Optimal. Leaf size=25 \[ \frac{1}{2} \sin (a) \text{Si}\left (\frac{b}{x^2}\right )-\frac{1}{2} \cos (a) \text{CosIntegral}\left (\frac{b}{x^2}\right ) \]
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Rubi [A] time = 0.0277989, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3378, 3376, 3375} \[ \frac{1}{2} \sin (a) \text{Si}\left (\frac{b}{x^2}\right )-\frac{1}{2} \cos (a) \text{CosIntegral}\left (\frac{b}{x^2}\right ) \]
Antiderivative was successfully verified.
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Rule 3378
Rule 3376
Rule 3375
Rubi steps
\begin{align*} \int \frac{\cos \left (a+\frac{b}{x^2}\right )}{x} \, dx &=\cos (a) \int \frac{\cos \left (\frac{b}{x^2}\right )}{x} \, dx-\sin (a) \int \frac{\sin \left (\frac{b}{x^2}\right )}{x} \, dx\\ &=-\frac{1}{2} \cos (a) \text{Ci}\left (\frac{b}{x^2}\right )+\frac{1}{2} \sin (a) \text{Si}\left (\frac{b}{x^2}\right )\\ \end{align*}
Mathematica [A] time = 0.0452284, size = 24, normalized size = 0.96 \[ \frac{1}{2} \left (\sin (a) \text{Si}\left (\frac{b}{x^2}\right )-\cos (a) \text{CosIntegral}\left (\frac{b}{x^2}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.037, size = 22, normalized size = 0.9 \begin{align*} -{\frac{\cos \left ( a \right ) }{2}{\it Ci} \left ({\frac{b}{{x}^{2}}} \right ) }+{\frac{\sin \left ( a \right ) }{2}{\it Si} \left ({\frac{b}{{x}^{2}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 1.35032, size = 58, normalized size = 2.32 \begin{align*} -\frac{1}{4} \,{\left ({\rm Ei}\left (\frac{i \, b}{x^{2}}\right ) +{\rm Ei}\left (-\frac{i \, b}{x^{2}}\right )\right )} \cos \left (a\right ) - \frac{1}{4} \,{\left (i \,{\rm Ei}\left (\frac{i \, b}{x^{2}}\right ) - i \,{\rm Ei}\left (-\frac{i \, b}{x^{2}}\right )\right )} \sin \left (a\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.58963, size = 123, normalized size = 4.92 \begin{align*} -\frac{1}{4} \,{\left (\operatorname{Ci}\left (\frac{b}{x^{2}}\right ) + \operatorname{Ci}\left (-\frac{b}{x^{2}}\right )\right )} \cos \left (a\right ) + \frac{1}{2} \, \sin \left (a\right ) \operatorname{Si}\left (\frac{b}{x^{2}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos{\left (a + \frac{b}{x^{2}} \right )}}{x}\, dx \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 (a + \frac{b}{x^{2}}\right )}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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