Integrand size = 20, antiderivative size = 74 \[ \int x^2 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx=\frac {x^2}{4 b \pi }-\frac {x \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{b^2 \pi }+\frac {\operatorname {FresnelC}(b x)^2}{2 b^3 \pi }+\frac {\sin \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2} \]
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Time = 0.04 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6598, 6576, 30, 3461, 2714} \[ \int x^2 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx=\frac {\operatorname {FresnelC}(b x)^2}{2 \pi b^3}-\frac {x \operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {\sin \left (\pi b^2 x^2\right )}{4 \pi ^2 b^3}+\frac {x^2}{4 \pi b} \]
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Rule 30
Rule 2714
Rule 3461
Rule 6576
Rule 6598
Rubi steps \begin{align*} \text {integral}& = -\frac {x \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{b^2 \pi }+\frac {\int \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x) \, dx}{b^2 \pi }+\frac {\int x \cos ^2\left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{b \pi } \\ & = -\frac {x \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{b^2 \pi }+\frac {\text {Subst}(\int x \, dx,x,\operatorname {FresnelC}(b x))}{b^3 \pi }+\frac {\text {Subst}\left (\int \cos ^2\left (\frac {1}{2} b^2 \pi x\right ) \, dx,x,x^2\right )}{2 b \pi } \\ & = \frac {x^2}{4 b \pi }-\frac {x \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{b^2 \pi }+\frac {\operatorname {FresnelC}(b x)^2}{2 b^3 \pi }+\frac {\sin \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.00 \[ \int x^2 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx=\frac {x^2}{4 b \pi }-\frac {x \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{b^2 \pi }+\frac {\operatorname {FresnelC}(b x)^2}{2 b^3 \pi }+\frac {\sin \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2} \]
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\[\int x^{2} \operatorname {FresnelC}\left (b x \right ) \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )d x\]
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none
Time = 0.27 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.91 \[ \int x^2 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx=\frac {\pi b^{2} x^{2} - 4 \, \pi b x \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {C}\left (b x\right ) + 2 \, \pi \operatorname {C}\left (b x\right )^{2} + 2 \, \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{4 \, \pi ^{2} b^{3}} \]
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Time = 0.46 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.54 \[ \int x^2 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx=\begin {cases} \frac {x^{2} \sin ^{2}{\left (\frac {\pi b^{2} x^{2}}{2} \right )}}{4 \pi b} + \frac {x^{2} \cos ^{2}{\left (\frac {\pi b^{2} x^{2}}{2} \right )}}{4 \pi b} - \frac {x \cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )} C\left (b x\right )}{\pi b^{2}} + \frac {\sin {\left (\frac {\pi b^{2} x^{2}}{2} \right )} \cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )}}{2 \pi ^{2} b^{3}} + \frac {C^{2}\left (b x\right )}{2 \pi b^{3}} & \text {for}\: b \neq 0 \\0 & \text {otherwise} \end {cases} \]
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\[ \int x^2 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx=\int { x^{2} \operatorname {C}\left (b x\right ) \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \,d x } \]
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\[ \int x^2 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx=\int { x^{2} \operatorname {C}\left (b x\right ) \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \,d x } \]
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Timed out. \[ \int x^2 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx=\int x^2\,\mathrm {FresnelC}\left (b\,x\right )\,\sin \left (\frac {\Pi \,b^2\,x^2}{2}\right ) \,d x \]
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