Optimal. Leaf size=46 \[ \frac{2 \sin \left (a+\frac{b}{x}\right )}{b^3}-\frac{2 \cos \left (a+\frac{b}{x}\right )}{b^2 x}-\frac{\sin \left (a+\frac{b}{x}\right )}{b x^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0467186, antiderivative size = 46, 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 = {3380, 3296, 2637} \[ \frac{2 \sin \left (a+\frac{b}{x}\right )}{b^3}-\frac{2 \cos \left (a+\frac{b}{x}\right )}{b^2 x}-\frac{\sin \left (a+\frac{b}{x}\right )}{b x^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3380
Rule 3296
Rule 2637
Rubi steps
\begin{align*} \int \frac{\cos \left (a+\frac{b}{x}\right )}{x^4} \, dx &=-\operatorname{Subst}\left (\int x^2 \cos (a+b x) \, dx,x,\frac{1}{x}\right )\\ &=-\frac{\sin \left (a+\frac{b}{x}\right )}{b x^2}+\frac{2 \operatorname{Subst}\left (\int x \sin (a+b x) \, dx,x,\frac{1}{x}\right )}{b}\\ &=-\frac{2 \cos \left (a+\frac{b}{x}\right )}{b^2 x}-\frac{\sin \left (a+\frac{b}{x}\right )}{b x^2}+\frac{2 \operatorname{Subst}\left (\int \cos (a+b x) \, dx,x,\frac{1}{x}\right )}{b^2}\\ &=-\frac{2 \cos \left (a+\frac{b}{x}\right )}{b^2 x}+\frac{2 \sin \left (a+\frac{b}{x}\right )}{b^3}-\frac{\sin \left (a+\frac{b}{x}\right )}{b x^2}\\ \end{align*}
Mathematica [A] time = 0.0041019, size = 46, normalized size = 1. \[ \frac{2 \sin \left (a+\frac{b}{x}\right )}{b^3}-\frac{2 \cos \left (a+\frac{b}{x}\right )}{b^2 x}-\frac{\sin \left (a+\frac{b}{x}\right )}{b x^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.033, size = 92, normalized size = 2. \begin{align*} -{\frac{1}{{b}^{3}} \left ( \left ( a+{\frac{b}{x}} \right ) ^{2}\sin \left ( a+{\frac{b}{x}} \right ) -2\,\sin \left ( a+{\frac{b}{x}} \right ) +2\, \left ( a+{\frac{b}{x}} \right ) \cos \left ( a+{\frac{b}{x}} \right ) -2\,a \left ( \cos \left ( a+{\frac{b}{x}} \right ) + \left ( a+{\frac{b}{x}} \right ) \sin \left ( a+{\frac{b}{x}} \right ) \right ) +{a}^{2}\sin \left ( a+{\frac{b}{x}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [C] time = 1.36795, size = 68, normalized size = 1.48 \begin{align*} \frac{{\left (i \, \Gamma \left (3, \frac{i \, b}{x}\right ) - i \, \Gamma \left (3, -\frac{i \, b}{x}\right )\right )} \cos \left (a\right ) +{\left (\Gamma \left (3, \frac{i \, b}{x}\right ) + \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.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.51528, size = 95, normalized size = 2.07 \begin{align*} -\frac{2 \, b x \cos \left (\frac{a x + b}{x}\right ) +{\left (b^{2} - 2 \, x^{2}\right )} \sin \left (\frac{a x + b}{x}\right )}{b^{3} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 6.18221, size = 46, normalized size = 1. \begin{align*} \begin{cases} - \frac{\sin{\left (a + \frac{b}{x} \right )}}{b x^{2}} - \frac{2 \cos{\left (a + \frac{b}{x} \right )}}{b^{2} x} + \frac{2 \sin{\left (a + \frac{b}{x} \right )}}{b^{3}} & \text{for}\: b \neq 0 \\- \frac{\cos{\left (a \right )}}{3 x^{3}} & \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{\cos \left (a + \frac{b}{x}\right )}{x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]