Optimal. Leaf size=108 \[ -\frac {5 S\left (\sqrt {2} b x\right )}{4 \sqrt {2} \pi ^2 b^4}+\frac {x^2 S(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {2 S(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi ^2 b^4}+\frac {x \sin \left (\pi b^2 x^2\right )}{4 \pi ^2 b^3}-\frac {x^3}{6 \pi b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.09, antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {6462, 3391, 30, 3386, 3351, 6452} \[ \frac {x^2 S(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {2 S(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi ^2 b^4}-\frac {5 S\left (\sqrt {2} b x\right )}{4 \sqrt {2} \pi ^2 b^4}+\frac {x \sin \left (\pi b^2 x^2\right )}{4 \pi ^2 b^3}-\frac {x^3}{6 \pi b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 30
Rule 3351
Rule 3386
Rule 3391
Rule 6452
Rule 6462
Rubi steps
\begin {align*} \int x^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x) \, dx &=\frac {x^2 S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^2 \pi }-\frac {2 \int x S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{b^2 \pi }-\frac {\int x^2 \sin ^2\left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{b \pi }\\ &=\frac {2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{b^4 \pi ^2}+\frac {x^2 S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^2 \pi }-\frac {\int \sin \left (b^2 \pi x^2\right ) \, dx}{b^3 \pi ^2}-\frac {\int x^2 \, dx}{2 b \pi }+\frac {\int x^2 \cos \left (b^2 \pi x^2\right ) \, dx}{2 b \pi }\\ &=-\frac {x^3}{6 b \pi }+\frac {2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{b^4 \pi ^2}-\frac {S\left (\sqrt {2} b x\right )}{\sqrt {2} b^4 \pi ^2}+\frac {x^2 S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^2 \pi }+\frac {x \sin \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}-\frac {\int \sin \left (b^2 \pi x^2\right ) \, dx}{4 b^3 \pi ^2}\\ &=-\frac {x^3}{6 b \pi }+\frac {2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{b^4 \pi ^2}-\frac {5 S\left (\sqrt {2} b x\right )}{4 \sqrt {2} b^4 \pi ^2}+\frac {x^2 S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^2 \pi }+\frac {x \sin \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.09, size = 90, normalized size = 0.83 \[ \frac {-4 \pi b^3 x^3+24 S(b x) \left (\pi b^2 x^2 \sin \left (\frac {1}{2} \pi b^2 x^2\right )+2 \cos \left (\frac {1}{2} \pi b^2 x^2\right )\right )+6 b x \sin \left (\pi b^2 x^2\right )-15 \sqrt {2} S\left (\sqrt {2} b x\right )}{24 \pi ^2 b^4} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{3} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) {\rm fresnels}\left (b x\right ), x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{3} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) {\rm fresnels}\left (b x\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 119, normalized size = 1.10 \[ \frac {\frac {\mathrm {S}\left (b x \right ) \left (\frac {b^{2} x^{2} \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi }+\frac {2 \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi ^{2}}\right )}{b^{3}}-\frac {\frac {\sqrt {2}\, \mathrm {S}\left (b x \sqrt {2}\right )}{2 \pi ^{2}}+\frac {b^{3} x^{3}}{6 \pi }-\frac {\frac {b x \sin \left (b^{2} \pi \,x^{2}\right )}{2 \pi }-\frac {\sqrt {2}\, \mathrm {S}\left (b x \sqrt {2}\right )}{4 \pi }}{2 \pi }}{b^{3}}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{3} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) {\rm fresnels}\left (b x\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x^3\,\mathrm {FresnelS}\left (b\,x\right )\,\cos \left (\frac {\Pi \,b^2\,x^2}{2}\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{3} \cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )} S\left (b x\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________