Optimal. Leaf size=30 \[ -\frac {\cos \left (a+\frac {b}{x}\right )}{b^2}-\frac {\sin \left (a+\frac {b}{x}\right )}{b x} \]
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Rubi [A]
time = 0.02, antiderivative size = 30, 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 = {3461, 3377,
2718} \begin {gather*} -\frac {\cos \left (a+\frac {b}{x}\right )}{b^2}-\frac {\sin \left (a+\frac {b}{x}\right )}{b x} \end {gather*}
Antiderivative was successfully verified.
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Rule 2718
Rule 3377
Rule 3461
Rubi steps
\begin {align*} \int \frac {\cos \left (a+\frac {b}{x}\right )}{x^3} \, dx &=-\text {Subst}\left (\int x \cos (a+b x) \, dx,x,\frac {1}{x}\right )\\ &=-\frac {\sin \left (a+\frac {b}{x}\right )}{b x}+\frac {\text {Subst}\left (\int \sin (a+b x) \, dx,x,\frac {1}{x}\right )}{b}\\ &=-\frac {\cos \left (a+\frac {b}{x}\right )}{b^2}-\frac {\sin \left (a+\frac {b}{x}\right )}{b x}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 29, normalized size = 0.97 \begin {gather*} -\frac {x \cos \left (a+\frac {b}{x}\right )+b \sin \left (a+\frac {b}{x}\right )}{b^2 x} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 42, normalized size = 1.40
method | result | size |
risch | \(-\frac {\cos \left (\frac {a x +b}{x}\right )}{b^{2}}-\frac {\sin \left (\frac {a x +b}{x}\right )}{b x}\) | \(35\) |
derivativedivides | \(-\frac {\cos \left (a +\frac {b}{x}\right )+\left (a +\frac {b}{x}\right ) \sin \left (a +\frac {b}{x}\right )-a \sin \left (a +\frac {b}{x}\right )}{b^{2}}\) | \(42\) |
default | \(-\frac {\cos \left (a +\frac {b}{x}\right )+\left (a +\frac {b}{x}\right ) \sin \left (a +\frac {b}{x}\right )-a \sin \left (a +\frac {b}{x}\right )}{b^{2}}\) | \(42\) |
norman | \(\frac {\frac {2 x^{2} \left (\tan ^{2}\left (\frac {a}{2}+\frac {b}{2 x}\right )\right )}{b^{2}}-\frac {2 x \tan \left (\frac {a}{2}+\frac {b}{2 x}\right )}{b}}{\left (1+\tan ^{2}\left (\frac {a}{2}+\frac {b}{2 x}\right )\right ) x^{2}}\) | \(61\) |
meijerg | \(-\frac {2 \sqrt {\pi }\, \cos \left (a \right ) \left (-\frac {1}{2 \sqrt {\pi }}+\frac {\cos \left (\frac {b}{x}\right )}{2 \sqrt {\pi }}+\frac {b \sin \left (\frac {b}{x}\right )}{2 \sqrt {\pi }\, x}\right )}{b^{2}}+\frac {2 \sqrt {\pi }\, \sin \left (a \right ) \left (-\frac {b \cos \left (\frac {b}{x}\right )}{2 \sqrt {\pi }\, x}+\frac {\sin \left (\frac {b}{x}\right )}{2 \sqrt {\pi }}\right )}{b^{2}}\) | \(81\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains higher order function than in optimal. Order 4 vs. order
3.
time = 0.32, size = 51, normalized size = 1.70 \begin {gather*} -\frac {{\left (\Gamma \left (2, \frac {i \, b}{x}\right ) + \Gamma \left (2, -\frac {i \, b}{x}\right )\right )} \cos \left (a\right ) - {\left (i \, \Gamma \left (2, \frac {i \, b}{x}\right ) - i \, \Gamma \left (2, -\frac {i \, b}{x}\right )\right )} \sin \left (a\right )}{2 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 33, normalized size = 1.10 \begin {gather*} -\frac {x \cos \left (\frac {a x + b}{x}\right ) + b \sin \left (\frac {a x + b}{x}\right )}{b^{2} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.47, size = 31, normalized size = 1.03 \begin {gather*} \begin {cases} - \frac {\sin {\left (a + \frac {b}{x} \right )}}{b x} - \frac {\cos {\left (a + \frac {b}{x} \right )}}{b^{2}} & \text {for}\: b \neq 0 \\- \frac {\cos {\left (a \right )}}{2 x^{2}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.45, size = 49, normalized size = 1.63 \begin {gather*} \frac {a \sin \left (\frac {a x + b}{x}\right ) - \frac {{\left (a x + b\right )} \sin \left (\frac {a x + b}{x}\right )}{x} - \cos \left (\frac {a x + b}{x}\right )}{b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.28, size = 30, normalized size = 1.00 \begin {gather*} -\frac {\cos \left (a+\frac {b}{x}\right )}{b^2}-\frac {\sin \left (a+\frac {b}{x}\right )}{b\,x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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