Optimal. Leaf size=17 \[ \frac{(a \sin (e+f x))^m}{f m} \]
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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.
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Rule 2592
Rule 30
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.
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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.
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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.
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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.
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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.
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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.
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