Optimal. Leaf size=59 \[ \frac {x^2 \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{3 \pi b}-\frac {2 \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{3 \pi ^2 b^3}+\frac {1}{3} x^3 S(b x) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6426, 3379, 3296, 2637} \[ -\frac {2 \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{3 \pi ^2 b^3}+\frac {x^2 \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{3 \pi b}+\frac {1}{3} x^3 S(b x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2637
Rule 3296
Rule 3379
Rule 6426
Rubi steps
\begin {align*} \int x^2 S(b x) \, dx &=\frac {1}{3} x^3 S(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 S(b x)-\frac {1}{6} b \operatorname {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 S(b x)-\frac {\operatorname {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 S(b x)-\frac {2 \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{3 b^3 \pi ^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 59, normalized size = 1.00 \[ \frac {x^2 \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{3 \pi b}-\frac {2 \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{3 \pi ^2 b^3}+\frac {1}{3} x^3 S(b x) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.41, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{2} {\rm fresnels}\left (b x\right ), x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{2} {\rm fresnels}\left (b x\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.02, size = 54, normalized size = 0.92 \[ \frac {\frac {b^{3} x^{3} \mathrm {S}\left (b x \right )}{3}+\frac {b^{2} x^{2} \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{3 \pi }-\frac {2 \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{3 \pi ^{2}}}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{2} {\rm fresnels}\left (b x\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int x^2\,\mathrm {FresnelS}\left (b\,x\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.80, size = 80, normalized size = 1.36 \[ \frac {x^{3} S\left (b x\right ) \Gamma \left (\frac {3}{4}\right )}{4 \Gamma \left (\frac {7}{4}\right )} + \frac {x^{2} \cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )} \Gamma \left (\frac {3}{4}\right )}{4 \pi b \Gamma \left (\frac {7}{4}\right )} - \frac {\sin {\left (\frac {\pi b^{2} x^{2}}{2} \right )} \Gamma \left (\frac {3}{4}\right )}{2 \pi ^{2} b^{3} \Gamma \left (\frac {7}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________