Optimal. Leaf size=10 \[ \frac {1}{x}-\tan \left (\frac {1}{x}\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3747, 3473, 8} \[ \frac {1}{x}-\tan \left (\frac {1}{x}\right ) \]
Antiderivative was successfully verified.
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Rule 8
Rule 3473
Rule 3747
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*}
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Mathematica [A] time = 0.02, size = 12, normalized size = 1.20 \[ \tan ^{-1}\left (\tan \left (\frac {1}{x}\right )\right )-\tan \left (\frac {1}{x}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.72, size = 13, normalized size = 1.30 \[ -\frac {x \tan \left (\frac {1}{x}\right ) - 1}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 10, normalized size = 1.00 \[ \frac {1}{x} - \tan \left (\frac {1}{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 11, normalized size = 1.10 \[ \frac {1}{x}-\tan \left (\frac {1}{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.33, size = 67, normalized size = 6.70 \[ \frac {\cos \left (\frac {2}{x}\right )^{2} - 2 \, x \sin \left (\frac {2}{x}\right ) + \sin \left (\frac {2}{x}\right )^{2} + 2 \, \cos \left (\frac {2}{x}\right ) + 1}{{\left (\cos \left (\frac {2}{x}\right )^{2} + \sin \left (\frac {2}{x}\right )^{2} + 2 \, \cos \left (\frac {2}{x}\right ) + 1\right )} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.92, size = 10, normalized size = 1.00 \[ \frac {1}{x}-\mathrm {tan}\left (\frac {1}{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.22, size = 7, normalized size = 0.70 \[ - \tan {\left (\frac {1}{x} \right )} + \frac {1}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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