Optimal. Leaf size=43 \[ -\frac{(a \sin (e+f x)+a)^{m+1} \, _2F_1(1,m+1;m+2;\sin (e+f x)+1)}{a f (m+1)} \]
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Rubi [A] time = 0.0439523, antiderivative size = 43, 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 = {2707, 65} \[ -\frac{(a \sin (e+f x)+a)^{m+1} \, _2F_1(1,m+1;m+2;\sin (e+f x)+1)}{a f (m+1)} \]
Antiderivative was successfully verified.
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Rule 2707
Rule 65
Rubi steps
\begin{align*} \int \cot (e+f x) (a+a \sin (e+f x))^m \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(a+x)^m}{x} \, dx,x,a \sin (e+f x)\right )}{f}\\ &=-\frac{\, _2F_1(1,1+m;2+m;1+\sin (e+f x)) (a+a \sin (e+f x))^{1+m}}{a f (1+m)}\\ \end{align*}
Mathematica [A] time = 0.0589381, size = 43, normalized size = 1. \[ -\frac{(a \sin (e+f x)+a)^{m+1} \, _2F_1(1,m+1;m+2;\sin (e+f x)+1)}{a f (m+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.821, size = 0, normalized size = 0. \begin{align*} \int \cot \left ( fx+e \right ) \left ( a+a\sin \left ( fx+e \right ) \right ) ^{m}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a \sin \left (f x + e\right ) + a\right )}^{m} \cot \left (f x + e\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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 ) + a\right )}^{m} \cot \left (f x + e\right ), x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a \left (\sin{\left (e + f x \right )} + 1\right )\right )^{m} \cot{\left (e + f x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a \sin \left (f x + e\right ) + a\right )}^{m} \cot \left (f x + e\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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