Optimal. Leaf size=5 \[ \tanh ^{-1}\left (\cos \left (\frac {1}{x}\right )\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 5, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4205, 3770} \[ \tanh ^{-1}\left (\cos \left (\frac {1}{x}\right )\right ) \]
Antiderivative was successfully verified.
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Rule 3770
Rule 4205
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*}
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Mathematica [B] time = 0.01, size = 21, normalized size = 4.20 \[ \log \left (\cos \left (\frac {1}{2 x}\right )\right )-\log \left (\sin \left (\frac {1}{2 x}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.65, size = 23, normalized size = 4.60 \[ \frac {1}{2} \, \log \left (\frac {1}{2} \, \cos \left (\frac {1}{x}\right ) + \frac {1}{2}\right ) - \frac {1}{2} \, \log \left (-\frac {1}{2} \, \cos \left (\frac {1}{x}\right ) + \frac {1}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.14, size = 43, normalized size = 8.60 \[ -\frac {1}{2} \, \log \left (\frac {4 \, \tan \left (\frac {1}{2 \, x}\right )^{2}}{\tan \left (\frac {1}{2 \, x}\right )^{2} + 1}\right ) + \frac {1}{2} \, \log \left (\frac {4}{\tan \left (\frac {1}{2 \, x}\right )^{2} + 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 11, normalized size = 2.20 \[ \ln \left (\csc \left (\frac {1}{x}\right )+\cot \left (\frac {1}{x}\right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 10, normalized size = 2.00 \[ \log \left (\cot \left (\frac {1}{x}\right ) + \csc \left (\frac {1}{x}\right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.68, size = 31, normalized size = 6.20 \[ \ln \left (-{\mathrm {e}}^{1{}\mathrm {i}/x}\,2{}\mathrm {i}-2{}\mathrm {i}\right )-\ln \left (-{\mathrm {e}}^{1{}\mathrm {i}/x}\,2{}\mathrm {i}+2{}\mathrm {i}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.26, size = 10, normalized size = 2.00 \[ \log {\left (\cot {\left (\frac {1}{x} \right )} + \csc {\left (\frac {1}{x} \right )} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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