\(\int (b \csc (e+f x))^n (a \sin (e+f x))^m \, dx\) [538]

Optimal result

Integrand size = 21, antiderivative size = 87 \[ \int (b \csc (e+f x))^n (a \sin (e+f x))^m \, dx=\frac {\cos (e+f x) (b \csc (e+f x))^n \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (1+m-n),\frac {1}{2} (3+m-n),\sin ^2(e+f x)\right ) (a \sin (e+f x))^{1+m}}{a f (1+m-n) \sqrt {\cos ^2(e+f x)}} \]

[Out]

cos(f*x+e)*(b*csc(f*x+e))^n*hypergeom([1/2, 1/2+1/2*m-1/2*n],[3/2+1/2*m-1/2*n],sin(f*x+e)^2)*(a*sin(f*x+e))^(1
+m)/a/f/(1+m-n)/(cos(f*x+e)^2)^(1/2)

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2668, 2722} \[ \int (b \csc (e+f x))^n (a \sin (e+f x))^m \, dx=\frac {\cos (e+f x) (a \sin (e+f x))^{m+1} (b \csc (e+f x))^n \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (m-n+1),\frac {1}{2} (m-n+3),\sin ^2(e+f x)\right )}{a f (m-n+1) \sqrt {\cos ^2(e+f x)}} \]

[In]

Int[(b*Csc[e + f*x])^n*(a*Sin[e + f*x])^m,x]

[Out]

(Cos[e + f*x]*(b*Csc[e + f*x])^n*Hypergeometric2F1[1/2, (1 + m - n)/2, (3 + m - n)/2, Sin[e + f*x]^2]*(a*Sin[e
+ f*x])^(1 + m))/(a*f*(1 + m - n)*Sqrt[Cos[e + f*x]^2])

Rule 2668

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[(a*b)^IntPar
t[n]*(a*Sin[e + f*x])^FracPart[n]*(b*Csc[e + f*x])^FracPart[n], Int[(a*Sin[e + f*x])^(m - n), x], x] /; FreeQ[
{a, b, e, f, m, n}, x] &&  !IntegerQ[m] &&  !IntegerQ[n]

Rule 2722

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]))*Hypergeometric2F1[1/2, (n + 1)/2, (n + 3)/2, Sin[c + d*x]^2], x] /; FreeQ[{b, c, d, n
}, x] &&  !IntegerQ[2*n]

Rubi steps \begin{align*} \text {integral}& = \left ((b \csc (e+f x))^n (a \sin (e+f x))^n\right ) \int (a \sin (e+f x))^{m-n} \, dx \\ & = \frac {\cos (e+f x) (b \csc (e+f x))^n \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (1+m-n),\frac {1}{2} (3+m-n),\sin ^2(e+f x)\right ) (a \sin (e+f x))^{1+m}}{a f (1+m-n) \sqrt {\cos ^2(e+f x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 8.10 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.17 \[ \int (b \csc (e+f x))^n (a \sin (e+f x))^m \, dx=\frac {2 (b \csc (e+f x))^n \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (1+m-n),1+m-n,\frac {1}{2} (3+m-n),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \sec ^2\left (\frac {1}{2} (e+f x)\right )^{m-n} (a \sin (e+f x))^m \tan \left (\frac {1}{2} (e+f x)\right )}{f (1+m-n)} \]

[In]

Integrate[(b*Csc[e + f*x])^n*(a*Sin[e + f*x])^m,x]

[Out]

(2*(b*Csc[e + f*x])^n*Hypergeometric2F1[(1 + m - n)/2, 1 + m - n, (3 + m - n)/2, -Tan[(e + f*x)/2]^2]*(Sec[(e
+ f*x)/2]^2)^(m - n)*(a*Sin[e + f*x])^m*Tan[(e + f*x)/2])/(f*(1 + m - n))

Maple [F]

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

[In]

int((b*csc(f*x+e))^n*(a*sin(f*x+e))^m,x)

[Out]

int((b*csc(f*x+e))^n*(a*sin(f*x+e))^m,x)

Fricas [F]

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

[In]

integrate((b*csc(f*x+e))^n*(a*sin(f*x+e))^m,x, algorithm="fricas")

[Out]

integral((b*csc(f*x + e))^n*(a*sin(f*x + e))^m, x)

Sympy [F]

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

[In]

integrate((b*csc(f*x+e))**n*(a*sin(f*x+e))**m,x)

[Out]

Integral((a*sin(e + f*x))**m*(b*csc(e + f*x))**n, x)

Maxima [F]

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

[In]

integrate((b*csc(f*x+e))^n*(a*sin(f*x+e))^m,x, algorithm="maxima")

[Out]

integrate((b*csc(f*x + e))^n*(a*sin(f*x + e))^m, x)

Giac [F]

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

[In]

integrate((b*csc(f*x+e))^n*(a*sin(f*x+e))^m,x, algorithm="giac")

[Out]

integrate((b*csc(f*x + e))^n*(a*sin(f*x + e))^m, x)

Mupad [F(-1)]

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

[In]

int((a*sin(e + f*x))^m*(b/sin(e + f*x))^n,x)

[Out]

int((a*sin(e + f*x))^m*(b/sin(e + f*x))^n, x)