Optimal. Leaf size=9 \[ \frac{1}{2} \log \left (\sin \left (x^2\right )\right ) \]
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Rubi [A] time = 0.0066126, 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 = {3748, 3475} \[ \frac{1}{2} \log \left (\sin \left (x^2\right )\right ) \]
Antiderivative was successfully verified.
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Rule 3748
Rule 3475
Rubi steps
\begin{align*} \int x \cot \left (x^2\right ) \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \cot (x) \, dx,x,x^2\right )\\ &=\frac{1}{2} \log \left (\sin \left (x^2\right )\right )\\ \end{align*}
Mathematica [B] time = 0.0086499, size = 19, normalized size = 2.11 \[ \frac{1}{2} \log \left (\tan \left (x^2\right )\right )+\frac{1}{2} \log \left (\cos \left (x^2\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 8, normalized size = 0.9 \begin{align*}{\frac{\ln \left ( \sin \left ({x}^{2} \right ) \right ) }{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.969597, size = 9, normalized size = 1. \begin{align*} \frac{1}{2} \, \log \left (\sin \left (x^{2}\right )\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.08113, size = 43, normalized size = 4.78 \begin{align*} \frac{1}{4} \, \log \left (-\frac{1}{2} \, \cos \left (2 \, x^{2}\right ) + \frac{1}{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.151614, size = 19, normalized size = 2.11 \begin{align*} - \frac{\log{\left (\tan ^{2}{\left (x^{2} \right )} + 1 \right )}}{4} + \frac{\log{\left (\tan{\left (x^{2} \right )} \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11234, size = 16, normalized size = 1.78 \begin{align*} \frac{1}{4} \, \log \left ({\left | \cos \left (x^{2}\right )^{2} - 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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