Integrand size = 12, antiderivative size = 61 \[ \int \frac {\sin \left (a+\frac {b}{x}\right )}{x^5} \, dx=\frac {\cos \left (a+\frac {b}{x}\right )}{b x^3}-\frac {6 \cos \left (a+\frac {b}{x}\right )}{b^3 x}+\frac {6 \sin \left (a+\frac {b}{x}\right )}{b^4}-\frac {3 \sin \left (a+\frac {b}{x}\right )}{b^2 x^2} \]
Time = 0.01 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00 \[ \int \frac {\sin \left (a+\frac {b}{x}\right )}{x^5} \, dx=\frac {\cos \left (a+\frac {b}{x}\right )}{b x^3}-\frac {6 \cos \left (a+\frac {b}{x}\right )}{b^3 x}+\frac {6 \sin \left (a+\frac {b}{x}\right )}{b^4}-\frac {3 \sin \left (a+\frac {b}{x}\right )}{b^2 x^2} \]
Cos[a + b/x]/(b*x^3) - (6*Cos[a + b/x])/(b^3*x) + (6*Sin[a + b/x])/b^4 - ( 3*Sin[a + b/x])/(b^2*x^2)
Time = 0.45 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.16, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {3860, 3042, 3777, 3042, 3777, 25, 3042, 3777, 3042, 3117}
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.
\(\displaystyle \int \frac {\sin \left (a+\frac {b}{x}\right )}{x^5} \, dx\) |
\(\Big \downarrow \) 3860 |
\(\displaystyle -\int \frac {\sin \left (a+\frac {b}{x}\right )}{x^3}d\frac {1}{x}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\int \frac {\sin \left (a+\frac {b}{x}\right )}{x^3}d\frac {1}{x}\) |
\(\Big \downarrow \) 3777 |
\(\displaystyle \frac {\cos \left (a+\frac {b}{x}\right )}{b x^3}-\frac {3 \int \frac {\cos \left (a+\frac {b}{x}\right )}{x^2}d\frac {1}{x}}{b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\cos \left (a+\frac {b}{x}\right )}{b x^3}-\frac {3 \int \frac {\sin \left (a+\frac {b}{x}+\frac {\pi }{2}\right )}{x^2}d\frac {1}{x}}{b}\) |
\(\Big \downarrow \) 3777 |
\(\displaystyle \frac {\cos \left (a+\frac {b}{x}\right )}{b x^3}-\frac {3 \left (\frac {2 \int -\frac {\sin \left (a+\frac {b}{x}\right )}{x}d\frac {1}{x}}{b}+\frac {\sin \left (a+\frac {b}{x}\right )}{b x^2}\right )}{b}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\cos \left (a+\frac {b}{x}\right )}{b x^3}-\frac {3 \left (\frac {\sin \left (a+\frac {b}{x}\right )}{b x^2}-\frac {2 \int \frac {\sin \left (a+\frac {b}{x}\right )}{x}d\frac {1}{x}}{b}\right )}{b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\cos \left (a+\frac {b}{x}\right )}{b x^3}-\frac {3 \left (\frac {\sin \left (a+\frac {b}{x}\right )}{b x^2}-\frac {2 \int \frac {\sin \left (a+\frac {b}{x}\right )}{x}d\frac {1}{x}}{b}\right )}{b}\) |
\(\Big \downarrow \) 3777 |
\(\displaystyle \frac {\cos \left (a+\frac {b}{x}\right )}{b x^3}-\frac {3 \left (\frac {\sin \left (a+\frac {b}{x}\right )}{b x^2}-\frac {2 \left (\frac {\int \cos \left (a+\frac {b}{x}\right )d\frac {1}{x}}{b}-\frac {\cos \left (a+\frac {b}{x}\right )}{b x}\right )}{b}\right )}{b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\cos \left (a+\frac {b}{x}\right )}{b x^3}-\frac {3 \left (\frac {\sin \left (a+\frac {b}{x}\right )}{b x^2}-\frac {2 \left (\frac {\int \sin \left (a+\frac {b}{x}+\frac {\pi }{2}\right )d\frac {1}{x}}{b}-\frac {\cos \left (a+\frac {b}{x}\right )}{b x}\right )}{b}\right )}{b}\) |
\(\Big \downarrow \) 3117 |
\(\displaystyle \frac {\cos \left (a+\frac {b}{x}\right )}{b x^3}-\frac {3 \left (\frac {\sin \left (a+\frac {b}{x}\right )}{b x^2}-\frac {2 \left (\frac {\sin \left (a+\frac {b}{x}\right )}{b^2}-\frac {\cos \left (a+\frac {b}{x}\right )}{b x}\right )}{b}\right )}{b}\) |
Cos[a + b/x]/(b*x^3) - (3*(Sin[a + b/x]/(b*x^2) - (2*(-(Cos[a + b/x]/(b*x) ) + Sin[a + b/x]/b^2))/b))/b
3.2.10.3.1 Defintions of rubi rules used
Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, 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[(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 = 0.26 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.90
method | result | size |
risch | \(\frac {\left (b^{2}-6 x^{2}\right ) \cos \left (\frac {a x +b}{x}\right )}{b^{3} x^{3}}-\frac {3 \left (b^{2}-2 x^{2}\right ) \sin \left (\frac {a x +b}{x}\right )}{x^{2} b^{4}}\) | \(55\) |
parallelrisch | \(\frac {6 x^{2} \left (\tan ^{2}\left (\frac {a x +b}{2 x}\right )\right ) b -\left (\tan ^{2}\left (\frac {a x +b}{2 x}\right )\right ) b^{3}+12 \tan \left (\frac {a x +b}{2 x}\right ) x^{3}-6 x \tan \left (\frac {a x +b}{2 x}\right ) b^{2}-6 b \,x^{2}+b^{3}}{x^{3} b^{4} \left (1+\tan ^{2}\left (\frac {a x +b}{2 x}\right )\right )}\) | \(105\) |
norman | \(\frac {\frac {x}{b}-\frac {6 x^{3}}{b^{3}}+\frac {12 x^{4} \tan \left (\frac {a}{2}+\frac {b}{2 x}\right )}{b^{4}}+\frac {6 x^{3} \left (\tan ^{2}\left (\frac {a}{2}+\frac {b}{2 x}\right )\right )}{b^{3}}-\frac {6 x^{2} \tan \left (\frac {a}{2}+\frac {b}{2 x}\right )}{b^{2}}-\frac {x \left (\tan ^{2}\left (\frac {a}{2}+\frac {b}{2 x}\right )\right )}{b}}{\left (1+\tan ^{2}\left (\frac {a}{2}+\frac {b}{2 x}\right )\right ) x^{4}}\) | \(114\) |
meijerg | \(-\frac {8 \sqrt {\pi }\, \cos \left (a \right ) \left (\frac {b \left (-\frac {5 b^{2}}{2 x^{2}}+15\right ) \cos \left (\frac {b}{x}\right )}{20 \sqrt {\pi }\, x}-\frac {\left (-\frac {15 b^{2}}{2 x^{2}}+15\right ) \sin \left (\frac {b}{x}\right )}{20 \sqrt {\pi }}\right )}{b^{4}}-\frac {8 \sqrt {\pi }\, \sin \left (a \right ) \left (\frac {3}{4 \sqrt {\pi }}-\frac {\left (-\frac {3 b^{2}}{2 x^{2}}+3\right ) \cos \left (\frac {b}{x}\right )}{4 \sqrt {\pi }}-\frac {b \left (-\frac {b^{2}}{2 x^{2}}+3\right ) \sin \left (\frac {b}{x}\right )}{4 \sqrt {\pi }\, x}\right )}{b^{4}}\) | \(121\) |
derivativedivides | \(-\frac {a^{3} \cos \left (a +\frac {b}{x}\right )+3 a^{2} \left (\sin \left (a +\frac {b}{x}\right )-\left (a +\frac {b}{x}\right ) \cos \left (a +\frac {b}{x}\right )\right )-3 a \left (-\left (a +\frac {b}{x}\right )^{2} \cos \left (a +\frac {b}{x}\right )+2 \cos \left (a +\frac {b}{x}\right )+2 \left (a +\frac {b}{x}\right ) \sin \left (a +\frac {b}{x}\right )\right )-\left (a +\frac {b}{x}\right )^{3} \cos \left (a +\frac {b}{x}\right )+3 \left (a +\frac {b}{x}\right )^{2} \sin \left (a +\frac {b}{x}\right )-6 \sin \left (a +\frac {b}{x}\right )+6 \left (a +\frac {b}{x}\right ) \cos \left (a +\frac {b}{x}\right )}{b^{4}}\) | \(165\) |
default | \(-\frac {a^{3} \cos \left (a +\frac {b}{x}\right )+3 a^{2} \left (\sin \left (a +\frac {b}{x}\right )-\left (a +\frac {b}{x}\right ) \cos \left (a +\frac {b}{x}\right )\right )-3 a \left (-\left (a +\frac {b}{x}\right )^{2} \cos \left (a +\frac {b}{x}\right )+2 \cos \left (a +\frac {b}{x}\right )+2 \left (a +\frac {b}{x}\right ) \sin \left (a +\frac {b}{x}\right )\right )-\left (a +\frac {b}{x}\right )^{3} \cos \left (a +\frac {b}{x}\right )+3 \left (a +\frac {b}{x}\right )^{2} \sin \left (a +\frac {b}{x}\right )-6 \sin \left (a +\frac {b}{x}\right )+6 \left (a +\frac {b}{x}\right ) \cos \left (a +\frac {b}{x}\right )}{b^{4}}\) | \(165\) |
Time = 0.25 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.85 \[ \int \frac {\sin \left (a+\frac {b}{x}\right )}{x^5} \, dx=\frac {{\left (b^{3} - 6 \, b x^{2}\right )} \cos \left (\frac {a x + b}{x}\right ) - 3 \, {\left (b^{2} x - 2 \, x^{3}\right )} \sin \left (\frac {a x + b}{x}\right )}{b^{4} x^{3}} \]
Time = 0.72 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00 \[ \int \frac {\sin \left (a+\frac {b}{x}\right )}{x^5} \, dx=\begin {cases} \frac {\cos {\left (a + \frac {b}{x} \right )}}{b x^{3}} - \frac {3 \sin {\left (a + \frac {b}{x} \right )}}{b^{2} x^{2}} - \frac {6 \cos {\left (a + \frac {b}{x} \right )}}{b^{3} x} + \frac {6 \sin {\left (a + \frac {b}{x} \right )}}{b^{4}} & \text {for}\: b \neq 0 \\- \frac {\sin {\left (a \right )}}{4 x^{4}} & \text {otherwise} \end {cases} \]
Piecewise((cos(a + b/x)/(b*x**3) - 3*sin(a + b/x)/(b**2*x**2) - 6*cos(a + b/x)/(b**3*x) + 6*sin(a + b/x)/b**4, Ne(b, 0)), (-sin(a)/(4*x**4), True))
Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.23 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.82 \[ \int \frac {\sin \left (a+\frac {b}{x}\right )}{x^5} \, dx=\frac {{\left (i \, \Gamma \left (4, \frac {i \, b}{x}\right ) - i \, \Gamma \left (4, -\frac {i \, b}{x}\right )\right )} \cos \left (a\right ) + {\left (\Gamma \left (4, \frac {i \, b}{x}\right ) + \Gamma \left (4, -\frac {i \, b}{x}\right )\right )} \sin \left (a\right )}{2 \, b^{4}} \]
1/2*((I*gamma(4, I*b/x) - I*gamma(4, -I*b/x))*cos(a) + (gamma(4, I*b/x) + gamma(4, -I*b/x))*sin(a))/b^4
Leaf count of result is larger than twice the leaf count of optimal. 191 vs. \(2 (61) = 122\).
Time = 0.31 (sec) , antiderivative size = 191, normalized size of antiderivative = 3.13 \[ \int \frac {\sin \left (a+\frac {b}{x}\right )}{x^5} \, dx=-\frac {a^{3} \cos \left (\frac {a x + b}{x}\right ) - \frac {3 \, {\left (a x + b\right )} a^{2} \cos \left (\frac {a x + b}{x}\right )}{x} + 3 \, a^{2} \sin \left (\frac {a x + b}{x}\right ) - 6 \, a \cos \left (\frac {a x + b}{x}\right ) + \frac {3 \, {\left (a x + b\right )}^{2} a \cos \left (\frac {a x + b}{x}\right )}{x^{2}} - \frac {6 \, {\left (a x + b\right )} a \sin \left (\frac {a x + b}{x}\right )}{x} - \frac {{\left (a x + b\right )}^{3} \cos \left (\frac {a x + b}{x}\right )}{x^{3}} + \frac {6 \, {\left (a x + b\right )} \cos \left (\frac {a x + b}{x}\right )}{x} + \frac {3 \, {\left (a x + b\right )}^{2} \sin \left (\frac {a x + b}{x}\right )}{x^{2}} - 6 \, \sin \left (\frac {a x + b}{x}\right )}{b^{4}} \]
-(a^3*cos((a*x + b)/x) - 3*(a*x + b)*a^2*cos((a*x + b)/x)/x + 3*a^2*sin((a *x + b)/x) - 6*a*cos((a*x + b)/x) + 3*(a*x + b)^2*a*cos((a*x + b)/x)/x^2 - 6*(a*x + b)*a*sin((a*x + b)/x)/x - (a*x + b)^3*cos((a*x + b)/x)/x^3 + 6*( a*x + b)*cos((a*x + b)/x)/x + 3*(a*x + b)^2*sin((a*x + b)/x)/x^2 - 6*sin(( a*x + b)/x))/b^4
Time = 5.99 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.05 \[ \int \frac {\sin \left (a+\frac {b}{x}\right )}{x^5} \, dx=\frac {6\,\sin \left (a+\frac {b}{x}\right )}{b^4}-\frac {6\,b\,x^2\,\cos \left (a+\frac {b}{x}\right )-b^3\,\cos \left (a+\frac {b}{x}\right )+3\,b^2\,x\,\sin \left (a+\frac {b}{x}\right )}{b^4\,x^3} \]