Integrand size = 12, antiderivative size = 45 \[ \int \frac {\sin \left (a+\frac {b}{x}\right )}{x^4} \, dx=-\frac {2 \cos \left (a+\frac {b}{x}\right )}{b^3}+\frac {\cos \left (a+\frac {b}{x}\right )}{b x^2}-\frac {2 \sin \left (a+\frac {b}{x}\right )}{b^2 x} \] Output:
-2*cos(a+b/x)/b^3+cos(a+b/x)/b/x^2-2*sin(a+b/x)/b^2/x
Time = 0.07 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.84 \[ \int \frac {\sin \left (a+\frac {b}{x}\right )}{x^4} \, dx=\frac {\left (b^2-2 x^2\right ) \cos \left (a+\frac {b}{x}\right )-2 b x \sin \left (a+\frac {b}{x}\right )}{b^3 x^2} \] Input:
Integrate[Sin[a + b/x]/x^4,x]
Output:
((b^2 - 2*x^2)*Cos[a + b/x] - 2*b*x*Sin[a + b/x])/(b^3*x^2)
Time = 0.36 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.09, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {3860, 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.
\(\displaystyle \int \frac {\sin \left (a+\frac {b}{x}\right )}{x^4} \, dx\) |
\(\Big \downarrow \) 3860 |
\(\displaystyle -\int \frac {\sin \left (a+\frac {b}{x}\right )}{x^2}d\frac {1}{x}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\int \frac {\sin \left (a+\frac {b}{x}\right )}{x^2}d\frac {1}{x}\) |
\(\Big \downarrow \) 3777 |
\(\displaystyle \frac {\cos \left (a+\frac {b}{x}\right )}{b x^2}-\frac {2 \int \frac {\cos \left (a+\frac {b}{x}\right )}{x}d\frac {1}{x}}{b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\cos \left (a+\frac {b}{x}\right )}{b x^2}-\frac {2 \int \frac {\sin \left (a+\frac {b}{x}+\frac {\pi }{2}\right )}{x}d\frac {1}{x}}{b}\) |
\(\Big \downarrow \) 3777 |
\(\displaystyle \frac {\cos \left (a+\frac {b}{x}\right )}{b x^2}-\frac {2 \left (\frac {\int -\sin \left (a+\frac {b}{x}\right )d\frac {1}{x}}{b}+\frac {\sin \left (a+\frac {b}{x}\right )}{b x}\right )}{b}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\cos \left (a+\frac {b}{x}\right )}{b x^2}-\frac {2 \left (\frac {\sin \left (a+\frac {b}{x}\right )}{b x}-\frac {\int \sin \left (a+\frac {b}{x}\right )d\frac {1}{x}}{b}\right )}{b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\cos \left (a+\frac {b}{x}\right )}{b x^2}-\frac {2 \left (\frac {\sin \left (a+\frac {b}{x}\right )}{b x}-\frac {\int \sin \left (a+\frac {b}{x}\right )d\frac {1}{x}}{b}\right )}{b}\) |
\(\Big \downarrow \) 3118 |
\(\displaystyle \frac {\cos \left (a+\frac {b}{x}\right )}{b x^2}-\frac {2 \left (\frac {\cos \left (a+\frac {b}{x}\right )}{b^2}+\frac {\sin \left (a+\frac {b}{x}\right )}{b x}\right )}{b}\) |
Input:
Int[Sin[a + b/x]/x^4,x]
Output:
Cos[a + b/x]/(b*x^2) - (2*(Cos[a + b/x]/b^2 + Sin[a + b/x]/(b*x)))/b
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[(x_)^(m_.)*((a_.) + (b_.)*Sin[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbol ] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*Sin[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 = 0.86 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.02
method | result | size |
risch | \(\frac {\left (b^{2}-2 x^{2}\right ) \cos \left (\frac {a x +b}{x}\right )}{b^{3} x^{2}}-\frac {2 \sin \left (\frac {a x +b}{x}\right )}{b^{2} x}\) | \(46\) |
parallelrisch | \(\frac {\left (b^{2}-2 x^{2}\right ) \cos \left (\frac {a x +b}{x}\right )-2 \sin \left (\frac {a x +b}{x}\right ) b x +2 x^{2}}{b^{3} x^{2}}\) | \(48\) |
orering | \(-\frac {2 \left (3 b^{2}-4 x^{2}\right ) \sin \left (a +\frac {b}{x}\right )}{x \,b^{4}}-\frac {\left (b^{2}-2 x^{2}\right ) x^{4} \left (-\frac {b \cos \left (a +\frac {b}{x}\right )}{x^{6}}-\frac {4 \sin \left (a +\frac {b}{x}\right )}{x^{5}}\right )}{b^{4}}\) | \(74\) |
norman | \(\frac {\frac {x}{b}+\frac {4 x^{3} \tan \left (\frac {a}{2}+\frac {b}{2 x}\right )^{2}}{b^{3}}-\frac {4 x^{2} \tan \left (\frac {a}{2}+\frac {b}{2 x}\right )}{b^{2}}-\frac {x \tan \left (\frac {a}{2}+\frac {b}{2 x}\right )^{2}}{b}}{\left (1+\tan \left (\frac {a}{2}+\frac {b}{2 x}\right )^{2}\right ) x^{3}}\) | \(87\) |
derivativedivides | \(-\frac {-a^{2} \cos \left (a +\frac {b}{x}\right )-2 a \left (\sin \left (a +\frac {b}{x}\right )-\left (a +\frac {b}{x}\right ) \cos \left (a +\frac {b}{x}\right )\right )-\left (a +\frac {b}{x}\right )^{2} \cos \left (a +\frac {b}{x}\right )+2 \cos \left (a +\frac {b}{x}\right )+2 \left (a +\frac {b}{x}\right ) \sin \left (a +\frac {b}{x}\right )}{b^{3}}\) | \(95\) |
default | \(-\frac {-a^{2} \cos \left (a +\frac {b}{x}\right )-2 a \left (\sin \left (a +\frac {b}{x}\right )-\left (a +\frac {b}{x}\right ) \cos \left (a +\frac {b}{x}\right )\right )-\left (a +\frac {b}{x}\right )^{2} \cos \left (a +\frac {b}{x}\right )+2 \cos \left (a +\frac {b}{x}\right )+2 \left (a +\frac {b}{x}\right ) \sin \left (a +\frac {b}{x}\right )}{b^{3}}\) | \(95\) |
meijerg | \(-\frac {4 \sqrt {\pi }\, \cos \left (a \right ) \left (-\frac {1}{2 \sqrt {\pi }}+\frac {\left (-\frac {b^{2}}{2 x^{2}}+1\right ) \cos \left (\frac {b}{x}\right )}{2 \sqrt {\pi }}+\frac {b \sin \left (\frac {b}{x}\right )}{2 \sqrt {\pi }\, x}\right )}{b^{3}}-\frac {4 \sqrt {\pi }\, \sin \left (a \right ) \sqrt {b^{2}}\, \left (\frac {\left (b^{2}\right )^{\frac {3}{2}} \cos \left (\frac {b}{x}\right )}{2 \sqrt {\pi }\, x \,b^{2}}-\frac {\left (b^{2}\right )^{\frac {3}{2}} \left (-\frac {3 b^{2}}{2 x^{2}}+3\right ) \sin \left (\frac {b}{x}\right )}{6 \sqrt {\pi }\, b^{3}}\right )}{b^{4}}\) | \(121\) |
Input:
int(sin(a+b/x)/x^4,x,method=_RETURNVERBOSE)
Output:
(b^2-2*x^2)/b^3/x^2*cos((a*x+b)/x)-2/b^2/x*sin((a*x+b)/x)
Time = 0.07 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.98 \[ \int \frac {\sin \left (a+\frac {b}{x}\right )}{x^4} \, dx=-\frac {2 \, b x \sin \left (\frac {a x + b}{x}\right ) - {\left (b^{2} - 2 \, x^{2}\right )} \cos \left (\frac {a x + b}{x}\right )}{b^{3} x^{2}} \] Input:
integrate(sin(a+b/x)/x^4,x, algorithm="fricas")
Output:
-(2*b*x*sin((a*x + b)/x) - (b^2 - 2*x^2)*cos((a*x + b)/x))/(b^3*x^2)
Time = 0.49 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.02 \[ \int \frac {\sin \left (a+\frac {b}{x}\right )}{x^4} \, dx=\begin {cases} \frac {\cos {\left (a + \frac {b}{x} \right )}}{b x^{2}} - \frac {2 \sin {\left (a + \frac {b}{x} \right )}}{b^{2} x} - \frac {2 \cos {\left (a + \frac {b}{x} \right )}}{b^{3}} & \text {for}\: b \neq 0 \\- \frac {\sin {\left (a \right )}}{3 x^{3}} & \text {otherwise} \end {cases} \] Input:
integrate(sin(a+b/x)/x**4,x)
Output:
Piecewise((cos(a + b/x)/(b*x**2) - 2*sin(a + b/x)/(b**2*x) - 2*cos(a + b/x )/b**3, Ne(b, 0)), (-sin(a)/(3*x**3), True))
Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.08 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.13 \[ \int \frac {\sin \left (a+\frac {b}{x}\right )}{x^4} \, dx=-\frac {{\left (\Gamma \left (3, \frac {i \, b}{x}\right ) + \Gamma \left (3, -\frac {i \, b}{x}\right )\right )} \cos \left (a\right ) - {\left (i \, \Gamma \left (3, \frac {i \, b}{x}\right ) - i \, \Gamma \left (3, -\frac {i \, b}{x}\right )\right )} \sin \left (a\right )}{2 \, b^{3}} \] Input:
integrate(sin(a+b/x)/x^4,x, algorithm="maxima")
Output:
-1/2*((gamma(3, I*b/x) + gamma(3, -I*b/x))*cos(a) - (I*gamma(3, I*b/x) - I *gamma(3, -I*b/x))*sin(a))/b^3
Leaf count of result is larger than twice the leaf count of optimal. 106 vs. \(2 (45) = 90\).
Time = 0.14 (sec) , antiderivative size = 106, normalized size of antiderivative = 2.36 \[ \int \frac {\sin \left (a+\frac {b}{x}\right )}{x^4} \, dx=\frac {a^{2} \cos \left (\frac {a x + b}{x}\right ) - \frac {2 \, {\left (a x + b\right )} a \cos \left (\frac {a x + b}{x}\right )}{x} + 2 \, a \sin \left (\frac {a x + b}{x}\right ) + \frac {{\left (a x + b\right )}^{2} \cos \left (\frac {a x + b}{x}\right )}{x^{2}} - \frac {2 \, {\left (a x + b\right )} \sin \left (\frac {a x + b}{x}\right )}{x} - 2 \, \cos \left (\frac {a x + b}{x}\right )}{b^{3}} \] Input:
integrate(sin(a+b/x)/x^4,x, algorithm="giac")
Output:
(a^2*cos((a*x + b)/x) - 2*(a*x + b)*a*cos((a*x + b)/x)/x + 2*a*sin((a*x + b)/x) + (a*x + b)^2*cos((a*x + b)/x)/x^2 - 2*(a*x + b)*sin((a*x + b)/x)/x - 2*cos((a*x + b)/x))/b^3
Time = 41.29 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.02 \[ \int \frac {\sin \left (a+\frac {b}{x}\right )}{x^4} \, dx=\frac {b^2\,\cos \left (a+\frac {b}{x}\right )-2\,b\,x\,\sin \left (a+\frac {b}{x}\right )}{b^3\,x^2}-\frac {2\,\cos \left (a+\frac {b}{x}\right )}{b^3} \] Input:
int(sin(a + b/x)/x^4,x)
Output:
(b^2*cos(a + b/x) - 2*b*x*sin(a + b/x))/(b^3*x^2) - (2*cos(a + b/x))/b^3
\[ \int \frac {\sin \left (a+\frac {b}{x}\right )}{x^4} \, dx=\frac {-4 \cos \left (\frac {a x +b}{x}\right )+\left (\int \frac {\sin \left (\frac {a x +b}{x}\right )}{x^{4}}d x \right ) b^{3}+4 \left (\int \frac {\sin \left (\frac {a x +b}{x}\right )}{x^{2}}d x \right ) b}{b^{3}} \] Input:
int(sin(a+b/x)/x^4,x)
Output:
( - 4*cos((a*x + b)/x) + int(sin((a*x + b)/x)/x**4,x)*b**3 + 4*int(sin((a* x + b)/x)/x**2,x)*b)/b**3