Integrand size = 19, antiderivative size = 85 \[ \int (d \cos (e+f x))^n \sin ^m(e+f x) \, dx=-\frac {(d \cos (e+f x))^{1+n} \operatorname {Hypergeometric2F1}\left (\frac {1-m}{2},\frac {1+n}{2},\frac {3+n}{2},\cos ^2(e+f x)\right ) \sin ^{-1+m}(e+f x) \sin ^2(e+f x)^{\frac {1-m}{2}}}{d f (1+n)} \] Output:
-(d*cos(f*x+e))^(1+n)*hypergeom([1/2+1/2*n, 1/2-1/2*m],[3/2+1/2*n],cos(f*x +e)^2)*sin(f*x+e)^(-1+m)*(sin(f*x+e)^2)^(1/2-1/2*m)/d/f/(1+n)
Time = 0.13 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.96 \[ \int (d \cos (e+f x))^n \sin ^m(e+f x) \, dx=\frac {d (d \cos (e+f x))^{-1+n} \cos ^2(e+f x)^{\frac {1-n}{2}} \operatorname {Hypergeometric2F1}\left (\frac {1+m}{2},\frac {1-n}{2},\frac {3+m}{2},\sin ^2(e+f x)\right ) \sin ^{1+m}(e+f x)}{f (1+m)} \] Input:
Integrate[(d*Cos[e + f*x])^n*Sin[e + f*x]^m,x]
Output:
(d*(d*Cos[e + f*x])^(-1 + n)*(Cos[e + f*x]^2)^((1 - n)/2)*Hypergeometric2F 1[(1 + m)/2, (1 - n)/2, (3 + m)/2, Sin[e + f*x]^2]*Sin[e + f*x]^(1 + m))/( f*(1 + m))
Time = 0.25 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {3042, 3056}
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.
\(\displaystyle \int \sin ^m(e+f x) (d \cos (e+f x))^n \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \sin (e+f x)^m (d \cos (e+f x))^ndx\) |
\(\Big \downarrow \) 3056 |
\(\displaystyle -\frac {\sin ^{m-1}(e+f x) \sin ^2(e+f x)^{\frac {1-m}{2}} (d \cos (e+f x))^{n+1} \operatorname {Hypergeometric2F1}\left (\frac {1-m}{2},\frac {n+1}{2},\frac {n+3}{2},\cos ^2(e+f x)\right )}{d f (n+1)}\) |
Input:
Int[(d*Cos[e + f*x])^n*Sin[e + f*x]^m,x]
Output:
-(((d*Cos[e + f*x])^(1 + n)*Hypergeometric2F1[(1 - m)/2, (1 + n)/2, (3 + n )/2, Cos[e + f*x]^2]*Sin[e + f*x]^(-1 + m)*(Sin[e + f*x]^2)^((1 - m)/2))/( d*f*(1 + n)))
Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n _), x_Symbol] :> Simp[(-b^(2*IntPart[(n - 1)/2] + 1))*(b*Sin[e + f*x])^(2*F racPart[(n - 1)/2])*((a*Cos[e + f*x])^(m + 1)/(a*f*(m + 1)*(Sin[e + f*x]^2) ^FracPart[(n - 1)/2]))*Hypergeometric2F1[(1 + m)/2, (1 - n)/2, (3 + m)/2, C os[e + f*x]^2], x] /; FreeQ[{a, b, e, f, m, n}, x] && SimplerQ[n, m]
\[\int \left (d \cos \left (f x +e \right )\right )^{n} \sin \left (f x +e \right )^{m}d x\]
Input:
int((d*cos(f*x+e))^n*sin(f*x+e)^m,x)
Output:
int((d*cos(f*x+e))^n*sin(f*x+e)^m,x)
\[ \int (d \cos (e+f x))^n \sin ^m(e+f x) \, dx=\int { \left (d \cos \left (f x + e\right )\right )^{n} \sin \left (f x + e\right )^{m} \,d x } \] Input:
integrate((d*cos(f*x+e))^n*sin(f*x+e)^m,x, algorithm="fricas")
Output:
integral((d*cos(f*x + e))^n*sin(f*x + e)^m, x)
\[ \int (d \cos (e+f x))^n \sin ^m(e+f x) \, dx=\int \left (d \cos {\left (e + f x \right )}\right )^{n} \sin ^{m}{\left (e + f x \right )}\, dx \] Input:
integrate((d*cos(f*x+e))**n*sin(f*x+e)**m,x)
Output:
Integral((d*cos(e + f*x))**n*sin(e + f*x)**m, x)
\[ \int (d \cos (e+f x))^n \sin ^m(e+f x) \, dx=\int { \left (d \cos \left (f x + e\right )\right )^{n} \sin \left (f x + e\right )^{m} \,d x } \] Input:
integrate((d*cos(f*x+e))^n*sin(f*x+e)^m,x, algorithm="maxima")
Output:
integrate((d*cos(f*x + e))^n*sin(f*x + e)^m, x)
\[ \int (d \cos (e+f x))^n \sin ^m(e+f x) \, dx=\int { \left (d \cos \left (f x + e\right )\right )^{n} \sin \left (f x + e\right )^{m} \,d x } \] Input:
integrate((d*cos(f*x+e))^n*sin(f*x+e)^m,x, algorithm="giac")
Output:
integrate((d*cos(f*x + e))^n*sin(f*x + e)^m, x)
Timed out. \[ \int (d \cos (e+f x))^n \sin ^m(e+f x) \, dx=\int {\sin \left (e+f\,x\right )}^m\,{\left (d\,\cos \left (e+f\,x\right )\right )}^n \,d x \] Input:
int(sin(e + f*x)^m*(d*cos(e + f*x))^n,x)
Output:
int(sin(e + f*x)^m*(d*cos(e + f*x))^n, x)
\[ \int (d \cos (e+f x))^n \sin ^m(e+f x) \, dx=d^{n} \left (\int \sin \left (f x +e \right )^{m} \cos \left (f x +e \right )^{n}d x \right ) \] Input:
int((d*cos(f*x+e))^n*sin(f*x+e)^m,x)
Output:
d**n*int(sin(e + f*x)**m*cos(e + f*x)**n,x)