Optimal. Leaf size=35 \[ -2 \sqrt{2 \pi } S\left (\sqrt{\frac{2}{\pi }} \sqrt{x}\right )-\frac{2 \cos (x)}{\sqrt{x}} \]
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Rubi [A] time = 0.0366712, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {3297, 3305, 3351} \[ -2 \sqrt{2 \pi } S\left (\sqrt{\frac{2}{\pi }} \sqrt{x}\right )-\frac{2 \cos (x)}{\sqrt{x}} \]
Antiderivative was successfully verified.
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Rule 3297
Rule 3305
Rule 3351
Rubi steps
\begin{align*} \int \frac{\cos (x)}{x^{3/2}} \, dx &=-\frac{2 \cos (x)}{\sqrt{x}}-2 \int \frac{\sin (x)}{\sqrt{x}} \, dx\\ &=-\frac{2 \cos (x)}{\sqrt{x}}-4 \operatorname{Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt{x}\right )\\ &=-\frac{2 \cos (x)}{\sqrt{x}}-2 \sqrt{2 \pi } S\left (\sqrt{\frac{2}{\pi }} \sqrt{x}\right )\\ \end{align*}
Mathematica [C] time = 0.0423232, size = 63, normalized size = 1.8 \[ \frac{\sqrt{-i x} \text{Gamma}\left (\frac{1}{2},-i x\right )+\sqrt{i x} \text{Gamma}\left (\frac{1}{2},i x\right )-e^{-i x} \left (1+e^{2 i x}\right )}{\sqrt{x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.028, size = 28, normalized size = 0.8 \begin{align*} -2\,{\it FresnelS} \left ({\frac{\sqrt{2}\sqrt{x}}{\sqrt{\pi }}} \right ) \sqrt{2}\sqrt{\pi }-2\,{\frac{\cos \left ( x \right ) }{\sqrt{x}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 2.43777, size = 28, normalized size = 0.8 \begin{align*} -\left (\frac{1}{4} i + \frac{1}{4}\right ) \, \sqrt{2} \Gamma \left (-\frac{1}{2}, i \, x\right ) + \left (\frac{1}{4} i - \frac{1}{4}\right ) \, \sqrt{2} \Gamma \left (-\frac{1}{2}, -i \, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.66506, size = 111, normalized size = 3.17 \begin{align*} -\frac{2 \,{\left (\sqrt{2} \sqrt{\pi } x \operatorname{S}\left (\frac{\sqrt{2} \sqrt{x}}{\sqrt{\pi }}\right ) + \sqrt{x} \cos \left (x\right )\right )}}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.17337, size = 61, normalized size = 1.74 \begin{align*} \frac{\sqrt{2} \sqrt{\pi } S\left (\frac{\sqrt{2} \sqrt{x}}{\sqrt{\pi }}\right ) \Gamma \left (- \frac{1}{4}\right )}{2 \Gamma \left (\frac{3}{4}\right )} + \frac{\cos{\left (x \right )} \Gamma \left (- \frac{1}{4}\right )}{2 \sqrt{x} \Gamma \left (\frac{3}{4}\right )} \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 (x\right )}{x^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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