Optimal. Leaf size=74 \[ a x+\frac{\sqrt{\frac{\pi }{2}} b \sin (c) \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{d} x\right )}{\sqrt{d}}+\frac{\sqrt{\frac{\pi }{2}} b \cos (c) S\left (\sqrt{d} \sqrt{\frac{2}{\pi }} x\right )}{\sqrt{d}} \]
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Rubi [A] time = 0.042957, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3353, 3352, 3351} \[ a x+\frac{\sqrt{\frac{\pi }{2}} b \sin (c) \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{d} x\right )}{\sqrt{d}}+\frac{\sqrt{\frac{\pi }{2}} b \cos (c) S\left (\sqrt{d} \sqrt{\frac{2}{\pi }} x\right )}{\sqrt{d}} \]
Antiderivative was successfully verified.
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Rule 3353
Rule 3352
Rule 3351
Rubi steps
\begin{align*} \int \left (a+b \sin \left (c+d x^2\right )\right ) \, dx &=a x+b \int \sin \left (c+d x^2\right ) \, dx\\ &=a x+(b \cos (c)) \int \sin \left (d x^2\right ) \, dx+(b \sin (c)) \int \cos \left (d x^2\right ) \, dx\\ &=a x+\frac{b \sqrt{\frac{\pi }{2}} \cos (c) S\left (\sqrt{d} \sqrt{\frac{2}{\pi }} x\right )}{\sqrt{d}}+\frac{b \sqrt{\frac{\pi }{2}} C\left (\sqrt{d} \sqrt{\frac{2}{\pi }} x\right ) \sin (c)}{\sqrt{d}}\\ \end{align*}
Mathematica [A] time = 0.142439, size = 61, normalized size = 0.82 \[ a x+\frac{\sqrt{\frac{\pi }{2}} b \left (\sin (c) \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{d} x\right )+\cos (c) S\left (\sqrt{d} \sqrt{\frac{2}{\pi }} x\right )\right )}{\sqrt{d}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 48, normalized size = 0.7 \begin{align*} ax+{\frac{b\sqrt{2}\sqrt{\pi }}{2} \left ( \cos \left ( c \right ){\it FresnelS} \left ({\frac{x\sqrt{2}}{\sqrt{\pi }}\sqrt{d}} \right ) +\sin \left ( c \right ){\it FresnelC} \left ({\frac{x\sqrt{2}}{\sqrt{\pi }}\sqrt{d}} \right ) \right ){\frac{1}{\sqrt{d}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 1.63441, size = 315, normalized size = 4.26 \begin{align*} a x - \frac{\sqrt{\pi }{\left ({\left ({\left (-i \, \cos \left (\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, d\right )\right ) - i \, \cos \left (-\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, d\right )\right ) - \sin \left (\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, d\right )\right ) + \sin \left (-\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, d\right )\right )\right )} \cos \left (c\right ) -{\left (\cos \left (\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, d\right )\right ) + \cos \left (-\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, d\right )\right ) - i \, \sin \left (\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, d\right )\right ) + i \, \sin \left (-\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, d\right )\right )\right )} \sin \left (c\right )\right )} \operatorname{erf}\left (\sqrt{i \, d} x\right ) +{\left ({\left (i \, \cos \left (\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, d\right )\right ) + i \, \cos \left (-\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, d\right )\right ) - \sin \left (\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, d\right )\right ) + \sin \left (-\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, d\right )\right )\right )} \cos \left (c\right ) -{\left (\cos \left (\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, d\right )\right ) + \cos \left (-\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, d\right )\right ) + i \, \sin \left (\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, d\right )\right ) - i \, \sin \left (-\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, d\right )\right )\right )} \sin \left (c\right )\right )} \operatorname{erf}\left (\sqrt{-i \, d} x\right )\right )} b}{8 \, \sqrt{{\left | d \right |}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.98925, size = 204, normalized size = 2.76 \begin{align*} \frac{\sqrt{2} \pi b \sqrt{\frac{d}{\pi }} \cos \left (c\right ) \operatorname{S}\left (\sqrt{2} x \sqrt{\frac{d}{\pi }}\right ) + \sqrt{2} \pi b \sqrt{\frac{d}{\pi }} \operatorname{C}\left (\sqrt{2} x \sqrt{\frac{d}{\pi }}\right ) \sin \left (c\right ) + 2 \, a d x}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.526034, size = 66, normalized size = 0.89 \begin{align*} a x + \frac{\sqrt{2} \sqrt{\pi } b \left (\sin{\left (c \right )} C\left (\frac{\sqrt{2} \sqrt{d} x}{\sqrt{\pi }}\right ) + \cos{\left (c \right )} S\left (\frac{\sqrt{2} \sqrt{d} x}{\sqrt{\pi }}\right )\right ) \sqrt{\frac{1}{d}}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] time = 1.11975, size = 138, normalized size = 1.86 \begin{align*} -\frac{1}{4} \,{\left (-\frac{i \, \sqrt{2} \sqrt{\pi } \operatorname{erf}\left (-\frac{1}{2} \, \sqrt{2} x{\left (-\frac{i \, d}{{\left | d \right |}} + 1\right )} \sqrt{{\left | d \right |}}\right ) e^{\left (i \, c\right )}}{{\left (-\frac{i \, d}{{\left | d \right |}} + 1\right )} \sqrt{{\left | d \right |}}} + \frac{i \, \sqrt{2} \sqrt{\pi } \operatorname{erf}\left (-\frac{1}{2} \, \sqrt{2} x{\left (\frac{i \, d}{{\left | d \right |}} + 1\right )} \sqrt{{\left | d \right |}}\right ) e^{\left (-i \, c\right )}}{{\left (\frac{i \, d}{{\left | d \right |}} + 1\right )} \sqrt{{\left | d \right |}}}\right )} b + a x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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