Integrand size = 18, antiderivative size = 60 \[ \int x \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx=\frac {x}{2 b \pi }-\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{b^2 \pi }+\frac {\operatorname {FresnelC}\left (\sqrt {2} b x\right )}{2 \sqrt {2} b^2 \pi } \]
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Time = 0.02 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6596, 3439, 3433} \[ \int x \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx=-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {\operatorname {FresnelC}\left (\sqrt {2} b x\right )}{2 \sqrt {2} \pi b^2}+\frac {x}{2 \pi b} \]
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Rule 3433
Rule 3439
Rule 6596
Rubi steps \begin{align*} \text {integral}& = -\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{b^2 \pi }+\frac {\int \cos ^2\left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{b \pi } \\ & = -\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{b^2 \pi }+\frac {\int \left (\frac {1}{2}+\frac {1}{2} \cos \left (b^2 \pi x^2\right )\right ) \, dx}{b \pi } \\ & = \frac {x}{2 b \pi }-\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{b^2 \pi }+\frac {\int \cos \left (b^2 \pi x^2\right ) \, dx}{2 b \pi } \\ & = \frac {x}{2 b \pi }-\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{b^2 \pi }+\frac {\operatorname {FresnelC}\left (\sqrt {2} b x\right )}{2 \sqrt {2} b^2 \pi } \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.80 \[ \int x \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx=\frac {2 b x-4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)+\sqrt {2} \operatorname {FresnelC}\left (\sqrt {2} b x\right )}{4 b^2 \pi } \]
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Time = 0.65 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.87
method | result | size |
default | \(\frac {-\frac {\operatorname {FresnelC}\left (b x \right ) \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{b \pi }+\frac {\frac {b x}{2}+\frac {\sqrt {2}\, \operatorname {FresnelC}\left (b x \sqrt {2}\right )}{4}}{b \pi }}{b}\) | \(52\) |
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Time = 0.27 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.87 \[ \int x \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx=\frac {2 \, b^{2} x - 4 \, b \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {C}\left (b x\right ) + \sqrt {2} \sqrt {b^{2}} \operatorname {C}\left (\sqrt {2} \sqrt {b^{2}} x\right )}{4 \, \pi b^{3}} \]
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\[ \int x \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx=\int x \sin {\left (\frac {\pi b^{2} x^{2}}{2} \right )} C\left (b x\right )\, dx \]
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\[ \int x \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx=\int { x \operatorname {C}\left (b x\right ) \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \,d x } \]
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\[ \int x \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx=\int { x \operatorname {C}\left (b x\right ) \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \,d x } \]
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Timed out. \[ \int x \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx=\int x\,\mathrm {FresnelC}\left (b\,x\right )\,\sin \left (\frac {\Pi \,b^2\,x^2}{2}\right ) \,d x \]
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