Optimal. Leaf size=13 \[ \frac {S(b x)^3}{3 b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {6440, 30} \[ \frac {S(b x)^3}{3 b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 30
Rule 6440
Rubi steps
\begin {align*} \int S(b x)^2 \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx &=\frac {\operatorname {Subst}\left (\int x^2 \, dx,x,S(b x)\right )}{b}\\ &=\frac {S(b x)^3}{3 b}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.00, size = 13, normalized size = 1.00 \[ \frac {S(b x)^3}{3 b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\rm fresnels}\left (b x\right )^{2} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ), x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\rm fresnels}\left (b x\right )^{2} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 12, normalized size = 0.92 \[ \frac {\mathrm {S}\left (b x \right )^{3}}{3 b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\rm fresnels}\left (b x\right )^{2} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.08 \[ \int {\mathrm {FresnelS}\left (b\,x\right )}^2\,\sin \left (\frac {\Pi \,b^2\,x^2}{2}\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 1.04, size = 10, normalized size = 0.77 \[ \begin {cases} \frac {S^{3}\left (b x\right )}{3 b} & \text {for}\: b \neq 0 \\0 & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________