\(\int (a \cos (c+d x))^m (b \cos (c+d x))^n (A+C \cos ^2(c+d x)) \, dx\) [182]

Optimal result

Integrand size = 33, antiderivative size = 144 \[ \int (a \cos (c+d x))^m (b \cos (c+d x))^n \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {C (a \cos (c+d x))^{1+m} (b \cos (c+d x))^n \sin (c+d x)}{a d (2+m+n)}-\frac {(C (1+m+n)+A (2+m+n)) (a \cos (c+d x))^{1+m} (b \cos (c+d x))^n \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (1+m+n),\frac {1}{2} (3+m+n),\cos ^2(c+d x)\right ) \sin (c+d x)}{a d (1+m+n) (2+m+n) \sqrt {\sin ^2(c+d x)}} \] Output:

C*(a*cos(d*x+c))^(1+m)*(b*cos(d*x+c))^n*sin(d*x+c)/a/d/(2+m+n)-(C*(1+m+n)+ 
A*(2+m+n))*(a*cos(d*x+c))^(1+m)*(b*cos(d*x+c))^n*hypergeom([1/2, 1/2+1/2*m 
+1/2*n],[3/2+1/2*m+1/2*n],cos(d*x+c)^2)*sin(d*x+c)/a/d/(1+m+n)/(2+m+n)/(si 
n(d*x+c)^2)^(1/2)
 

Mathematica [A] (verified)

Time = 0.27 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.92 \[ \int (a \cos (c+d x))^m (b \cos (c+d x))^n \left (A+C \cos ^2(c+d x)\right ) \, dx=-\frac {(a \cos (c+d x))^m (b \cos (c+d x))^n \cot (c+d x) \left (A (3+m+n) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (1+m+n),\frac {1}{2} (3+m+n),\cos ^2(c+d x)\right )+C (1+m+n) \cos ^2(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (3+m+n),\frac {1}{2} (5+m+n),\cos ^2(c+d x)\right )\right ) \sqrt {\sin ^2(c+d x)}}{d (1+m+n) (3+m+n)} \] Input:

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

Output:

-(((a*Cos[c + d*x])^m*(b*Cos[c + d*x])^n*Cot[c + d*x]*(A*(3 + m + n)*Hyper 
geometric2F1[1/2, (1 + m + n)/2, (3 + m + n)/2, Cos[c + d*x]^2] + C*(1 + m 
 + n)*Cos[c + d*x]^2*Hypergeometric2F1[1/2, (3 + m + n)/2, (5 + m + n)/2, 
Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(d*(1 + m + n)*(3 + m + n)))
 

Rubi [A] (verified)

Time = 0.45 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {2034, 3042, 3493, 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.

\(\displaystyle \int (a \cos (c+d x))^m \left (A+C \cos ^2(c+d x)\right ) (b \cos (c+d x))^n \, dx\)

\(\Big \downarrow \) 2034

\(\displaystyle (a \cos (c+d x))^{-n} (b \cos (c+d x))^n \int (a \cos (c+d x))^{m+n} \left (C \cos ^2(c+d x)+A\right )dx\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3493

\(\displaystyle (a \cos (c+d x))^{-n} (b \cos (c+d x))^n \left (\left (A+\frac {C (m+n+1)}{m+n+2}\right ) \int (a \cos (c+d x))^{m+n}dx+\frac {C \sin (c+d x) (a \cos (c+d x))^{m+n+1}}{a d (m+n+2)}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle (a \cos (c+d x))^{-n} (b \cos (c+d x))^n \left (\left (A+\frac {C (m+n+1)}{m+n+2}\right ) \int \left (a \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{m+n}dx+\frac {C \sin (c+d x) (a \cos (c+d x))^{m+n+1}}{a d (m+n+2)}\right )\)

\(\Big \downarrow \) 3122

\(\displaystyle (a \cos (c+d x))^{-n} (b \cos (c+d x))^n \left (\frac {C \sin (c+d x) (a \cos (c+d x))^{m+n+1}}{a d (m+n+2)}-\frac {\left (A+\frac {C (m+n+1)}{m+n+2}\right ) \sin (c+d x) (a \cos (c+d x))^{m+n+1} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (m+n+1),\frac {1}{2} (m+n+3),\cos ^2(c+d x)\right )}{a d (m+n+1) \sqrt {\sin ^2(c+d x)}}\right )\)

Input:

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

Output:

((b*Cos[c + d*x])^n*((C*(a*Cos[c + d*x])^(1 + m + n)*Sin[c + d*x])/(a*d*(2 
 + m + n)) - ((A + (C*(1 + m + n))/(2 + m + n))*(a*Cos[c + d*x])^(1 + m + 
n)*Hypergeometric2F1[1/2, (1 + m + n)/2, (3 + m + n)/2, Cos[c + d*x]^2]*Si 
n[c + d*x])/(a*d*(1 + m + n)*Sqrt[Sin[c + d*x]^2])))/(a*Cos[c + d*x])^n
 

Defintions of rubi rules used

rule 2034

Int[(Fx_.)*((a_.)*(v_))^(m_)*((b_.)*(v_))^(n_), x_Symbol] :> Simp[b^IntPart 
[n]*((b*v)^FracPart[n]/(a^IntPart[n]*(a*v)^FracPart[n]))   Int[(a*v)^(m + n 
)*Fx, x], x] /; FreeQ[{a, b, m, n}, x] &&  !IntegerQ[m] &&  !IntegerQ[n] && 
  !IntegerQ[m + n]
 

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 3493

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral((C*cos(d*x + c)^2 + A)*(a*cos(d*x + c))^m*(b*cos(d*x + c))^n, x)
 

Sympy [F]

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

integrate((a*cos(d*x+c))**m*(b*cos(d*x+c))**n*(A+C*cos(d*x+c)**2),x)
 

Output:

Integral((a*cos(c + d*x))**m*(b*cos(c + d*x))**n*(A + C*cos(c + d*x)**2), 
x)
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

b**n*a**m*(int(cos(c + d*x)**(m + n),x)*a + int(cos(c + d*x)**(m + n)*cos( 
c + d*x)**2,x)*c)