3.43 \(\int \frac{\cos (a+\frac{b}{x^2})}{x^3} \, dx\)

Optimal. Leaf size=15 \[ -\frac{\sin \left (a+\frac{b}{x^2}\right )}{2 b} \]

[Out]

-Sin[a + b/x^2]/(2*b)

________________________________________________________________________________________

Rubi [A]  time = 0.0150678, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3380, 2637} \[ -\frac{\sin \left (a+\frac{b}{x^2}\right )}{2 b} \]

Antiderivative was successfully verified.

[In]

Int[Cos[a + b/x^2]/x^3,x]

[Out]

-Sin[a + b/x^2]/(2*b)

Rule 3380

Int[((a_.) + Cos[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplif
y[(m + 1)/n] - 1)*(a + b*Cos[c + d*x])^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IntegerQ[Simpl
ify[(m + 1)/n]] && (EqQ[p, 1] || EqQ[m, n - 1] || (IntegerQ[p] && GtQ[Simplify[(m + 1)/n], 0]))

Rule 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int \frac{\cos \left (a+\frac{b}{x^2}\right )}{x^3} \, dx &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \cos (a+b x) \, dx,x,\frac{1}{x^2}\right )\right )\\ &=-\frac{\sin \left (a+\frac{b}{x^2}\right )}{2 b}\\ \end{align*}

Mathematica [A]  time = 0.0031249, size = 15, normalized size = 1. \[ -\frac{\sin \left (a+\frac{b}{x^2}\right )}{2 b} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[a + b/x^2]/x^3,x]

[Out]

-Sin[a + b/x^2]/(2*b)

________________________________________________________________________________________

Maple [A]  time = 0.024, size = 14, normalized size = 0.9 \begin{align*} -{\frac{1}{2\,b}\sin \left ( a+{\frac{b}{{x}^{2}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(a+b/x^2)/x^3,x)

[Out]

-1/2*sin(a+b/x^2)/b

________________________________________________________________________________________

Maxima [A]  time = 1.14863, size = 18, normalized size = 1.2 \begin{align*} -\frac{\sin \left (a + \frac{b}{x^{2}}\right )}{2 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(a+b/x^2)/x^3,x, algorithm="maxima")

[Out]

-1/2*sin(a + b/x^2)/b

________________________________________________________________________________________

Fricas [A]  time = 1.55821, size = 39, normalized size = 2.6 \begin{align*} -\frac{\sin \left (\frac{a x^{2} + b}{x^{2}}\right )}{2 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(a+b/x^2)/x^3,x, algorithm="fricas")

[Out]

-1/2*sin((a*x^2 + b)/x^2)/b

________________________________________________________________________________________

Sympy [A]  time = 6.09938, size = 22, normalized size = 1.47 \begin{align*} \begin{cases} - \frac{\sin{\left (a + \frac{b}{x^{2}} \right )}}{2 b} & \text{for}\: b \neq 0 \\- \frac{\cos{\left (a \right )}}{2 x^{2}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(a+b/x**2)/x**3,x)

[Out]

Piecewise((-sin(a + b/x**2)/(2*b), Ne(b, 0)), (-cos(a)/(2*x**2), True))

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos \left (a + \frac{b}{x^{2}}\right )}{x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(a+b/x^2)/x^3,x, algorithm="giac")

[Out]

integrate(cos(a + b/x^2)/x^3, x)