Optimal. Leaf size=17 \[ \frac{\csc ^4(x)}{4}-\frac{\csc ^6(x)}{6} \]
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Rubi [A] time = 0.0259722, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {2606, 14} \[ \frac{\csc ^4(x)}{4}-\frac{\csc ^6(x)}{6} \]
Antiderivative was successfully verified.
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Rule 2606
Rule 14
Rubi steps
\begin{align*} \int \cot ^3(x) \csc ^4(x) \, dx &=-\operatorname{Subst}\left (\int x^3 \left (-1+x^2\right ) \, dx,x,\csc (x)\right )\\ &=-\operatorname{Subst}\left (\int \left (-x^3+x^5\right ) \, dx,x,\csc (x)\right )\\ &=\frac{\csc ^4(x)}{4}-\frac{\csc ^6(x)}{6}\\ \end{align*}
Mathematica [A] time = 0.0073385, size = 17, normalized size = 1. \[ \frac{\csc ^4(x)}{4}-\frac{\csc ^6(x)}{6} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 22, normalized size = 1.3 \begin{align*} -{\frac{ \left ( \cos \left ( x \right ) \right ) ^{4}}{6\, \left ( \sin \left ( x \right ) \right ) ^{6}}}-{\frac{ \left ( \cos \left ( x \right ) \right ) ^{4}}{12\, \left ( \sin \left ( x \right ) \right ) ^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.940499, size = 19, normalized size = 1.12 \begin{align*} \frac{3 \, \sin \left (x\right )^{2} - 2}{12 \, \sin \left (x\right )^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.01787, size = 86, normalized size = 5.06 \begin{align*} \frac{3 \, \cos \left (x\right )^{2} - 1}{12 \,{\left (\cos \left (x\right )^{6} - 3 \, \cos \left (x\right )^{4} + 3 \, \cos \left (x\right )^{2} - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.103146, size = 14, normalized size = 0.82 \begin{align*} \frac{3 \sin ^{2}{\left (x \right )} - 2}{12 \sin ^{6}{\left (x \right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.05266, size = 24, normalized size = 1.41 \begin{align*} \frac{3 \, \cos \left (x\right )^{2} - 1}{12 \,{\left (\cos \left (x\right )^{2} - 1\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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