Optimal. Leaf size=39 \[ \frac{\sqrt{\frac{\pi }{2}} S\left (\sqrt{b} \sqrt{\frac{2}{\pi }} (c+d x)\right )}{\sqrt{b} d} \]
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Rubi [A] time = 0.0080351, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {3351} \[ \frac{\sqrt{\frac{\pi }{2}} S\left (\sqrt{b} \sqrt{\frac{2}{\pi }} (c+d x)\right )}{\sqrt{b} d} \]
Antiderivative was successfully verified.
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Rule 3351
Rubi steps
\begin{align*} \int \sin \left (b (c+d x)^2\right ) \, dx &=\frac{\sqrt{\frac{\pi }{2}} S\left (\sqrt{b} \sqrt{\frac{2}{\pi }} (c+d x)\right )}{\sqrt{b} d}\\ \end{align*}
Mathematica [A] time = 0.0151665, size = 39, normalized size = 1. \[ \frac{\sqrt{\frac{\pi }{2}} S\left (\sqrt{b} \sqrt{\frac{2}{\pi }} (c+d x)\right )}{\sqrt{b} d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 42, normalized size = 1.1 \begin{align*}{\frac{\sqrt{2}\sqrt{\pi }}{2}{\it FresnelS} \left ({\frac{\sqrt{2} \left ( b{d}^{2}x+bcd \right ) }{\sqrt{\pi }}{\frac{1}{\sqrt{{d}^{2}b}}}} \right ){\frac{1}{\sqrt{{d}^{2}b}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 1.83639, size = 193, normalized size = 4.95 \begin{align*} \frac{\sqrt{\pi }{\left ({\left (i \, \cos \left (\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, b\right )\right ) + i \, \cos \left (-\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, b\right )\right ) + \sin \left (\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, b\right )\right ) - \sin \left (-\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, b\right )\right )\right )} \operatorname{erf}\left (\frac{i \, b d x + i \, b c}{\sqrt{i \, b}}\right ) +{\left (i \, \cos \left (\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, b\right )\right ) + i \, \cos \left (-\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, b\right )\right ) - \sin \left (\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, b\right )\right ) + \sin \left (-\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, b\right )\right )\right )} \operatorname{erf}\left (\frac{i \, b d x + i \, b c}{\sqrt{-i \, b}}\right )\right )}}{8 \, d \sqrt{{\left | b \right |}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.5917, size = 117, normalized size = 3. \begin{align*} \frac{\sqrt{2} \pi \sqrt{\frac{b d^{2}}{\pi }} \operatorname{S}\left (\frac{\sqrt{2} \sqrt{\frac{b d^{2}}{\pi }}{\left (d x + c\right )}}{d}\right )}{2 \, b d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sin{\left (b \left (c + d x\right )^{2} \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] time = 1.10767, size = 193, normalized size = 4.95 \begin{align*} -\frac{i \, \sqrt{2} \sqrt{\pi } \operatorname{erf}\left (-\frac{1}{2} \, \sqrt{2} \sqrt{b d^{2}}{\left (\frac{i \, b d^{2}}{\sqrt{b^{2} d^{4}}} + 1\right )}{\left (x + \frac{c}{d}\right )}\right )}{4 \, \sqrt{b d^{2}}{\left (\frac{i \, b d^{2}}{\sqrt{b^{2} d^{4}}} + 1\right )}} + \frac{i \, \sqrt{2} \sqrt{\pi } \operatorname{erf}\left (-\frac{1}{2} \, \sqrt{2} \sqrt{b d^{2}}{\left (-\frac{i \, b d^{2}}{\sqrt{b^{2} d^{4}}} + 1\right )}{\left (x + \frac{c}{d}\right )}\right )}{4 \, \sqrt{b d^{2}}{\left (-\frac{i \, b d^{2}}{\sqrt{b^{2} d^{4}}} + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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