Integrand size = 8, antiderivative size = 143 \[ \int x \operatorname {FresnelS}(b x)^2 \, dx=\frac {\cos \left (b^2 \pi x^2\right )}{4 b^2 \pi ^2}+\frac {x \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{b \pi }-\frac {\operatorname {FresnelC}(b x) \operatorname {FresnelS}(b x)}{2 b^2 \pi }+\frac {1}{2} x^2 \operatorname {FresnelS}(b x)^2+\frac {i x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-\frac {1}{2} i b^2 \pi x^2\right )}{8 \pi }-\frac {i x^2 \, _2F_2\left (1,1;\frac {3}{2},2;\frac {1}{2} i b^2 \pi x^2\right )}{8 \pi } \]
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Time = 0.06 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {6565, 6589, 6581, 3460, 2718} \[ \int x \operatorname {FresnelS}(b x)^2 \, dx=\frac {i x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-\frac {1}{2} i b^2 \pi x^2\right )}{8 \pi }-\frac {i x^2 \, _2F_2\left (1,1;\frac {3}{2},2;\frac {1}{2} i b^2 \pi x^2\right )}{8 \pi }-\frac {\operatorname {FresnelC}(b x) \operatorname {FresnelS}(b x)}{2 \pi b^2}+\frac {x \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b}+\frac {\cos \left (\pi b^2 x^2\right )}{4 \pi ^2 b^2}+\frac {1}{2} x^2 \operatorname {FresnelS}(b x)^2 \]
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Rule 2718
Rule 3460
Rule 6565
Rule 6581
Rule 6589
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} x^2 \operatorname {FresnelS}(b x)^2-b \int x^2 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx \\ & = \frac {x \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{b \pi }+\frac {1}{2} x^2 \operatorname {FresnelS}(b x)^2-\frac {\int x \sin \left (b^2 \pi x^2\right ) \, dx}{2 \pi }-\frac {\int \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x) \, dx}{b \pi } \\ & = \frac {x \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{b \pi }-\frac {\operatorname {FresnelC}(b x) \operatorname {FresnelS}(b x)}{2 b^2 \pi }+\frac {1}{2} x^2 \operatorname {FresnelS}(b x)^2+\frac {i x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-\frac {1}{2} i b^2 \pi x^2\right )}{8 \pi }-\frac {i x^2 \, _2F_2\left (1,1;\frac {3}{2},2;\frac {1}{2} i b^2 \pi x^2\right )}{8 \pi }-\frac {\text {Subst}\left (\int \sin \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{4 \pi } \\ & = \frac {\cos \left (b^2 \pi x^2\right )}{4 b^2 \pi ^2}+\frac {x \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{b \pi }-\frac {\operatorname {FresnelC}(b x) \operatorname {FresnelS}(b x)}{2 b^2 \pi }+\frac {1}{2} x^2 \operatorname {FresnelS}(b x)^2+\frac {i x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-\frac {1}{2} i b^2 \pi x^2\right )}{8 \pi }-\frac {i x^2 \, _2F_2\left (1,1;\frac {3}{2},2;\frac {1}{2} i b^2 \pi x^2\right )}{8 \pi } \\ \end{align*}
\[ \int x \operatorname {FresnelS}(b x)^2 \, dx=\int x \operatorname {FresnelS}(b x)^2 \, dx \]
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\[\int x \operatorname {FresnelS}\left (b x \right )^{2}d x\]
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\[ \int x \operatorname {FresnelS}(b x)^2 \, dx=\int { x \operatorname {S}\left (b x\right )^{2} \,d x } \]
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\[ \int x \operatorname {FresnelS}(b x)^2 \, dx=\int x S^{2}\left (b x\right )\, dx \]
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\[ \int x \operatorname {FresnelS}(b x)^2 \, dx=\int { x \operatorname {S}\left (b x\right )^{2} \,d x } \]
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\[ \int x \operatorname {FresnelS}(b x)^2 \, dx=\int { x \operatorname {S}\left (b x\right )^{2} \,d x } \]
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Timed out. \[ \int x \operatorname {FresnelS}(b x)^2 \, dx=\int x\,{\mathrm {FresnelS}\left (b\,x\right )}^2 \,d x \]
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