Optimal. Leaf size=75 \[ -\frac{\sqrt{\frac{\pi }{2}} \sin (a) \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{b}}{x}\right )}{\sqrt{b}}-\frac{\sqrt{\frac{\pi }{2}} \cos (a) S\left (\frac{\sqrt{b} \sqrt{\frac{2}{\pi }}}{x}\right )}{\sqrt{b}} \]
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Rubi [A] time = 0.030979, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {3383, 3353, 3352, 3351} \[ -\frac{\sqrt{\frac{\pi }{2}} \sin (a) \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{b}}{x}\right )}{\sqrt{b}}-\frac{\sqrt{\frac{\pi }{2}} \cos (a) S\left (\frac{\sqrt{b} \sqrt{\frac{2}{\pi }}}{x}\right )}{\sqrt{b}} \]
Antiderivative was successfully verified.
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Rule 3383
Rule 3353
Rule 3352
Rule 3351
Rubi steps
\begin{align*} \int \frac{\sin \left (a+\frac{b}{x^2}\right )}{x^2} \, dx &=-\operatorname{Subst}\left (\int \sin \left (a+b x^2\right ) \, dx,x,\frac{1}{x}\right )\\ &=-\left (\cos (a) \operatorname{Subst}\left (\int \sin \left (b x^2\right ) \, dx,x,\frac{1}{x}\right )\right )-\sin (a) \operatorname{Subst}\left (\int \cos \left (b x^2\right ) \, dx,x,\frac{1}{x}\right )\\ &=-\frac{\sqrt{\frac{\pi }{2}} \cos (a) S\left (\frac{\sqrt{b} \sqrt{\frac{2}{\pi }}}{x}\right )}{\sqrt{b}}-\frac{\sqrt{\frac{\pi }{2}} C\left (\frac{\sqrt{b} \sqrt{\frac{2}{\pi }}}{x}\right ) \sin (a)}{\sqrt{b}}\\ \end{align*}
Mathematica [A] time = 0.10151, size = 61, normalized size = 0.81 \[ -\frac{\sqrt{\frac{\pi }{2}} \left (\sin (a) \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{b}}{x}\right )+\cos (a) S\left (\frac{\sqrt{b} \sqrt{\frac{2}{\pi }}}{x}\right )\right )}{\sqrt{b}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 47, normalized size = 0.6 \begin{align*} -{\frac{\sqrt{2}\sqrt{\pi }}{2} \left ( \cos \left ( a \right ){\it FresnelS} \left ({\frac{\sqrt{2}}{\sqrt{\pi }x}\sqrt{b}} \right ) +\sin \left ( a \right ){\it FresnelC} \left ({\frac{\sqrt{2}}{\sqrt{\pi }x}\sqrt{b}} \right ) \right ){\frac{1}{\sqrt{b}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 1.17396, size = 495, normalized size = 6.6 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.70581, size = 186, normalized size = 2.48 \begin{align*} -\frac{\sqrt{2} \pi \sqrt{\frac{b}{\pi }} \cos \left (a\right ) \operatorname{S}\left (\frac{\sqrt{2} \sqrt{\frac{b}{\pi }}}{x}\right ) + \sqrt{2} \pi \sqrt{\frac{b}{\pi }} \operatorname{C}\left (\frac{\sqrt{2} \sqrt{\frac{b}{\pi }}}{x}\right ) \sin \left (a\right )}{2 \, b} \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 \frac{\sin{\left (a + \frac{b}{x^{2}} \right )}}{x^{2}}\, 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 \frac{\sin \left (a + \frac{b}{x^{2}}\right )}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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