Optimal. Leaf size=25 \[ \frac {1}{2} \sin (a) \text {Si}\left (\frac {b}{x^2}\right )-\frac {1}{2} \cos (a) \text {Ci}\left (\frac {b}{x^2}\right ) \]
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Rubi [A] time = 0.03, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, 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 3375
Rule 3376
Rule 3378
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*}
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Mathematica [A] time = 0.04, size = 24, normalized size = 0.96 \[ \frac {1}{2} \left (\sin (a) \text {Si}\left (\frac {b}{x^2}\right )-\cos (a) \text {Ci}\left (\frac {b}{x^2}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.57, size = 29, normalized size = 1.16 \[ -\frac {1}{4} \, {\left (\operatorname {Ci}\left (\frac {b}{x^{2}}\right ) + \operatorname {Ci}\left (-\frac {b}{x^{2}}\right )\right )} \cos \relax (a) + \frac {1}{2} \, \sin \relax (a) \operatorname {Si}\left (\frac {b}{x^{2}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cos \left (a + \frac {b}{x^{2}}\right )}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 22, normalized size = 0.88 \[ -\frac {\Ci \left (\frac {b}{x^{2}}\right ) \cos \relax (a )}{2}+\frac {\Si \left (\frac {b}{x^{2}}\right ) \sin \relax (a )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 1.34, size = 43, normalized size = 1.72 \[ -\frac {1}{4} \, {\left ({\rm Ei}\left (\frac {i \, b}{x^{2}}\right ) + {\rm Ei}\left (-\frac {i \, b}{x^{2}}\right )\right )} \cos \relax (a) - \frac {1}{4} \, {\left (i \, {\rm Ei}\left (\frac {i \, b}{x^{2}}\right ) - i \, {\rm Ei}\left (-\frac {i \, b}{x^{2}}\right )\right )} \sin \relax (a) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.04 \[ \frac {\sin \relax (a)\,\mathrm {sinint}\left (\frac {b}{x^2}\right )}{2}-\frac {\cos \relax (a)\,\mathrm {cosint}\left (\frac {b}{x^2}\right )}{2} \]
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 \[ \int \frac {\cos {\left (a + \frac {b}{x^{2}} \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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