Optimal. Leaf size=109 \[ \frac{2 \text{FresnelC}(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{\pi ^2 b^4}-\frac{x^2 \text{FresnelC}(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac{5 S\left (\sqrt{2} b x\right )}{4 \sqrt{2} \pi ^2 b^4}+\frac{x \sin \left (\pi b^2 x^2\right )}{4 \pi ^2 b^3}+\frac{x^3}{6 \pi b} \]
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Rubi [A] time = 0.0893288, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {6463, 6453, 3351, 3392, 30, 3386} \[ \frac{2 \text{FresnelC}(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{\pi ^2 b^4}-\frac{x^2 \text{FresnelC}(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac{5 S\left (\sqrt{2} b x\right )}{4 \sqrt{2} \pi ^2 b^4}+\frac{x \sin \left (\pi b^2 x^2\right )}{4 \pi ^2 b^3}+\frac{x^3}{6 \pi b} \]
Antiderivative was successfully verified.
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Rule 6463
Rule 6453
Rule 3351
Rule 3392
Rule 30
Rule 3386
Rubi steps
\begin{align*} \int x^3 C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right ) \, dx &=-\frac{x^2 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{b^2 \pi }+\frac{2 \int x \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x) \, dx}{b^2 \pi }+\frac{\int x^2 \cos ^2\left (\frac{1}{2} b^2 \pi x^2\right ) \, dx}{b \pi }\\ &=-\frac{x^2 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{b^2 \pi }+\frac{2 C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{b^4 \pi ^2}-\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{x^2 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{b^2 \pi }-\frac{S\left (\sqrt{2} b x\right )}{\sqrt{2} b^4 \pi ^2}+\frac{2 C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{b^4 \pi ^2}+\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{x^2 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{b^2 \pi }-\frac{5 S\left (\sqrt{2} b x\right )}{4 \sqrt{2} b^4 \pi ^2}+\frac{2 C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{b^4 \pi ^2}+\frac{x \sin \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}\\ \end{align*}
Mathematica [A] time = 0.0841006, size = 90, normalized size = 0.83 \[ \frac{-24 \text{FresnelC}(b x) \left (\pi b^2 x^2 \cos \left (\frac{1}{2} \pi b^2 x^2\right )-2 \sin \left (\frac{1}{2} \pi b^2 x^2\right )\right )+4 \pi b^3 x^3+6 b x \sin \left (\pi b^2 x^2\right )-15 \sqrt{2} S\left (\sqrt{2} b x\right )}{24 \pi ^2 b^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.08, size = 120, normalized size = 1.1 \begin{align*}{\frac{1}{b} \left ({\frac{{\it FresnelC} \left ( bx \right ) }{{b}^{3}} \left ( -{\frac{{b}^{2}{x}^{2}}{\pi }\cos \left ({\frac{{b}^{2}\pi \,{x}^{2}}{2}} \right ) }+2\,{\frac{\sin \left ( 1/2\,{b}^{2}\pi \,{x}^{2} \right ) }{{\pi }^{2}}} \right ) }-{\frac{1}{{b}^{3}} \left ({\frac{\sqrt{2}{\it FresnelS} \left ( bx\sqrt{2} \right ) }{2\,{\pi }^{2}}}-{\frac{{x}^{3}{b}^{3}}{6\,\pi }}-{\frac{1}{2\,\pi } \left ({\frac{bx\sin \left ({b}^{2}\pi \,{x}^{2} \right ) }{2\,\pi }}-{\frac{\sqrt{2}{\it FresnelS} \left ( bx\sqrt{2} \right ) }{4\,\pi }} \right ) } \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{3}{\rm fresnelc}\left (b x\right ) \sin \left (\frac{1}{2} \, \pi b^{2} x^{2}\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (x^{3}{\rm fresnelc}\left (b x\right ) \sin \left (\frac{1}{2} \, \pi b^{2} x^{2}\right ), x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{3} \sin{\left (\frac{\pi b^{2} x^{2}}{2} \right )} C\left (b x\right )\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{3}{\rm fresnelc}\left (b x\right ) \sin \left (\frac{1}{2} \, \pi b^{2} x^{2}\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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