\(\int (c \cos (e+f x))^m (a+b \cos (e+f x))^3 (A+B \cos (e+f x)) \, dx\) [451]

Optimal result

Integrand size = 33, antiderivative size = 406 \[ \int (c \cos (e+f x))^m (a+b \cos (e+f x))^3 (A+B \cos (e+f x)) \, dx=\frac {b \left (b^2 B (3+m)+3 a A b (4+m)+2 a^2 B (5+m)\right ) (c \cos (e+f x))^{1+m} \sin (e+f x)}{c f (2+m) (4+m)}+\frac {b^2 (A b (4+m)+a B (6+m)) \cos (e+f x) (c \cos (e+f x))^{1+m} \sin (e+f x)}{c f (3+m) (4+m)}+\frac {b B (c \cos (e+f x))^{1+m} (a+b \cos (e+f x))^2 \sin (e+f x)}{c f (4+m)}-\frac {\left (a^2 (2+m) (b B (1+m)+a A (4+m))+b (1+m) \left (b^2 B (3+m)+3 a A b (4+m)+2 a^2 B (5+m)\right )\right ) (c \cos (e+f x))^{1+m} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},\cos ^2(e+f x)\right ) \sin (e+f x)}{c f (1+m) (2+m) (4+m) \sqrt {\sin ^2(e+f x)}}-\frac {\left (A b^3 (2+m)+3 a b^2 B (2+m)+3 a^2 A b (3+m)+a^3 B (3+m)\right ) (c \cos (e+f x))^{2+m} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},\cos ^2(e+f x)\right ) \sin (e+f x)}{c^2 f (2+m) (3+m) \sqrt {\sin ^2(e+f x)}} \] Output:

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

Mathematica [A] (verified)

Time = 3.25 (sec) , antiderivative size = 263, normalized size of antiderivative = 0.65 \[ \int (c \cos (e+f x))^m (a+b \cos (e+f x))^3 (A+B \cos (e+f x)) \, dx=\frac {(c \cos (e+f x))^m \cot (e+f x) \left (-\frac {a^3 A \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},\cos ^2(e+f x)\right )}{1+m}+\cos (e+f x) \left (-\frac {a^2 (3 A b+a B) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},\cos ^2(e+f x)\right )}{2+m}+b \cos (e+f x) \left (-\frac {3 a (A b+a B) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3+m}{2},\frac {5+m}{2},\cos ^2(e+f x)\right )}{3+m}+b \cos (e+f x) \left (-\frac {(A b+3 a B) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {4+m}{2},\frac {6+m}{2},\cos ^2(e+f x)\right )}{4+m}-\frac {b B \cos (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {5+m}{2},\frac {7+m}{2},\cos ^2(e+f x)\right )}{5+m}\right )\right )\right )\right ) \sqrt {\sin ^2(e+f x)}}{f} \] Input:

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

Output:

((c*Cos[e + f*x])^m*Cot[e + f*x]*(-((a^3*A*Hypergeometric2F1[1/2, (1 + m)/ 
2, (3 + m)/2, Cos[e + f*x]^2])/(1 + m)) + Cos[e + f*x]*(-((a^2*(3*A*b + a* 
B)*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, Cos[e + f*x]^2])/(2 + m)) 
+ b*Cos[e + f*x]*((-3*a*(A*b + a*B)*Hypergeometric2F1[1/2, (3 + m)/2, (5 + 
 m)/2, Cos[e + f*x]^2])/(3 + m) + b*Cos[e + f*x]*(-(((A*b + 3*a*B)*Hyperge 
ometric2F1[1/2, (4 + m)/2, (6 + m)/2, Cos[e + f*x]^2])/(4 + m)) - (b*B*Cos 
[e + f*x]*Hypergeometric2F1[1/2, (5 + m)/2, (7 + m)/2, Cos[e + f*x]^2])/(5 
 + m)))))*Sqrt[Sin[e + f*x]^2])/f
 

Rubi [A] (verified)

Time = 1.77 (sec) , antiderivative size = 408, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.303, Rules used = {3042, 3469, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3469

\(\displaystyle \frac {\int (c \cos (e+f x))^m (a+b \cos (e+f x)) \left (b c (A b (m+4)+a B (m+6)) \cos ^2(e+f x)+c \left (B (m+3) b^2+a (2 A b+a B) (m+4)\right ) \cos (e+f x)+a c (b B (m+1)+a A (m+4))\right )dx}{c (m+4)}+\frac {b B \sin (e+f x) (a+b \cos (e+f x))^2 (c \cos (e+f x))^{m+1}}{c f (m+4)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \left (c \sin \left (e+f x+\frac {\pi }{2}\right )\right )^m \left (a+b \sin \left (e+f x+\frac {\pi }{2}\right )\right ) \left (b c (A b (m+4)+a B (m+6)) \sin \left (e+f x+\frac {\pi }{2}\right )^2+c \left (B (m+3) b^2+a (2 A b+a B) (m+4)\right ) \sin \left (e+f x+\frac {\pi }{2}\right )+a c (b B (m+1)+a A (m+4))\right )dx}{c (m+4)}+\frac {b B \sin (e+f x) (a+b \cos (e+f x))^2 (c \cos (e+f x))^{m+1}}{c f (m+4)}\)

\(\Big \downarrow \) 3512

\(\displaystyle \frac {\frac {\int (c \cos (e+f x))^m \left (b (m+3) \left (2 B (m+5) a^2+3 A b (m+4) a+b^2 B (m+3)\right ) \cos ^2(e+f x) c^2+a^2 (m+3) (b B (m+1)+a A (m+4)) c^2+(m+4) \left (B (m+3) a^3+3 A b (m+3) a^2+3 b^2 B (m+2) a+A b^3 (m+2)\right ) \cos (e+f x) c^2\right )dx}{c (m+3)}+\frac {b^2 \sin (e+f x) \cos (e+f x) (a B (m+6)+A b (m+4)) (c \cos (e+f x))^{m+1}}{f (m+3)}}{c (m+4)}+\frac {b B \sin (e+f x) (a+b \cos (e+f x))^2 (c \cos (e+f x))^{m+1}}{c f (m+4)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\int \left (c \sin \left (e+f x+\frac {\pi }{2}\right )\right )^m \left (b (m+3) \left (2 B (m+5) a^2+3 A b (m+4) a+b^2 B (m+3)\right ) \sin \left (e+f x+\frac {\pi }{2}\right )^2 c^2+a^2 (m+3) (b B (m+1)+a A (m+4)) c^2+(m+4) \left (B (m+3) a^3+3 A b (m+3) a^2+3 b^2 B (m+2) a+A b^3 (m+2)\right ) \sin \left (e+f x+\frac {\pi }{2}\right ) c^2\right )dx}{c (m+3)}+\frac {b^2 \sin (e+f x) \cos (e+f x) (a B (m+6)+A b (m+4)) (c \cos (e+f x))^{m+1}}{f (m+3)}}{c (m+4)}+\frac {b B \sin (e+f x) (a+b \cos (e+f x))^2 (c \cos (e+f x))^{m+1}}{c f (m+4)}\)

\(\Big \downarrow \) 3502

\(\displaystyle \frac {\frac {\frac {\int (c \cos (e+f x))^m \left ((m+3) \left ((m+2) (b B (m+1)+a A (m+4)) a^2+b (m+1) \left (2 B (m+5) a^2+3 A b (m+4) a+b^2 B (m+3)\right )\right ) c^3+(m+2) (m+4) \left (B (m+3) a^3+3 A b (m+3) a^2+3 b^2 B (m+2) a+A b^3 (m+2)\right ) \cos (e+f x) c^3\right )dx}{c (m+2)}+\frac {b c (m+3) \sin (e+f x) \left (2 a^2 B (m+5)+3 a A b (m+4)+b^2 B (m+3)\right ) (c \cos (e+f x))^{m+1}}{f (m+2)}}{c (m+3)}+\frac {b^2 \sin (e+f x) \cos (e+f x) (a B (m+6)+A b (m+4)) (c \cos (e+f x))^{m+1}}{f (m+3)}}{c (m+4)}+\frac {b B \sin (e+f x) (a+b \cos (e+f x))^2 (c \cos (e+f x))^{m+1}}{c f (m+4)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {\int \left (c \sin \left (e+f x+\frac {\pi }{2}\right )\right )^m \left ((m+3) \left ((m+2) (b B (m+1)+a A (m+4)) a^2+b (m+1) \left (2 B (m+5) a^2+3 A b (m+4) a+b^2 B (m+3)\right )\right ) c^3+(m+2) (m+4) \left (B (m+3) a^3+3 A b (m+3) a^2+3 b^2 B (m+2) a+A b^3 (m+2)\right ) \sin \left (e+f x+\frac {\pi }{2}\right ) c^3\right )dx}{c (m+2)}+\frac {b c (m+3) \sin (e+f x) \left (2 a^2 B (m+5)+3 a A b (m+4)+b^2 B (m+3)\right ) (c \cos (e+f x))^{m+1}}{f (m+2)}}{c (m+3)}+\frac {b^2 \sin (e+f x) \cos (e+f x) (a B (m+6)+A b (m+4)) (c \cos (e+f x))^{m+1}}{f (m+3)}}{c (m+4)}+\frac {b B \sin (e+f x) (a+b \cos (e+f x))^2 (c \cos (e+f x))^{m+1}}{c f (m+4)}\)

\(\Big \downarrow \) 3227

\(\displaystyle \frac {\frac {\frac {c^3 (m+3) \left (b (m+1) \left (2 a^2 B (m+5)+3 a A b (m+4)+b^2 B (m+3)\right )+a^2 (m+2) (a A (m+4)+b B (m+1))\right ) \int (c \cos (e+f x))^mdx+c^2 (m+2) (m+4) \left (a^3 B (m+3)+3 a^2 A b (m+3)+3 a b^2 B (m+2)+A b^3 (m+2)\right ) \int (c \cos (e+f x))^{m+1}dx}{c (m+2)}+\frac {b c (m+3) \sin (e+f x) \left (2 a^2 B (m+5)+3 a A b (m+4)+b^2 B (m+3)\right ) (c \cos (e+f x))^{m+1}}{f (m+2)}}{c (m+3)}+\frac {b^2 \sin (e+f x) \cos (e+f x) (a B (m+6)+A b (m+4)) (c \cos (e+f x))^{m+1}}{f (m+3)}}{c (m+4)}+\frac {b B \sin (e+f x) (a+b \cos (e+f x))^2 (c \cos (e+f x))^{m+1}}{c f (m+4)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {c^3 (m+3) \left (b (m+1) \left (2 a^2 B (m+5)+3 a A b (m+4)+b^2 B (m+3)\right )+a^2 (m+2) (a A (m+4)+b B (m+1))\right ) \int \left (c \sin \left (e+f x+\frac {\pi }{2}\right )\right )^mdx+c^2 (m+2) (m+4) \left (a^3 B (m+3)+3 a^2 A b (m+3)+3 a b^2 B (m+2)+A b^3 (m+2)\right ) \int \left (c \sin \left (e+f x+\frac {\pi }{2}\right )\right )^{m+1}dx}{c (m+2)}+\frac {b c (m+3) \sin (e+f x) \left (2 a^2 B (m+5)+3 a A b (m+4)+b^2 B (m+3)\right ) (c \cos (e+f x))^{m+1}}{f (m+2)}}{c (m+3)}+\frac {b^2 \sin (e+f x) \cos (e+f x) (a B (m+6)+A b (m+4)) (c \cos (e+f x))^{m+1}}{f (m+3)}}{c (m+4)}+\frac {b B \sin (e+f x) (a+b \cos (e+f x))^2 (c \cos (e+f x))^{m+1}}{c f (m+4)}\)

\(\Big \downarrow \) 3122

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

Input:

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

Output:

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Sympy [F(-1)]

Timed out. \[ \int (c \cos (e+f x))^m (a+b \cos (e+f x))^3 (A+B \cos (e+f x)) \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

Reduce [F]

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

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

Output:

c**m*(int(cos(e + f*x)**m,x)*a**4 + 4*int(cos(e + f*x)**m*cos(e + f*x),x)* 
a**3*b + int(cos(e + f*x)**m*cos(e + f*x)**4,x)*b**4 + 4*int(cos(e + f*x)* 
*m*cos(e + f*x)**3,x)*a*b**3 + 6*int(cos(e + f*x)**m*cos(e + f*x)**2,x)*a* 
*2*b**2)