Optimal. Leaf size=43 \[ \frac{(b \csc (e+f x))^m}{f m}-\frac{(b \csc (e+f x))^{m+2}}{b^2 f (m+2)} \]
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Rubi [A] time = 0.0472561, antiderivative size = 43, 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 = {2606, 14} \[ \frac{(b \csc (e+f x))^m}{f m}-\frac{(b \csc (e+f x))^{m+2}}{b^2 f (m+2)} \]
Antiderivative was successfully verified.
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Rule 2606
Rule 14
Rubi steps
\begin{align*} \int \cot ^3(e+f x) (b \csc (e+f x))^m \, dx &=-\frac{b \operatorname{Subst}\left (\int (b x)^{-1+m} \left (-1+x^2\right ) \, dx,x,\csc (e+f x)\right )}{f}\\ &=-\frac{b \operatorname{Subst}\left (\int \left (-(b x)^{-1+m}+\frac{(b x)^{1+m}}{b^2}\right ) \, dx,x,\csc (e+f x)\right )}{f}\\ &=\frac{(b \csc (e+f x))^m}{f m}-\frac{(b \csc (e+f x))^{2+m}}{b^2 f (2+m)}\\ \end{align*}
Mathematica [A] time = 0.0819227, size = 36, normalized size = 0.84 \[ \frac{\left (-m \csc ^2(e+f x)+m+2\right ) (b \csc (e+f x))^m}{f m (m+2)} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.601, size = 6612, normalized size = 153.8 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.968742, size = 68, normalized size = 1.58 \begin{align*} \frac{\frac{b^{m} \sin \left (f x + e\right )^{-m}}{m} - \frac{b^{m} \sin \left (f x + e\right )^{-m}}{{\left (m + 2\right )} \sin \left (f x + e\right )^{2}}}{f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.66319, size = 134, normalized size = 3.12 \begin{align*} -\frac{{\left ({\left (m + 2\right )} \cos \left (f x + e\right )^{2} - 2\right )} \left (\frac{b}{\sin \left (f x + e\right )}\right )^{m}}{f m^{2} -{\left (f m^{2} + 2 \, f m\right )} \cos \left (f x + e\right )^{2} + 2 \, f m} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \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 )^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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