Optimal. Leaf size=52 \[ \frac{b (b \csc (e+f x))^{n-1}}{f (1-n)}-\frac{b^3 (b \csc (e+f x))^{n-3}}{f (3-n)} \]
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Rubi [A] time = 0.0516861, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {2621, 14} \[ \frac{b (b \csc (e+f x))^{n-1}}{f (1-n)}-\frac{b^3 (b \csc (e+f x))^{n-3}}{f (3-n)} \]
Antiderivative was successfully verified.
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Rule 2621
Rule 14
Rubi steps
\begin{align*} \int \cos ^3(e+f x) (b \csc (e+f x))^n \, dx &=-\frac{b^3 \operatorname{Subst}\left (\int x^{-4+n} \left (-1+\frac{x^2}{b^2}\right ) \, dx,x,b \csc (e+f x)\right )}{f}\\ &=-\frac{b^3 \operatorname{Subst}\left (\int \left (-x^{-4+n}+\frac{x^{-2+n}}{b^2}\right ) \, dx,x,b \csc (e+f x)\right )}{f}\\ &=-\frac{b^3 (b \csc (e+f x))^{-3+n}}{f (3-n)}+\frac{b (b \csc (e+f x))^{-1+n}}{f (1-n)}\\ \end{align*}
Mathematica [A] time = 0.126489, size = 45, normalized size = 0.87 \[ -\frac{b ((n-1) \cos (2 (e+f x))+n-5) (b \csc (e+f x))^{n-1}}{2 f (n-3) (n-1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.153, size = 0, normalized size = 0. \begin{align*} \int \left ( \cos \left ( fx+e \right ) \right ) ^{3} \left ( b\csc \left ( fx+e \right ) \right ) ^{n}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.1554, size = 78, normalized size = 1.5 \begin{align*} \frac{\frac{b^{n} \sin \left (f x + e\right )^{-n} \sin \left (f x + e\right )^{3}}{n - 3} - \frac{b^{n} \sin \left (f x + e\right )^{-n} \sin \left (f x + e\right )}{n - 1}}{f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.7469, size = 115, normalized size = 2.21 \begin{align*} -\frac{{\left ({\left (n - 1\right )} \cos \left (f x + e\right )^{2} - 2\right )} \left (\frac{b}{\sin \left (f x + e\right )}\right )^{n} \sin \left (f x + e\right )}{f n^{2} - 4 \, f n + 3 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 )^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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