3.285 \(\int (a \cos (e+f x))^m \csc ^n(e+f x) \, dx\)

Optimal. Leaf size=88 \[ \frac{a \cos ^2(e+f x)^{\frac{1-m}{2}} \csc ^{n-1}(e+f x) (a \cos (e+f x))^{m-1} \, _2F_1\left (\frac{1-m}{2},\frac{1-n}{2};\frac{3-n}{2};\sin ^2(e+f x)\right )}{f (1-n)} \]

[Out]

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

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Rubi [A]  time = 0.0912086, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {2587, 2577} \[ \frac{a \cos ^2(e+f x)^{\frac{1-m}{2}} \csc ^{n-1}(e+f x) (a \cos (e+f x))^{m-1} \, _2F_1\left (\frac{1-m}{2},\frac{1-n}{2};\frac{3-n}{2};\sin ^2(e+f x)\right )}{f (1-n)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2587

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

Rule 2577

Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(b^(2*IntPart
[(n - 1)/2] + 1)*(b*Cos[e + f*x])^(2*FracPart[(n - 1)/2])*(a*Sin[e + f*x])^(m + 1)*Hypergeometric2F1[(1 + m)/2
, (1 - n)/2, (3 + m)/2, Sin[e + f*x]^2])/(a*f*(m + 1)*(Cos[e + f*x]^2)^FracPart[(n - 1)/2]), x] /; FreeQ[{a, b
, e, f, m, n}, x]

Rubi steps

\begin{align*} \int (a \cos (e+f x))^m \csc ^n(e+f x) \, dx &=\left (\csc ^n(e+f x) \sin ^n(e+f x)\right ) \int (a \cos (e+f x))^m \sin ^{-n}(e+f x) \, dx\\ &=\frac{a (a \cos (e+f x))^{-1+m} \cos ^2(e+f x)^{\frac{1-m}{2}} \csc ^{-1+n}(e+f x) \, _2F_1\left (\frac{1-m}{2},\frac{1-n}{2};\frac{3-n}{2};\sin ^2(e+f x)\right )}{f (1-n)}\\ \end{align*}

Mathematica [C]  time = 0.300098, size = 314, normalized size = 3.57 \[ -\frac{2 (n-3) \sin \left (\frac{1}{2} (e+f x)\right ) \cos ^3\left (\frac{1}{2} (e+f x)\right ) \csc ^n(e+f x) (a \cos (e+f x))^m F_1\left (\frac{1}{2}-\frac{n}{2};-m,m-n+1;\frac{3}{2}-\frac{n}{2};\tan ^2\left (\frac{1}{2} (e+f x)\right ),-\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )}{f (n-1) \left (2 \sin ^2\left (\frac{1}{2} (e+f x)\right ) \left (m F_1\left (\frac{3}{2}-\frac{n}{2};1-m,m-n+1;\frac{5}{2}-\frac{n}{2};\tan ^2\left (\frac{1}{2} (e+f x)\right ),-\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )+(m-n+1) F_1\left (\frac{3}{2}-\frac{n}{2};-m,m-n+2;\frac{5}{2}-\frac{n}{2};\tan ^2\left (\frac{1}{2} (e+f x)\right ),-\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )\right )+(n-3) \cos ^2\left (\frac{1}{2} (e+f x)\right ) F_1\left (\frac{1}{2}-\frac{n}{2};-m,m-n+1;\frac{3}{2}-\frac{n}{2};\tan ^2\left (\frac{1}{2} (e+f x)\right ),-\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [F]  time = 1.004, size = 0, normalized size = 0. \begin{align*} \int \left ( a\cos \left ( fx+e \right ) \right ) ^{m} \left ( \csc \left ( fx+e \right ) \right ) ^{n}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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