3.281 \(\int \csc ^n(e+f x) (a \sec (e+f x))^m \, dx\)

Optimal. Leaf size=86 \[ \frac{\cos ^2(e+f x)^{\frac{m+1}{2}} \csc ^{n-1}(e+f x) (a \sec (e+f x))^{m+1} \text{Hypergeometric2F1}\left (\frac{m+1}{2},\frac{1-n}{2},\frac{3-n}{2},\sin ^2(e+f x)\right )}{a f (1-n)} \]

[Out]

((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]*(a*Sec[e + f*x])^(1 + m))/(a*f*(1 - n))

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Rubi [A]  time = 0.0864896, antiderivative size = 86, 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 = {2631, 2577} \[ \frac{\cos ^2(e+f x)^{\frac{m+1}{2}} \csc ^{n-1}(e+f x) (a \sec (e+f x))^{m+1} \, _2F_1\left (\frac{m+1}{2},\frac{1-n}{2};\frac{3-n}{2};\sin ^2(e+f x)\right )}{a f (1-n)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((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]*(a*Sec[e + f*x])^(1 + m))/(a*f*(1 - n))

Rule 2631

Int[(csc[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*((b_.)*sec[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[(a^2*(a*Csc[e
+ f*x])^(m - 1)*(b*Sec[e + f*x])^(n + 1)*(a*Sin[e + f*x])^(m - 1)*(b*Cos[e + f*x])^(n + 1))/b^2, Int[1/((a*Si
n[e + f*x])^m*(b*Cos[e + f*x])^n), x], x] /; FreeQ[{a, b, e, f, m, n}, x] &&  !SimplerQ[-m, -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 \csc ^n(e+f x) (a \sec (e+f x))^m \, dx &=\frac{\left ((a \cos (e+f x))^{1+m} \csc ^{-1+n}(e+f x) (a \sec (e+f x))^{1+m} \sin ^{-1+n}(e+f x)\right ) \int (a \cos (e+f x))^{-m} \sin ^{-n}(e+f x) \, dx}{a^2}\\ &=\frac{\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 ) (a \sec (e+f x))^{1+m}}{a f (1-n)}\\ \end{align*}

Mathematica [C]  time = 0.549284, size = 280, normalized size = 3.26 \[ -\frac{(n-3) \csc ^{n-1}(e+f x) (a \sec (e+f x))^m F_1\left (\frac{1}{2}-\frac{n}{2};m,-m-n+1;\frac{3}{2}-\frac{n}{2};\tan ^2\left (\frac{1}{2} (e+f x)\right ),-\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )}{f (n-1) \left ((n-3) F_1\left (\frac{1}{2}-\frac{n}{2};m,-m-n+1;\frac{3}{2}-\frac{n}{2};\tan ^2\left (\frac{1}{2} (e+f x)\right ),-\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )-2 \tan ^2\left (\frac{1}{2} (e+f x)\right ) \left ((m+n-1) F_1\left (\frac{3}{2}-\frac{n}{2};m,-m-n+2;\frac{5}{2}-\frac{n}{2};\tan ^2\left (\frac{1}{2} (e+f x)\right ),-\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )+m F_1\left (\frac{3}{2}-\frac{n}{2};m+1,-m-n+1;\frac{5}{2}-\frac{n}{2};\tan ^2\left (\frac{1}{2} (e+f x)\right ),-\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [F]  time = 0.509, size = 0, normalized size = 0. \begin{align*} \int \left ( \csc \left ( fx+e \right ) \right ) ^{n} \left ( a\sec \left ( fx+e \right ) \right ) ^{m}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a \sec \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(csc(f*x+e)^n*(a*sec(f*x+e))^m,x, algorithm="maxima")

[Out]

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

[Out]

integral((a*sec(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 \sec{\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(csc(f*x+e)**n*(a*sec(f*x+e))**m,x)

[Out]

Integral((a*sec(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 \sec \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(csc(f*x+e)^n*(a*sec(f*x+e))^m,x, algorithm="giac")

[Out]

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