Optimal. Leaf size=97 \[ -\frac{\sqrt{\frac{\pi }{2}} \cos (a) \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{b}}{x}\right )}{2 b^{3/2}}+\frac{\sqrt{\frac{\pi }{2}} \sin (a) S\left (\frac{\sqrt{b} \sqrt{\frac{2}{\pi }}}{x}\right )}{2 b^{3/2}}+\frac{\cos \left (a+\frac{b}{x^2}\right )}{2 b x} \]
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Rubi [A] time = 0.0585725, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {3409, 3385, 3354, 3352, 3351} \[ -\frac{\sqrt{\frac{\pi }{2}} \cos (a) \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{b}}{x}\right )}{2 b^{3/2}}+\frac{\sqrt{\frac{\pi }{2}} \sin (a) S\left (\frac{\sqrt{b} \sqrt{\frac{2}{\pi }}}{x}\right )}{2 b^{3/2}}+\frac{\cos \left (a+\frac{b}{x^2}\right )}{2 b x} \]
Antiderivative was successfully verified.
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Rule 3409
Rule 3385
Rule 3354
Rule 3352
Rule 3351
Rubi steps
\begin{align*} \int \frac{\sin \left (a+\frac{b}{x^2}\right )}{x^4} \, dx &=-\operatorname{Subst}\left (\int x^2 \sin \left (a+b x^2\right ) \, dx,x,\frac{1}{x}\right )\\ &=\frac{\cos \left (a+\frac{b}{x^2}\right )}{2 b x}-\frac{\operatorname{Subst}\left (\int \cos \left (a+b x^2\right ) \, dx,x,\frac{1}{x}\right )}{2 b}\\ &=\frac{\cos \left (a+\frac{b}{x^2}\right )}{2 b x}-\frac{\cos (a) \operatorname{Subst}\left (\int \cos \left (b x^2\right ) \, dx,x,\frac{1}{x}\right )}{2 b}+\frac{\sin (a) \operatorname{Subst}\left (\int \sin \left (b x^2\right ) \, dx,x,\frac{1}{x}\right )}{2 b}\\ &=\frac{\cos \left (a+\frac{b}{x^2}\right )}{2 b x}-\frac{\sqrt{\frac{\pi }{2}} \cos (a) C\left (\frac{\sqrt{b} \sqrt{\frac{2}{\pi }}}{x}\right )}{2 b^{3/2}}+\frac{\sqrt{\frac{\pi }{2}} S\left (\frac{\sqrt{b} \sqrt{\frac{2}{\pi }}}{x}\right ) \sin (a)}{2 b^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.155992, size = 89, normalized size = 0.92 \[ \frac{-\sqrt{2 \pi } x \cos (a) \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{b}}{x}\right )+\sqrt{2 \pi } x \sin (a) S\left (\frac{\sqrt{b} \sqrt{\frac{2}{\pi }}}{x}\right )+2 \sqrt{b} \cos \left (a+\frac{b}{x^2}\right )}{4 b^{3/2} x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 65, normalized size = 0.7 \begin{align*}{\frac{1}{2\,bx}\cos \left ( a+{\frac{b}{{x}^{2}}} \right ) }-{\frac{\sqrt{2}\sqrt{\pi }}{4} \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 ){b}^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 1.16273, size = 359, normalized size = 3.7 \begin{align*} -\frac{{\left ({\left (-i \, \Gamma \left (\frac{3}{2}, \frac{i \, b}{x^{2}}\right ) + i \, \Gamma \left (\frac{3}{2}, -\frac{i \, b}{x^{2}}\right )\right )} \cos \left (\frac{3}{4} \, \pi + \frac{3}{2} \, \arctan \left (0, b\right )\right ) +{\left (-i \, \Gamma \left (\frac{3}{2}, \frac{i \, b}{x^{2}}\right ) + i \, \Gamma \left (\frac{3}{2}, -\frac{i \, b}{x^{2}}\right )\right )} \cos \left (-\frac{3}{4} \, \pi + \frac{3}{2} \, \arctan \left (0, b\right )\right ) -{\left (\Gamma \left (\frac{3}{2}, \frac{i \, b}{x^{2}}\right ) + \Gamma \left (\frac{3}{2}, -\frac{i \, b}{x^{2}}\right )\right )} \sin \left (\frac{3}{4} \, \pi + \frac{3}{2} \, \arctan \left (0, b\right )\right ) +{\left (\Gamma \left (\frac{3}{2}, \frac{i \, b}{x^{2}}\right ) + \Gamma \left (\frac{3}{2}, -\frac{i \, b}{x^{2}}\right )\right )} \sin \left (-\frac{3}{4} \, \pi + \frac{3}{2} \, \arctan \left (0, b\right )\right )\right )} \cos \left (a\right ) -{\left ({\left (\Gamma \left (\frac{3}{2}, \frac{i \, b}{x^{2}}\right ) + \Gamma \left (\frac{3}{2}, -\frac{i \, b}{x^{2}}\right )\right )} \cos \left (\frac{3}{4} \, \pi + \frac{3}{2} \, \arctan \left (0, b\right )\right ) +{\left (\Gamma \left (\frac{3}{2}, \frac{i \, b}{x^{2}}\right ) + \Gamma \left (\frac{3}{2}, -\frac{i \, b}{x^{2}}\right )\right )} \cos \left (-\frac{3}{4} \, \pi + \frac{3}{2} \, \arctan \left (0, b\right )\right ) -{\left (i \, \Gamma \left (\frac{3}{2}, \frac{i \, b}{x^{2}}\right ) - i \, \Gamma \left (\frac{3}{2}, -\frac{i \, b}{x^{2}}\right )\right )} \sin \left (\frac{3}{4} \, \pi + \frac{3}{2} \, \arctan \left (0, b\right )\right ) -{\left (-i \, \Gamma \left (\frac{3}{2}, \frac{i \, b}{x^{2}}\right ) + i \, \Gamma \left (\frac{3}{2}, -\frac{i \, b}{x^{2}}\right )\right )} \sin \left (-\frac{3}{4} \, \pi + \frac{3}{2} \, \arctan \left (0, b\right )\right )\right )} \sin \left (a\right )}{8 \, x^{3} \left (\frac{{\left | b \right |}}{x^{2}}\right )^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.79547, size = 236, normalized size = 2.43 \begin{align*} -\frac{\sqrt{2} \pi x \sqrt{\frac{b}{\pi }} \cos \left (a\right ) \operatorname{C}\left (\frac{\sqrt{2} \sqrt{\frac{b}{\pi }}}{x}\right ) - \sqrt{2} \pi x \sqrt{\frac{b}{\pi }} \operatorname{S}\left (\frac{\sqrt{2} \sqrt{\frac{b}{\pi }}}{x}\right ) \sin \left (a\right ) - 2 \, b \cos \left (\frac{a x^{2} + b}{x^{2}}\right )}{4 \, b^{2} x} \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^{4}}\, 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^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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