Optimal. Leaf size=80 \[ \sqrt{2 \pi } \left (-\sqrt{b}\right ) \cos (a) \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{b}}{x}\right )+\sqrt{2 \pi } \sqrt{b} \sin (a) S\left (\frac{\sqrt{b} \sqrt{\frac{2}{\pi }}}{x}\right )+x \sin \left (a+\frac{b}{x^2}\right ) \]
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Rubi [A] time = 0.0577005, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.625, Rules used = {3359, 3387, 3354, 3352, 3351} \[ \sqrt{2 \pi } \left (-\sqrt{b}\right ) \cos (a) \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{b}}{x}\right )+\sqrt{2 \pi } \sqrt{b} \sin (a) S\left (\frac{\sqrt{b} \sqrt{\frac{2}{\pi }}}{x}\right )+x \sin \left (a+\frac{b}{x^2}\right ) \]
Antiderivative was successfully verified.
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Rule 3359
Rule 3387
Rule 3354
Rule 3352
Rule 3351
Rubi steps
\begin{align*} \int \sin \left (a+\frac{b}{x^2}\right ) \, dx &=-\operatorname{Subst}\left (\int \frac{\sin \left (a+b x^2\right )}{x^2} \, dx,x,\frac{1}{x}\right )\\ &=x \sin \left (a+\frac{b}{x^2}\right )-(2 b) \operatorname{Subst}\left (\int \cos \left (a+b x^2\right ) \, dx,x,\frac{1}{x}\right )\\ &=x \sin \left (a+\frac{b}{x^2}\right )-(2 b \cos (a)) \operatorname{Subst}\left (\int \cos \left (b x^2\right ) \, dx,x,\frac{1}{x}\right )+(2 b \sin (a)) \operatorname{Subst}\left (\int \sin \left (b x^2\right ) \, dx,x,\frac{1}{x}\right )\\ &=-\sqrt{b} \sqrt{2 \pi } \cos (a) C\left (\frac{\sqrt{b} \sqrt{\frac{2}{\pi }}}{x}\right )+\sqrt{b} \sqrt{2 \pi } S\left (\frac{\sqrt{b} \sqrt{\frac{2}{\pi }}}{x}\right ) \sin (a)+x \sin \left (a+\frac{b}{x^2}\right )\\ \end{align*}
Mathematica [A] time = 0.134061, size = 81, normalized size = 1.01 \[ -\sqrt{2 \pi } \sqrt{b} \left (\cos (a) \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{b}}{x}\right )-\sin (a) S\left (\frac{\sqrt{b} \sqrt{\frac{2}{\pi }}}{x}\right )\right )+x \sin (a) \cos \left (\frac{b}{x^2}\right )+x \cos (a) \sin \left (\frac{b}{x^2}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 59, normalized size = 0.7 \begin{align*} x\sin \left ( a+{\frac{b}{{x}^{2}}} \right ) -\sqrt{b}\sqrt{2}\sqrt{\pi } \left ( \cos \left ( a \right ){\it FresnelC} \left ({\frac{\sqrt{2}}{\sqrt{\pi }x}\sqrt{b}} \right ) -\sin \left ( a \right ){\it FresnelS} \left ({\frac{\sqrt{2}}{\sqrt{\pi }x}\sqrt{b}} \right ) \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 1.41088, size = 539, normalized size = 6.74 \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 = 1.72826, size = 209, normalized size = 2.61 \begin{align*} -\sqrt{2} \pi \sqrt{\frac{b}{\pi }} \cos \left (a\right ) \operatorname{C}\left (\frac{\sqrt{2} \sqrt{\frac{b}{\pi }}}{x}\right ) + \sqrt{2} \pi \sqrt{\frac{b}{\pi }} \operatorname{S}\left (\frac{\sqrt{2} \sqrt{\frac{b}{\pi }}}{x}\right ) \sin \left (a\right ) + x \sin \left (\frac{a x^{2} + b}{x^{2}}\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 \sin{\left (a + \frac{b}{x^{2}} \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 \sin \left (a + \frac{b}{x^{2}}\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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