Optimal. Leaf size=105 \[ \frac{5 \text{FresnelC}\left (\sqrt{2} b x\right )}{4 \sqrt{2} \pi ^2 b^4}+\frac{2 S(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{\pi ^2 b^4}-\frac{x^2 S(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac{x \cos \left (\pi b^2 x^2\right )}{4 \pi ^2 b^3}-\frac{x}{\pi ^2 b^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.081234, antiderivative size = 105, 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 = {6454, 6460, 3357, 3352, 3385} \[ \frac{5 \text{FresnelC}\left (\sqrt{2} b x\right )}{4 \sqrt{2} \pi ^2 b^4}+\frac{2 S(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{\pi ^2 b^4}-\frac{x^2 S(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac{x \cos \left (\pi b^2 x^2\right )}{4 \pi ^2 b^3}-\frac{x}{\pi ^2 b^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6454
Rule 6460
Rule 3357
Rule 3352
Rule 3385
Rubi steps
\begin{align*} \int x^3 S(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 ) S(b x)}{b^2 \pi }+\frac{2 \int x \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x) \, 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{x^2 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{b^2 \pi }+\frac{2 S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{b^4 \pi ^2}+\frac{\int \cos \left (b^2 \pi x^2\right ) \, dx}{4 b^3 \pi ^2}-\frac{2 \int \sin ^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{C\left (\sqrt{2} b x\right )}{4 \sqrt{2} b^4 \pi ^2}-\frac{x^2 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{b^2 \pi }+\frac{2 S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{b^4 \pi ^2}-\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{C\left (\sqrt{2} b x\right )}{4 \sqrt{2} b^4 \pi ^2}-\frac{x^2 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{b^2 \pi }+\frac{2 S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{b^4 \pi ^2}+\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{5 C\left (\sqrt{2} b x\right )}{4 \sqrt{2} b^4 \pi ^2}-\frac{x^2 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{b^2 \pi }+\frac{2 S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{b^4 \pi ^2}\\ \end{align*}
Mathematica [A] time = 0.102146, size = 83, normalized size = 0.79 \[ \frac{-8 S(b x) \left (\pi b^2 x^2 \cos \left (\frac{1}{2} \pi b^2 x^2\right )-2 \sin \left (\frac{1}{2} \pi b^2 x^2\right )\right )-2 b x \left (\cos \left (\pi b^2 x^2\right )+4\right )+5 \sqrt{2} \text{FresnelC}\left (\sqrt{2} b x\right )}{8 \pi ^2 b^4} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.083, size = 115, normalized size = 1.1 \begin{align*}{\frac{1}{b} \left ({\frac{{\it FresnelS} \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{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.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{3}{\rm fresnels}\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.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (x^{3}{\rm fresnels}\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.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{3} \sin{\left (\frac{\pi b^{2} x^{2}}{2} \right )} S\left (b x\right )\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{3}{\rm fresnels}\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.
[In]
[Out]