\right )\right )^{m-4} \left ((b B (1-m)+a A (4-m)) (2-m) b^2+a (2-m) \left (A (2-m) a^2+3 b B (1-m) a-2 A b^2 m\right ) \csc \left (e+f x+\frac {\pi }{2}\right )^2\right )dx+\frac {(1-m) \left (a^3 B (3-m)+3 a^2 A b (3-m)+3 a b^2 B (2-m)+A b^3 (2-m)\right ) \int \left (c \csc \left (e+f x+\frac {\pi }{2}\right )\right )^{m-3}dx}{c}}{2-m}-\frac {a^2 \tan (e+f x) \sec (e+f x) (a B (1-m)-A b (m+1)) (c \sec (e+f x))^{m-4}}{f (2-m)}}{1-m}-\frac {a A \tan (e+f x) (a \sec (e+f x)+b)^2 (c \sec (e+f x))^{m-4}}{f (1-m)}\right )\)

\(\Big \downarrow \) 4259

\(\displaystyle c^4 \left (\frac {\frac {\int \left (c \csc \left (e+f x+\frac {\pi }{2}\right )\right )^{m-4} \left ((b B (1-m)+a A (4-m)) (2-m) b^2+a (2-m) \left (A (2-m) a^2+3 b B (1-m) a-2 A b^2 m\right ) \csc \left (e+f x+\frac {\pi }{2}\right )^2\right )dx+\frac {(1-m) \left (a^3 B (3-m)+3 a^2 A b (3-m)+3 a b^2 B (2-m)+A b^3 (2-m)\right ) \left (\frac {\cos (e+f x)}{c}\right )^m (c \sec (e+f x))^m \int \left (\frac {\cos (e+f x)}{c}\right )^{3-m}dx}{c}}{2-m}-\frac {a^2 \tan (e+f x) \sec (e+f x) (a B (1-m)-A b (m+1)) (c \sec (e+f x))^{m-4}}{f (2-m)}}{1-m}-\frac {a A \tan (e+f x) (a \sec (e+f x)+b)^2 (c \sec (e+f x))^{m-4}}{f (1-m)}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle c^4 \left (\frac {\frac {\int \left (c \csc \left (e+f x+\frac {\pi }{2}\right )\right )^{m-4} \left ((b B (1-m)+a A (4-m)) (2-m) b^2+a (2-m) \left (A (2-m) a^2+3 b B (1-m) a-2 A b^2 m\right ) \csc \left (e+f x+\frac {\pi }{2}\right )^2\right )dx+\frac {(1-m) \left (a^3 B (3-m)+3 a^2 A b (3-m)+3 a b^2 B (2-m)+A b^3 (2-m)\right ) \left (\frac {\cos (e+f x)}{c}\right )^m (c \sec (e+f x))^m \int \left (\frac {\sin \left (e+f x+\frac {\pi }{2}\right )}{c}\right )^{3-m}dx}{c}}{2-m}-\frac {a^2 \tan (e+f x) \sec (e+f x) (a B (1-m)-A b (m+1)) (c \sec (e+f x))^{m-4}}{f (2-m)}}{1-m}-\frac {a A \tan (e+f x) (a \sec (e+f x)+b)^2 (c \sec (e+f x))^{m-4}}{f (1-m)}\right )\)

\(\Big \downarrow \) 3122

\(\displaystyle c^4 \left (\frac {\frac {\int \left (c \csc \left (e+f x+\frac {\pi }{2}\right )\right )^{m-4} \left ((b B (1-m)+a A (4-m)) (2-m) b^2+a (2-m) \left (A (2-m) a^2+3 b B (1-m) a-2 A b^2 m\right ) \csc \left (e+f x+\frac {\pi }{2}\right )^2\right )dx-\frac {(1-m) \sin (e+f x) \left (a^3 B (3-m)+3 a^2 A b (3-m)+3 a b^2 B (2-m)+A b^3 (2-m)\right ) (c \sec (e+f x))^{m-4} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {4-m}{2},\frac {6-m}{2},\cos ^2(e+f x)\right )}{f (4-m) \sqrt {\sin ^2(e+f x)}}}{2-m}-\frac {a^2 \tan (e+f x) \sec (e+f x) (a B (1-m)-A b (m+1)) (c \sec (e+f x))^{m-4}}{f (2-m)}}{1-m}-\frac {a A \tan (e+f x) (a \sec (e+f x)+b)^2 (c \sec (e+f x))^{m-4}}{f (1-m)}\right )\)

\(\Big \downarrow \) 4534

\(\displaystyle c^4 \left (\frac {\frac {\frac {(2-m) \left (a^3 A \left (m^2-6 m+8\right )+3 a^2 b B \left (m^2-5 m+4\right )+3 a A b^2 \left (m^2-5 m+4\right )+b^3 B \left (m^2-4 m+3\right )\right ) \int (c \sec (e+f x))^{m-4}dx}{3-m}-\frac {a (2-m) \tan (e+f x) \left (a^2 A (2-m)+3 a b B (1-m)-2 A b^2 m\right ) (c \sec (e+f x))^{m-4}}{f (3-m)}-\frac {(1-m) \sin (e+f x) \left (a^3 B (3-m)+3 a^2 A b (3-m)+3 a b^2 B (2-m)+A b^3 (2-m)\right ) (c \sec (e+f x))^{m-4} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {4-m}{2},\frac {6-m}{2},\cos ^2(e+f x)\right )}{f (4-m) \sqrt {\sin ^2(e+f x)}}}{2-m}-\frac {a^2 \tan (e+f x) \sec (e+f x) (a B (1-m)-A b (m+1)) (c \sec (e+f x))^{m-4}}{f (2-m)}}{1-m}-\frac {a A \tan (e+f x) (a \sec (e+f x)+b)^2 (c \sec (e+f x))^{m-4}}{f (1-m)}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle c^4 \left (\frac {\frac {\frac {(2-m) \left (a^3 A \left (m^2-6 m+8\right )+3 a^2 b B \left (m^2-5 m+4\right )+3 a A b^2 \left (m^2-5 m+4\right )+b^3 B \left (m^2-4 m+3\right )\right ) \int \left (c \csc \left (e+f x+\frac {\pi }{2}\right )\right )^{m-4}dx}{3-m}-\frac {a (2-m) \tan (e+f x) \left (a^2 A (2-m)+3 a b B (1-m)-2 A b^2 m\right ) (c \sec (e+f x))^{m-4}}{f (3-m)}-\frac {(1-m) \sin (e+f x) \left (a^3 B (3-m)+3 a^2 A b (3-m)+3 a b^2 B (2-m)+A b^3 (2-m)\right ) (c \sec (e+f x))^{m-4} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {4-m}{2},\frac {6-m}{2},\cos ^2(e+f x)\right )}{f (4-m) \sqrt {\sin ^2(e+f x)}}}{2-m}-\frac {a^2 \tan (e+f x) \sec (e+f x) (a B (1-m)-A b (m+1)) (c \sec (e+f x))^{m-4}}{f (2-m)}}{1-m}-\frac {a A \tan (e+f x) (a \sec (e+f x)+b)^2 (c \sec (e+f x))^{m-4}}{f (1-m)}\right )\)

\(\Big \downarrow \) 4259

\(\displaystyle c^4 \left (\frac {\frac {\frac {(2-m) \left (a^3 A \left (m^2-6 m+8\right )+3 a^2 b B \left (m^2-5 m+4\right )+3 a A b^2 \left (m^2-5 m+4\right )+b^3 B \left (m^2-4 m+3\right )\right ) \left (\frac {\cos (e+f x)}{c}\right )^m (c \sec (e+f x))^m \int \left (\frac {\cos (e+f x)}{c}\right )^{4-m}dx}{3-m}-\frac {a (2-m) \tan (e+f x) \left (a^2 A (2-m)+3 a b B (1-m)-2 A b^2 m\right ) (c \sec (e+f x))^{m-4}}{f (3-m)}-\frac {(1-m) \sin (e+f x) \left (a^3 B (3-m)+3 a^2 A b (3-m)+3 a b^2 B (2-m)+A b^3 (2-m)\right ) (c \sec (e+f x))^{m-4} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {4-m}{2},\frac {6-m}{2},\cos ^2(e+f x)\right )}{f (4-m) \sqrt {\sin ^2(e+f x)}}}{2-m}-\frac {a^2 \tan (e+f x) \sec (e+f x) (a B (1-m)-A b (m+1)) (c \sec (e+f x))^{m-4}}{f (2-m)}}{1-m}-\frac {a A \tan (e+f x) (a \sec (e+f x)+b)^2 (c \sec (e+f x))^{m-4}}{f (1-m)}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle c^4 \left (\frac {\frac {\frac {(2-m) \left (a^3 A \left (m^2-6 m+8\right )+3 a^2 b B \left (m^2-5 m+4\right )+3 a A b^2 \left (m^2-5 m+4\right )+b^3 B \left (m^2-4 m+3\right )\right ) \left (\frac {\cos (e+f x)}{c}\right )^m (c \sec (e+f x))^m \int \left (\frac {\sin \left (e+f x+\frac {\pi }{2}\right )}{c}\right )^{4-m}dx}{3-m}-\frac {a (2-m) \tan (e+f x) \left (a^2 A (2-m)+3 a b B (1-m)-2 A b^2 m\right ) (c \sec (e+f x))^{m-4}}{f (3-m)}-\frac {(1-m) \sin (e+f x) \left (a^3 B (3-m)+3 a^2 A b (3-m)+3 a b^2 B (2-m)+A b^3 (2-m)\right ) (c \sec (e+f x))^{m-4} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {4-m}{2},\frac {6-m}{2},\cos ^2(e+f x)\right )}{f (4-m) \sqrt {\sin ^2(e+f x)}}}{2-m}-\frac {a^2 \tan (e+f x) \sec (e+f x) (a B (1-m)-A b (m+1)) (c \sec (e+f x))^{m-4}}{f (2-m)}}{1-m}-\frac {a A \tan (e+f x) (a \sec (e+f x)+b)^2 (c \sec (e+f x))^{m-4}}{f (1-m)}\right )\)

\(\Big \downarrow \) 3122

\(\displaystyle c^4 \left (\frac {\frac {-\frac {a (2-m) \tan (e+f x) \left (a^2 A (2-m)+3 a b B (1-m)-2 A b^2 m\right ) (c \sec (e+f x))^{m-4}}{f (3-m)}-\frac {c (2-m) \sin (e+f x) \left (a^3 A \left (m^2-6 m+8\right )+3 a^2 b B \left (m^2-5 m+4\right )+3 a A b^2 \left (m^2-5 m+4\right )+b^3 B \left (m^2-4 m+3\right )\right ) (c \sec (e+f x))^{m-5} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {5-m}{2},\frac {7-m}{2},\cos ^2(e+f x)\right )}{f (3-m) (5-m) \sqrt {\sin ^2(e+f x)}}-\frac {(1-m) \sin (e+f x) \left (a^3 B (3-m)+3 a^2 A b (3-m)+3 a b^2 B (2-m)+A b^3 (2-m)\right ) (c \sec (e+f x))^{m-4} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {4-m}{2},\frac {6-m}{2},\cos ^2(e+f x)\right )}{f (4-m) \sqrt {\sin ^2(e+f x)}}}{2-m}-\frac {a^2 \tan (e+f x) \sec (e+f x) (a B (1-m)-A b (m+1)) (c \sec (e+f x))^{m-4}}{f (2-m)}}{1-m}-\frac {a A \tan (e+f x) (a \sec (e+f x)+b)^2 (c \sec (e+f x))^{m-4}}{f (1-m)}\right )\)

input

Int[(a + b*Cos[e + f*x])^3*(A + B*Cos[e + f*x])*(c*Sec[e + f*x])^m,x]
 

output

c^4*(-((a*A*(c*Sec[e + f*x])^(-4 + m)*(b + a*Sec[e + f*x])^2*Tan[e + f*x]) 
/(f*(1 - m))) + (-((a^2*(a*B*(1 - m) - A*b*(1 + m))*Sec[e + f*x]*(c*Sec[e 
+ f*x])^(-4 + m)*Tan[e + f*x])/(f*(2 - m))) + (-((c*(2 - m)*(a^3*A*(8 - 6* 
m + m^2) + 3*a*A*b^2*(4 - 5*m + m^2) + 3*a^2*b*B*(4 - 5*m + m^2) + b^3*B*( 
3 - 4*m + m^2))*Hypergeometric2F1[1/2, (5 - m)/2, (7 - m)/2, Cos[e + f*x]^ 
2]*(c*Sec[e + f*x])^(-5 + m)*Sin[e + f*x])/(f*(3 - m)*(5 - m)*Sqrt[Sin[e + 
 f*x]^2])) - ((A*b^3*(2 - m) + 3*a*b^2*B*(2 - m) + 3*a^2*A*b*(3 - m) + a^3 
*B*(3 - m))*(1 - m)*Hypergeometric2F1[1/2, (4 - m)/2, (6 - m)/2, Cos[e + f 
*x]^2]*(c*Sec[e + f*x])^(-4 + m)*Sin[e + f*x])/(f*(4 - m)*Sqrt[Sin[e + f*x 
]^2]) - (a*(2 - m)*(3*a*b*B*(1 - m) + a^2*A*(2 - m) - 2*A*b^2*m)*(c*Sec[e 
+ f*x])^(-4 + m)*Tan[e + f*x])/(f*(3 - m)))/(2 - m))/(1 - m))
 

3.7.37.3.1 Defintions of rubi rules used

rule 25

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

rule 3042

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

rule 3122

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[Cos[c + d*x]*(( 
b*Sin[c + d*x])^(n + 1)/(b*d*(n + 1)*Sqrt[Cos[c + d*x]^2]))*Hypergeometric2 
F1[1/2, (n + 1)/2, (n + 3)/2, Sin[c + d*x]^2], x] /; FreeQ[{b, c, d, n}, x] 
 &&  !IntegerQ[2*n]
 

rule 3439

Int[(csc[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*((a_.) + (b_.)*sin[(e_.) + (f_.)* 
(x_)])^(m_.)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Sim 
p[g^(m + n)   Int[(g*Csc[e + f*x])^(p - m - n)*(b + a*Csc[e + f*x])^m*(d + 
c*Csc[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[b*c 
- a*d, 0] &&  !IntegerQ[p] && IntegerQ[m] && IntegerQ[n]
 

rule 4259

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(b*Csc[c + d*x] 
)^(n - 1)*((Sin[c + d*x]/b)^(n - 1)   Int[1/(Sin[c + d*x]/b)^n, x]), x] /; 
FreeQ[{b, c, d, n}, x] &&  !IntegerQ[n]
 

rule 4514

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

rule 4534

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(m_.)*(csc[(e_.) + (f_.)*(x_)]^2*(C_.) 
+ (A_)), x_Symbol] :> Simp[(-C)*Cot[e + f*x]*((b*Csc[e + f*x])^m/(f*(m + 1) 
)), x] + Simp[(C*m + A*(m + 1))/(m + 1)   Int[(b*Csc[e + f*x])^m, x], x] /; 
 FreeQ[{b, e, f, A, C, m}, x] && NeQ[C*m + A*(m + 1), 0] &&  !LeQ[m, -1]
 

rule 4535

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(m_.)*((A_.) + csc[(e_.) + (f_.)*(x_)]* 
(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.)), x_Symbol] :> Simp[B/b   Int[(b*Cs 
c[e + f*x])^(m + 1), x], x] + Int[(b*Csc[e + f*x])^m*(A + C*Csc[e + f*x]^2) 
, x] /; FreeQ[{b, e, f, A, B, C, m}, x]
 

rule 4564

Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. 
))*(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + ( 
a_)), x_Symbol] :> Simp[(-b)*C*Csc[e + f*x]*Cot[e + f*x]*((d*Csc[e + f*x])^ 
n/(f*(n + 2))), x] + Simp[1/(n + 2)   Int[(d*Csc[e + f*x])^n*Simp[A*a*(n + 
2) + (B*a*(n + 2) + b*(C*(n + 1) + A*(n + 2)))*Csc[e + f*x] + (a*C + B*b)*( 
n + 2)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, B, C, n}, x] 
&&  !LtQ[n, -1]
 

3.7.37.4 Maple [F]

\[\int \left (a +b \cos \left (f x +e \right )\right )^{3} \left (A +\cos \left (f x +e \right ) B \right ) \left (c \sec \left (f x +e \right )\right )^{m}d x\]

input

int((a+b*cos(f*x+e))^3*(A+cos(f*x+e)*B)*(c*sec(f*x+e))^m,x)
 

output

int((a+b*cos(f*x+e))^3*(A+cos(f*x+e)*B)*(c*sec(f*x+e))^m,x)
 

3.7.37.5 Fricas [F]

\[ \int (a+b \cos (e+f x))^3 (A+B \cos (e+f x)) (c \sec (e+f x))^m \, dx=\int { {\left (B \cos \left (f x + e\right ) + A\right )} {\left (b \cos \left (f x + e\right ) + a\right )}^{3} \left (c \sec \left (f x + e\right )\right )^{m} \,d x } \]

input

integrate((a+b*cos(f*x+e))^3*(A+B*cos(f*x+e))*(c*sec(f*x+e))^m,x, algorith 
m="fricas")
 

output

integral((B*b^3*cos(f*x + e)^4 + A*a^3 + (3*B*a*b^2 + A*b^3)*cos(f*x + e)^ 
3 + 3*(B*a^2*b + A*a*b^2)*cos(f*x + e)^2 + (B*a^3 + 3*A*a^2*b)*cos(f*x + e 
))*(c*sec(f*x + e))^m, x)
 

3.7.37.6 Sympy [F]

\[ \int (a+b \cos (e+f x))^3 (A+B \cos (e+f x)) (c \sec (e+f x))^m \, dx=\int \left (c \sec {\left (e + f x \right )}\right )^{m} \left (A + B \cos {\left (e + f x \right )}\right ) \left (a + b \cos {\left (e + f x \right )}\right )^{3}\, dx \]

input

integrate((a+b*cos(f*x+e))**3*(A+B*cos(f*x+e))*(c*sec(f*x+e))**m,x)
 

output

Integral((c*sec(e + f*x))**m*(A + B*cos(e + f*x))*(a + b*cos(e + f*x))**3, 
 x)
 

3.7.37.7 Maxima [F]

\[ \int (a+b \cos (e+f x))^3 (A+B \cos (e+f x)) (c \sec (e+f x))^m \, dx=\int { {\left (B \cos \left (f x + e\right ) + A\right )} {\left (b \cos \left (f x + e\right ) + a\right )}^{3} \left (c \sec \left (f x + e\right )\right )^{m} \,d x } \]

input

integrate((a+b*cos(f*x+e))^3*(A+B*cos(f*x+e))*(c*sec(f*x+e))^m,x, algorith 
m="maxima")
 

output

integrate((B*cos(f*x + e) + A)*(b*cos(f*x + e) + a)^3*(c*sec(f*x + e))^m, 
x)
 

3.7.37.8 Giac [F]

\[ \int (a+b \cos (e+f x))^3 (A+B \cos (e+f x)) (c \sec (e+f x))^m \, dx=\int { {\left (B \cos \left (f x + e\right ) + A\right )} {\left (b \cos \left (f x + e\right ) + a\right )}^{3} \left (c \sec \left (f x + e\right )\right )^{m} \,d x } \]

input

integrate((a+b*cos(f*x+e))^3*(A+B*cos(f*x+e))*(c*sec(f*x+e))^m,x, algorith 
m="giac")
 

output

integrate((B*cos(f*x + e) + A)*(b*cos(f*x + e) + a)^3*(c*sec(f*x + e))^m, 
x)
 

3.7.37.9 Mupad [F(-1)]

Timed out. \[ \int (a+b \cos (e+f x))^3 (A+B \cos (e+f x)) (c \sec (e+f x))^m \, dx=\int {\left (\frac {c}{\cos \left (e+f\,x\right )}\right )}^m\,\left (A+B\,\cos \left (e+f\,x\right )\right )\,{\left (a+b\,\cos \left (e+f\,x\right )\right )}^3 \,d x \]

input

int((c/cos(e + f*x))^m*(A + B*cos(e + f*x))*(a + b*cos(e + f*x))^3,x)
 

output

int((c/cos(e + f*x))^m*(A + B*cos(e + f*x))*(a + b*cos(e + f*x))^3, x)