Optimal. Leaf size=9 \[ \log (\sin (x))-x \cot (x) \]
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Rubi [A] time = 0.0156671, antiderivative size = 9, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {4184, 3475} \[ \log (\sin (x))-x \cot (x) \]
Antiderivative was successfully verified.
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Rule 4184
Rule 3475
Rubi steps
\begin{align*} \int x \csc ^2(x) \, dx &=-x \cot (x)+\int \cot (x) \, dx\\ &=-x \cot (x)+\log (\sin (x))\\ \end{align*}
Mathematica [A] time = 0.0152348, size = 9, normalized size = 1. \[ \log (\sin (x))-x \cot (x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 10, normalized size = 1.1 \begin{align*} -x\cot \left ( x \right ) +\ln \left ( \sin \left ( x \right ) \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.944753, size = 140, normalized size = 15.56 \begin{align*} \frac{{\left (\cos \left (2 \, x\right )^{2} + \sin \left (2 \, x\right )^{2} - 2 \, \cos \left (2 \, x\right ) + 1\right )} \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} + 2 \, \cos \left (x\right ) + 1\right ) +{\left (\cos \left (2 \, x\right )^{2} + \sin \left (2 \, x\right )^{2} - 2 \, \cos \left (2 \, x\right ) + 1\right )} \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} - 2 \, \cos \left (x\right ) + 1\right ) - 4 \, x \sin \left (2 \, x\right )}{2 \,{\left (\cos \left (2 \, x\right )^{2} + \sin \left (2 \, x\right )^{2} - 2 \, \cos \left (2 \, x\right ) + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.61758, size = 61, normalized size = 6.78 \begin{align*} -\frac{x \cos \left (x\right ) - \log \left (\frac{1}{2} \, \sin \left (x\right )\right ) \sin \left (x\right )}{\sin \left (x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x \csc ^{2}{\left (x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.0947, size = 70, normalized size = 7.78 \begin{align*} \frac{x \tan \left (\frac{1}{2} \, x\right )^{2} + \log \left (\frac{16 \, \tan \left (\frac{1}{2} \, x\right )^{2}}{\tan \left (\frac{1}{2} \, x\right )^{4} + 2 \, \tan \left (\frac{1}{2} \, x\right )^{2} + 1}\right ) \tan \left (\frac{1}{2} \, x\right ) - x}{2 \, \tan \left (\frac{1}{2} \, x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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