Optimal. Leaf size=46 \[ \frac{\sqrt{2 \pi } S\left (\frac{\sqrt{f} \sqrt{\frac{2}{\pi }} \sqrt{d x}}{\sqrt{d}}\right )}{\sqrt{d} \sqrt{f}} \]
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Rubi [A] time = 0.0344381, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3305, 3351} \[ \frac{\sqrt{2 \pi } S\left (\frac{\sqrt{f} \sqrt{\frac{2}{\pi }} \sqrt{d x}}{\sqrt{d}}\right )}{\sqrt{d} \sqrt{f}} \]
Antiderivative was successfully verified.
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Rule 3305
Rule 3351
Rubi steps
\begin{align*} \int \frac{\sin (f x)}{\sqrt{d x}} \, dx &=\frac{2 \operatorname{Subst}\left (\int \sin \left (\frac{f x^2}{d}\right ) \, dx,x,\sqrt{d x}\right )}{d}\\ &=\frac{\sqrt{2 \pi } S\left (\frac{\sqrt{f} \sqrt{\frac{2}{\pi }} \sqrt{d x}}{\sqrt{d}}\right )}{\sqrt{d} \sqrt{f}}\\ \end{align*}
Mathematica [C] time = 0.0076831, size = 59, normalized size = 1.28 \[ \frac{-\sqrt{-i f x} \text{Gamma}\left (\frac{1}{2},-i f x\right )-\sqrt{i f x} \text{Gamma}\left (\frac{1}{2},i f x\right )}{2 f \sqrt{d x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 42, normalized size = 0.9 \begin{align*}{\frac{\sqrt{2}\sqrt{\pi }}{d}{\it FresnelS} \left ({\frac{\sqrt{2}f}{\sqrt{\pi }d}\sqrt{dx}{\frac{1}{\sqrt{{\frac{f}{d}}}}}} \right ){\frac{1}{\sqrt{{\frac{f}{d}}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 1.66492, size = 344, normalized size = 7.48 \begin{align*} \frac{{\left (i \, \sqrt{\pi } \cos \left (\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, f\right ) + \frac{1}{2} \, \arctan \left (0, \frac{d}{\sqrt{d^{2}}}\right )\right ) + i \, \sqrt{\pi } \cos \left (-\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, f\right ) + \frac{1}{2} \, \arctan \left (0, \frac{d}{\sqrt{d^{2}}}\right )\right ) + \sqrt{\pi } \sin \left (\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, f\right ) + \frac{1}{2} \, \arctan \left (0, \frac{d}{\sqrt{d^{2}}}\right )\right ) - \sqrt{\pi } \sin \left (-\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, f\right ) + \frac{1}{2} \, \arctan \left (0, \frac{d}{\sqrt{d^{2}}}\right )\right )\right )} \operatorname{erf}\left (\sqrt{d x} \sqrt{\frac{i \, f}{d}}\right ) +{\left (-i \, \sqrt{\pi } \cos \left (\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, f\right ) + \frac{1}{2} \, \arctan \left (0, \frac{d}{\sqrt{d^{2}}}\right )\right ) - i \, \sqrt{\pi } \cos \left (-\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, f\right ) + \frac{1}{2} \, \arctan \left (0, \frac{d}{\sqrt{d^{2}}}\right )\right ) + \sqrt{\pi } \sin \left (\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, f\right ) + \frac{1}{2} \, \arctan \left (0, \frac{d}{\sqrt{d^{2}}}\right )\right ) - \sqrt{\pi } \sin \left (-\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, f\right ) + \frac{1}{2} \, \arctan \left (0, \frac{d}{\sqrt{d^{2}}}\right )\right )\right )} \operatorname{erf}\left (\sqrt{d x} \sqrt{-\frac{i \, f}{d}}\right )}{4 \, d \sqrt{\frac{{\left | f \right |}}{{\left | d \right |}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.04256, size = 101, normalized size = 2.2 \begin{align*} \frac{\sqrt{2} \pi \sqrt{\frac{f}{\pi d}} \operatorname{S}\left (\sqrt{2} \sqrt{d x} \sqrt{\frac{f}{\pi d}}\right )}{f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.38134, size = 54, normalized size = 1.17 \begin{align*} \frac{3 \sqrt{2} \sqrt{\pi } S\left (\frac{\sqrt{2} \sqrt{f} \sqrt{x}}{\sqrt{\pi }}\right ) \Gamma \left (\frac{3}{4}\right )}{4 \sqrt{d} \sqrt{f} \Gamma \left (\frac{7}{4}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] time = 1.15571, size = 184, normalized size = 4. \begin{align*} -\frac{\frac{i \, \sqrt{2} \sqrt{\pi } d \operatorname{erf}\left (-\frac{\sqrt{2} \sqrt{d f} \sqrt{d x}{\left (\frac{i \, d f}{\sqrt{d^{2} f^{2}}} + 1\right )}}{2 \, d}\right )}{\sqrt{d f}{\left (\frac{i \, d f}{\sqrt{d^{2} f^{2}}} + 1\right )}} - \frac{i \, \sqrt{2} \sqrt{\pi } d \operatorname{erf}\left (-\frac{\sqrt{2} \sqrt{d f} \sqrt{d x}{\left (-\frac{i \, d f}{\sqrt{d^{2} f^{2}}} + 1\right )}}{2 \, d}\right )}{\sqrt{d f}{\left (-\frac{i \, d f}{\sqrt{d^{2} f^{2}}} + 1\right )}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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