Optimal. Leaf size=10 \[ \frac{1}{x}-\tan \left (\frac{1}{x}\right ) \]
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Rubi [A] time = 0.0178687, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {3747, 3473, 8} \[ \frac{1}{x}-\tan \left (\frac{1}{x}\right ) \]
Antiderivative was successfully verified.
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Rule 3747
Rule 3473
Rule 8
Rubi steps
\begin{align*} \int \frac{\tan ^2\left (\frac{1}{x}\right )}{x^2} \, dx &=-\operatorname{Subst}\left (\int \tan ^2(x) \, dx,x,\frac{1}{x}\right )\\ &=-\tan \left (\frac{1}{x}\right )+\operatorname{Subst}\left (\int 1 \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{x}-\tan \left (\frac{1}{x}\right )\\ \end{align*}
Mathematica [A] time = 0.0233596, size = 12, normalized size = 1.2 \[ \tan ^{-1}\left (\tan \left (\frac{1}{x}\right )\right )-\tan \left (\frac{1}{x}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 11, normalized size = 1.1 \begin{align*}{x}^{-1}-\tan \left ({x}^{-1} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.961496, size = 90, normalized size = 9. \begin{align*} \frac{\cos \left (\frac{2}{x}\right )^{2} - 2 \, x \sin \left (\frac{2}{x}\right ) + \sin \left (\frac{2}{x}\right )^{2} + 2 \, \cos \left (\frac{2}{x}\right ) + 1}{{\left (\cos \left (\frac{2}{x}\right )^{2} + \sin \left (\frac{2}{x}\right )^{2} + 2 \, \cos \left (\frac{2}{x}\right ) + 1\right )} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.03941, size = 28, normalized size = 2.8 \begin{align*} -\frac{x \tan \left (\frac{1}{x}\right ) - 1}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.382807, size = 7, normalized size = 0.7 \begin{align*} - \tan{\left (\frac{1}{x} \right )} + \frac{1}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09455, size = 14, normalized size = 1.4 \begin{align*} \frac{1}{x} - \tan \left (\frac{1}{x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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