Optimal. Leaf size=15 \[ \frac{\log (x)}{2}-\frac{1}{2} \text{CosIntegral}(2 x) \]
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Rubi [A] time = 0.0382972, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3312, 3302} \[ \frac{\log (x)}{2}-\frac{1}{2} \text{CosIntegral}(2 x) \]
Antiderivative was successfully verified.
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Rule 3312
Rule 3302
Rubi steps
\begin{align*} \int \frac{\sin ^2(x)}{x} \, dx &=\int \left (\frac{1}{2 x}-\frac{\cos (2 x)}{2 x}\right ) \, dx\\ &=\frac{\log (x)}{2}-\frac{1}{2} \int \frac{\cos (2 x)}{x} \, dx\\ &=-\frac{\text{Ci}(2 x)}{2}+\frac{\log (x)}{2}\\ \end{align*}
Mathematica [A] time = 0.0115446, size = 15, normalized size = 1. \[ \frac{\log (x)}{2}-\frac{1}{2} \text{CosIntegral}(2 x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 12, normalized size = 0.8 \begin{align*} -{\frac{{\it Ci} \left ( 2\,x \right ) }{2}}+{\frac{\ln \left ( x \right ) }{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 1.06165, size = 23, normalized size = 1.53 \begin{align*} -\frac{1}{4} \,{\rm Ei}\left (2 i \, x\right ) - \frac{1}{4} \,{\rm Ei}\left (-2 i \, x\right ) + \frac{1}{2} \, \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.86062, size = 84, normalized size = 5.6 \begin{align*} -\frac{1}{4} \, \operatorname{Ci}\left (2 \, x\right ) - \frac{1}{4} \, \operatorname{Ci}\left (-2 \, x\right ) + \frac{1}{2} \, \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.0865, size = 10, normalized size = 0.67 \begin{align*} \frac{\log{\left (x \right )}}{2} - \frac{\operatorname{Ci}{\left (2 x \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.08195, size = 15, normalized size = 1. \begin{align*} -\frac{1}{2} \, \operatorname{Ci}\left (2 \, x\right ) + \frac{1}{2} \, \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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