\(\int (b \cot (e+f x))^n (a \sin (e+f x))^m \, dx\) [39]

Optimal result

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

[Out]

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

Rubi [A] (verified)

Time = 0.11 (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 = {2683, 2697} \[ \int (b \cot (e+f x))^n (a \sin (e+f x))^m \, dx=-\frac {(a \sin (e+f x))^m (b \cot (e+f x))^{n+1} \sin ^2(e+f x)^{\frac {1}{2} (-m+n+1)} \operatorname {Hypergeometric2F1}\left (\frac {n+1}{2},\frac {1}{2} (-m+n+1),\frac {n+3}{2},\cos ^2(e+f x)\right )}{b f (n+1)} \]

[In]

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

[Out]

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

Rule 2683

Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[(a*Cos[e + f*
x])^FracPart[m]*(Sec[e + f*x]/a)^FracPart[m], Int[(b*Tan[e + f*x])^n/(Sec[e + f*x]/a)^m, x], x] /; FreeQ[{a, b
, e, f, m, n}, x] &&  !IntegerQ[m] &&  !IntegerQ[n]

Rule 2697

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(a*Sec[e + f
*x])^m*(b*Tan[e + f*x])^(n + 1)*((Cos[e + f*x]^2)^((m + n + 1)/2)/(b*f*(n + 1)))*Hypergeometric2F1[(n + 1)/2,
(m + n + 1)/2, (n + 3)/2, Sin[e + f*x]^2], x] /; FreeQ[{a, b, e, f, m, n}, x] &&  !IntegerQ[(n - 1)/2] &&  !In
tegerQ[m/2]

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

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.

Time = 2.69 (sec) , antiderivative size = 289, normalized size of antiderivative = 3.32 \[ \int (b \cot (e+f x))^n (a \sin (e+f x))^m \, dx=\frac {(3+m-n) \operatorname {AppellF1}\left (\frac {1}{2} (1+m-n),-n,1+m,\frac {1}{2} (3+m-n),\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) (b \cot (e+f x))^n \sin (e+f x) (a \sin (e+f x))^m}{f (1+m-n) \left ((3+m-n) \operatorname {AppellF1}\left (\frac {1}{2} (1+m-n),-n,1+m,\frac {1}{2} (3+m-n),\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )-2 \left (n \operatorname {AppellF1}\left (\frac {1}{2} (3+m-n),1-n,1+m,\frac {1}{2} (5+m-n),\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )+(1+m) \operatorname {AppellF1}\left (\frac {1}{2} (3+m-n),-n,2+m,\frac {1}{2} (5+m-n),\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )\right ) \tan ^2\left (\frac {1}{2} (e+f x)\right )\right )} \]

[In]

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

[Out]

((3 + m - n)*AppellF1[(1 + m - n)/2, -n, 1 + m, (3 + m - n)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(b*Cot
[e + f*x])^n*Sin[e + f*x]*(a*Sin[e + f*x])^m)/(f*(1 + m - n)*((3 + m - n)*AppellF1[(1 + m - n)/2, -n, 1 + m, (
3 + m - n)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - 2*(n*AppellF1[(3 + m - n)/2, 1 - n, 1 + m, (5 + m - n
)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (1 + m)*AppellF1[(3 + m - n)/2, -n, 2 + m, (5 + m - n)/2, Tan[
(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2))

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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