Integrand size = 8, antiderivative size = 34 \[ \int \frac {\cos \left (\frac {1}{x}\right )}{x^5} \, dx=6 \cos \left (\frac {1}{x}\right )-\frac {3 \cos \left (\frac {1}{x}\right )}{x^2}-\frac {\sin \left (\frac {1}{x}\right )}{x^3}+\frac {6 \sin \left (\frac {1}{x}\right )}{x} \] Output:
6*cos(1/x)-3*cos(1/x)/x^2-sin(1/x)/x^3+6*sin(1/x)/x
Time = 0.02 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.94 \[ \int \frac {\cos \left (\frac {1}{x}\right )}{x^5} \, dx=\frac {3 \left (-1+2 x^2\right ) \cos \left (\frac {1}{x}\right )}{x^2}+\frac {\left (-1+6 x^2\right ) \sin \left (\frac {1}{x}\right )}{x^3} \] Input:
Integrate[Cos[x^(-1)]/x^5,x]
Output:
(3*(-1 + 2*x^2)*Cos[x^(-1)])/x^2 + ((-1 + 6*x^2)*Sin[x^(-1)])/x^3
Time = 0.37 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.09, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.375, Rules used = {3861, 3042, 3777, 25, 3042, 3777, 3042, 3777, 25, 3042, 3118}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\cos \left (\frac {1}{x}\right )}{x^5} \, dx\) |
\(\Big \downarrow \) 3861 |
\(\displaystyle -\int \frac {\cos \left (\frac {1}{x}\right )}{x^3}d\frac {1}{x}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\int \frac {\sin \left (\frac {\pi }{2}+\frac {1}{x}\right )}{x^3}d\frac {1}{x}\) |
\(\Big \downarrow \) 3777 |
\(\displaystyle -3 \int -\frac {\sin \left (\frac {1}{x}\right )}{x^2}d\frac {1}{x}-\frac {\sin \left (\frac {1}{x}\right )}{x^3}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle 3 \int \frac {\sin \left (\frac {1}{x}\right )}{x^2}d\frac {1}{x}-\frac {\sin \left (\frac {1}{x}\right )}{x^3}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle 3 \int \frac {\sin \left (\frac {1}{x}\right )}{x^2}d\frac {1}{x}-\frac {\sin \left (\frac {1}{x}\right )}{x^3}\) |
\(\Big \downarrow \) 3777 |
\(\displaystyle 3 \left (2 \int \frac {\cos \left (\frac {1}{x}\right )}{x}d\frac {1}{x}-\frac {\cos \left (\frac {1}{x}\right )}{x^2}\right )-\frac {\sin \left (\frac {1}{x}\right )}{x^3}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle 3 \left (2 \int \frac {\sin \left (\frac {\pi }{2}+\frac {1}{x}\right )}{x}d\frac {1}{x}-\frac {\cos \left (\frac {1}{x}\right )}{x^2}\right )-\frac {\sin \left (\frac {1}{x}\right )}{x^3}\) |
\(\Big \downarrow \) 3777 |
\(\displaystyle 3 \left (2 \left (\int -\sin \left (\frac {1}{x}\right )d\frac {1}{x}+\frac {\sin \left (\frac {1}{x}\right )}{x}\right )-\frac {\cos \left (\frac {1}{x}\right )}{x^2}\right )-\frac {\sin \left (\frac {1}{x}\right )}{x^3}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle 3 \left (2 \left (\frac {\sin \left (\frac {1}{x}\right )}{x}-\int \sin \left (\frac {1}{x}\right )d\frac {1}{x}\right )-\frac {\cos \left (\frac {1}{x}\right )}{x^2}\right )-\frac {\sin \left (\frac {1}{x}\right )}{x^3}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle 3 \left (2 \left (\frac {\sin \left (\frac {1}{x}\right )}{x}-\int \sin \left (\frac {1}{x}\right )d\frac {1}{x}\right )-\frac {\cos \left (\frac {1}{x}\right )}{x^2}\right )-\frac {\sin \left (\frac {1}{x}\right )}{x^3}\) |
\(\Big \downarrow \) 3118 |
\(\displaystyle 3 \left (2 \left (\frac {\sin \left (\frac {1}{x}\right )}{x}+\cos \left (\frac {1}{x}\right )\right )-\frac {\cos \left (\frac {1}{x}\right )}{x^2}\right )-\frac {\sin \left (\frac {1}{x}\right )}{x^3}\) |
Input:
Int[Cos[x^(-1)]/x^5,x]
Output:
-(Sin[x^(-1)]/x^3) + 3*(-(Cos[x^(-1)]/x^2) + 2*(Cos[x^(-1)] + Sin[x^(-1)]/ x))
Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[( -(c + d*x)^m)*(Cos[e + f*x]/f), x] + Simp[d*(m/f) Int[(c + d*x)^(m - 1)*C os[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
Int[((a_.) + Cos[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol ] :> Simp[1/n Subst[Int[x^(Simplify[(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[Simplify[ (m + 1)/n]] && (EqQ[p, 1] || EqQ[m, n - 1] || (IntegerQ[p] && GtQ[Simplify[ (m + 1)/n], 0]))
Time = 0.20 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.97
method | result | size |
risch | \(\frac {3 \left (2 x^{2}-1\right ) \cos \left (\frac {1}{x}\right )}{x^{2}}+\frac {\left (6 x^{2}-1\right ) \sin \left (\frac {1}{x}\right )}{x^{3}}\) | \(33\) |
derivativedivides | \(6 \cos \left (\frac {1}{x}\right )-\frac {3 \cos \left (\frac {1}{x}\right )}{x^{2}}-\frac {\sin \left (\frac {1}{x}\right )}{x^{3}}+\frac {6 \sin \left (\frac {1}{x}\right )}{x}\) | \(35\) |
default | \(6 \cos \left (\frac {1}{x}\right )-\frac {3 \cos \left (\frac {1}{x}\right )}{x^{2}}-\frac {\sin \left (\frac {1}{x}\right )}{x^{3}}+\frac {6 \sin \left (\frac {1}{x}\right )}{x}\) | \(35\) |
parallelrisch | \(\frac {\left (6 x^{3}-3 x \right ) \cos \left (\frac {1}{x}\right )-6 x^{3}+6 \sin \left (\frac {1}{x}\right ) x^{2}-\sin \left (\frac {1}{x}\right )}{x^{3}}\) | \(40\) |
meijerg | \(-8 \sqrt {\pi }\, \left (\frac {3}{4 \sqrt {\pi }}-\frac {\left (-\frac {3}{2 x^{2}}+3\right ) \cos \left (\frac {1}{x}\right )}{4 \sqrt {\pi }}-\frac {\left (-\frac {1}{2 x^{2}}+3\right ) \sin \left (\frac {1}{x}\right )}{4 \sqrt {\pi }\, x}\right )\) | \(47\) |
orering | \(\frac {\left (36 x^{5}-8 x^{3}\right ) \cos \left (\frac {1}{x}\right )}{x^{5}}+\left (6 x^{2}-1\right ) x^{4} \left (\frac {\sin \left (\frac {1}{x}\right )}{x^{7}}-\frac {5 \cos \left (\frac {1}{x}\right )}{x^{6}}\right )\) | \(50\) |
norman | \(\frac {12 x^{4}-3 x^{2}-2 x \tan \left (\frac {1}{2 x}\right )+3 x^{2} \tan \left (\frac {1}{2 x}\right )^{2}+12 x^{3} \tan \left (\frac {1}{2 x}\right )}{\left (1+\tan \left (\frac {1}{2 x}\right )^{2}\right ) x^{4}}\) | \(61\) |
Input:
int(cos(1/x)/x^5,x,method=_RETURNVERBOSE)
Output:
3/x^2*(2*x^2-1)*cos(1/x)+(6*x^2-1)/x^3*sin(1/x)
Time = 0.06 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.94 \[ \int \frac {\cos \left (\frac {1}{x}\right )}{x^5} \, dx=\frac {3 \, {\left (2 \, x^{3} - x\right )} \cos \left (\frac {1}{x}\right ) + {\left (6 \, x^{2} - 1\right )} \sin \left (\frac {1}{x}\right )}{x^{3}} \] Input:
integrate(cos(1/x)/x^5,x, algorithm="fricas")
Output:
(3*(2*x^3 - x)*cos(1/x) + (6*x^2 - 1)*sin(1/x))/x^3
Time = 0.50 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.94 \[ \int \frac {\cos \left (\frac {1}{x}\right )}{x^5} \, dx=6 \cos {\left (\frac {1}{x} \right )} + \frac {6 \sin {\left (\frac {1}{x} \right )}}{x} - \frac {3 \cos {\left (\frac {1}{x} \right )}}{x^{2}} - \frac {\sin {\left (\frac {1}{x} \right )}}{x^{3}} \] Input:
integrate(cos(1/x)/x**5,x)
Output:
6*cos(1/x) + 6*sin(1/x)/x - 3*cos(1/x)/x**2 - sin(1/x)/x**3
Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.06 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.56 \[ \int \frac {\cos \left (\frac {1}{x}\right )}{x^5} \, dx=\frac {1}{2} \, \Gamma \left (4, \frac {i}{x}\right ) + \frac {1}{2} \, \Gamma \left (4, -\frac {i}{x}\right ) \] Input:
integrate(cos(1/x)/x^5,x, algorithm="maxima")
Output:
1/2*gamma(4, I/x) + 1/2*gamma(4, -I/x)
Time = 0.13 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00 \[ \int \frac {\cos \left (\frac {1}{x}\right )}{x^5} \, dx=\frac {6 \, \sin \left (\frac {1}{x}\right )}{x} - \frac {3 \, \cos \left (\frac {1}{x}\right )}{x^{2}} - \frac {\sin \left (\frac {1}{x}\right )}{x^{3}} + 6 \, \cos \left (\frac {1}{x}\right ) \] Input:
integrate(cos(1/x)/x^5,x, algorithm="giac")
Output:
6*sin(1/x)/x - 3*cos(1/x)/x^2 - sin(1/x)/x^3 + 6*cos(1/x)
Time = 17.12 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.97 \[ \int \frac {\cos \left (\frac {1}{x}\right )}{x^5} \, dx=6\,\cos \left (\frac {1}{x}\right )-\frac {\sin \left (\frac {1}{x}\right )+3\,x\,\cos \left (\frac {1}{x}\right )-6\,x^2\,\sin \left (\frac {1}{x}\right )}{x^3} \] Input:
int(cos(1/x)/x^5,x)
Output:
6*cos(1/x) - (sin(1/x) + 3*x*cos(1/x) - 6*x^2*sin(1/x))/x^3
Time = 0.16 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.06 \[ \int \frac {\cos \left (\frac {1}{x}\right )}{x^5} \, dx=\frac {6 \cos \left (\frac {1}{x}\right ) x^{3}-3 \cos \left (\frac {1}{x}\right ) x +6 \sin \left (\frac {1}{x}\right ) x^{2}-\sin \left (\frac {1}{x}\right )}{x^{3}} \] Input:
int(cos(1/x)/x^5,x)
Output:
(6*cos(1/x)*x**3 - 3*cos(1/x)*x + 6*sin(1/x)*x**2 - sin(1/x))/x**3