Optimal. Leaf size=18 \[ -\frac{(b \csc (e+f x))^m}{f m} \]
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Rubi [A] time = 0.0214582, antiderivative size = 18, 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 = {2606, 32} \[ -\frac{(b \csc (e+f x))^m}{f m} \]
Antiderivative was successfully verified.
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Rule 2606
Rule 32
Rubi steps
\begin{align*} \int \cot (e+f x) (b \csc (e+f x))^m \, dx &=-\frac{b \operatorname{Subst}\left (\int (b x)^{-1+m} \, dx,x,\csc (e+f x)\right )}{f}\\ &=-\frac{(b \csc (e+f x))^m}{f m}\\ \end{align*}
Mathematica [A] time = 0.0174604, size = 18, normalized size = 1. \[ -\frac{(b \csc (e+f x))^m}{f m} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 19, normalized size = 1.1 \begin{align*} -{\frac{ \left ( b\csc \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 = 0.969372, size = 28, normalized size = 1.56 \begin{align*} -\frac{b^{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 = 1.60248, size = 36, normalized size = 2. \begin{align*} -\frac{\left (\frac{b}{\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 [A] time = 0.523902, size = 56, normalized size = 3.11 \begin{align*} \begin{cases} x \cot{\left (e \right )} & \text{for}\: f = 0 \wedge m = 0 \\x \left (b \csc{\left (e \right )}\right )^{m} \cot{\left (e \right )} & \text{for}\: f = 0 \\- \frac{\log{\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{2 f} + \frac{\log{\left (\tan{\left (e + f x \right )} \right )}}{f} & \text{for}\: m = 0 \\- \frac{b^{m} \csc ^{m}{\left (e + f x \right )}}{f m} & \text{otherwise} \end{cases} \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 )^{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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