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

Optimal. Leaf size=85 \[ \frac {\cos ^{m-1}(e+f x) \cos ^2(e+f x)^{\frac {1-m}{2}} \csc ^{n-1}(e+f x) \, _2F_1\left (\frac {1-m}{2},\frac {1-n}{2};\frac {3-n}{2};\sin ^2(e+f x)\right )}{f (1-n)} \]

[Out]

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

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Rubi [A]  time = 0.08, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2587, 2577} \[ \frac {\cos ^{m-1}(e+f x) \cos ^2(e+f x)^{\frac {1-m}{2}} \csc ^{n-1}(e+f x) \, _2F_1\left (\frac {1-m}{2},\frac {1-n}{2};\frac {3-n}{2};\sin ^2(e+f x)\right )}{f (1-n)} \]

Antiderivative was successfully verified.

[In]

Int[Cos[e + f*x]^m*Csc[e + f*x]^n,x]

[Out]

(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 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]

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]

Rubi steps

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

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Mathematica [C]  time = 1.97, size = 312, normalized size = 3.67 \[ -\frac {2 (n-3) \sin \left (\frac {1}{2} (e+f x)\right ) \cos ^3\left (\frac {1}{2} (e+f x)\right ) \cos ^m(e+f x) \csc ^n(e+f x) F_1\left (\frac {1}{2}-\frac {n}{2};-m,m-n+1;\frac {3}{2}-\frac {n}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )}{f (n-1) \left (2 \sin ^2\left (\frac {1}{2} (e+f x)\right ) \left (m F_1\left (\frac {3}{2}-\frac {n}{2};1-m,m-n+1;\frac {5}{2}-\frac {n}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )+(m-n+1) F_1\left (\frac {3}{2}-\frac {n}{2};-m,m-n+2;\frac {5}{2}-\frac {n}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )\right )+(n-3) \cos ^2\left (\frac {1}{2} (e+f x)\right ) F_1\left (\frac {1}{2}-\frac {n}{2};-m,m-n+1;\frac {3}{2}-\frac {n}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[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*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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fricas [F]  time = 0.66, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\cos \left (f x + e\right )^{m} \csc \left (f x + e\right )^{n}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \cos \left (f x + e\right )^{m} \csc \left (f x + e\right )^{n}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.65, size = 0, normalized size = 0.00 \[ \int \left (\cos ^{m}\left (f x +e \right )\right ) \left (\csc ^{n}\left (f x +e \right )\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \cos \left (f x + e\right )^{m} \csc \left (f x + e\right )^{n}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int {\cos \left (e+f\,x\right )}^m\,{\left (\frac {1}{\sin \left (e+f\,x\right )}\right )}^n \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(e + f*x)^m*(1/sin(e + f*x))^n,x)

[Out]

int(cos(e + f*x)^m*(1/sin(e + f*x))^n, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \cos ^{m}{\left (e + f x \right )} \csc ^{n}{\left (e + f x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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