Integrand size = 8, antiderivative size = 59 \[ \int x^2 \operatorname {FresnelS}(b x) \, dx=\frac {x^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{3 b \pi }+\frac {1}{3} x^3 \operatorname {FresnelS}(b x)-\frac {2 \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{3 b^3 \pi ^2} \]
[Out]
Time = 0.04 (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 = {6561, 3460, 3377, 2717} \[ \int x^2 \operatorname {FresnelS}(b x) \, dx=\frac {x^2 \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{3 \pi b}-\frac {2 \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{3 \pi ^2 b^3}+\frac {1}{3} x^3 \operatorname {FresnelS}(b x) \]
[In]
[Out]
Rule 2717
Rule 3377
Rule 3460
Rule 6561
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} x^3 \operatorname {FresnelS}(b x)-\frac {1}{3} b \int x^3 \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx \\ & = \frac {1}{3} x^3 \operatorname {FresnelS}(b x)-\frac {1}{6} b \text {Subst}\left (\int x \sin \left (\frac {1}{2} b^2 \pi x\right ) \, dx,x,x^2\right ) \\ & = \frac {x^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{3 b \pi }+\frac {1}{3} x^3 \operatorname {FresnelS}(b x)-\frac {\text {Subst}\left (\int \cos \left (\frac {1}{2} b^2 \pi x\right ) \, dx,x,x^2\right )}{3 b \pi } \\ & = \frac {x^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{3 b \pi }+\frac {1}{3} x^3 \operatorname {FresnelS}(b x)-\frac {2 \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{3 b^3 \pi ^2} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00 \[ \int x^2 \operatorname {FresnelS}(b x) \, dx=\frac {x^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{3 b \pi }+\frac {1}{3} x^3 \operatorname {FresnelS}(b x)-\frac {2 \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{3 b^3 \pi ^2} \]
[In]
[Out]
Time = 0.41 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.92
method | result | size |
derivativedivides | \(\frac {\frac {\operatorname {FresnelS}\left (b x \right ) b^{3} x^{3}}{3}+\frac {b^{2} x^{2} \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{3 \pi }-\frac {2 \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{3 \pi ^{2}}}{b^{3}}\) | \(54\) |
default | \(\frac {\frac {\operatorname {FresnelS}\left (b x \right ) b^{3} x^{3}}{3}+\frac {b^{2} x^{2} \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{3 \pi }-\frac {2 \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{3 \pi ^{2}}}{b^{3}}\) | \(54\) |
parts | \(\frac {x^{3} \operatorname {FresnelS}\left (b x \right )}{3}-\frac {b \left (-\frac {x^{2} \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{b^{2} \pi }+\frac {2 \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{b^{4} \pi ^{2}}\right )}{3}\) | \(54\) |
meijerg | \(\frac {\frac {\sqrt {\pi }\, x^{2} b^{2} \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{3}-\frac {2 \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{3 \sqrt {\pi }}+\frac {\pi ^{\frac {3}{2}} x^{3} b^{3} \operatorname {FresnelS}\left (b x \right )}{3}}{b^{3} \pi ^{\frac {3}{2}}}\) | \(60\) |
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.90 \[ \int x^2 \operatorname {FresnelS}(b x) \, dx=\frac {\pi ^{2} b^{3} x^{3} \operatorname {S}\left (b x\right ) + \pi b^{2} x^{2} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) - 2 \, \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{3 \, \pi ^{2} b^{3}} \]
[In]
[Out]
Time = 0.53 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.36 \[ \int x^2 \operatorname {FresnelS}(b x) \, dx=\frac {x^{3} S\left (b x\right ) \Gamma \left (\frac {3}{4}\right )}{4 \Gamma \left (\frac {7}{4}\right )} + \frac {x^{2} \cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )} \Gamma \left (\frac {3}{4}\right )}{4 \pi b \Gamma \left (\frac {7}{4}\right )} - \frac {\sin {\left (\frac {\pi b^{2} x^{2}}{2} \right )} \Gamma \left (\frac {3}{4}\right )}{2 \pi ^{2} b^{3} \Gamma \left (\frac {7}{4}\right )} \]
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.83 \[ \int x^2 \operatorname {FresnelS}(b x) \, dx=\frac {1}{3} \, x^{3} \operatorname {S}\left (b x\right ) + \frac {\pi b^{2} x^{2} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) - 2 \, \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{3 \, \pi ^{2} b^{3}} \]
[In]
[Out]
\[ \int x^2 \operatorname {FresnelS}(b x) \, dx=\int { x^{2} \operatorname {S}\left (b x\right ) \,d x } \]
[In]
[Out]
Timed out. \[ \int x^2 \operatorname {FresnelS}(b x) \, dx=\int x^2\,\mathrm {FresnelS}\left (b\,x\right ) \,d x \]
[In]
[Out]