Optimal. Leaf size=70 \[ \frac{\sqrt{\frac{\pi }{2}} \cos (a) \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{b} x\right )}{\sqrt{b}}-\frac{\sqrt{\frac{\pi }{2}} \sin (a) S\left (\sqrt{b} \sqrt{\frac{2}{\pi }} x\right )}{\sqrt{b}} \]
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Rubi [A] time = 0.0214534, antiderivative size = 70, 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 = {3354, 3352, 3351} \[ \frac{\sqrt{\frac{\pi }{2}} \cos (a) \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{b} x\right )}{\sqrt{b}}-\frac{\sqrt{\frac{\pi }{2}} \sin (a) S\left (\sqrt{b} \sqrt{\frac{2}{\pi }} x\right )}{\sqrt{b}} \]
Antiderivative was successfully verified.
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Rule 3354
Rule 3352
Rule 3351
Rubi steps
\begin{align*} \int \cos \left (a+b x^2\right ) \, dx &=\cos (a) \int \cos \left (b x^2\right ) \, dx-\sin (a) \int \sin \left (b x^2\right ) \, dx\\ &=\frac{\sqrt{\frac{\pi }{2}} \cos (a) C\left (\sqrt{b} \sqrt{\frac{2}{\pi }} x\right )}{\sqrt{b}}-\frac{\sqrt{\frac{\pi }{2}} S\left (\sqrt{b} \sqrt{\frac{2}{\pi }} x\right ) \sin (a)}{\sqrt{b}}\\ \end{align*}
Mathematica [A] time = 0.090349, size = 57, normalized size = 0.81 \[ \frac{\sqrt{\frac{\pi }{2}} \left (\cos (a) \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{b} x\right )-\sin (a) S\left (\sqrt{b} \sqrt{\frac{2}{\pi }} x\right )\right )}{\sqrt{b}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.025, size = 44, normalized size = 0.6 \begin{align*}{\frac{\sqrt{2}\sqrt{\pi }}{2} \left ( \cos \left ( a \right ){\it FresnelC} \left ({\frac{x\sqrt{2}}{\sqrt{\pi }}\sqrt{b}} \right ) -\sin \left ( a \right ){\it FresnelS} \left ({\frac{x\sqrt{2}}{\sqrt{\pi }}\sqrt{b}} \right ) \right ){\frac{1}{\sqrt{b}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 2.25014, size = 308, normalized size = 4.4 \begin{align*} \frac{\sqrt{\pi }{\left ({\left ({\left (\cos \left (\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, b\right )\right ) + \cos \left (-\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, b\right )\right ) - i \, \sin \left (\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, b\right )\right ) + i \, \sin \left (-\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, b\right )\right )\right )} \cos \left (a\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 )} \sin \left (a\right )\right )} \operatorname{erf}\left (\sqrt{i \, b} x\right ) +{\left ({\left (\cos \left (\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, b\right )\right ) + \cos \left (-\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, b\right )\right ) + i \, \sin \left (\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, b\right )\right ) - i \, \sin \left (-\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, b\right )\right )\right )} \cos \left (a\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 )} \sin \left (a\right )\right )} \operatorname{erf}\left (\sqrt{-i \, b} x\right )\right )}}{8 \, \sqrt{{\left | b \right |}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.59267, size = 185, normalized size = 2.64 \begin{align*} \frac{\sqrt{2} \pi \sqrt{\frac{b}{\pi }} \cos \left (a\right ) \operatorname{C}\left (\sqrt{2} x \sqrt{\frac{b}{\pi }}\right ) - \sqrt{2} \pi \sqrt{\frac{b}{\pi }} \operatorname{S}\left (\sqrt{2} x \sqrt{\frac{b}{\pi }}\right ) \sin \left (a\right )}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.564621, size = 61, normalized size = 0.87 \begin{align*} \frac{\sqrt{2} \sqrt{\pi } \left (- \sin{\left (a \right )} S\left (\frac{\sqrt{2} \sqrt{b} x}{\sqrt{\pi }}\right ) + \cos{\left (a \right )} C\left (\frac{\sqrt{2} \sqrt{b} x}{\sqrt{\pi }}\right )\right ) \sqrt{\frac{1}{b}}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] time = 1.16209, size = 128, normalized size = 1.83 \begin{align*} -\frac{\sqrt{2} \sqrt{\pi } \operatorname{erf}\left (-\frac{1}{2} \, \sqrt{2} x{\left (-\frac{i \, b}{{\left | b \right |}} + 1\right )} \sqrt{{\left | b \right |}}\right ) e^{\left (i \, a\right )}}{4 \,{\left (-\frac{i \, b}{{\left | b \right |}} + 1\right )} \sqrt{{\left | b \right |}}} - \frac{\sqrt{2} \sqrt{\pi } \operatorname{erf}\left (-\frac{1}{2} \, \sqrt{2} x{\left (\frac{i \, b}{{\left | b \right |}} + 1\right )} \sqrt{{\left | b \right |}}\right ) e^{\left (-i \, a\right )}}{4 \,{\left (\frac{i \, b}{{\left | b \right |}} + 1\right )} \sqrt{{\left | b \right |}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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