Optimal. Leaf size=6 \[ -\tan \left (\frac {1}{x}\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 6, 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 = {4204, 3767, 8} \[ -\tan \left (\frac {1}{x}\right ) \]
Antiderivative was successfully verified.
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Rule 8
Rule 3767
Rule 4204
Rubi steps
\begin {align*} \int \frac {\sec ^2\left (\frac {1}{x}\right )}{x^2} \, dx &=-\operatorname {Subst}\left (\int \sec ^2(x) \, dx,x,\frac {1}{x}\right )\\ &=\operatorname {Subst}\left (\int 1 \, dx,x,-\tan \left (\frac {1}{x}\right )\right )\\ &=-\tan \left (\frac {1}{x}\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 6, normalized size = 1.00 \[ -\tan \left (\frac {1}{x}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 1.04, size = 12, normalized size = 2.00 \[ -\frac {\sin \left (\frac {1}{x}\right )}{\cos \left (\frac {1}{x}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.13, size = 20, normalized size = 3.33 \[ \frac {2 \, \tan \left (\frac {1}{2 \, x}\right )}{\tan \left (\frac {1}{2 \, x}\right )^{2} - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 7, normalized size = 1.17 \[ -\tan \left (\frac {1}{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.38, size = 36, normalized size = 6.00 \[ -\frac {2 \, \sin \left (\frac {2}{x}\right )}{\cos \left (\frac {2}{x}\right )^{2} + \sin \left (\frac {2}{x}\right )^{2} + 2 \, \cos \left (\frac {2}{x}\right ) + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.03, size = 14, normalized size = 2.33 \[ -\frac {2{}\mathrm {i}}{{\mathrm {e}}^{\frac {2{}\mathrm {i}}{x}}+1} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sec ^{2}{\left (\frac {1}{x} \right )}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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