Optimal. Leaf size=45 \[ -\frac{2 \sin \left (a+\frac{b}{x}\right )}{b^2 x}-\frac{2 \cos \left (a+\frac{b}{x}\right )}{b^3}+\frac{\cos \left (a+\frac{b}{x}\right )}{b x^2} \]
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Rubi [A] time = 0.0462511, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3379, 3296, 2638} \[ -\frac{2 \sin \left (a+\frac{b}{x}\right )}{b^2 x}-\frac{2 \cos \left (a+\frac{b}{x}\right )}{b^3}+\frac{\cos \left (a+\frac{b}{x}\right )}{b x^2} \]
Antiderivative was successfully verified.
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Rule 3379
Rule 3296
Rule 2638
Rubi steps
\begin{align*} \int \frac{\sin \left (a+\frac{b}{x}\right )}{x^4} \, dx &=-\operatorname{Subst}\left (\int x^2 \sin (a+b x) \, dx,x,\frac{1}{x}\right )\\ &=\frac{\cos \left (a+\frac{b}{x}\right )}{b x^2}-\frac{2 \operatorname{Subst}\left (\int x \cos (a+b x) \, dx,x,\frac{1}{x}\right )}{b}\\ &=\frac{\cos \left (a+\frac{b}{x}\right )}{b x^2}-\frac{2 \sin \left (a+\frac{b}{x}\right )}{b^2 x}+\frac{2 \operatorname{Subst}\left (\int \sin (a+b x) \, dx,x,\frac{1}{x}\right )}{b^2}\\ &=-\frac{2 \cos \left (a+\frac{b}{x}\right )}{b^3}+\frac{\cos \left (a+\frac{b}{x}\right )}{b x^2}-\frac{2 \sin \left (a+\frac{b}{x}\right )}{b^2 x}\\ \end{align*}
Mathematica [A] time = 0.0522672, size = 38, normalized size = 0.84 \[ \frac{\left (b^2-2 x^2\right ) \cos \left (a+\frac{b}{x}\right )-2 b x \sin \left (a+\frac{b}{x}\right )}{b^3 x^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.009, size = 95, normalized size = 2.1 \begin{align*} -{\frac{1}{{b}^{3}} \left ( - \left ( a+{\frac{b}{x}} \right ) ^{2}\cos \left ( a+{\frac{b}{x}} \right ) +2\,\cos \left ( a+{\frac{b}{x}} \right ) +2\, \left ( a+{\frac{b}{x}} \right ) \sin \left ( a+{\frac{b}{x}} \right ) -2\,a \left ( \sin \left ( a+{\frac{b}{x}} \right ) - \left ( a+{\frac{b}{x}} \right ) \cos \left ( a+{\frac{b}{x}} \right ) \right ) -{a}^{2}\cos \left ( a+{\frac{b}{x}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 1.3807, size = 69, normalized size = 1.53 \begin{align*} -\frac{{\left (\Gamma \left (3, \frac{i \, b}{x}\right ) + \Gamma \left (3, -\frac{i \, b}{x}\right )\right )} \cos \left (a\right ) -{\left (i \, \Gamma \left (3, \frac{i \, b}{x}\right ) - i \, \Gamma \left (3, -\frac{i \, b}{x}\right )\right )} \sin \left (a\right )}{2 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.67475, size = 95, normalized size = 2.11 \begin{align*} -\frac{2 \, b x \sin \left (\frac{a x + b}{x}\right ) -{\left (b^{2} - 2 \, x^{2}\right )} \cos \left (\frac{a x + b}{x}\right )}{b^{3} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.45554, size = 46, normalized size = 1.02 \begin{align*} \begin{cases} \frac{\cos{\left (a + \frac{b}{x} \right )}}{b x^{2}} - \frac{2 \sin{\left (a + \frac{b}{x} \right )}}{b^{2} x} - \frac{2 \cos{\left (a + \frac{b}{x} \right )}}{b^{3}} & \text{for}\: b \neq 0 \\- \frac{\sin{\left (a \right )}}{3 x^{3}} & \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}\right )}{x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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