Optimal. Leaf size=104 \[ \frac{x^2 \text{FresnelC}(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac{2 \text{FresnelC}(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{\pi ^2 b^4}-\frac{5 \text{FresnelC}\left (\sqrt{2} b x\right )}{4 \sqrt{2} \pi ^2 b^4}+\frac{x \cos \left (\pi b^2 x^2\right )}{4 \pi ^2 b^3}-\frac{x}{\pi ^2 b^3} \]
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Rubi [A] time = 0.0782079, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {6455, 6461, 3358, 3352, 3385} \[ \frac{x^2 \text{FresnelC}(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac{2 \text{FresnelC}(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{\pi ^2 b^4}-\frac{5 \text{FresnelC}\left (\sqrt{2} b x\right )}{4 \sqrt{2} \pi ^2 b^4}+\frac{x \cos \left (\pi b^2 x^2\right )}{4 \pi ^2 b^3}-\frac{x}{\pi ^2 b^3} \]
Antiderivative was successfully verified.
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Rule 6455
Rule 6461
Rule 3358
Rule 3352
Rule 3385
Rubi steps
\begin{align*} \int x^3 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x) \, dx &=\frac{x^2 C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{b^2 \pi }-\frac{2 \int x C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right ) \, dx}{b^2 \pi }-\frac{\int x^2 \sin \left (b^2 \pi x^2\right ) \, dx}{2 b \pi }\\ &=\frac{x \cos \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}+\frac{2 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{b^4 \pi ^2}+\frac{x^2 C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{b^2 \pi }-\frac{\int \cos \left (b^2 \pi x^2\right ) \, dx}{4 b^3 \pi ^2}-\frac{2 \int \cos ^2\left (\frac{1}{2} b^2 \pi x^2\right ) \, dx}{b^3 \pi ^2}\\ &=\frac{x \cos \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}+\frac{2 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{b^4 \pi ^2}-\frac{C\left (\sqrt{2} b x\right )}{4 \sqrt{2} b^4 \pi ^2}+\frac{x^2 C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{b^2 \pi }-\frac{2 \int \left (\frac{1}{2}+\frac{1}{2} \cos \left (b^2 \pi x^2\right )\right ) \, dx}{b^3 \pi ^2}\\ &=-\frac{x}{b^3 \pi ^2}+\frac{x \cos \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}+\frac{2 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{b^4 \pi ^2}-\frac{C\left (\sqrt{2} b x\right )}{4 \sqrt{2} b^4 \pi ^2}+\frac{x^2 C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{b^2 \pi }-\frac{\int \cos \left (b^2 \pi x^2\right ) \, dx}{b^3 \pi ^2}\\ &=-\frac{x}{b^3 \pi ^2}+\frac{x \cos \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}+\frac{2 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{b^4 \pi ^2}-\frac{5 C\left (\sqrt{2} b x\right )}{4 \sqrt{2} b^4 \pi ^2}+\frac{x^2 C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{b^2 \pi }\\ \end{align*}
Mathematica [A] time = 0.084731, size = 83, normalized size = 0.8 \[ \frac{8 \text{FresnelC}(b x) \left (\pi b^2 x^2 \sin \left (\frac{1}{2} \pi b^2 x^2\right )+2 \cos \left (\frac{1}{2} \pi b^2 x^2\right )\right )+2 b x \left (\cos \left (\pi b^2 x^2\right )-4\right )-5 \sqrt{2} \text{FresnelC}\left (\sqrt{2} b x\right )}{8 \pi ^2 b^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.076, size = 114, 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 }\sin \left ({\frac{{b}^{2}\pi \,{x}^{2}}{2}} \right ) }+2\,{\frac{\cos \left ( 1/2\,{b}^{2}\pi \,{x}^{2} \right ) }{{\pi }^{2}}} \right ) }-{\frac{1}{{b}^{3}} \left ({\frac{bx}{{\pi }^{2}}}+{\frac{\sqrt{2}{\it FresnelC} \left ( bx\sqrt{2} \right ) }{2\,{\pi }^{2}}}+{\frac{1}{2\,\pi } \left ( -{\frac{bx\cos \left ({b}^{2}\pi \,{x}^{2} \right ) }{2\,\pi }}+{\frac{\sqrt{2}{\it FresnelC} \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} \cos \left (\frac{1}{2} \, \pi b^{2} x^{2}\right ){\rm fresnelc}\left (b x\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} \cos \left (\frac{1}{2} \, \pi b^{2} x^{2}\right ){\rm fresnelc}\left (b x\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} \cos{\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} \cos \left (\frac{1}{2} \, \pi b^{2} x^{2}\right ){\rm fresnelc}\left (b x\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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