Optimal. Leaf size=29 \[ \frac {\cos \left (a+\frac {b}{x}\right )}{b x}-\frac {\sin \left (a+\frac {b}{x}\right )}{b^2} \]
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Rubi [A] time = 0.02, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3379, 3296, 2637} \[ \frac {\cos \left (a+\frac {b}{x}\right )}{b x}-\frac {\sin \left (a+\frac {b}{x}\right )}{b^2} \]
Antiderivative was successfully verified.
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Rule 2637
Rule 3296
Rule 3379
Rubi steps
\begin {align*} \int \frac {\sin \left (a+\frac {b}{x}\right )}{x^3} \, dx &=-\operatorname {Subst}\left (\int x \sin (a+b x) \, dx,x,\frac {1}{x}\right )\\ &=\frac {\cos \left (a+\frac {b}{x}\right )}{b x}-\frac {\operatorname {Subst}\left (\int \cos (a+b x) \, dx,x,\frac {1}{x}\right )}{b}\\ &=\frac {\cos \left (a+\frac {b}{x}\right )}{b x}-\frac {\sin \left (a+\frac {b}{x}\right )}{b^2}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 29, normalized size = 1.00 \[ \frac {\cos \left (a+\frac {b}{x}\right )}{b x}-\frac {\sin \left (a+\frac {b}{x}\right )}{b^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 33, normalized size = 1.14 \[ \frac {b \cos \left (\frac {a x + b}{x}\right ) - x \sin \left (\frac {a x + b}{x}\right )}{b^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.59, size = 48, normalized size = 1.66 \[ -\frac {a \cos \left (\frac {a x + b}{x}\right ) - \frac {{\left (a x + b\right )} \cos \left (\frac {a x + b}{x}\right )}{x} + \sin \left (\frac {a x + b}{x}\right )}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 42, normalized size = 1.45 \[ -\frac {\sin \left (a +\frac {b}{x}\right )-\left (a +\frac {b}{x}\right ) \cos \left (a +\frac {b}{x}\right )+a \cos \left (a +\frac {b}{x}\right )}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.36, size = 50, normalized size = 1.72 \[ -\frac {{\left (i \, \Gamma \left (2, \frac {i \, b}{x}\right ) - i \, \Gamma \left (2, -\frac {i \, b}{x}\right )\right )} \cos \relax (a) + {\left (\Gamma \left (2, \frac {i \, b}{x}\right ) + \Gamma \left (2, -\frac {i \, b}{x}\right )\right )} \sin \relax (a)}{2 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.54, size = 29, normalized size = 1.00 \[ \frac {\cos \left (a+\frac {b}{x}\right )}{b\,x}-\frac {\sin \left (a+\frac {b}{x}\right )}{b^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.27, size = 29, normalized size = 1.00 \[ \begin {cases} \frac {\cos {\left (a + \frac {b}{x} \right )}}{b x} - \frac {\sin {\left (a + \frac {b}{x} \right )}}{b^{2}} & \text {for}\: b \neq 0 \\- \frac {\sin {\relax (a )}}{2 x^{2}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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