\(\int (a+b \cos (e+f x))^m (A+(A+C) \cos (e+f x)+C \cos ^2(e+f x)) \, dx\) [392]

Optimal result

Integrand size = 35, antiderivative size = 215 \[ \int (a+b \cos (e+f x))^m \left (A+(A+C) \cos (e+f x)+C \cos ^2(e+f x)\right ) \, dx=\frac {4 \sqrt {2} C \operatorname {AppellF1}\left (\frac {1}{2},-\frac {3}{2},-m,\frac {3}{2},\frac {1}{2} (1-\cos (e+f x)),\frac {b (1-\cos (e+f x))}{a+b}\right ) (a+b \cos (e+f x))^m \left (\frac {a+b \cos (e+f x)}{a+b}\right )^{-m} \sin (e+f x)}{f \sqrt {1+\cos (e+f x)}}+\frac {2 \sqrt {2} (A-C) \operatorname {AppellF1}\left (\frac {1}{2},-\frac {1}{2},-m,\frac {3}{2},\frac {1}{2} (1-\cos (e+f x)),\frac {b (1-\cos (e+f x))}{a+b}\right ) (a+b \cos (e+f x))^m \left (\frac {a+b \cos (e+f x)}{a+b}\right )^{-m} \sin (e+f x)}{f \sqrt {1+\cos (e+f x)}} \] Output:

4*2^(1/2)*C*AppellF1(1/2,-m,-3/2,3/2,b*(1-cos(f*x+e))/(a+b),1/2-1/2*cos(f* 
x+e))*(a+b*cos(f*x+e))^m*sin(f*x+e)/f/(1+cos(f*x+e))^(1/2)/(((a+b*cos(f*x+ 
e))/(a+b))^m)+2*2^(1/2)*(A-C)*AppellF1(1/2,-m,-1/2,3/2,b*(1-cos(f*x+e))/(a 
+b),1/2-1/2*cos(f*x+e))*(a+b*cos(f*x+e))^m*sin(f*x+e)/f/(1+cos(f*x+e))^(1/ 
2)/(((a+b*cos(f*x+e))/(a+b))^m)
                                                                                    
                                                                                    
 

Mathematica [F]

\[ \int (a+b \cos (e+f x))^m \left (A+(A+C) \cos (e+f x)+C \cos ^2(e+f x)\right ) \, dx=\int (a+b \cos (e+f x))^m \left (A+(A+C) \cos (e+f x)+C \cos ^2(e+f x)\right ) \, dx \] Input:

Integrate[(a + b*Cos[e + f*x])^m*(A + (A + C)*Cos[e + f*x] + C*Cos[e + f*x 
]^2),x]
 

Output:

Integrate[(a + b*Cos[e + f*x])^m*(A + (A + C)*Cos[e + f*x] + C*Cos[e + f*x 
]^2), x]
 

Rubi [A] (verified)

Time = 0.57 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.257, Rules used = {3042, 3496, 3042, 3234, 156, 155, 3263, 156, 155}

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.

\(\displaystyle \int \left ((A+C) \cos (e+f x)+A+C \cos ^2(e+f x)\right ) (a+b \cos (e+f x))^m \, dx\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3496

\(\displaystyle (A-C) \int (\cos (e+f x)+1) (a+b \cos (e+f x))^mdx+C \int (\cos (e+f x)+1)^2 (a+b \cos (e+f x))^mdx\)

\(\Big \downarrow \) 3042

\(\displaystyle (A-C) \int \left (\sin \left (e+f x+\frac {\pi }{2}\right )+1\right ) \left (a+b \sin \left (e+f x+\frac {\pi }{2}\right )\right )^mdx+C \int \left (\sin \left (e+f x+\frac {\pi }{2}\right )+1\right )^2 \left (a+b \sin \left (e+f x+\frac {\pi }{2}\right )\right )^mdx\)

\(\Big \downarrow \) 3234

\(\displaystyle C \int \left (\sin \left (e+f x+\frac {\pi }{2}\right )+1\right )^2 \left (a+b \sin \left (e+f x+\frac {\pi }{2}\right )\right )^mdx-\frac {(A-C) \sin (e+f x) \int \frac {\sqrt {\cos (e+f x)+1} (a+b \cos (e+f x))^m}{\sqrt {1-\cos (e+f x)}}d\cos (e+f x)}{f \sqrt {1-\cos (e+f x)} \sqrt {\cos (e+f x)+1}}\)

\(\Big \downarrow \) 156

\(\displaystyle C \int \left (\sin \left (e+f x+\frac {\pi }{2}\right )+1\right )^2 \left (a+b \sin \left (e+f x+\frac {\pi }{2}\right )\right )^mdx-\frac {(A-C) \sin (e+f x) (a+b \cos (e+f x))^m \left (\frac {a+b \cos (e+f x)}{a+b}\right )^{-m} \int \frac {\sqrt {\cos (e+f x)+1} \left (\frac {a}{a+b}+\frac {b \cos (e+f x)}{a+b}\right )^m}{\sqrt {1-\cos (e+f x)}}d\cos (e+f x)}{f \sqrt {1-\cos (e+f x)} \sqrt {\cos (e+f x)+1}}\)

\(\Big \downarrow \) 155

\(\displaystyle C \int \left (\sin \left (e+f x+\frac {\pi }{2}\right )+1\right )^2 \left (a+b \sin \left (e+f x+\frac {\pi }{2}\right )\right )^mdx+\frac {2 \sqrt {2} (A-C) \sin (e+f x) (a+b \cos (e+f x))^m \left (\frac {a+b \cos (e+f x)}{a+b}\right )^{-m} \operatorname {AppellF1}\left (\frac {1}{2},-\frac {1}{2},-m,\frac {3}{2},\frac {1}{2} (1-\cos (e+f x)),\frac {b (1-\cos (e+f x))}{a+b}\right )}{f \sqrt {\cos (e+f x)+1}}\)

\(\Big \downarrow \) 3263

\(\displaystyle \frac {2 \sqrt {2} (A-C) \sin (e+f x) (a+b \cos (e+f x))^m \left (\frac {a+b \cos (e+f x)}{a+b}\right )^{-m} \operatorname {AppellF1}\left (\frac {1}{2},-\frac {1}{2},-m,\frac {3}{2},\frac {1}{2} (1-\cos (e+f x)),\frac {b (1-\cos (e+f x))}{a+b}\right )}{f \sqrt {\cos (e+f x)+1}}-\frac {C \sin (e+f x) \int \frac {(\cos (e+f x)+1)^{3/2} (a+b \cos (e+f x))^m}{\sqrt {1-\cos (e+f x)}}d\cos (e+f x)}{f \sqrt {1-\cos (e+f x)} \sqrt {\cos (e+f x)+1}}\)

\(\Big \downarrow \) 156

\(\displaystyle \frac {2 \sqrt {2} (A-C) \sin (e+f x) (a+b \cos (e+f x))^m \left (\frac {a+b \cos (e+f x)}{a+b}\right )^{-m} \operatorname {AppellF1}\left (\frac {1}{2},-\frac {1}{2},-m,\frac {3}{2},\frac {1}{2} (1-\cos (e+f x)),\frac {b (1-\cos (e+f x))}{a+b}\right )}{f \sqrt {\cos (e+f x)+1}}-\frac {C \sin (e+f x) (a+b \cos (e+f x))^m \left (\frac {a+b \cos (e+f x)}{a+b}\right )^{-m} \int \frac {(\cos (e+f x)+1)^{3/2} \left (\frac {a}{a+b}+\frac {b \cos (e+f x)}{a+b}\right )^m}{\sqrt {1-\cos (e+f x)}}d\cos (e+f x)}{f \sqrt {1-\cos (e+f x)} \sqrt {\cos (e+f x)+1}}\)

\(\Big \downarrow \) 155

\(\displaystyle \frac {2 \sqrt {2} (A-C) \sin (e+f x) (a+b \cos (e+f x))^m \left (\frac {a+b \cos (e+f x)}{a+b}\right )^{-m} \operatorname {AppellF1}\left (\frac {1}{2},-\frac {1}{2},-m,\frac {3}{2},\frac {1}{2} (1-\cos (e+f x)),\frac {b (1-\cos (e+f x))}{a+b}\right )}{f \sqrt {\cos (e+f x)+1}}+\frac {4 \sqrt {2} C \sin (e+f x) (a+b \cos (e+f x))^m \left (\frac {a+b \cos (e+f x)}{a+b}\right )^{-m} \operatorname {AppellF1}\left (\frac {1}{2},-\frac {3}{2},-m,\frac {3}{2},\frac {1}{2} (1-\cos (e+f x)),\frac {b (1-\cos (e+f x))}{a+b}\right )}{f \sqrt {\cos (e+f x)+1}}\)

Input:

Int[(a + b*Cos[e + f*x])^m*(A + (A + C)*Cos[e + f*x] + C*Cos[e + f*x]^2),x 
]
 

Output:

(4*Sqrt[2]*C*AppellF1[1/2, -3/2, -m, 3/2, (1 - Cos[e + f*x])/2, (b*(1 - Co 
s[e + f*x]))/(a + b)]*(a + b*Cos[e + f*x])^m*Sin[e + f*x])/(f*Sqrt[1 + Cos 
[e + f*x]]*((a + b*Cos[e + f*x])/(a + b))^m) + (2*Sqrt[2]*(A - C)*AppellF1 
[1/2, -1/2, -m, 3/2, (1 - Cos[e + f*x])/2, (b*(1 - Cos[e + f*x]))/(a + b)] 
*(a + b*Cos[e + f*x])^m*Sin[e + f*x])/(f*Sqrt[1 + Cos[e + f*x]]*((a + b*Co 
s[e + f*x])/(a + b))^m)
 

Defintions of rubi rules used

rule 155

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_)) 
^(p_), x_] :> Simp[((a + b*x)^(m + 1)/(b*(m + 1)*Simplify[b/(b*c - a*d)]^n* 
Simplify[b/(b*e - a*f)]^p))*AppellF1[m + 1, -n, -p, m + 2, (-d)*((a + b*x)/ 
(b*c - a*d)), (-f)*((a + b*x)/(b*e - a*f))], x] /; FreeQ[{a, b, c, d, e, f, 
 m, n, p}, x] &&  !IntegerQ[m] &&  !IntegerQ[n] &&  !IntegerQ[p] && GtQ[Sim 
plify[b/(b*c - a*d)], 0] && GtQ[Simplify[b/(b*e - a*f)], 0] &&  !(GtQ[Simpl 
ify[d/(d*a - c*b)], 0] && GtQ[Simplify[d/(d*e - c*f)], 0] && SimplerQ[c + d 
*x, a + b*x]) &&  !(GtQ[Simplify[f/(f*a - e*b)], 0] && GtQ[Simplify[f/(f*c 
- e*d)], 0] && SimplerQ[e + f*x, a + b*x])
 

rule 156

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_)) 
^(p_), x_] :> Simp[(e + f*x)^FracPart[p]/(Simplify[b/(b*e - a*f)]^IntPart[p 
]*(b*((e + f*x)/(b*e - a*f)))^FracPart[p])   Int[(a + b*x)^m*(c + d*x)^n*Si 
mp[b*(e/(b*e - a*f)) + b*f*(x/(b*e - a*f)), x]^p, x], x] /; FreeQ[{a, b, c, 
 d, e, f, m, n, p}, x] &&  !IntegerQ[m] &&  !IntegerQ[n] &&  !IntegerQ[p] & 
& GtQ[Simplify[b/(b*c - a*d)], 0] &&  !GtQ[Simplify[b/(b*e - a*f)], 0]
 

rule 3042

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

rule 3234

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + ( 
f_.)*(x_)]), x_Symbol] :> Simp[c*(Cos[e + f*x]/(f*Sqrt[1 + Sin[e + f*x]]*Sq 
rt[1 - Sin[e + f*x]]))   Subst[Int[(a + b*x)^m*(Sqrt[1 + (d/c)*x]/Sqrt[1 - 
(d/c)*x]), x], x, Sin[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && N 
eQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] &&  !IntegerQ[2*m] && EqQ[c^2 - d^2, 
0]
 

rule 3263

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + 
(f_.)*(x_)])^(n_.), x_Symbol] :> Simp[a^m*(Cos[e + f*x]/(f*Sqrt[1 + Sin[e + 
 f*x]]*Sqrt[1 - Sin[e + f*x]]))   Subst[Int[(1 + (b/a)*x)^(m - 1/2)*((c + d 
*x)^n/Sqrt[1 - (b/a)*x]), x], x, Sin[e + f*x]], x] /; FreeQ[{a, b, c, d, e, 
 f, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] & 
& IntegerQ[m]
 

rule 3496

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + 
 (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(A - C) 
  Int[(a + b*Sin[e + f*x])^m*(1 + Sin[e + f*x]), x], x] + Simp[C   Int[(a + 
 b*Sin[e + f*x])^m*(1 + Sin[e + f*x])^2, x], x] /; FreeQ[{a, b, e, f, A, B, 
 C, m}, x] && EqQ[A - B + C, 0] &&  !IntegerQ[2*m]
 

Maple [F]

\[\int \left (a +b \cos \left (f x +e \right )\right )^{m} \left (A +\left (A +C \right ) \cos \left (f x +e \right )+C \cos \left (f x +e \right )^{2}\right )d x\]

Input:

int((a+b*cos(f*x+e))^m*(A+(A+C)*cos(f*x+e)+C*cos(f*x+e)^2),x)
 

Output:

int((a+b*cos(f*x+e))^m*(A+(A+C)*cos(f*x+e)+C*cos(f*x+e)^2),x)
 

Fricas [F]

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

integrate((a+b*cos(f*x+e))^m*(A+(A+C)*cos(f*x+e)+C*cos(f*x+e)^2),x, algori 
thm="fricas")
 

Output:

integral((C*cos(f*x + e)^2 + (A + C)*cos(f*x + e) + A)*(b*cos(f*x + e) + a 
)^m, x)
 

Sympy [F(-1)]

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

integrate((a+b*cos(f*x+e))**m*(A+(A+C)*cos(f*x+e)+C*cos(f*x+e)**2),x)
 

Output:

Timed out
 

Maxima [F]

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

integrate((a+b*cos(f*x+e))^m*(A+(A+C)*cos(f*x+e)+C*cos(f*x+e)^2),x, algori 
thm="maxima")
 

Output:

integrate((C*cos(f*x + e)^2 + (A + C)*cos(f*x + e) + A)*(b*cos(f*x + e) + 
a)^m, x)
 

Giac [F]

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

integrate((a+b*cos(f*x+e))^m*(A+(A+C)*cos(f*x+e)+C*cos(f*x+e)^2),x, algori 
thm="giac")
 

Output:

integrate((C*cos(f*x + e)^2 + (A + C)*cos(f*x + e) + A)*(b*cos(f*x + e) + 
a)^m, x)
 

Mupad [F(-1)]

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

int((a + b*cos(e + f*x))^m*(A + C*cos(e + f*x)^2 + cos(e + f*x)*(A + C)),x 
)
 

Output:

int((a + b*cos(e + f*x))^m*(A + C*cos(e + f*x)^2 + cos(e + f*x)*(A + C)), 
x)
 

Reduce [F]

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

int((a+b*cos(f*x+e))^m*(A+(A+C)*cos(f*x+e)+C*cos(f*x+e)^2),x)
                                                                                    
                                                                                    
 

Output:

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