Optimal. Leaf size=74 \[ -\frac{2^{m+\frac{1}{2}} \cot (e+f x) (\csc (e+f x)+1)^{-m-\frac{1}{2}} (a \csc (e+f x)+a)^m \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{1}{2}-m,\frac{3}{2},\frac{1}{2} (1-\csc (e+f x))\right )}{f} \]
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Rubi [A] time = 0.0564489, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {3828, 3827, 69} \[ -\frac{2^{m+\frac{1}{2}} \cot (e+f x) (\csc (e+f x)+1)^{-m-\frac{1}{2}} (a \csc (e+f x)+a)^m \, _2F_1\left (\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};\frac{1}{2} (1-\csc (e+f x))\right )}{f} \]
Antiderivative was successfully verified.
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Rule 3828
Rule 3827
Rule 69
Rubi steps
\begin{align*} \int \csc (e+f x) (a+a \csc (e+f x))^m \, dx &=\left ((1+\csc (e+f x))^{-m} (a+a \csc (e+f x))^m\right ) \int \csc (e+f x) (1+\csc (e+f x))^m \, dx\\ &=\frac{\left (\cot (e+f x) (1+\csc (e+f x))^{-\frac{1}{2}-m} (a+a \csc (e+f x))^m\right ) \operatorname{Subst}\left (\int \frac{(1+x)^{-\frac{1}{2}+m}}{\sqrt{1-x}} \, dx,x,\csc (e+f x)\right )}{f \sqrt{1-\csc (e+f x)}}\\ &=-\frac{2^{\frac{1}{2}+m} \cot (e+f x) (1+\csc (e+f x))^{-\frac{1}{2}-m} (a+a \csc (e+f x))^m \, _2F_1\left (\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};\frac{1}{2} (1-\csc (e+f x))\right )}{f}\\ \end{align*}
Mathematica [A] time = 0.202145, size = 60, normalized size = 0.81 \[ -\frac{\left (\tan \left (\frac{1}{2} (e+f x)\right )+1\right )^{-2 m} (a (\csc (e+f x)+1))^m \text{Hypergeometric2F1}\left (-2 m,-m,1-m,-\tan \left (\frac{1}{2} (e+f x)\right )\right )}{f m} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.429, size = 0, normalized size = 0. \begin{align*} \int \csc \left ( fx+e \right ) \left ( a+a\csc \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 \csc \left (f x + e\right ) + a\right )}^{m} \csc \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 \csc \left (f x + e\right ) + a\right )}^{m} \csc \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 (\csc{\left (e + f x \right )} + 1\right )\right )^{m} \csc{\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 \csc \left (f x + e\right ) + a\right )}^{m} \csc \left (f x + e\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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