Optimal. Leaf size=109 \[ -\frac{S(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{3 x^3}-\frac{\pi b^2 S(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{3 x}-\frac{1}{6} \pi ^2 b^3 S(b x)^2+\frac{1}{6} \pi b^3 \text{Si}\left (b^2 \pi x^2\right )+\frac{b \cos \left (\pi b^2 x^2\right )}{12 x^2}-\frac{b}{12 x^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.116202, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {6456, 6464, 6440, 30, 3375, 3380, 3297, 3299} \[ -\frac{S(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{3 x^3}-\frac{\pi b^2 S(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{3 x}-\frac{1}{6} \pi ^2 b^3 S(b x)^2+\frac{1}{6} \pi b^3 \text{Si}\left (b^2 \pi x^2\right )+\frac{b \cos \left (\pi b^2 x^2\right )}{12 x^2}-\frac{b}{12 x^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6456
Rule 6464
Rule 6440
Rule 30
Rule 3375
Rule 3380
Rule 3297
Rule 3299
Rubi steps
\begin{align*} \int \frac{S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{x^4} \, dx &=-\frac{b}{12 x^2}-\frac{S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{3 x^3}-\frac{1}{6} b \int \frac{\cos \left (b^2 \pi x^2\right )}{x^3} \, dx+\frac{1}{3} \left (b^2 \pi \right ) \int \frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{x^2} \, dx\\ &=-\frac{b}{12 x^2}-\frac{b^2 \pi \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{3 x}-\frac{S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{3 x^3}-\frac{1}{12} b \operatorname{Subst}\left (\int \frac{\cos \left (b^2 \pi x\right )}{x^2} \, dx,x,x^2\right )+\frac{1}{6} \left (b^3 \pi \right ) \int \frac{\sin \left (b^2 \pi x^2\right )}{x} \, dx-\frac{1}{3} \left (b^4 \pi ^2\right ) \int S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right ) \, dx\\ &=-\frac{b}{12 x^2}+\frac{b \cos \left (b^2 \pi x^2\right )}{12 x^2}-\frac{b^2 \pi \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{3 x}-\frac{S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{3 x^3}+\frac{1}{12} b^3 \pi \text{Si}\left (b^2 \pi x^2\right )+\frac{1}{12} \left (b^3 \pi \right ) \operatorname{Subst}\left (\int \frac{\sin \left (b^2 \pi x\right )}{x} \, dx,x,x^2\right )-\frac{1}{3} \left (b^3 \pi ^2\right ) \operatorname{Subst}(\int x \, dx,x,S(b x))\\ &=-\frac{b}{12 x^2}+\frac{b \cos \left (b^2 \pi x^2\right )}{12 x^2}-\frac{b^2 \pi \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{3 x}-\frac{1}{6} b^3 \pi ^2 S(b x)^2-\frac{S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{3 x^3}+\frac{1}{6} b^3 \pi \text{Si}\left (b^2 \pi x^2\right )\\ \end{align*}
Mathematica [A] time = 0.007643, size = 109, normalized size = 1. \[ -\frac{S(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{3 x^3}-\frac{\pi b^2 S(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{3 x}-\frac{1}{6} \pi ^2 b^3 S(b x)^2+\frac{1}{6} \pi b^3 \text{Si}\left (b^2 \pi x^2\right )+\frac{b \cos \left (\pi b^2 x^2\right )}{12 x^2}-\frac{b}{12 x^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.062, size = 0, normalized size = 0. \begin{align*} \int{\frac{{\it FresnelS} \left ( bx \right ) }{{x}^{4}}\sin \left ({\frac{{b}^{2}\pi \,{x}^{2}}{2}} \right ) }\, dx \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 \frac{{\rm fresnels}\left (b x\right ) \sin \left (\frac{1}{2} \, \pi b^{2} x^{2}\right )}{x^{4}}\,{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 (\frac{{\rm fresnels}\left (b x\right ) \sin \left (\frac{1}{2} \, \pi b^{2} x^{2}\right )}{x^{4}}, 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 \frac{\sin{\left (\frac{\pi b^{2} x^{2}}{2} \right )} S\left (b x\right )}{x^{4}}\, 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 \frac{{\rm fresnels}\left (b x\right ) \sin \left (\frac{1}{2} \, \pi b^{2} x^{2}\right )}{x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]