Integrand size = 33, antiderivative size = 144 \[ \int (a \cos (c+d x))^m (b \cos (c+d x))^n \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {C (a \cos (c+d x))^{1+m} (b \cos (c+d x))^n \sin (c+d x)}{a d (2+m+n)}-\frac {(C (1+m+n)+A (2+m+n)) (a \cos (c+d x))^{1+m} (b \cos (c+d x))^n \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (1+m+n),\frac {1}{2} (3+m+n),\cos ^2(c+d x)\right ) \sin (c+d x)}{a d (1+m+n) (2+m+n) \sqrt {\sin ^2(c+d x)}} \]
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)
Time = 0.19 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.92 \[ \int (a \cos (c+d x))^m (b \cos (c+d x))^n \left (A+C \cos ^2(c+d x)\right ) \, dx=-\frac {(a \cos (c+d x))^m (b \cos (c+d x))^n \cot (c+d x) \left (A (3+m+n) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (1+m+n),\frac {1}{2} (3+m+n),\cos ^2(c+d x)\right )+C (1+m+n) \cos ^2(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (3+m+n),\frac {1}{2} (5+m+n),\cos ^2(c+d x)\right )\right ) \sqrt {\sin ^2(c+d x)}}{d (1+m+n) (3+m+n)} \]
-(((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)))
Time = 0.43 (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 |
| \(\displaystyle (a \cos (c+d x))^{-n} (b \cos (c+d x))^n \left (\frac {C \sin (c+d x) (a \cos (c+d x))^{m+n+1}}{a d (m+n+2)}-\frac {\left (A+\frac {C (m+n+1)}{m+n+2}\right ) \sin (c+d x) (a \cos (c+d x))^{m+n+1} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (m+n+1),\frac {1}{2} (m+n+3),\cos ^2(c+d x)\right )}{a d (m+n+1) \sqrt {\sin ^2(c+d x)}}\right )\) |
((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
3.2.82.3.1 Defintions of rubi rules used
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]
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]
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]
\[\int \left (\cos \left (d x +c \right ) a \right )^{m} \left (\cos \left (d x +c \right ) b \right )^{n} \left (A +C \left (\cos ^{2}\left (d x +c \right )\right )\right )d x\]
\[ \int (a \cos (c+d x))^m (b \cos (c+d x))^n \left (A+C \cos ^2(c+d x)\right ) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} \left (a \cos \left (d x + c\right )\right )^{m} \left (b \cos \left (d x + c\right )\right )^{n} \,d x } \]
\[ \int (a \cos (c+d x))^m (b \cos (c+d x))^n \left (A+C \cos ^2(c+d x)\right ) \, dx=\int \left (a \cos {\left (c + d x \right )}\right )^{m} \left (b \cos {\left (c + d x \right )}\right )^{n} \left (A + C \cos ^{2}{\left (c + d x \right )}\right )\, dx \]
\[ \int (a \cos (c+d x))^m (b \cos (c+d x))^n \left (A+C \cos ^2(c+d x)\right ) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} \left (a \cos \left (d x + c\right )\right )^{m} \left (b \cos \left (d x + c\right )\right )^{n} \,d x } \]
\[ \int (a \cos (c+d x))^m (b \cos (c+d x))^n \left (A+C \cos ^2(c+d x)\right ) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} \left (a \cos \left (d x + c\right )\right )^{m} \left (b \cos \left (d x + c\right )\right )^{n} \,d x } \]
Timed out. \[ \int (a \cos (c+d x))^m (b \cos (c+d x))^n \left (A+C \cos ^2(c+d x)\right ) \, dx=\int \left (C\,{\cos \left (c+d\,x\right )}^2+A\right )\,{\left (a\,\cos \left (c+d\,x\right )\right )}^m\,{\left (b\,\cos \left (c+d\,x\right )\right )}^n \,d x \]