Optimal. Leaf size=24 \[ \frac{b (b \csc (e+f x))^{n-1}}{f (1-n)} \]
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Rubi [A] time = 0.0335722, antiderivative size = 24, 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 = {2621, 30} \[ \frac{b (b \csc (e+f x))^{n-1}}{f (1-n)} \]
Antiderivative was successfully verified.
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Rule 2621
Rule 30
Rubi steps
\begin{align*} \int \cos (e+f x) (b \csc (e+f x))^n \, dx &=-\frac{b \operatorname{Subst}\left (\int x^{-2+n} \, dx,x,b \csc (e+f x)\right )}{f}\\ &=\frac{b (b \csc (e+f x))^{-1+n}}{f (1-n)}\\ \end{align*}
Mathematica [A] time = 0.0249491, size = 23, normalized size = 0.96 \[ -\frac{b (b \csc (e+f x))^{n-1}}{f (n-1)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.07, size = 66, normalized size = 2.8 \begin{align*} -2\,{\frac{\tan \left ( 1/2\,fx+e/2 \right ) }{f \left ( -1+n \right ) \left ( 1+ \left ( \tan \left ( 1/2\,fx+e/2 \right ) \right ) ^{2} \right ) }{{\rm e}^{n\ln \left ( 1/2\,{\frac{b \left ( 1+ \left ( \tan \left ( 1/2\,fx+e/2 \right ) \right ) ^{2} \right ) }{\tan \left ( 1/2\,fx+e/2 \right ) }} \right ) }}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.14941, size = 39, normalized size = 1.62 \begin{align*} -\frac{b^{n} \sin \left (f x + e\right )^{-n} \sin \left (f x + e\right )}{f{\left (n - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.69949, size = 59, normalized size = 2.46 \begin{align*} -\frac{\left (\frac{b}{\sin \left (f x + e\right )}\right )^{n} \sin \left (f x + e\right )}{f n - f} \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 (b \csc{\left (e + f x \right )}\right )^{n} \cos{\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 (b \csc \left (f x + e\right )\right )^{n} \cos \left (f x + e\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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