Optimal. Leaf size=13 \[ -\frac{1}{3} \cot ^3(x)-\cot (x) \]
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Rubi [A] time = 0.0065964, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 4, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3767} \[ -\frac{1}{3} \cot ^3(x)-\cot (x) \]
Antiderivative was successfully verified.
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Rule 3767
Rubi steps
\begin{align*} \int \csc ^4(x) \, dx &=-\operatorname{Subst}\left (\int \left (1+x^2\right ) \, dx,x,\cot (x)\right )\\ &=-\cot (x)-\frac{\cot ^3(x)}{3}\\ \end{align*}
Mathematica [A] time = 0.0023914, size = 17, normalized size = 1.31 \[ -\frac{2 \cot (x)}{3}-\frac{1}{3} \cot (x) \csc ^2(x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.026, size = 12, normalized size = 0.9 \begin{align*} \left ( -{\frac{2}{3}}-{\frac{ \left ( \csc \left ( x \right ) \right ) ^{2}}{3}} \right ) \cot \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.929989, size = 19, normalized size = 1.46 \begin{align*} -\frac{3 \, \tan \left (x\right )^{2} + 1}{3 \, \tan \left (x\right )^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.96192, size = 73, normalized size = 5.62 \begin{align*} -\frac{2 \, \cos \left (x\right )^{3} - 3 \, \cos \left (x\right )}{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 = 0.057034, size = 20, normalized size = 1.54 \begin{align*} - \frac{2 \cos{\left (x \right )}}{3 \sin{\left (x \right )}} - \frac{\cos{\left (x \right )}}{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 = 1.05834, size = 19, normalized size = 1.46 \begin{align*} -\frac{3 \, \tan \left (x\right )^{2} + 1}{3 \, \tan \left (x\right )^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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