Optimal. Leaf size=24 \[ \sqrt{2 \pi } \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{x}\right ) \]
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Rubi [A] time = 0.019832, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3304, 3352} \[ \sqrt{2 \pi } \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{x}\right ) \]
Antiderivative was successfully verified.
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Rule 3304
Rule 3352
Rubi steps
\begin{align*} \int \frac{\cos (x)}{\sqrt{x}} \, dx &=2 \operatorname{Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt{x}\right )\\ &=\sqrt{2 \pi } C\left (\sqrt{\frac{2}{\pi }} \sqrt{x}\right )\\ \end{align*}
Mathematica [C] time = 0.0066152, size = 51, normalized size = 2.12 \[ -\frac{i \left (\sqrt{-i x} \text{Gamma}\left (\frac{1}{2},-i x\right )-\sqrt{i x} \text{Gamma}\left (\frac{1}{2},i x\right )\right )}{2 \sqrt{x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.026, size = 19, normalized size = 0.8 \begin{align*}{\it FresnelC} \left ({\frac{\sqrt{2}}{\sqrt{\pi }}\sqrt{x}} \right ) \sqrt{2}\sqrt{\pi } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 1.8776, size = 81, normalized size = 3.38 \begin{align*} -\frac{1}{8} \, \sqrt{\pi }{\left (\left (i - 1\right ) \, \sqrt{2} \operatorname{erf}\left (\left (\frac{1}{2} i + \frac{1}{2}\right ) \, \sqrt{2} \sqrt{x}\right ) + \left (i + 1\right ) \, \sqrt{2} \operatorname{erf}\left (\left (\frac{1}{2} i - \frac{1}{2}\right ) \, \sqrt{2} \sqrt{x}\right ) - \left (i + 1\right ) \, \sqrt{2} \operatorname{erf}\left (\sqrt{-i} \sqrt{x}\right ) + \left (i - 1\right ) \, \sqrt{2} \operatorname{erf}\left (\left (-1\right )^{\frac{1}{4}} \sqrt{x}\right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.6563, size = 76, normalized size = 3.17 \begin{align*} \sqrt{2} \sqrt{\pi } \operatorname{C}\left (\frac{\sqrt{2} \sqrt{x}}{\sqrt{\pi }}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.996044, size = 37, normalized size = 1.54 \begin{align*} \frac{\sqrt{2} \sqrt{\pi } C\left (\frac{\sqrt{2} \sqrt{x}}{\sqrt{\pi }}\right ) \Gamma \left (\frac{1}{4}\right )}{4 \Gamma \left (\frac{5}{4}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] time = 1.13746, size = 47, normalized size = 1.96 \begin{align*} -\left (\frac{1}{4} i + \frac{1}{4}\right ) \, \sqrt{2} \sqrt{\pi } \operatorname{erf}\left (\left (\frac{1}{2} i - \frac{1}{2}\right ) \, \sqrt{2} \sqrt{x}\right ) + \left (\frac{1}{4} i - \frac{1}{4}\right ) \, \sqrt{2} \sqrt{\pi } \operatorname{erf}\left (-\left (\frac{1}{2} i + \frac{1}{2}\right ) \, \sqrt{2} \sqrt{x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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