Optimal. Leaf size=29 \[ \frac{\cos \left (a+\frac{b}{x}\right )}{b x}-\frac{\sin \left (a+\frac{b}{x}\right )}{b^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0246476, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, 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.
[In]
[Out]
Rule 3379
Rule 3296
Rule 2637
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*}
Mathematica [A] time = 0.0040076, size = 29, normalized size = 1. \[ \frac{\cos \left (a+\frac{b}{x}\right )}{b x}-\frac{\sin \left (a+\frac{b}{x}\right )}{b^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.007, size = 42, normalized size = 1.5 \begin{align*} -{\frac{1}{{b}^{2}} \left ( \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 ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [C] time = 1.13393, size = 68, normalized size = 2.34 \begin{align*} -\frac{{\left (i \, \Gamma \left (2, \frac{i \, b}{x}\right ) - i \, \Gamma \left (2, -\frac{i \, b}{x}\right )\right )} \cos \left (a\right ) +{\left (\Gamma \left (2, \frac{i \, b}{x}\right ) + \Gamma \left (2, -\frac{i \, b}{x}\right )\right )} \sin \left (a\right )}{2 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.56024, size = 69, normalized size = 2.38 \begin{align*} \frac{b \cos \left (\frac{a x + b}{x}\right ) - x \sin \left (\frac{a x + b}{x}\right )}{b^{2} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 2.42722, size = 29, normalized size = 1. \begin{align*} \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{\left (a \right )}}{2 x^{2}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin \left (a + \frac{b}{x}\right )}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]