\(\int (c \cos (e+f x))^m (a+b \cos (e+f x))^4 (A+B \cos (e+f x)) \, dx\) [450]

Optimal result

Integrand size = 33, antiderivative size = 595 \[ \int (c \cos (e+f x))^m (a+b \cos (e+f x))^4 (A+B \cos (e+f x)) \, dx=\frac {b \left (A b^3 \left (15+8 m+m^2\right )+4 a b^2 B \left (15+8 m+m^2\right )+2 a^3 B \left (28+10 m+m^2\right )+a^2 A b \left (110+47 m+5 m^2\right )\right ) (c \cos (e+f x))^{1+m} \sin (e+f x)}{c f (2+m) (4+m) (5+m)}+\frac {b^2 \left (b^2 B (4+m)^2+2 a A b (5+m)^2+a^2 B \left (36+11 m+m^2\right )\right ) \cos (e+f x) (c \cos (e+f x))^{1+m} \sin (e+f x)}{c f (3+m) (4+m) (5+m)}+\frac {b (A b (5+m)+a B (8+m)) (c \cos (e+f x))^{1+m} (a+b \cos (e+f x))^2 \sin (e+f x)}{c f (4+m) (5+m)}+\frac {b B (c \cos (e+f x))^{1+m} (a+b \cos (e+f x))^3 \sin (e+f x)}{c f (5+m)}-\frac {\left (A b^4 \left (3+4 m+m^2\right )+4 a b^3 B \left (3+4 m+m^2\right )+6 a^2 A b^2 \left (4+5 m+m^2\right )+4 a^3 b B \left (4+5 m+m^2\right )+a^4 A \left (8+6 m+m^2\right )\right ) (c \cos (e+f x))^{1+m} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},\cos ^2(e+f x)\right ) \sin (e+f x)}{c f (1+m) (2+m) (4+m) \sqrt {\sin ^2(e+f x)}}-\frac {\left (b^4 B \left (8+6 m+m^2\right )+4 a A b^3 \left (10+7 m+m^2\right )+6 a^2 b^2 B \left (10+7 m+m^2\right )+4 a^3 A b \left (15+8 m+m^2\right )+a^4 B \left (15+8 m+m^2\right )\right ) (c \cos (e+f x))^{2+m} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},\cos ^2(e+f x)\right ) \sin (e+f x)}{c^2 f (2+m) (3+m) (5+m) \sqrt {\sin ^2(e+f x)}} \] Output:

b*(A*b^3*(m^2+8*m+15)+4*a*b^2*B*(m^2+8*m+15)+2*a^3*B*(m^2+10*m+28)+a^2*A*b 
*(5*m^2+47*m+110))*(c*cos(f*x+e))^(1+m)*sin(f*x+e)/c/f/(2+m)/(4+m)/(5+m)+b 
^2*(b^2*B*(4+m)^2+2*a*A*b*(5+m)^2+a^2*B*(m^2+11*m+36))*cos(f*x+e)*(c*cos(f 
*x+e))^(1+m)*sin(f*x+e)/c/f/(3+m)/(4+m)/(5+m)+b*(A*b*(5+m)+a*B*(8+m))*(c*c 
os(f*x+e))^(1+m)*(a+b*cos(f*x+e))^2*sin(f*x+e)/c/f/(4+m)/(5+m)+b*B*(c*cos( 
f*x+e))^(1+m)*(a+b*cos(f*x+e))^3*sin(f*x+e)/c/f/(5+m)-(A*b^4*(m^2+4*m+3)+4 
*a*b^3*B*(m^2+4*m+3)+6*a^2*A*b^2*(m^2+5*m+4)+4*a^3*b*B*(m^2+5*m+4)+a^4*A*( 
m^2+6*m+8))*(c*cos(f*x+e))^(1+m)*hypergeom([1/2, 1/2+1/2*m],[3/2+1/2*m],co 
s(f*x+e)^2)*sin(f*x+e)/c/f/(1+m)/(2+m)/(4+m)/(sin(f*x+e)^2)^(1/2)-(b^4*B*( 
m^2+6*m+8)+4*a*A*b^3*(m^2+7*m+10)+6*a^2*b^2*B*(m^2+7*m+10)+4*a^3*A*b*(m^2+ 
8*m+15)+a^4*B*(m^2+8*m+15))*(c*cos(f*x+e))^(2+m)*hypergeom([1/2, 1+1/2*m], 
[2+1/2*m],cos(f*x+e)^2)*sin(f*x+e)/c^2/f/(2+m)/(3+m)/(5+m)/(sin(f*x+e)^2)^ 
(1/2)
 

Mathematica [A] (verified)

Time = 6.24 (sec) , antiderivative size = 479, normalized size of antiderivative = 0.81 \[ \int (c \cos (e+f x))^m (a+b \cos (e+f x))^4 (A+B \cos (e+f x)) \, dx=-\frac {a^4 A (c \cos (e+f x))^m \cot (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},\cos ^2(e+f x)\right ) \sqrt {\sin ^2(e+f x)}}{f (1+m)}-\frac {a^3 (4 A b+a B) \cos (e+f x) (c \cos (e+f x))^m \cot (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},\cos ^2(e+f x)\right ) \sqrt {\sin ^2(e+f x)}}{f (2+m)}-\frac {2 a^2 b (3 A b+2 a B) \cos ^2(e+f x) (c \cos (e+f x))^m \cot (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3+m}{2},\frac {5+m}{2},\cos ^2(e+f x)\right ) \sqrt {\sin ^2(e+f x)}}{f (3+m)}-\frac {2 a b^2 (2 A b+3 a B) \cos ^3(e+f x) (c \cos (e+f x))^m \cot (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {4+m}{2},\frac {6+m}{2},\cos ^2(e+f x)\right ) \sqrt {\sin ^2(e+f x)}}{f (4+m)}-\frac {b^3 (A b+4 a B) \cos ^4(e+f x) (c \cos (e+f x))^m \cot (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {5+m}{2},\frac {7+m}{2},\cos ^2(e+f x)\right ) \sqrt {\sin ^2(e+f x)}}{f (5+m)}-\frac {b^4 B \cos ^5(e+f x) (c \cos (e+f x))^m \cot (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {6+m}{2},\frac {8+m}{2},\cos ^2(e+f x)\right ) \sqrt {\sin ^2(e+f x)}}{f (6+m)} \] Input:

Integrate[(c*Cos[e + f*x])^m*(a + b*Cos[e + f*x])^4*(A + B*Cos[e + f*x]),x 
]
 

Output:

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

Rubi [A] (verified)

Time = 3.01 (sec) , antiderivative size = 614, normalized size of antiderivative = 1.03, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {3042, 3469, 3042, 3528, 3042, 3512, 3042, 3502, 3042, 3227, 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.

\(\displaystyle \int (a+b \cos (e+f x))^4 (A+B \cos (e+f x)) (c \cos (e+f x))^m \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \left (a+b \sin \left (e+f x+\frac {\pi }{2}\right )\right )^4 \left (A+B \sin \left (e+f x+\frac {\pi }{2}\right )\right ) \left (c \sin \left (e+f x+\frac {\pi }{2}\right )\right )^mdx\)

\(\Big \downarrow \) 3469

\(\displaystyle \frac {\int (c \cos (e+f x))^m (a+b \cos (e+f x))^2 \left (b c (A b (m+5)+a B (m+8)) \cos ^2(e+f x)+c \left (B (m+4) b^2+a (2 A b+a B) (m+5)\right ) \cos (e+f x)+a c (b B (m+1)+a A (m+5))\right )dx}{c (m+5)}+\frac {b B \sin (e+f x) (a+b \cos (e+f x))^3 (c \cos (e+f x))^{m+1}}{c f (m+5)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \left (c \sin \left (e+f x+\frac {\pi }{2}\right )\right )^m \left (a+b \sin \left (e+f x+\frac {\pi }{2}\right )\right )^2 \left (b c (A b (m+5)+a B (m+8)) \sin \left (e+f x+\frac {\pi }{2}\right )^2+c \left (B (m+4) b^2+a (2 A b+a B) (m+5)\right ) \sin \left (e+f x+\frac {\pi }{2}\right )+a c (b B (m+1)+a A (m+5))\right )dx}{c (m+5)}+\frac {b B \sin (e+f x) (a+b \cos (e+f x))^3 (c \cos (e+f x))^{m+1}}{c f (m+5)}\)

\(\Big \downarrow \) 3528

\(\displaystyle \frac {\frac {\int (c \cos (e+f x))^m (a+b \cos (e+f x)) \left (b \left (B \left (m^2+11 m+36\right ) a^2+2 A b (m+5)^2 a+b^2 B (m+4)^2\right ) \cos ^2(e+f x) c^2+a (a (m+4) (b B (m+1)+a A (m+5))+b (m+1) (A b (m+5)+a B (m+8))) c^2+\left ((m+3) (A b (m+5)+a B (m+8)) b^2+a (m+4) \left (B (m+5) a^2+3 A b (m+5) a+b^2 B (2 m+5)\right )\right ) \cos (e+f x) c^2\right )dx}{c (m+4)}+\frac {b \sin (e+f x) (a B (m+8)+A b (m+5)) (a+b \cos (e+f x))^2 (c \cos (e+f x))^{m+1}}{f (m+4)}}{c (m+5)}+\frac {b B \sin (e+f x) (a+b \cos (e+f x))^3 (c \cos (e+f x))^{m+1}}{c f (m+5)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\int \left (c \sin \left (e+f x+\frac {\pi }{2}\right )\right )^m \left (a+b \sin \left (e+f x+\frac {\pi }{2}\right )\right ) \left (b \left (B \left (m^2+11 m+36\right ) a^2+2 A b (m+5)^2 a+b^2 B (m+4)^2\right ) \sin \left (e+f x+\frac {\pi }{2}\right )^2 c^2+a (a (m+4) (b B (m+1)+a A (m+5))+b (m+1) (A b (m+5)+a B (m+8))) c^2+\left ((m+3) (A b (m+5)+a B (m+8)) b^2+a (m+4) \left (B (m+5) a^2+3 A b (m+5) a+b^2 B (2 m+5)\right )\right ) \sin \left (e+f x+\frac {\pi }{2}\right ) c^2\right )dx}{c (m+4)}+\frac {b \sin (e+f x) (a B (m+8)+A b (m+5)) (a+b \cos (e+f x))^2 (c \cos (e+f x))^{m+1}}{f (m+4)}}{c (m+5)}+\frac {b B \sin (e+f x) (a+b \cos (e+f x))^3 (c \cos (e+f x))^{m+1}}{c f (m+5)}\)

\(\Big \downarrow \) 3512

\(\displaystyle \frac {\frac {\frac {\int (c \cos (e+f x))^m \left (b (m+3) \left (2 B \left (m^2+10 m+28\right ) a^3+A b \left (5 m^2+47 m+110\right ) a^2+4 b^2 B \left (m^2+8 m+15\right ) a+A b^3 \left (m^2+8 m+15\right )\right ) \cos ^2(e+f x) c^3+a^2 (m+3) (a (m+4) (b B (m+1)+a A (m+5))+b (m+1) (A b (m+5)+a B (m+8))) c^3+(m+4) \left (B \left (m^2+8 m+15\right ) a^4+4 A b \left (m^2+8 m+15\right ) a^3+6 b^2 B \left (m^2+7 m+10\right ) a^2+4 A b^3 \left (m^2+7 m+10\right ) a+b^4 B \left (m^2+6 m+8\right )\right ) \cos (e+f x) c^3\right )dx}{c (m+3)}+\frac {b^2 c \sin (e+f x) \cos (e+f x) \left (a^2 B \left (m^2+11 m+36\right )+2 a A b (m+5)^2+b^2 B (m+4)^2\right ) (c \cos (e+f x))^{m+1}}{f (m+3)}}{c (m+4)}+\frac {b \sin (e+f x) (a B (m+8)+A b (m+5)) (a+b \cos (e+f x))^2 (c \cos (e+f x))^{m+1}}{f (m+4)}}{c (m+5)}+\frac {b B \sin (e+f x) (a+b \cos (e+f x))^3 (c \cos (e+f x))^{m+1}}{c f (m+5)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {\int \left (c \sin \left (e+f x+\frac {\pi }{2}\right )\right )^m \left (b (m+3) \left (2 B \left (m^2+10 m+28\right ) a^3+A b \left (5 m^2+47 m+110\right ) a^2+4 b^2 B \left (m^2+8 m+15\right ) a+A b^3 \left (m^2+8 m+15\right )\right ) \sin \left (e+f x+\frac {\pi }{2}\right )^2 c^3+a^2 (m+3) (a (m+4) (b B (m+1)+a A (m+5))+b (m+1) (A b (m+5)+a B (m+8))) c^3+(m+4) \left (B \left (m^2+8 m+15\right ) a^4+4 A b \left (m^2+8 m+15\right ) a^3+6 b^2 B \left (m^2+7 m+10\right ) a^2+4 A b^3 \left (m^2+7 m+10\right ) a+b^4 B \left (m^2+6 m+8\right )\right ) \sin \left (e+f x+\frac {\pi }{2}\right ) c^3\right )dx}{c (m+3)}+\frac {b^2 c \sin (e+f x) \cos (e+f x) \left (a^2 B \left (m^2+11 m+36\right )+2 a A b (m+5)^2+b^2 B (m+4)^2\right ) (c \cos (e+f x))^{m+1}}{f (m+3)}}{c (m+4)}+\frac {b \sin (e+f x) (a B (m+8)+A b (m+5)) (a+b \cos (e+f x))^2 (c \cos (e+f x))^{m+1}}{f (m+4)}}{c (m+5)}+\frac {b B \sin (e+f x) (a+b \cos (e+f x))^3 (c \cos (e+f x))^{m+1}}{c f (m+5)}\)

\(\Big \downarrow \) 3502

\(\displaystyle \frac {\frac {\frac {\frac {\int (c \cos (e+f x))^m \left ((m+3) \left (A \left (m^3+11 m^2+38 m+40\right ) a^4+4 b B \left (m^3+10 m^2+29 m+20\right ) a^3+6 A b^2 \left (m^3+10 m^2+29 m+20\right ) a^2+4 b^3 B \left (m^3+9 m^2+23 m+15\right ) a+A b^4 \left (m^3+9 m^2+23 m+15\right )\right ) c^4+(m+2) (m+4) \left (B \left (m^2+8 m+15\right ) a^4+4 A b \left (m^2+8 m+15\right ) a^3+6 b^2 B \left (m^2+7 m+10\right ) a^2+4 A b^3 \left (m^2+7 m+10\right ) a+b^4 B \left (m^2+6 m+8\right )\right ) \cos (e+f x) c^4\right )dx}{c (m+2)}+\frac {b c^2 (m+3) \sin (e+f x) \left (2 a^3 B \left (m^2+10 m+28\right )+a^2 A b \left (5 m^2+47 m+110\right )+4 a b^2 B \left (m^2+8 m+15\right )+A b^3 \left (m^2+8 m+15\right )\right ) (c \cos (e+f x))^{m+1}}{f (m+2)}}{c (m+3)}+\frac {b^2 c \sin (e+f x) \cos (e+f x) \left (a^2 B \left (m^2+11 m+36\right )+2 a A b (m+5)^2+b^2 B (m+4)^2\right ) (c \cos (e+f x))^{m+1}}{f (m+3)}}{c (m+4)}+\frac {b \sin (e+f x) (a B (m+8)+A b (m+5)) (a+b \cos (e+f x))^2 (c \cos (e+f x))^{m+1}}{f (m+4)}}{c (m+5)}+\frac {b B \sin (e+f x) (a+b \cos (e+f x))^3 (c \cos (e+f x))^{m+1}}{c f (m+5)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {\frac {\int \left (c \sin \left (e+f x+\frac {\pi }{2}\right )\right )^m \left ((m+3) \left (A \left (m^3+11 m^2+38 m+40\right ) a^4+4 b B \left (m^3+10 m^2+29 m+20\right ) a^3+6 A b^2 \left (m^3+10 m^2+29 m+20\right ) a^2+4 b^3 B \left (m^3+9 m^2+23 m+15\right ) a+A b^4 \left (m^3+9 m^2+23 m+15\right )\right ) c^4+(m+2) (m+4) \left (B \left (m^2+8 m+15\right ) a^4+4 A b \left (m^2+8 m+15\right ) a^3+6 b^2 B \left (m^2+7 m+10\right ) a^2+4 A b^3 \left (m^2+7 m+10\right ) a+b^4 B \left (m^2+6 m+8\right )\right ) \sin \left (e+f x+\frac {\pi }{2}\right ) c^4\right )dx}{c (m+2)}+\frac {b c^2 (m+3) \sin (e+f x) \left (2 a^3 B \left (m^2+10 m+28\right )+a^2 A b \left (5 m^2+47 m+110\right )+4 a b^2 B \left (m^2+8 m+15\right )+A b^3 \left (m^2+8 m+15\right )\right ) (c \cos (e+f x))^{m+1}}{f (m+2)}}{c (m+3)}+\frac {b^2 c \sin (e+f x) \cos (e+f x) \left (a^2 B \left (m^2+11 m+36\right )+2 a A b (m+5)^2+b^2 B (m+4)^2\right ) (c \cos (e+f x))^{m+1}}{f (m+3)}}{c (m+4)}+\frac {b \sin (e+f x) (a B (m+8)+A b (m+5)) (a+b \cos (e+f x))^2 (c \cos (e+f x))^{m+1}}{f (m+4)}}{c (m+5)}+\frac {b B \sin (e+f x) (a+b \cos (e+f x))^3 (c \cos (e+f x))^{m+1}}{c f (m+5)}\)

\(\Big \downarrow \) 3227

\(\displaystyle \frac {\frac {\frac {\frac {c^4 (m+3) \left (a^4 A \left (m^3+11 m^2+38 m+40\right )+4 a^3 b B \left (m^3+10 m^2+29 m+20\right )+6 a^2 A b^2 \left (m^3+10 m^2+29 m+20\right )+4 a b^3 B \left (m^3+9 m^2+23 m+15\right )+A b^4 \left (m^3+9 m^2+23 m+15\right )\right ) \int (c \cos (e+f x))^mdx+c^3 (m+2) (m+4) \left (a^4 B \left (m^2+8 m+15\right )+4 a^3 A b \left (m^2+8 m+15\right )+6 a^2 b^2 B \left (m^2+7 m+10\right )+4 a A b^3 \left (m^2+7 m+10\right )+b^4 B \left (m^2+6 m+8\right )\right ) \int (c \cos (e+f x))^{m+1}dx}{c (m+2)}+\frac {b c^2 (m+3) \sin (e+f x) \left (2 a^3 B \left (m^2+10 m+28\right )+a^2 A b \left (5 m^2+47 m+110\right )+4 a b^2 B \left (m^2+8 m+15\right )+A b^3 \left (m^2+8 m+15\right )\right ) (c \cos (e+f x))^{m+1}}{f (m+2)}}{c (m+3)}+\frac {b^2 c \sin (e+f x) \cos (e+f x) \left (a^2 B \left (m^2+11 m+36\right )+2 a A b (m+5)^2+b^2 B (m+4)^2\right ) (c \cos (e+f x))^{m+1}}{f (m+3)}}{c (m+4)}+\frac {b \sin (e+f x) (a B (m+8)+A b (m+5)) (a+b \cos (e+f x))^2 (c \cos (e+f x))^{m+1}}{f (m+4)}}{c (m+5)}+\frac {b B \sin (e+f x) (a+b \cos (e+f x))^3 (c \cos (e+f x))^{m+1}}{c f (m+5)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {\frac {c^4 (m+3) \left (a^4 A \left (m^3+11 m^2+38 m+40\right )+4 a^3 b B \left (m^3+10 m^2+29 m+20\right )+6 a^2 A b^2 \left (m^3+10 m^2+29 m+20\right )+4 a b^3 B \left (m^3+9 m^2+23 m+15\right )+A b^4 \left (m^3+9 m^2+23 m+15\right )\right ) \int \left (c \sin \left (e+f x+\frac {\pi }{2}\right )\right )^mdx+c^3 (m+2) (m+4) \left (a^4 B \left (m^2+8 m+15\right )+4 a^3 A b \left (m^2+8 m+15\right )+6 a^2 b^2 B \left (m^2+7 m+10\right )+4 a A b^3 \left (m^2+7 m+10\right )+b^4 B \left (m^2+6 m+8\right )\right ) \int \left (c \sin \left (e+f x+\frac {\pi }{2}\right )\right )^{m+1}dx}{c (m+2)}+\frac {b c^2 (m+3) \sin (e+f x) \left (2 a^3 B \left (m^2+10 m+28\right )+a^2 A b \left (5 m^2+47 m+110\right )+4 a b^2 B \left (m^2+8 m+15\right )+A b^3 \left (m^2+8 m+15\right )\right ) (c \cos (e+f x))^{m+1}}{f (m+2)}}{c (m+3)}+\frac {b^2 c \sin (e+f x) \cos (e+f x) \left (a^2 B \left (m^2+11 m+36\right )+2 a A b (m+5)^2+b^2 B (m+4)^2\right ) (c \cos (e+f x))^{m+1}}{f (m+3)}}{c (m+4)}+\frac {b \sin (e+f x) (a B (m+8)+A b (m+5)) (a+b \cos (e+f x))^2 (c \cos (e+f x))^{m+1}}{f (m+4)}}{c (m+5)}+\frac {b B \sin (e+f x) (a+b \cos (e+f x))^3 (c \cos (e+f x))^{m+1}}{c f (m+5)}\)

\(\Big \downarrow \) 3122

\(\displaystyle \frac {\frac {\frac {b^2 c \sin (e+f x) \cos (e+f x) \left (a^2 B \left (m^2+11 m+36\right )+2 a A b (m+5)^2+b^2 B (m+4)^2\right ) (c \cos (e+f x))^{m+1}}{f (m+3)}+\frac {\frac {b c^2 (m+3) \sin (e+f x) \left (2 a^3 B \left (m^2+10 m+28\right )+a^2 A b \left (5 m^2+47 m+110\right )+4 a b^2 B \left (m^2+8 m+15\right )+A b^3 \left (m^2+8 m+15\right )\right ) (c \cos (e+f x))^{m+1}}{f (m+2)}+\frac {-\frac {c^3 (m+3) \sin (e+f x) \left (a^4 A \left (m^3+11 m^2+38 m+40\right )+4 a^3 b B \left (m^3+10 m^2+29 m+20\right )+6 a^2 A b^2 \left (m^3+10 m^2+29 m+20\right )+4 a b^3 B \left (m^3+9 m^2+23 m+15\right )+A b^4 \left (m^3+9 m^2+23 m+15\right )\right ) (c \cos (e+f x))^{m+1} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{2},\frac {m+3}{2},\cos ^2(e+f x)\right )}{f (m+1) \sqrt {\sin ^2(e+f x)}}-\frac {c^2 (m+4) \sin (e+f x) \left (a^4 B \left (m^2+8 m+15\right )+4 a^3 A b \left (m^2+8 m+15\right )+6 a^2 b^2 B \left (m^2+7 m+10\right )+4 a A b^3 \left (m^2+7 m+10\right )+b^4 B \left (m^2+6 m+8\right )\right ) (c \cos (e+f x))^{m+2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+2}{2},\frac {m+4}{2},\cos ^2(e+f x)\right )}{f \sqrt {\sin ^2(e+f x)}}}{c (m+2)}}{c (m+3)}}{c (m+4)}+\frac {b \sin (e+f x) (a B (m+8)+A b (m+5)) (a+b \cos (e+f x))^2 (c \cos (e+f x))^{m+1}}{f (m+4)}}{c (m+5)}+\frac {b B \sin (e+f x) (a+b \cos (e+f x))^3 (c \cos (e+f x))^{m+1}}{c f (m+5)}\)

Input:

Int[(c*Cos[e + f*x])^m*(a + b*Cos[e + f*x])^4*(A + B*Cos[e + f*x]),x]
 

Output:

(b*B*(c*Cos[e + f*x])^(1 + m)*(a + b*Cos[e + f*x])^3*Sin[e + f*x])/(c*f*(5 
 + m)) + ((b*(A*b*(5 + m) + a*B*(8 + m))*(c*Cos[e + f*x])^(1 + m)*(a + b*C 
os[e + f*x])^2*Sin[e + f*x])/(f*(4 + m)) + ((b^2*c*(b^2*B*(4 + m)^2 + 2*a* 
A*b*(5 + m)^2 + a^2*B*(36 + 11*m + m^2))*Cos[e + f*x]*(c*Cos[e + f*x])^(1 
+ m)*Sin[e + f*x])/(f*(3 + m)) + ((b*c^2*(3 + m)*(A*b^3*(15 + 8*m + m^2) + 
 4*a*b^2*B*(15 + 8*m + m^2) + 2*a^3*B*(28 + 10*m + m^2) + a^2*A*b*(110 + 4 
7*m + 5*m^2))*(c*Cos[e + f*x])^(1 + m)*Sin[e + f*x])/(f*(2 + m)) + (-((c^3 
*(3 + m)*(A*b^4*(15 + 23*m + 9*m^2 + m^3) + 4*a*b^3*B*(15 + 23*m + 9*m^2 + 
 m^3) + 6*a^2*A*b^2*(20 + 29*m + 10*m^2 + m^3) + 4*a^3*b*B*(20 + 29*m + 10 
*m^2 + m^3) + a^4*A*(40 + 38*m + 11*m^2 + m^3))*(c*Cos[e + f*x])^(1 + m)*H 
ypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Cos[e + f*x]^2]*Sin[e + f*x])/ 
(f*(1 + m)*Sqrt[Sin[e + f*x]^2])) - (c^2*(4 + m)*(b^4*B*(8 + 6*m + m^2) + 
4*a*A*b^3*(10 + 7*m + m^2) + 6*a^2*b^2*B*(10 + 7*m + m^2) + 4*a^3*A*b*(15 
+ 8*m + m^2) + a^4*B*(15 + 8*m + m^2))*(c*Cos[e + f*x])^(2 + m)*Hypergeome 
tric2F1[1/2, (2 + m)/2, (4 + m)/2, Cos[e + f*x]^2]*Sin[e + f*x])/(f*Sqrt[S 
in[e + f*x]^2]))/(c*(2 + m)))/(c*(3 + m)))/(c*(4 + m)))/(c*(5 + m))
 

Defintions of rubi rules used

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 3227

Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x 
_)]), x_Symbol] :> Simp[c   Int[(b*Sin[e + f*x])^m, x], x] + Simp[d/b   Int 
[(b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x]
 

rule 3469

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

rule 3502

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) 
+ (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Co 
s[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2))), x] + Simp[1/(b*(m 
+ 2))   Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m 
 + 2) - a*C)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] 
 &&  !LtQ[m, -1]
 

rule 3512

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_) + (d_.)*sin[(e_.) + 
 (f_.)*(x_)])*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f 
_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*d*Cos[e + f*x]*Sin[e + f*x]*((a + b*Si 
n[e + f*x])^(m + 1)/(b*f*(m + 3))), x] + Simp[1/(b*(m + 3))   Int[(a + b*Si 
n[e + f*x])^m*Simp[a*C*d + A*b*c*(m + 3) + b*(B*c*(m + 3) + d*(C*(m + 2) + 
A*(m + 3)))*Sin[e + f*x] - (2*a*C*d - b*(c*C + B*d)*(m + 3))*Sin[e + f*x]^2 
, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m}, x] && NeQ[b*c - a*d, 
0] && NeQ[a^2 - b^2, 0] &&  !LtQ[m, -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])))
 

Maple [F]

\[\int \left (c \cos \left (f x +e \right )\right )^{m} \left (a +\cos \left (f x +e \right ) b \right )^{4} \left (A +B \cos \left (f x +e \right )\right )d x\]

Input:

int((c*cos(f*x+e))^m*(a+cos(f*x+e)*b)^4*(A+B*cos(f*x+e)),x)
 

Output:

int((c*cos(f*x+e))^m*(a+cos(f*x+e)*b)^4*(A+B*cos(f*x+e)),x)
 

Fricas [F]

\[ \int (c \cos (e+f x))^m (a+b \cos (e+f x))^4 (A+B \cos (e+f x)) \, dx=\int { {\left (B \cos \left (f x + e\right ) + A\right )} {\left (b \cos \left (f x + e\right ) + a\right )}^{4} \left (c \cos \left (f x + e\right )\right )^{m} \,d x } \] Input:

integrate((c*cos(f*x+e))^m*(a+b*cos(f*x+e))^4*(A+B*cos(f*x+e)),x, algorith 
m="fricas")
 

Output:

integral((B*b^4*cos(f*x + e)^5 + A*a^4 + (4*B*a*b^3 + A*b^4)*cos(f*x + e)^ 
4 + 2*(3*B*a^2*b^2 + 2*A*a*b^3)*cos(f*x + e)^3 + 2*(2*B*a^3*b + 3*A*a^2*b^ 
2)*cos(f*x + e)^2 + (B*a^4 + 4*A*a^3*b)*cos(f*x + e))*(c*cos(f*x + e))^m, 
x)
 

Sympy [F(-1)]

Timed out. \[ \int (c \cos (e+f x))^m (a+b \cos (e+f x))^4 (A+B \cos (e+f x)) \, dx=\text {Timed out} \] Input:

integrate((c*cos(f*x+e))**m*(a+b*cos(f*x+e))**4*(A+B*cos(f*x+e)),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int (c \cos (e+f x))^m (a+b \cos (e+f x))^4 (A+B \cos (e+f x)) \, dx=\int { {\left (B \cos \left (f x + e\right ) + A\right )} {\left (b \cos \left (f x + e\right ) + a\right )}^{4} \left (c \cos \left (f x + e\right )\right )^{m} \,d x } \] Input:

integrate((c*cos(f*x+e))^m*(a+b*cos(f*x+e))^4*(A+B*cos(f*x+e)),x, algorith 
m="maxima")
 

Output:

integrate((B*cos(f*x + e) + A)*(b*cos(f*x + e) + a)^4*(c*cos(f*x + e))^m, 
x)
 

Giac [F]

\[ \int (c \cos (e+f x))^m (a+b \cos (e+f x))^4 (A+B \cos (e+f x)) \, dx=\int { {\left (B \cos \left (f x + e\right ) + A\right )} {\left (b \cos \left (f x + e\right ) + a\right )}^{4} \left (c \cos \left (f x + e\right )\right )^{m} \,d x } \] Input:

integrate((c*cos(f*x+e))^m*(a+b*cos(f*x+e))^4*(A+B*cos(f*x+e)),x, algorith 
m="giac")
 

Output:

integrate((B*cos(f*x + e) + A)*(b*cos(f*x + e) + a)^4*(c*cos(f*x + e))^m, 
x)
 

Mupad [F(-1)]

Timed out. \[ \int (c \cos (e+f x))^m (a+b \cos (e+f x))^4 (A+B \cos (e+f x)) \, dx=\int {\left (c\,\cos \left (e+f\,x\right )\right )}^m\,\left (A+B\,\cos \left (e+f\,x\right )\right )\,{\left (a+b\,\cos \left (e+f\,x\right )\right )}^4 \,d x \] Input:

int((c*cos(e + f*x))^m*(A + B*cos(e + f*x))*(a + b*cos(e + f*x))^4,x)
 

Output:

int((c*cos(e + f*x))^m*(A + B*cos(e + f*x))*(a + b*cos(e + f*x))^4, x)
 

Reduce [F]

\[ \int (c \cos (e+f x))^m (a+b \cos (e+f x))^4 (A+B \cos (e+f x)) \, dx=c^{m} \left (\left (\int \cos \left (f x +e \right )^{m}d x \right ) a^{5}+5 \left (\int \cos \left (f x +e \right )^{m} \cos \left (f x +e \right )d x \right ) a^{4} b +\left (\int \cos \left (f x +e \right )^{m} \cos \left (f x +e \right )^{5}d x \right ) b^{5}+5 \left (\int \cos \left (f x +e \right )^{m} \cos \left (f x +e \right )^{4}d x \right ) a \,b^{4}+10 \left (\int \cos \left (f x +e \right )^{m} \cos \left (f x +e \right )^{3}d x \right ) a^{2} b^{3}+10 \left (\int \cos \left (f x +e \right )^{m} \cos \left (f x +e \right )^{2}d x \right ) a^{3} b^{2}\right ) \] Input:

int((c*cos(f*x+e))^m*(a+b*cos(f*x+e))^4*(A+B*cos(f*x+e)),x)
 

Output:

c**m*(int(cos(e + f*x)**m,x)*a**5 + 5*int(cos(e + f*x)**m*cos(e + f*x),x)* 
a**4*b + int(cos(e + f*x)**m*cos(e + f*x)**5,x)*b**5 + 5*int(cos(e + f*x)* 
*m*cos(e + f*x)**4,x)*a*b**4 + 10*int(cos(e + f*x)**m*cos(e + f*x)**3,x)*a 
**2*b**3 + 10*int(cos(e + f*x)**m*cos(e + f*x)**2,x)*a**3*b**2)