Integrand size = 6, antiderivative size = 55 \[ \int \operatorname {FresnelS}(b x)^2 \, dx=\frac {2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{b \pi }+x \operatorname {FresnelS}(b x)^2-\frac {\operatorname {FresnelS}\left (\sqrt {2} b x\right )}{\sqrt {2} b \pi } \]
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Time = 0.03 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {6555, 12, 6587, 3432} \[ \int \operatorname {FresnelS}(b x)^2 \, dx=\frac {2 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b}+x \operatorname {FresnelS}(b x)^2-\frac {\operatorname {FresnelS}\left (\sqrt {2} b x\right )}{\sqrt {2} \pi b} \]
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Rule 12
Rule 3432
Rule 6555
Rule 6587
Rubi steps \begin{align*} \text {integral}& = x \operatorname {FresnelS}(b x)^2-2 \int b x \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx \\ & = x \operatorname {FresnelS}(b x)^2-(2 b) \int x \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx \\ & = \frac {2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{b \pi }+x \operatorname {FresnelS}(b x)^2-\frac {\int \sin \left (b^2 \pi x^2\right ) \, dx}{\pi } \\ & = \frac {2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{b \pi }+x \operatorname {FresnelS}(b x)^2-\frac {\operatorname {FresnelS}\left (\sqrt {2} b x\right )}{\sqrt {2} b \pi } \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00 \[ \int \operatorname {FresnelS}(b x)^2 \, dx=\frac {2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{b \pi }+x \operatorname {FresnelS}(b x)^2-\frac {\operatorname {FresnelS}\left (\sqrt {2} b x\right )}{\sqrt {2} b \pi } \]
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Time = 0.51 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.89
method | result | size |
derivativedivides | \(\frac {\operatorname {FresnelS}\left (b x \right )^{2} b x +\frac {2 \,\operatorname {FresnelS}\left (b x \right ) \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi }-\frac {\sqrt {2}\, \operatorname {FresnelS}\left (b x \sqrt {2}\right )}{2 \pi }}{b}\) | \(49\) |
default | \(\frac {\operatorname {FresnelS}\left (b x \right )^{2} b x +\frac {2 \,\operatorname {FresnelS}\left (b x \right ) \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi }-\frac {\sqrt {2}\, \operatorname {FresnelS}\left (b x \sqrt {2}\right )}{2 \pi }}{b}\) | \(49\) |
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Time = 0.26 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.09 \[ \int \operatorname {FresnelS}(b x)^2 \, dx=\frac {2 \, \pi b^{2} x \operatorname {S}\left (b x\right )^{2} + 4 \, b \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {S}\left (b x\right ) - \sqrt {2} \sqrt {b^{2}} \operatorname {S}\left (\sqrt {2} \sqrt {b^{2}} x\right )}{2 \, \pi b^{2}} \]
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\[ \int \operatorname {FresnelS}(b x)^2 \, dx=\int S^{2}\left (b x\right )\, dx \]
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\[ \int \operatorname {FresnelS}(b x)^2 \, dx=\int { \operatorname {S}\left (b x\right )^{2} \,d x } \]
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\[ \int \operatorname {FresnelS}(b x)^2 \, dx=\int { \operatorname {S}\left (b x\right )^{2} \,d x } \]
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Timed out. \[ \int \operatorname {FresnelS}(b x)^2 \, dx=\int {\mathrm {FresnelS}\left (b\,x\right )}^2 \,d x \]
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