Integrand size = 8, antiderivative size = 59 \[ \int x^2 \operatorname {FresnelC}(b x) \, dx=-\frac {2 \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{3 b^3 \pi ^2}+\frac {1}{3} x^3 \operatorname {FresnelC}(b x)-\frac {x^2 \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{3 b \pi } \]
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Time = 0.03 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6562, 3461, 3377, 2718} \[ \int x^2 \operatorname {FresnelC}(b x) \, dx=-\frac {x^2 \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{3 \pi b}-\frac {2 \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{3 \pi ^2 b^3}+\frac {1}{3} x^3 \operatorname {FresnelC}(b x) \]
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Rule 2718
Rule 3377
Rule 3461
Rule 6562
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} x^3 \operatorname {FresnelC}(b x)-\frac {1}{3} b \int x^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx \\ & = \frac {1}{3} x^3 \operatorname {FresnelC}(b x)-\frac {1}{6} b \text {Subst}\left (\int x \cos \left (\frac {1}{2} b^2 \pi x\right ) \, dx,x,x^2\right ) \\ & = \frac {1}{3} x^3 \operatorname {FresnelC}(b x)-\frac {x^2 \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{3 b \pi }+\frac {\text {Subst}\left (\int \sin \left (\frac {1}{2} b^2 \pi x\right ) \, dx,x,x^2\right )}{3 b \pi } \\ & = -\frac {2 \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{3 b^3 \pi ^2}+\frac {1}{3} x^3 \operatorname {FresnelC}(b x)-\frac {x^2 \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{3 b \pi } \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00 \[ \int x^2 \operatorname {FresnelC}(b x) \, dx=-\frac {2 \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{3 b^3 \pi ^2}+\frac {1}{3} x^3 \operatorname {FresnelC}(b x)-\frac {x^2 \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{3 b \pi } \]
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Result contains higher order function than in optimal. Order 5 vs. order 4.
Time = 0.38 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.44
method | result | size |
meijerg | \(\frac {b \,x^{4} \operatorname {hypergeom}\left (\left [\frac {1}{4}, 1\right ], \left [\frac {1}{2}, \frac {5}{4}, 2\right ], -\frac {x^{4} \pi ^{2} b^{4}}{16}\right )}{4}\) | \(26\) |
parts | \(\frac {x^{3} \operatorname {FresnelC}\left (b x \right )}{3}-\frac {b \left (\frac {x^{2} \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{b^{2} \pi }+\frac {2 \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{b^{4} \pi ^{2}}\right )}{3}\) | \(53\) |
derivativedivides | \(\frac {\frac {\operatorname {FresnelC}\left (b x \right ) b^{3} x^{3}}{3}-\frac {b^{2} x^{2} \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{3 \pi }-\frac {2 \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{3 \pi ^{2}}}{b^{3}}\) | \(54\) |
default | \(\frac {\frac {\operatorname {FresnelC}\left (b x \right ) b^{3} x^{3}}{3}-\frac {b^{2} x^{2} \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{3 \pi }-\frac {2 \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{3 \pi ^{2}}}{b^{3}}\) | \(54\) |
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Time = 0.27 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.92 \[ \int x^2 \operatorname {FresnelC}(b x) \, dx=\frac {\pi ^{2} b^{3} x^{3} \operatorname {C}\left (b x\right ) - \pi b^{2} x^{2} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) - 2 \, \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{3 \, \pi ^{2} b^{3}} \]
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Time = 0.58 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.36 \[ \int x^2 \operatorname {FresnelC}(b x) \, dx=\frac {x^{3} C\left (b x\right ) \Gamma \left (\frac {1}{4}\right )}{12 \Gamma \left (\frac {5}{4}\right )} - \frac {x^{2} \sin {\left (\frac {\pi b^{2} x^{2}}{2} \right )} \Gamma \left (\frac {1}{4}\right )}{12 \pi b \Gamma \left (\frac {5}{4}\right )} - \frac {\cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )} \Gamma \left (\frac {1}{4}\right )}{6 \pi ^{2} b^{3} \Gamma \left (\frac {5}{4}\right )} \]
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Time = 0.21 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.83 \[ \int x^2 \operatorname {FresnelC}(b x) \, dx=\frac {1}{3} \, x^{3} \operatorname {C}\left (b x\right ) - \frac {\pi b^{2} x^{2} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) + 2 \, \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{3 \, \pi ^{2} b^{3}} \]
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\[ \int x^2 \operatorname {FresnelC}(b x) \, dx=\int { x^{2} \operatorname {C}\left (b x\right ) \,d x } \]
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Timed out. \[ \int x^2 \operatorname {FresnelC}(b x) \, dx=\int x^2\,\mathrm {FresnelC}\left (b\,x\right ) \,d x \]
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