Optimal. Leaf size=34 \[ -\frac {\sin \left (\frac {1}{x}\right )}{x^3}-\frac {3 \cos \left (\frac {1}{x}\right )}{x^2}+\frac {6 \sin \left (\frac {1}{x}\right )}{x}+6 \cos \left (\frac {1}{x}\right ) \]
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Rubi [A] time = 0.05, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {3380, 3296, 2638} \[ -\frac {\sin \left (\frac {1}{x}\right )}{x^3}-\frac {3 \cos \left (\frac {1}{x}\right )}{x^2}+\frac {6 \sin \left (\frac {1}{x}\right )}{x}+6 \cos \left (\frac {1}{x}\right ) \]
Antiderivative was successfully verified.
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Rule 2638
Rule 3296
Rule 3380
Rubi steps
\begin {align*} \int \frac {\cos \left (\frac {1}{x}\right )}{x^5} \, dx &=-\operatorname {Subst}\left (\int x^3 \cos (x) \, dx,x,\frac {1}{x}\right )\\ &=-\frac {\sin \left (\frac {1}{x}\right )}{x^3}+3 \operatorname {Subst}\left (\int x^2 \sin (x) \, dx,x,\frac {1}{x}\right )\\ &=-\frac {3 \cos \left (\frac {1}{x}\right )}{x^2}-\frac {\sin \left (\frac {1}{x}\right )}{x^3}+6 \operatorname {Subst}\left (\int x \cos (x) \, dx,x,\frac {1}{x}\right )\\ &=-\frac {3 \cos \left (\frac {1}{x}\right )}{x^2}-\frac {\sin \left (\frac {1}{x}\right )}{x^3}+\frac {6 \sin \left (\frac {1}{x}\right )}{x}-6 \operatorname {Subst}\left (\int \sin (x) \, dx,x,\frac {1}{x}\right )\\ &=6 \cos \left (\frac {1}{x}\right )-\frac {3 \cos \left (\frac {1}{x}\right )}{x^2}-\frac {\sin \left (\frac {1}{x}\right )}{x^3}+\frac {6 \sin \left (\frac {1}{x}\right )}{x}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 32, normalized size = 0.94 \[ \frac {3 \left (2 x^2-1\right ) \cos \left (\frac {1}{x}\right )}{x^2}+\frac {\left (6 x^2-1\right ) \sin \left (\frac {1}{x}\right )}{x^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.87, size = 32, normalized size = 0.94 \[ \frac {3 \, {\left (2 \, x^{3} - x\right )} \cos \left (\frac {1}{x}\right ) + {\left (6 \, x^{2} - 1\right )} \sin \left (\frac {1}{x}\right )}{x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 34, normalized size = 1.00 \[ \frac {6 \, \sin \left (\frac {1}{x}\right )}{x} - \frac {3 \, \cos \left (\frac {1}{x}\right )}{x^{2}} - \frac {\sin \left (\frac {1}{x}\right )}{x^{3}} + 6 \, \cos \left (\frac {1}{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 35, normalized size = 1.03 \[ 6 \cos \left (\frac {1}{x}\right )-\frac {3 \cos \left (\frac {1}{x}\right )}{x^{2}}-\frac {\sin \left (\frac {1}{x}\right )}{x^{3}}+\frac {6 \sin \left (\frac {1}{x}\right )}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.37, size = 19, normalized size = 0.56 \[ \frac {1}{2} \, \Gamma \left (4, \frac {i}{x}\right ) + \frac {1}{2} \, \Gamma \left (4, -\frac {i}{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.00, size = 33, normalized size = 0.97 \[ 6\,\cos \left (\frac {1}{x}\right )-\frac {\sin \left (\frac {1}{x}\right )+3\,x\,\cos \left (\frac {1}{x}\right )-6\,x^2\,\sin \left (\frac {1}{x}\right )}{x^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.68, size = 32, normalized size = 0.94 \[ 6 \cos {\left (\frac {1}{x} \right )} + \frac {6 \sin {\left (\frac {1}{x} \right )}}{x} - \frac {3 \cos {\left (\frac {1}{x} \right )}}{x^{2}} - \frac {\sin {\left (\frac {1}{x} \right )}}{x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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