Optimal. Leaf size=5 \[ \tanh ^{-1}\left (\cos \left (\frac{1}{x}\right )\right ) \]
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Rubi [A] time = 0.0090749, antiderivative size = 5, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {4205, 3770} \[ \tanh ^{-1}\left (\cos \left (\frac{1}{x}\right )\right ) \]
Antiderivative was successfully verified.
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Rule 4205
Rule 3770
Rubi steps
\begin{align*} \int \frac{\csc \left (\frac{1}{x}\right )}{x^2} \, dx &=-\operatorname{Subst}\left (\int \csc (x) \, dx,x,\frac{1}{x}\right )\\ &=\tanh ^{-1}\left (\cos \left (\frac{1}{x}\right )\right )\\ \end{align*}
Mathematica [B] time = 0.0148051, size = 21, normalized size = 4.2 \[ \log \left (\cos \left (\frac{1}{2 x}\right )\right )-\log \left (\sin \left (\frac{1}{2 x}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 11, normalized size = 2.2 \begin{align*} \ln \left ( \csc \left ({x}^{-1} \right ) +\cot \left ({x}^{-1} \right ) \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.96226, size = 14, normalized size = 2.8 \begin{align*} \log \left (\cot \left (\frac{1}{x}\right ) + \csc \left (\frac{1}{x}\right )\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.29585, size = 81, normalized size = 16.2 \begin{align*} \frac{1}{2} \, \log \left (\frac{1}{2} \, \cos \left (\frac{1}{x}\right ) + \frac{1}{2}\right ) - \frac{1}{2} \, \log \left (-\frac{1}{2} \, \cos \left (\frac{1}{x}\right ) + \frac{1}{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.35122, size = 10, normalized size = 2. \begin{align*} \log{\left (\cot{\left (\frac{1}{x} \right )} + \csc{\left (\frac{1}{x} \right )} \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14418, size = 14, normalized size = 2.8 \begin{align*} -\log \left ({\left | \tan \left (\frac{1}{2 \, x}\right ) \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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