\(\int (d \cos (e+f x))^n \sin ^m(e+f x) \, dx\) [337]

Optimal result

Integrand size = 19, antiderivative size = 85 \[ \int (d \cos (e+f x))^n \sin ^m(e+f x) \, dx=-\frac {(d \cos (e+f x))^{1+n} \operatorname {Hypergeometric2F1}\left (\frac {1-m}{2},\frac {1+n}{2},\frac {3+n}{2},\cos ^2(e+f x)\right ) \sin ^{-1+m}(e+f x) \sin ^2(e+f x)^{\frac {1-m}{2}}}{d f (1+n)} \]

[Out]

-(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)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {2656} \[ \int (d \cos (e+f x))^n \sin ^m(e+f x) \, dx=-\frac {\sin ^{m-1}(e+f x) \sin ^2(e+f x)^{\frac {1-m}{2}} (d \cos (e+f x))^{n+1} \operatorname {Hypergeometric2F1}\left (\frac {1-m}{2},\frac {n+1}{2},\frac {n+3}{2},\cos ^2(e+f x)\right )}{d f (n+1)} \]

[In]

Int[(d*Cos[e + f*x])^n*Sin[e + f*x]^m,x]

[Out]

-(((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)))

Rule 2656

Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b^(2*IntPar
t[(n - 1)/2] + 1))*(b*Sin[e + f*x])^(2*FracPart[(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, Cos[e + f*x]^2], x] /; FreeQ[{a
, b, e, f, m, n}, x] && SimplerQ[n, m]

Rubi steps \begin{align*} \text {integral}& = -\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)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.96 \[ \int (d \cos (e+f x))^n \sin ^m(e+f x) \, dx=\frac {d (d \cos (e+f x))^{-1+n} \cos ^2(e+f x)^{\frac {1-n}{2}} \operatorname {Hypergeometric2F1}\left (\frac {1+m}{2},\frac {1-n}{2},\frac {3+m}{2},\sin ^2(e+f x)\right ) \sin ^{1+m}(e+f x)}{f (1+m)} \]

[In]

Integrate[(d*Cos[e + f*x])^n*Sin[e + f*x]^m,x]

[Out]

(d*(d*Cos[e + f*x])^(-1 + n)*(Cos[e + f*x]^2)^((1 - n)/2)*Hypergeometric2F1[(1 + m)/2, (1 - n)/2, (3 + m)/2, S
in[e + f*x]^2]*Sin[e + f*x]^(1 + m))/(f*(1 + m))

Maple [F]

\[\int \left (d \cos \left (f x +e \right )\right )^{n} \left (\sin ^{m}\left (f x +e \right )\right )d x\]

[In]

int((d*cos(f*x+e))^n*sin(f*x+e)^m,x)

[Out]

int((d*cos(f*x+e))^n*sin(f*x+e)^m,x)

Fricas [F]

\[ \int (d \cos (e+f x))^n \sin ^m(e+f x) \, dx=\int { \left (d \cos \left (f x + e\right )\right )^{n} \sin \left (f x + e\right )^{m} \,d x } \]

[In]

integrate((d*cos(f*x+e))^n*sin(f*x+e)^m,x, algorithm="fricas")

[Out]

integral((d*cos(f*x + e))^n*sin(f*x + e)^m, x)

Sympy [F]

\[ \int (d \cos (e+f x))^n \sin ^m(e+f x) \, dx=\int \left (d \cos {\left (e + f x \right )}\right )^{n} \sin ^{m}{\left (e + f x \right )}\, dx \]

[In]

integrate((d*cos(f*x+e))**n*sin(f*x+e)**m,x)

[Out]

Integral((d*cos(e + f*x))**n*sin(e + f*x)**m, x)

Maxima [F]

\[ \int (d \cos (e+f x))^n \sin ^m(e+f x) \, dx=\int { \left (d \cos \left (f x + e\right )\right )^{n} \sin \left (f x + e\right )^{m} \,d x } \]

[In]

integrate((d*cos(f*x+e))^n*sin(f*x+e)^m,x, algorithm="maxima")

[Out]

integrate((d*cos(f*x + e))^n*sin(f*x + e)^m, x)

Giac [F]

\[ \int (d \cos (e+f x))^n \sin ^m(e+f x) \, dx=\int { \left (d \cos \left (f x + e\right )\right )^{n} \sin \left (f x + e\right )^{m} \,d x } \]

[In]

integrate((d*cos(f*x+e))^n*sin(f*x+e)^m,x, algorithm="giac")

[Out]

integrate((d*cos(f*x + e))^n*sin(f*x + e)^m, x)

Mupad [F(-1)]

Timed out. \[ \int (d \cos (e+f x))^n \sin ^m(e+f x) \, dx=\int {\sin \left (e+f\,x\right )}^m\,{\left (d\,\cos \left (e+f\,x\right )\right )}^n \,d x \]

[In]

int(sin(e + f*x)^m*(d*cos(e + f*x))^n,x)

[Out]

int(sin(e + f*x)^m*(d*cos(e + f*x))^n, x)