Optimal. Leaf size=16 \[ \text{CosIntegral}(2 x)-\frac{\sin (2 x)}{2 x} \]
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Rubi [A] time = 0.0457541, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {4406, 12, 3297, 3302} \[ \text{CosIntegral}(2 x)-\frac{\sin (2 x)}{2 x} \]
Antiderivative was successfully verified.
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Rule 4406
Rule 12
Rule 3297
Rule 3302
Rubi steps
\begin{align*} \int \frac{\cos (x) \sin (x)}{x^2} \, dx &=\int \frac{\sin (2 x)}{2 x^2} \, dx\\ &=\frac{1}{2} \int \frac{\sin (2 x)}{x^2} \, dx\\ &=-\frac{\sin (2 x)}{2 x}+\int \frac{\cos (2 x)}{x} \, dx\\ &=\text{Ci}(2 x)-\frac{\sin (2 x)}{2 x}\\ \end{align*}
Mathematica [A] time = 0.0060185, size = 16, normalized size = 1. \[ \text{CosIntegral}(2 x)-\frac{\sin (2 x)}{2 x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.026, size = 15, normalized size = 0.9 \begin{align*}{\it Ci} \left ( 2\,x \right ) -{\frac{\sin \left ( 2\,x \right ) }{2\,x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 1.20794, size = 20, normalized size = 1.25 \begin{align*} \frac{1}{2} \, \Gamma \left (-1, 2 i \, x\right ) + \frac{1}{2} \, \Gamma \left (-1, -2 i \, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.459234, size = 95, normalized size = 5.94 \begin{align*} \frac{x \operatorname{Ci}\left (2 \, x\right ) + x \operatorname{Ci}\left (-2 \, x\right ) - 2 \, \cos \left (x\right ) \sin \left (x\right )}{2 \, x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.15956, size = 22, normalized size = 1.38 \begin{align*} - \log{\left (x \right )} + \frac{\log{\left (x^{2} \right )}}{2} + \operatorname{Ci}{\left (2 x \right )} - \frac{\sin{\left (2 x \right )}}{2 x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12137, size = 26, normalized size = 1.62 \begin{align*} \frac{2 \, x \operatorname{Ci}\left (2 \, x\right ) - \sin \left (2 \, x\right )}{2 \, x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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