Optimal. Leaf size=74 \[ \frac {x^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{4 b \pi }+\frac {3 S(b x)}{4 b^4 \pi ^2}+\frac {1}{4} x^4 S(b x)-\frac {3 x \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{4 b^3 \pi ^2} \]
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Rubi [A]
time = 0.03, antiderivative size = 74, 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 = {6561, 3466,
3467, 3432} \begin {gather*} \frac {3 S(b x)}{4 \pi ^2 b^4}+\frac {x^3 \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{4 \pi b}-\frac {3 x \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{4 \pi ^2 b^3}+\frac {1}{4} x^4 S(b x) \end {gather*}
Antiderivative was successfully verified.
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Rule 3432
Rule 3466
Rule 3467
Rule 6561
Rubi steps
\begin {align*} \int x^3 S(b x) \, dx &=\frac {1}{4} x^4 S(b x)-\frac {1}{4} b \int x^4 \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx\\ &=\frac {x^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{4 b \pi }+\frac {1}{4} x^4 S(b x)-\frac {3 \int x^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{4 b \pi }\\ &=\frac {x^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{4 b \pi }+\frac {1}{4} x^4 S(b x)-\frac {3 x \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{4 b^3 \pi ^2}+\frac {3 \int \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{4 b^3 \pi ^2}\\ &=\frac {x^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{4 b \pi }+\frac {3 S(b x)}{4 b^4 \pi ^2}+\frac {1}{4} x^4 S(b x)-\frac {3 x \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{4 b^3 \pi ^2}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 74, normalized size = 1.00 \begin {gather*} \frac {x^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{4 b \pi }+\frac {3 S(b x)}{4 b^4 \pi ^2}+\frac {1}{4} x^4 S(b x)-\frac {3 x \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{4 b^3 \pi ^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.33, size = 70, normalized size = 0.95
method | result | size |
meijerg | \(\frac {\frac {\pi \,x^{3} b^{3} \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{4}-\frac {3 \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right ) b x}{4}+\frac {\left (21 x^{4} \pi ^{2} b^{4}+63\right ) \mathrm {S}\left (b x \right )}{84}}{\pi ^{2} b^{4}}\) | \(62\) |
derivativedivides | \(\frac {\frac {\mathrm {S}\left (b x \right ) b^{4} x^{4}}{4}+\frac {b^{3} x^{3} \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{4 \pi }-\frac {3 \left (\frac {b x \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi }-\frac {\mathrm {S}\left (b x \right )}{\pi }\right )}{4 \pi }}{b^{4}}\) | \(70\) |
default | \(\frac {\frac {\mathrm {S}\left (b x \right ) b^{4} x^{4}}{4}+\frac {b^{3} x^{3} \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{4 \pi }-\frac {3 \left (\frac {b x \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi }-\frac {\mathrm {S}\left (b x \right )}{\pi }\right )}{4 \pi }}{b^{4}}\) | \(70\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 0.48, size = 94, normalized size = 1.27 \begin {gather*} \frac {1}{4} \, x^{4} \operatorname {S}\left (b x\right ) + \frac {\sqrt {\frac {1}{2}} {\left (4 \, \sqrt {\frac {1}{2}} \pi ^{2} b^{3} x^{3} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) - 12 \, \sqrt {\frac {1}{2}} \pi b x \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) + \left (3 i + 3\right ) \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \pi \operatorname {erf}\left (\sqrt {\frac {1}{2} i \, \pi } b x\right ) - \left (3 i - 3\right ) \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \pi \operatorname {erf}\left (\sqrt {-\frac {1}{2} i \, \pi } b x\right )\right )}}{8 \, \pi ^{3} b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 58, normalized size = 0.78 \begin {gather*} \frac {\pi b^{3} x^{3} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) - 3 \, b x \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) + {\left (\pi ^{2} b^{4} x^{4} + 3\right )} \operatorname {S}\left (b x\right )}{4 \, \pi ^{2} b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.52, size = 112, normalized size = 1.51 \begin {gather*} \frac {21 x^{4} S\left (b x\right ) \Gamma \left (\frac {3}{4}\right )}{64 \Gamma \left (\frac {11}{4}\right )} + \frac {21 x^{3} \cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )} \Gamma \left (\frac {3}{4}\right )}{64 \pi b \Gamma \left (\frac {11}{4}\right )} - \frac {63 x \sin {\left (\frac {\pi b^{2} x^{2}}{2} \right )} \Gamma \left (\frac {3}{4}\right )}{64 \pi ^{2} b^{3} \Gamma \left (\frac {11}{4}\right )} + \frac {63 S\left (b x\right ) \Gamma \left (\frac {3}{4}\right )}{64 \pi ^{2} b^{4} \Gamma \left (\frac {11}{4}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x^3\,\mathrm {FresnelS}\left (b\,x\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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