3.40 \(\int (a \cos (e+f x))^m (b \cot (e+f x))^n \, dx\)

Optimal. Leaf size=84 \[ -\frac{\sin ^2(e+f x)^{\frac{n+1}{2}} (a \cos (e+f x))^m (b \cot (e+f x))^{n+1} \text{Hypergeometric2F1}\left (\frac{n+1}{2},\frac{1}{2} (m+n+1),\frac{1}{2} (m+n+3),\cos ^2(e+f x)\right )}{b f (m+n+1)} \]

[Out]

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

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Rubi [A]  time = 0.101382, antiderivative size = 84, 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 = {2602, 2576} \[ -\frac{\sin ^2(e+f x)^{\frac{n+1}{2}} (a \cos (e+f x))^m (b \cot (e+f x))^{n+1} \, _2F_1\left (\frac{n+1}{2},\frac{1}{2} (m+n+1);\frac{1}{2} (m+n+3);\cos ^2(e+f x)\right )}{b f (m+n+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2602

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

Rule 2576

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

Rubi steps

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

Mathematica [A]  time = 0.510996, size = 83, normalized size = 0.99 \[ -\frac{b \sec ^2(e+f x)^{m/2} (a \cos (e+f x))^m (b \cot (e+f x))^{n-1} \text{Hypergeometric2F1}\left (\frac{m+2}{2},\frac{1-n}{2},\frac{3-n}{2},-\tan ^2(e+f x)\right )}{f (n-1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 1.02, size = 0, normalized size = 0. \begin{align*} \int \left ( a\cos \left ( fx+e \right ) \right ) ^{m} \left ( b\cot \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*(b*cot(f*x+e))^n,x)

[Out]

int((a*cos(f*x+e))^m*(b*cot(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} \left (b \cot \left (f x + e\right )\right )^{n}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((a*cos(f*x + e))^m*(b*cot(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} \left (b \cot \left (f x + e\right )\right )^{n}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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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} \left (b \cot \left (f x + e\right )\right )^{n}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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