Optimal. Leaf size=61 \[ \frac {6 \sin \left (a+\frac {b}{x}\right )}{b^4}-\frac {6 \cos \left (a+\frac {b}{x}\right )}{b^3 x}-\frac {3 \sin \left (a+\frac {b}{x}\right )}{b^2 x^2}+\frac {\cos \left (a+\frac {b}{x}\right )}{b x^3} \]
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Rubi [A] time = 0.07, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3379, 3296, 2637} \[ -\frac {3 \sin \left (a+\frac {b}{x}\right )}{b^2 x^2}+\frac {6 \sin \left (a+\frac {b}{x}\right )}{b^4}-\frac {6 \cos \left (a+\frac {b}{x}\right )}{b^3 x}+\frac {\cos \left (a+\frac {b}{x}\right )}{b x^3} \]
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^5} \, dx &=-\operatorname {Subst}\left (\int x^3 \sin (a+b x) \, dx,x,\frac {1}{x}\right )\\ &=\frac {\cos \left (a+\frac {b}{x}\right )}{b x^3}-\frac {3 \operatorname {Subst}\left (\int x^2 \cos (a+b x) \, dx,x,\frac {1}{x}\right )}{b}\\ &=\frac {\cos \left (a+\frac {b}{x}\right )}{b x^3}-\frac {3 \sin \left (a+\frac {b}{x}\right )}{b^2 x^2}+\frac {6 \operatorname {Subst}\left (\int x \sin (a+b x) \, dx,x,\frac {1}{x}\right )}{b^2}\\ &=\frac {\cos \left (a+\frac {b}{x}\right )}{b x^3}-\frac {6 \cos \left (a+\frac {b}{x}\right )}{b^3 x}-\frac {3 \sin \left (a+\frac {b}{x}\right )}{b^2 x^2}+\frac {6 \operatorname {Subst}\left (\int \cos (a+b x) \, dx,x,\frac {1}{x}\right )}{b^3}\\ &=\frac {\cos \left (a+\frac {b}{x}\right )}{b x^3}-\frac {6 \cos \left (a+\frac {b}{x}\right )}{b^3 x}+\frac {6 \sin \left (a+\frac {b}{x}\right )}{b^4}-\frac {3 \sin \left (a+\frac {b}{x}\right )}{b^2 x^2}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 61, normalized size = 1.00 \[ \frac {6 \sin \left (a+\frac {b}{x}\right )}{b^4}-\frac {6 \cos \left (a+\frac {b}{x}\right )}{b^3 x}-\frac {3 \sin \left (a+\frac {b}{x}\right )}{b^2 x^2}+\frac {\cos \left (a+\frac {b}{x}\right )}{b x^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.67, size = 52, normalized size = 0.85 \[ \frac {{\left (b^{3} - 6 \, b x^{2}\right )} \cos \left (\frac {a x + b}{x}\right ) - 3 \, {\left (b^{2} x - 2 \, x^{3}\right )} \sin \left (\frac {a x + b}{x}\right )}{b^{4} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.15, size = 191, normalized size = 3.13 \[ -\frac {a^{3} \cos \left (\frac {a x + b}{x}\right ) - \frac {3 \, {\left (a x + b\right )} a^{2} \cos \left (\frac {a x + b}{x}\right )}{x} + 3 \, a^{2} \sin \left (\frac {a x + b}{x}\right ) - 6 \, a \cos \left (\frac {a x + b}{x}\right ) + \frac {3 \, {\left (a x + b\right )}^{2} a \cos \left (\frac {a x + b}{x}\right )}{x^{2}} - \frac {6 \, {\left (a x + b\right )} a \sin \left (\frac {a x + b}{x}\right )}{x} - \frac {{\left (a x + b\right )}^{3} \cos \left (\frac {a x + b}{x}\right )}{x^{3}} + \frac {6 \, {\left (a x + b\right )} \cos \left (\frac {a x + b}{x}\right )}{x} + \frac {3 \, {\left (a x + b\right )}^{2} \sin \left (\frac {a x + b}{x}\right )}{x^{2}} - 6 \, \sin \left (\frac {a x + b}{x}\right )}{b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.03, size = 165, normalized size = 2.70 \[ -\frac {-\left (a +\frac {b}{x}\right )^{3} \cos \left (a +\frac {b}{x}\right )+3 \left (a +\frac {b}{x}\right )^{2} \sin \left (a +\frac {b}{x}\right )-6 \sin \left (a +\frac {b}{x}\right )+6 \left (a +\frac {b}{x}\right ) \cos \left (a +\frac {b}{x}\right )-3 a \left (-\left (a +\frac {b}{x}\right )^{2} \cos \left (a +\frac {b}{x}\right )+2 \cos \left (a +\frac {b}{x}\right )+2 \left (a +\frac {b}{x}\right ) \sin \left (a +\frac {b}{x}\right )\right )+3 a^{2} \left (\sin \left (a +\frac {b}{x}\right )-\left (a +\frac {b}{x}\right ) \cos \left (a +\frac {b}{x}\right )\right )+a^{3} \cos \left (a +\frac {b}{x}\right )}{b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.42, size = 50, normalized size = 0.82 \[ \frac {{\left (i \, \Gamma \left (4, \frac {i \, b}{x}\right ) - i \, \Gamma \left (4, -\frac {i \, b}{x}\right )\right )} \cos \relax (a) + {\left (\Gamma \left (4, \frac {i \, b}{x}\right ) + \Gamma \left (4, -\frac {i \, b}{x}\right )\right )} \sin \relax (a)}{2 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.77, size = 64, normalized size = 1.05 \[ \frac {6\,\sin \left (a+\frac {b}{x}\right )}{b^4}-\frac {6\,b\,x^2\,\cos \left (a+\frac {b}{x}\right )-b^3\,\cos \left (a+\frac {b}{x}\right )+3\,b^2\,x\,\sin \left (a+\frac {b}{x}\right )}{b^4\,x^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 6.42, size = 61, normalized size = 1.00 \[ \begin {cases} \frac {\cos {\left (a + \frac {b}{x} \right )}}{b x^{3}} - \frac {3 \sin {\left (a + \frac {b}{x} \right )}}{b^{2} x^{2}} - \frac {6 \cos {\left (a + \frac {b}{x} \right )}}{b^{3} x} + \frac {6 \sin {\left (a + \frac {b}{x} \right )}}{b^{4}} & \text {for}\: b \neq 0 \\- \frac {\sin {\relax (a )}}{4 x^{4}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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