Optimal. Leaf size=11 \[ \csc (x)-\frac{\csc ^3(x)}{3} \]
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Rubi [A] time = 0.0156247, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2606} \[ \csc (x)-\frac{\csc ^3(x)}{3} \]
Antiderivative was successfully verified.
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Rule 2606
Rubi steps
\begin{align*} \int \cot ^3(x) \csc (x) \, dx &=-\operatorname{Subst}\left (\int \left (-1+x^2\right ) \, dx,x,\csc (x)\right )\\ &=\csc (x)-\frac{\csc ^3(x)}{3}\\ \end{align*}
Mathematica [A] time = 0.00721, size = 11, normalized size = 1. \[ \csc (x)-\frac{\csc ^3(x)}{3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.009, size = 32, normalized size = 2.9 \begin{align*} -{\frac{ \left ( \cos \left ( x \right ) \right ) ^{4}}{3\, \left ( \sin \left ( x \right ) \right ) ^{3}}}+{\frac{ \left ( \cos \left ( x \right ) \right ) ^{4}}{3\,\sin \left ( x \right ) }}+{\frac{ \left ( 2+ \left ( \cos \left ( x \right ) \right ) ^{2} \right ) \sin \left ( x \right ) }{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.943858, size = 19, normalized size = 1.73 \begin{align*} \frac{3 \, \sin \left (x\right )^{2} - 1}{3 \, \sin \left (x\right )^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.86833, size = 62, normalized size = 5.64 \begin{align*} \frac{3 \, \cos \left (x\right )^{2} - 2}{3 \,{\left (\cos \left (x\right )^{2} - 1\right )} \sin \left (x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.089148, size = 14, normalized size = 1.27 \begin{align*} \frac{3 \sin ^{2}{\left (x \right )} - 1}{3 \sin ^{3}{\left (x \right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.08197, size = 19, normalized size = 1.73 \begin{align*} \frac{3 \, \sin \left (x\right )^{2} - 1}{3 \, \sin \left (x\right )^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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