Optimal. Leaf size=15 \[ \frac{\cos \left (a+\frac{b}{x^2}\right )}{2 b} \]
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Rubi [A] time = 0.0157894, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3379, 2638} \[ \frac{\cos \left (a+\frac{b}{x^2}\right )}{2 b} \]
Antiderivative was successfully verified.
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Rule 3379
Rule 2638
Rubi steps
\begin{align*} \int \frac{\sin \left (a+\frac{b}{x^2}\right )}{x^3} \, dx &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \sin (a+b x) \, dx,x,\frac{1}{x^2}\right )\right )\\ &=\frac{\cos \left (a+\frac{b}{x^2}\right )}{2 b}\\ \end{align*}
Mathematica [A] time = 0.0140532, size = 15, normalized size = 1. \[ \frac{\cos \left (a+\frac{b}{x^2}\right )}{2 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 14, normalized size = 0.9 \begin{align*}{\frac{1}{2\,b}\cos \left ( a+{\frac{b}{{x}^{2}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.942449, size = 18, normalized size = 1.2 \begin{align*} \frac{\cos \left (a + \frac{b}{x^{2}}\right )}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.67289, size = 38, normalized size = 2.53 \begin{align*} \frac{\cos \left (\frac{a x^{2} + b}{x^{2}}\right )}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.29878, size = 20, normalized size = 1.33 \begin{align*} \begin{cases} \frac{\cos{\left (a + \frac{b}{x^{2}} \right )}}{2 b} & \text{for}\: b \neq 0 \\- \frac{\sin{\left (a \right )}}{2 x^{2}} & \text{otherwise} \end{cases} \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^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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