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

Optimal. Leaf size=17 \[ \frac{(a \sin (e+f x))^m}{f m} \]

[Out]

(a*Sin[e + f*x])^m/(f*m)

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Rubi [A]  time = 0.0286763, antiderivative size = 17, normalized size of antiderivative = 1., 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 = {2592, 30} \[ \frac{(a \sin (e+f x))^m}{f m} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*Sin[e + f*x])^m/(f*m)

Rule 2592

Int[((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*tan[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> With[{ff = FreeFactors[S
in[e + f*x], x]}, Dist[ff/f, Subst[Int[(ff*x)^(m + n)/(a^2 - ff^2*x^2)^((n + 1)/2), x], x, (a*Sin[e + f*x])/ff
], x]] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n + 1)/2]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \cot (e+f x) (a \sin (e+f x))^m \, dx &=\frac{\operatorname{Subst}\left (\int x^{-1+m} \, dx,x,a \sin (e+f x)\right )}{f}\\ &=\frac{(a \sin (e+f x))^m}{f m}\\ \end{align*}

Mathematica [A]  time = 0.0093166, size = 17, normalized size = 1. \[ \frac{(a \sin (e+f x))^m}{f m} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*Sin[e + f*x])^m/(f*m)

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Maple [A]  time = 0.011, size = 18, normalized size = 1.1 \begin{align*}{\frac{ \left ( a\sin \left ( fx+e \right ) \right ) ^{m}}{fm}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a*sin(f*x+e))^m/f/m

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Maxima [A]  time = 0.945853, size = 24, normalized size = 1.41 \begin{align*} \frac{a^{m} \sin \left (f x + e\right )^{m}}{f m} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a^m*sin(f*x + e)^m/(f*m)

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Fricas [A]  time = 1.58187, size = 35, normalized size = 2.06 \begin{align*} \frac{\left (a \sin \left (f x + e\right )\right )^{m}}{f m} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a*sin(f*x + e))^m/(f*m)

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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{\left (e + f x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.17099, size = 24, normalized size = 1.41 \begin{align*} \frac{\left (a \sin \left (f x + e\right )\right )^{m}}{f m} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a*sin(f*x + e))^m/(f*m)