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 = 0.0485441, antiderivative size = 34, normalized size of antiderivative = 1., 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 3380
Rule 3296
Rule 2638
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*}
Mathematica [A] time = 0.0252488, size = 32, normalized size = 0.94 \[ \frac{\left (6 x^2-1\right ) \sin \left (\frac{1}{x}\right )}{x^3}+\frac{3 \left (2 x^2-1\right ) \cos \left (\frac{1}{x}\right )}{x^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 35, normalized size = 1. \begin{align*} 6\,\cos \left ({x}^{-1} \right ) -3\,{\frac{\cos \left ({x}^{-1} \right ) }{{x}^{2}}}-{\frac{\sin \left ({x}^{-1} \right ) }{{x}^{3}}}+6\,{\frac{\sin \left ({x}^{-1} \right ) }{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 1.07376, size = 26, normalized size = 0.76 \begin{align*} \frac{1}{2} \, \Gamma \left (4, \frac{i}{x}\right ) + \frac{1}{2} \, \Gamma \left (4, -\frac{i}{x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.02326, size = 72, normalized size = 2.12 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.17184, size = 32, normalized size = 0.94 \begin{align*} 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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos \left (\frac{1}{x}\right )}{x^{5}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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