3.168 \(\int \cot ^2(e+f x) (a \sin (e+f x))^m \, dx\)

Optimal. Leaf size=69 \[ -\frac{a \cos (e+f x) (a \sin (e+f x))^{m-1} \, _2F_1\left (-\frac{1}{2},\frac{m-1}{2};\frac{m+1}{2};\sin ^2(e+f x)\right )}{f (1-m) \sqrt{\cos ^2(e+f x)}} \]

[Out]

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

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Rubi [A]  time = 0.0921402, antiderivative size = 69, 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 = {2600, 2577} \[ -\frac{a \cos (e+f x) (a \sin (e+f x))^{m-1} \, _2F_1\left (-\frac{1}{2},\frac{m-1}{2};\frac{m+1}{2};\sin ^2(e+f x)\right )}{f (1-m) \sqrt{\cos ^2(e+f x)}} \]

Antiderivative was successfully verified.

[In]

Int[Cot[e + f*x]^2*(a*Sin[e + f*x])^m,x]

[Out]

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

Rule 2600

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

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

Mathematica [A]  time = 0.0797154, size = 66, normalized size = 0.96 \[ \frac{a \sqrt{\cos ^2(e+f x)} \sec (e+f x) (a \sin (e+f x))^{m-1} \, _2F_1\left (-\frac{1}{2},\frac{m-1}{2};\frac{m+1}{2};\sin ^2(e+f x)\right )}{f (m-1)} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[e + f*x]^2*(a*Sin[e + f*x])^m,x]

[Out]

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

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Maple [F]  time = 0.206, size = 0, normalized size = 0. \begin{align*} \int \left ( \cot \left ( fx+e \right ) \right ) ^{2} \left ( a\sin \left ( fx+e \right ) \right ) ^{m}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(f*x+e)^2*(a*sin(f*x+e))^m,x)

[Out]

int(cot(f*x+e)^2*(a*sin(f*x+e))^m,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a \sin \left (f x + e\right )\right )^{m} \cot \left (f x + e\right )^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(f*x+e)^2*(a*sin(f*x+e))^m,x, algorithm="maxima")

[Out]

integrate((a*sin(f*x + e))^m*cot(f*x + e)^2, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\left (a \sin \left (f x + e\right )\right )^{m} \cot \left (f x + e\right )^{2}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(f*x+e)^2*(a*sin(f*x+e))^m,x, algorithm="fricas")

[Out]

integral((a*sin(f*x + e))^m*cot(f*x + e)^2, 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} \cot ^{2}{\left (e + f x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(f*x+e)**2*(a*sin(f*x+e))**m,x)

[Out]

Integral((a*sin(e + f*x))**m*cot(e + f*x)**2, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a \sin \left (f x + e\right )\right )^{m} \cot \left (f x + e\right )^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(f*x+e)^2*(a*sin(f*x+e))^m,x, algorithm="giac")

[Out]

integrate((a*sin(f*x + e))^m*cot(f*x + e)^2, x)