Integrand size = 21, antiderivative size = 82 \[ \int (a \cos (e+f x))^m (b \cos (e+f x))^n \, dx=-\frac {(a \cos (e+f x))^{1+m} (b \cos (e+f x))^n \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (1+m+n),\frac {1}{2} (3+m+n),\cos ^2(e+f x)\right ) \sin (e+f x)}{a f (1+m+n) \sqrt {\sin ^2(e+f x)}} \] Output:
-(a*cos(f*x+e))^(1+m)*(b*cos(f*x+e))^n*hypergeom([1/2, 1/2+1/2*m+1/2*n],[3 /2+1/2*m+1/2*n],cos(f*x+e)^2)*sin(f*x+e)/a/f/(1+m+n)/(sin(f*x+e)^2)^(1/2)
Time = 0.12 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.94 \[ \int (a \cos (e+f x))^m (b \cos (e+f x))^n \, dx=-\frac {(a \cos (e+f x))^m (b \cos (e+f x))^n \cot (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (1+m+n),\frac {1}{2} (3+m+n),\cos ^2(e+f x)\right ) \sqrt {\sin ^2(e+f x)}}{f (1+m+n)} \] Input:
Integrate[(a*Cos[e + f*x])^m*(b*Cos[e + f*x])^n,x]
Output:
-(((a*Cos[e + f*x])^m*(b*Cos[e + f*x])^n*Cot[e + f*x]*Hypergeometric2F1[1/ 2, (1 + m + n)/2, (3 + m + n)/2, Cos[e + f*x]^2]*Sqrt[Sin[e + f*x]^2])/(f* (1 + m + n)))
Time = 0.26 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2034, 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.
\(\displaystyle \int (a \cos (e+f x))^m (b \cos (e+f x))^n \, dx\) |
\(\Big \downarrow \) 2034 |
\(\displaystyle (a \cos (e+f x))^{-n} (b \cos (e+f x))^n \int (a \cos (e+f x))^{m+n}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle (a \cos (e+f x))^{-n} (b \cos (e+f x))^n \int \left (a \sin \left (e+f x+\frac {\pi }{2}\right )\right )^{m+n}dx\) |
\(\Big \downarrow \) 3122 |
\(\displaystyle -\frac {\sin (e+f x) (a \cos (e+f x))^{m+1} (b \cos (e+f x))^n \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (m+n+1),\frac {1}{2} (m+n+3),\cos ^2(e+f x)\right )}{a f (m+n+1) \sqrt {\sin ^2(e+f x)}}\) |
Input:
Int[(a*Cos[e + f*x])^m*(b*Cos[e + f*x])^n,x]
Output:
-(((a*Cos[e + f*x])^(1 + m)*(b*Cos[e + f*x])^n*Hypergeometric2F1[1/2, (1 + m + n)/2, (3 + m + n)/2, Cos[e + f*x]^2]*Sin[e + f*x])/(a*f*(1 + m + n)*S qrt[Sin[e + f*x]^2]))
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]
\[\int \left (\cos \left (f x +e \right ) a \right )^{m} \left (\cos \left (f x +e \right ) b \right )^{n}d x\]
Input:
int((cos(f*x+e)*a)^m*(cos(f*x+e)*b)^n,x)
Output:
int((cos(f*x+e)*a)^m*(cos(f*x+e)*b)^n,x)
\[ \int (a \cos (e+f x))^m (b \cos (e+f x))^n \, dx=\int { \left (a \cos \left (f x + e\right )\right )^{m} \left (b \cos \left (f x + e\right )\right )^{n} \,d x } \] Input:
integrate((a*cos(f*x+e))^m*(b*cos(f*x+e))^n,x, algorithm="fricas")
Output:
integral((a*cos(f*x + e))^m*(b*cos(f*x + e))^n, x)
\[ \int (a \cos (e+f x))^m (b \cos (e+f x))^n \, dx=\int \left (a \cos {\left (e + f x \right )}\right )^{m} \left (b \cos {\left (e + f x \right )}\right )^{n}\, dx \] Input:
integrate((a*cos(f*x+e))**m*(b*cos(f*x+e))**n,x)
Output:
Integral((a*cos(e + f*x))**m*(b*cos(e + f*x))**n, x)
\[ \int (a \cos (e+f x))^m (b \cos (e+f x))^n \, dx=\int { \left (a \cos \left (f x + e\right )\right )^{m} \left (b \cos \left (f x + e\right )\right )^{n} \,d x } \] Input:
integrate((a*cos(f*x+e))^m*(b*cos(f*x+e))^n,x, algorithm="maxima")
Output:
integrate((a*cos(f*x + e))^m*(b*cos(f*x + e))^n, x)
\[ \int (a \cos (e+f x))^m (b \cos (e+f x))^n \, dx=\int { \left (a \cos \left (f x + e\right )\right )^{m} \left (b \cos \left (f x + e\right )\right )^{n} \,d x } \] Input:
integrate((a*cos(f*x+e))^m*(b*cos(f*x+e))^n,x, algorithm="giac")
Output:
integrate((a*cos(f*x + e))^m*(b*cos(f*x + e))^n, x)
Timed out. \[ \int (a \cos (e+f x))^m (b \cos (e+f x))^n \, dx=\int {\left (a\,\cos \left (e+f\,x\right )\right )}^m\,{\left (b\,\cos \left (e+f\,x\right )\right )}^n \,d x \] Input:
int((a*cos(e + f*x))^m*(b*cos(e + f*x))^n,x)
Output:
int((a*cos(e + f*x))^m*(b*cos(e + f*x))^n, x)
\[ \int (a \cos (e+f x))^m (b \cos (e+f x))^n \, dx=b^{n} a^{m} \left (\int \cos \left (f x +e \right )^{m +n}d x \right ) \] Input:
int((a*cos(f*x+e))^m*(b*cos(f*x+e))^n,x)
Output:
b**n*a**m*int(cos(e + f*x)**(m + n),x)