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

Optimal. Leaf size=87 \[ \frac{\cos (e+f x) (a \sin (e+f x))^{m+1} (b \csc (e+f x))^n \, _2F_1\left (\frac{1}{2},\frac{1}{2} (m-n+1);\frac{1}{2} (m-n+3);\sin ^2(e+f x)\right )}{a f (m-n+1) \sqrt{\cos ^2(e+f x)}} \]

[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])

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Rubi [A]  time = 0.0636076, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2588, 2643} \[ \frac{\cos (e+f x) (a \sin (e+f x))^{m+1} (b \csc (e+f x))^n \, _2F_1\left (\frac{1}{2},\frac{1}{2} (m-n+1);\frac{1}{2} (m-n+3);\sin ^2(e+f x)\right )}{a f (m-n+1) \sqrt{\cos ^2(e+f x)}} \]

Antiderivative was successfully verified.

[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 2588

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 2643

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(Cos[c + d*x]*(b*Sin[c + d*x])^(n + 1)*Hypergeomet
ric2F1[1/2, (n + 1)/2, (n + 3)/2, Sin[c + d*x]^2])/(b*d*(n + 1)*Sqrt[Cos[c + d*x]^2]), x] /; FreeQ[{b, c, d, n
}, x] &&  !IntegerQ[2*n]

Rubi steps

\begin{align*} \int (b \csc (e+f x))^n (a \sin (e+f x))^m \, dx &=\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 \, _2F_1\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]  time = 9.79536, size = 102, normalized size = 1.17 \[ \frac{2 \tan \left (\frac{1}{2} (e+f x)\right ) (a \sin (e+f x))^m (b \csc (e+f x))^n \sec ^2\left (\frac{1}{2} (e+f x)\right )^{m-n} \, _2F_1\left (\frac{1}{2} (m-n+1),m-n+1;\frac{1}{2} (m-n+3);-\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )}{f (m-n+1)} \]

Antiderivative was successfully verified.

[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))

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Maple [F]  time = 0.781, size = 0, normalized size = 0. \begin{align*} \int \left ( b\csc \left ( fx+e \right ) \right ) ^{n} \left ( a\sin \left ( fx+e \right ) \right ) ^{m}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (b \csc \left (f x + e\right )\right )^{n} \left (a \sin \left (f x + e\right )\right )^{m}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\left (b \csc \left (f x + e\right )\right )^{n} \left (a \sin \left (f x + e\right )\right )^{m}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a \sin{\left (e + f x \right )}\right )^{m} \left (b \csc{\left (e + f x \right )}\right )^{n}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (b \csc \left (f x + e\right )\right )^{n} \left (a \sin \left (f x + e\right )\right )^{m}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)