Optimal. Leaf size=49 \[ \frac {1}{2} x^2 \text {FresnelC}(b x)+\frac {S(b x)}{2 b^2 \pi }-\frac {x \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{2 b \pi } \]
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Rubi [A]
time = 0.02, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6562, 3467,
3432} \begin {gather*} \frac {S(b x)}{2 \pi b^2}-\frac {x \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{2 \pi b}+\frac {1}{2} x^2 \text {FresnelC}(b x) \end {gather*}
Antiderivative was successfully verified.
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Rule 3432
Rule 3467
Rule 6562
Rubi steps
\begin {align*} \int x C(b x) \, dx &=\frac {1}{2} x^2 C(b x)-\frac {1}{2} b \int x^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx\\ &=\frac {1}{2} x^2 C(b x)-\frac {x \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{2 b \pi }+\frac {\int \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{2 b \pi }\\ &=\frac {1}{2} x^2 C(b x)+\frac {S(b x)}{2 b^2 \pi }-\frac {x \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{2 b \pi }\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 49, normalized size = 1.00 \begin {gather*} \frac {1}{2} x^2 \text {FresnelC}(b x)+\frac {S(b x)}{2 b^2 \pi }-\frac {x \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{2 b \pi } \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.30, size = 44, normalized size = 0.90
method | result | size |
meijerg | \(\frac {b \,x^{3} \hypergeom \left (\left [\frac {1}{4}, \frac {3}{4}\right ], \left [\frac {1}{2}, \frac {5}{4}, \frac {7}{4}\right ], -\frac {x^{4} \pi ^{2} b^{4}}{16}\right )}{3}\) | \(26\) |
derivativedivides | \(\frac {\frac {\FresnelC \left (b x \right ) b^{2} x^{2}}{2}-\frac {b x \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{2 \pi }+\frac {\mathrm {S}\left (b x \right )}{2 \pi }}{b^{2}}\) | \(44\) |
default | \(\frac {\frac {\FresnelC \left (b x \right ) b^{2} x^{2}}{2}-\frac {b x \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{2 \pi }+\frac {\mathrm {S}\left (b x \right )}{2 \pi }}{b^{2}}\) | \(44\) |
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 = 70, normalized size = 1.43 \begin {gather*} \frac {1}{2} \, x^{2} \operatorname {C}\left (b x\right ) - \frac {\sqrt {\frac {1}{2}} {\left (4 \, \sqrt {\frac {1}{2}} \pi b x \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) - \left (i + 1\right ) \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \pi \operatorname {erf}\left (\sqrt {\frac {1}{2} i \, \pi } b x\right ) + \left (i - 1\right ) \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \pi \operatorname {erf}\left (\sqrt {-\frac {1}{2} i \, \pi } b x\right )\right )}}{4 \, \pi ^{2} b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 51, normalized size = 1.04 \begin {gather*} \frac {\pi b^{3} x^{2} \operatorname {C}\left (b x\right ) - b^{2} x \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) + \sqrt {b^{2}} \operatorname {S}\left (\sqrt {b^{2}} x\right )}{2 \, \pi b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.30, size = 49, normalized size = 1.00 \begin {gather*} \frac {b x^{3} \Gamma \left (\frac {1}{4}\right ) \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{3}\left (\begin {matrix} \frac {1}{4}, \frac {3}{4} \\ \frac {1}{2}, \frac {5}{4}, \frac {7}{4} \end {matrix}\middle | {- \frac {\pi ^{2} b^{4} x^{4}}{16}} \right )}}{16 \Gamma \left (\frac {5}{4}\right ) \Gamma \left (\frac {7}{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 = -0.02 \begin {gather*} \int x\,\mathrm {FresnelC}\left (b\,x\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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