Integrand size = 19, antiderivative size = 101 \[ \int \operatorname {FresnelS}(b x) \sin \left (c+\frac {1}{2} b^2 \pi x^2\right ) \, dx=\frac {\cos (c) \operatorname {FresnelS}(b x)^2}{2 b}+\frac {\operatorname {FresnelC}(b x) \operatorname {FresnelS}(b x) \sin (c)}{2 b}-\frac {1}{8} i b x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-\frac {1}{2} i b^2 \pi x^2\right ) \sin (c)+\frac {1}{8} i b x^2 \, _2F_2\left (1,1;\frac {3}{2},2;\frac {1}{2} i b^2 \pi x^2\right ) \sin (c) \]
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Time = 0.04 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {6577, 6581, 6575, 30} \[ \int \operatorname {FresnelS}(b x) \sin \left (c+\frac {1}{2} b^2 \pi x^2\right ) \, dx=-\frac {1}{8} i b x^2 \sin (c) \, _2F_2\left (1,1;\frac {3}{2},2;-\frac {1}{2} i b^2 \pi x^2\right )+\frac {1}{8} i b x^2 \sin (c) \, _2F_2\left (1,1;\frac {3}{2},2;\frac {1}{2} i b^2 \pi x^2\right )+\frac {\sin (c) \operatorname {FresnelC}(b x) \operatorname {FresnelS}(b x)}{2 b}+\frac {\cos (c) \operatorname {FresnelS}(b x)^2}{2 b} \]
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Rule 30
Rule 6575
Rule 6577
Rule 6581
Rubi steps \begin{align*} \text {integral}& = \cos (c) \int \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx+\sin (c) \int \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x) \, dx \\ & = \frac {\operatorname {FresnelC}(b x) \operatorname {FresnelS}(b x) \sin (c)}{2 b}-\frac {1}{8} i b x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-\frac {1}{2} i b^2 \pi x^2\right ) \sin (c)+\frac {1}{8} i b x^2 \, _2F_2\left (1,1;\frac {3}{2},2;\frac {1}{2} i b^2 \pi x^2\right ) \sin (c)+\frac {\cos (c) \text {Subst}(\int x \, dx,x,\operatorname {FresnelS}(b x))}{b} \\ & = \frac {\cos (c) \operatorname {FresnelS}(b x)^2}{2 b}+\frac {\operatorname {FresnelC}(b x) \operatorname {FresnelS}(b x) \sin (c)}{2 b}-\frac {1}{8} i b x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-\frac {1}{2} i b^2 \pi x^2\right ) \sin (c)+\frac {1}{8} i b x^2 \, _2F_2\left (1,1;\frac {3}{2},2;\frac {1}{2} i b^2 \pi x^2\right ) \sin (c) \\ \end{align*}
\[ \int \operatorname {FresnelS}(b x) \sin \left (c+\frac {1}{2} b^2 \pi x^2\right ) \, dx=\int \operatorname {FresnelS}(b x) \sin \left (c+\frac {1}{2} b^2 \pi x^2\right ) \, dx \]
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\[\int \operatorname {FresnelS}\left (b x \right ) \sin \left (c +\frac {b^{2} \pi \,x^{2}}{2}\right )d x\]
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\[ \int \operatorname {FresnelS}(b x) \sin \left (c+\frac {1}{2} b^2 \pi x^2\right ) \, dx=\int { \operatorname {S}\left (b x\right ) \sin \left (\frac {1}{2} \, \pi b^{2} x^{2} + c\right ) \,d x } \]
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\[ \int \operatorname {FresnelS}(b x) \sin \left (c+\frac {1}{2} b^2 \pi x^2\right ) \, dx=\int \sin {\left (\frac {\pi b^{2} x^{2}}{2} + c \right )} S\left (b x\right )\, dx \]
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\[ \int \operatorname {FresnelS}(b x) \sin \left (c+\frac {1}{2} b^2 \pi x^2\right ) \, dx=\int { \operatorname {S}\left (b x\right ) \sin \left (\frac {1}{2} \, \pi b^{2} x^{2} + c\right ) \,d x } \]
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\[ \int \operatorname {FresnelS}(b x) \sin \left (c+\frac {1}{2} b^2 \pi x^2\right ) \, dx=\int { \operatorname {S}\left (b x\right ) \sin \left (\frac {1}{2} \, \pi b^{2} x^{2} + c\right ) \,d x } \]
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Timed out. \[ \int \operatorname {FresnelS}(b x) \sin \left (c+\frac {1}{2} b^2 \pi x^2\right ) \, dx=\int \sin \left (\frac {\Pi \,b^2\,x^2}{2}+c\right )\,\mathrm {FresnelS}\left (b\,x\right ) \,d x \]
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