Optimal. Leaf size=29 \[ \frac{\sqrt{\frac{\pi }{2}} \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} (a+b x)\right )}{b} \]
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Rubi [A] time = 0.0058369, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {3352} \[ \frac{\sqrt{\frac{\pi }{2}} \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} (a+b x)\right )}{b} \]
Antiderivative was successfully verified.
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Rule 3352
Rubi steps
\begin{align*} \int \cos \left ((a+b x)^2\right ) \, dx &=\frac{\sqrt{\frac{\pi }{2}} C\left (\sqrt{\frac{2}{\pi }} (a+b x)\right )}{b}\\ \end{align*}
Mathematica [A] time = 0.0080303, size = 29, normalized size = 1. \[ \frac{\sqrt{\frac{\pi }{2}} \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} (a+b x)\right )}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.028, size = 36, normalized size = 1.2 \begin{align*}{\frac{\sqrt{2}\sqrt{\pi }}{2}{\it FresnelC} \left ({\frac{\sqrt{2} \left ({b}^{2}x+ab \right ) }{\sqrt{\pi }}{\frac{1}{\sqrt{{b}^{2}}}}} \right ){\frac{1}{\sqrt{{b}^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 1.76279, size = 113, normalized size = 3.9 \begin{align*} -\frac{\sqrt{\pi }{\left (\left (i - 1\right ) \, \sqrt{2} \operatorname{erf}\left (-\left (-1\right )^{\frac{3}{4}}{\left (i \, b x + i \, a\right )}\right ) + \left (i - 1\right ) \, \sqrt{2} \operatorname{erf}\left (-\left (\frac{1}{4} i - \frac{1}{4}\right ) \, \sqrt{2}{\left (2 i \, b x + 2 i \, a\right )}\right ) - \left (i + 1\right ) \, \sqrt{2} \operatorname{erf}\left (-\left (\frac{1}{4} i + \frac{1}{4}\right ) \, \sqrt{2}{\left (2 i \, b x + 2 i \, a\right )}\right ) + \left (i + 1\right ) \, \sqrt{2} \operatorname{erf}\left (\frac{i \, b x + i \, a}{\sqrt{-i}}\right )\right )}}{16 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.54214, size = 107, normalized size = 3.69 \begin{align*} \frac{\sqrt{2} \pi \sqrt{\frac{b^{2}}{\pi }} \operatorname{C}\left (\frac{\sqrt{2}{\left (b x + a\right )} \sqrt{\frac{b^{2}}{\pi }}}{b}\right )}{2 \, b^{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 \cos{\left (\left (a + b x\right )^{2} \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] time = 1.09037, size = 74, normalized size = 2.55 \begin{align*} -\frac{\left (i + 1\right ) \, \sqrt{2} \sqrt{\pi } \operatorname{erf}\left (\left (\frac{1}{2} i - \frac{1}{2}\right ) \, \sqrt{2}{\left (x + \frac{a}{b}\right )}{\left | b \right |}\right )}{8 \,{\left | b \right |}} + \frac{\left (i - 1\right ) \, \sqrt{2} \sqrt{\pi } \operatorname{erf}\left (-\left (\frac{1}{2} i + \frac{1}{2}\right ) \, \sqrt{2}{\left (x + \frac{a}{b}\right )}{\left | b \right |}\right )}{8 \,{\left | b \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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