Integrand size = 14, antiderivative size = 51 \[ \int \frac {\sin ^2\left (a+\frac {b}{x}\right )}{x^3} \, dx=-\frac {1}{4 x^2}+\frac {\cos \left (a+\frac {b}{x}\right ) \sin \left (a+\frac {b}{x}\right )}{2 b x}-\frac {\sin ^2\left (a+\frac {b}{x}\right )}{4 b^2} \] Output:
-1/4/x^2+1/2*cos(a+b/x)*sin(a+b/x)/b/x-1/4*sin(a+b/x)^2/b^2
Time = 0.09 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.84 \[ \int \frac {\sin ^2\left (a+\frac {b}{x}\right )}{x^3} \, dx=\frac {x^2 \cos \left (2 \left (a+\frac {b}{x}\right )\right )-2 b \left (b-x \sin \left (2 \left (a+\frac {b}{x}\right )\right )\right )}{8 b^2 x^2} \] Input:
Integrate[Sin[a + b/x]^2/x^3,x]
Output:
(x^2*Cos[2*(a + b/x)] - 2*b*(b - x*Sin[2*(a + b/x)]))/(8*b^2*x^2)
Time = 0.25 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3860, 3042, 3791, 15}
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.
\(\displaystyle \int \frac {\sin ^2\left (a+\frac {b}{x}\right )}{x^3} \, dx\) |
\(\Big \downarrow \) 3860 |
\(\displaystyle -\int \frac {\sin ^2\left (a+\frac {b}{x}\right )}{x}d\frac {1}{x}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\int \frac {\sin \left (a+\frac {b}{x}\right )^2}{x}d\frac {1}{x}\) |
\(\Big \downarrow \) 3791 |
\(\displaystyle -\frac {1}{2} \int \frac {1}{x}d\frac {1}{x}-\frac {\sin ^2\left (a+\frac {b}{x}\right )}{4 b^2}+\frac {\sin \left (a+\frac {b}{x}\right ) \cos \left (a+\frac {b}{x}\right )}{2 b x}\) |
\(\Big \downarrow \) 15 |
\(\displaystyle -\frac {\sin ^2\left (a+\frac {b}{x}\right )}{4 b^2}+\frac {\sin \left (a+\frac {b}{x}\right ) \cos \left (a+\frac {b}{x}\right )}{2 b x}-\frac {1}{4 x^2}\) |
Input:
Int[Sin[a + b/x]^2/x^3,x]
Output:
-1/4*1/x^2 + (Cos[a + b/x]*Sin[a + b/x])/(2*b*x) - Sin[a + b/x]^2/(4*b^2)
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[((c_.) + (d_.)*(x_))*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[d*((b*Sin[e + f*x])^n/(f^2*n^2)), x] + (-Simp[b*(c + d*x)*Cos[e + f*x ]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x] + Simp[b^2*((n - 1)/n) Int[(c + d* x)*(b*Sin[e + f*x])^(n - 2), x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1]
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 = 1.06 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.82
method | result | size |
risch | \(-\frac {1}{4 x^{2}}+\frac {\cos \left (\frac {2 a x +2 b}{x}\right )}{8 b^{2}}+\frac {\sin \left (\frac {2 a x +2 b}{x}\right )}{4 b x}\) | \(42\) |
parallelrisch | \(\frac {2 b x \sin \left (\frac {2 a x +2 b}{x}\right )+x^{2} \cos \left (\frac {2 a x +2 b}{x}\right )-2 b^{2}-x^{2}}{8 x^{2} b^{2}}\) | \(54\) |
derivativedivides | \(-\frac {\left (a +\frac {b}{x}\right ) \left (-\frac {\cos \left (a +\frac {b}{x}\right ) \sin \left (a +\frac {b}{x}\right )}{2}+\frac {a}{2}+\frac {b}{2 x}\right )-\frac {\left (a +\frac {b}{x}\right )^{2}}{4}+\frac {\sin \left (a +\frac {b}{x}\right )^{2}}{4}-a \left (-\frac {\cos \left (a +\frac {b}{x}\right ) \sin \left (a +\frac {b}{x}\right )}{2}+\frac {a}{2}+\frac {b}{2 x}\right )}{b^{2}}\) | \(97\) |
default | \(-\frac {\left (a +\frac {b}{x}\right ) \left (-\frac {\cos \left (a +\frac {b}{x}\right ) \sin \left (a +\frac {b}{x}\right )}{2}+\frac {a}{2}+\frac {b}{2 x}\right )-\frac {\left (a +\frac {b}{x}\right )^{2}}{4}+\frac {\sin \left (a +\frac {b}{x}\right )^{2}}{4}-a \left (-\frac {\cos \left (a +\frac {b}{x}\right ) \sin \left (a +\frac {b}{x}\right )}{2}+\frac {a}{2}+\frac {b}{2 x}\right )}{b^{2}}\) | \(97\) |
norman | \(\frac {-\frac {1}{4}+\frac {x \tan \left (\frac {a}{2}+\frac {b}{2 x}\right )}{b}-\frac {x^{2} \tan \left (\frac {a}{2}+\frac {b}{2 x}\right )^{2}}{b^{2}}-\frac {\tan \left (\frac {a}{2}+\frac {b}{2 x}\right )^{2}}{2}-\frac {\tan \left (\frac {a}{2}+\frac {b}{2 x}\right )^{4}}{4}-\frac {x \tan \left (\frac {a}{2}+\frac {b}{2 x}\right )^{3}}{b}}{\left (1+\tan \left (\frac {a}{2}+\frac {b}{2 x}\right )^{2}\right )^{2} x^{2}}\) | \(110\) |
orering | \(-\frac {\left (b^{2}+5 x^{2}\right ) \sin \left (a +\frac {b}{x}\right )^{2}}{2 x^{2} b^{2}}-\frac {5 x^{4} \left (-\frac {2 \sin \left (a +\frac {b}{x}\right ) b \cos \left (a +\frac {b}{x}\right )}{x^{5}}-\frac {3 \sin \left (a +\frac {b}{x}\right )^{2}}{x^{4}}\right )}{4 b^{2}}-\frac {x^{5} \left (\frac {2 b^{2} \cos \left (a +\frac {b}{x}\right )^{2}}{x^{7}}+\frac {16 \sin \left (a +\frac {b}{x}\right ) b \cos \left (a +\frac {b}{x}\right )}{x^{6}}-\frac {2 \sin \left (a +\frac {b}{x}\right )^{2} b^{2}}{x^{7}}+\frac {12 \sin \left (a +\frac {b}{x}\right )^{2}}{x^{5}}\right )}{8 b^{2}}\) | \(157\) |
Input:
int(sin(a+b/x)^2/x^3,x,method=_RETURNVERBOSE)
Output:
-1/4/x^2+1/8/b^2*cos(2*(a*x+b)/x)+1/4/b/x*sin(2*(a*x+b)/x)
Time = 0.08 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.18 \[ \int \frac {\sin ^2\left (a+\frac {b}{x}\right )}{x^3} \, dx=\frac {2 \, x^{2} \cos \left (\frac {a x + b}{x}\right )^{2} + 4 \, b x \cos \left (\frac {a x + b}{x}\right ) \sin \left (\frac {a x + b}{x}\right ) - 2 \, b^{2} - x^{2}}{8 \, b^{2} x^{2}} \] Input:
integrate(sin(a+b/x)^2/x^3,x, algorithm="fricas")
Output:
1/8*(2*x^2*cos((a*x + b)/x)^2 + 4*b*x*cos((a*x + b)/x)*sin((a*x + b)/x) - 2*b^2 - x^2)/(b^2*x^2)
Leaf count of result is larger than twice the leaf count of optimal. 391 vs. \(2 (37) = 74\).
Time = 1.29 (sec) , antiderivative size = 391, normalized size of antiderivative = 7.67 \[ \int \frac {\sin ^2\left (a+\frac {b}{x}\right )}{x^3} \, dx=\begin {cases} - \frac {b^{2} \tan ^{4}{\left (\frac {a}{2} + \frac {b}{2 x} \right )}}{4 b^{2} x^{2} \tan ^{4}{\left (\frac {a}{2} + \frac {b}{2 x} \right )} + 8 b^{2} x^{2} \tan ^{2}{\left (\frac {a}{2} + \frac {b}{2 x} \right )} + 4 b^{2} x^{2}} - \frac {2 b^{2} \tan ^{2}{\left (\frac {a}{2} + \frac {b}{2 x} \right )}}{4 b^{2} x^{2} \tan ^{4}{\left (\frac {a}{2} + \frac {b}{2 x} \right )} + 8 b^{2} x^{2} \tan ^{2}{\left (\frac {a}{2} + \frac {b}{2 x} \right )} + 4 b^{2} x^{2}} - \frac {b^{2}}{4 b^{2} x^{2} \tan ^{4}{\left (\frac {a}{2} + \frac {b}{2 x} \right )} + 8 b^{2} x^{2} \tan ^{2}{\left (\frac {a}{2} + \frac {b}{2 x} \right )} + 4 b^{2} x^{2}} - \frac {4 b x \tan ^{3}{\left (\frac {a}{2} + \frac {b}{2 x} \right )}}{4 b^{2} x^{2} \tan ^{4}{\left (\frac {a}{2} + \frac {b}{2 x} \right )} + 8 b^{2} x^{2} \tan ^{2}{\left (\frac {a}{2} + \frac {b}{2 x} \right )} + 4 b^{2} x^{2}} + \frac {4 b x \tan {\left (\frac {a}{2} + \frac {b}{2 x} \right )}}{4 b^{2} x^{2} \tan ^{4}{\left (\frac {a}{2} + \frac {b}{2 x} \right )} + 8 b^{2} x^{2} \tan ^{2}{\left (\frac {a}{2} + \frac {b}{2 x} \right )} + 4 b^{2} x^{2}} - \frac {4 x^{2} \tan ^{2}{\left (\frac {a}{2} + \frac {b}{2 x} \right )}}{4 b^{2} x^{2} \tan ^{4}{\left (\frac {a}{2} + \frac {b}{2 x} \right )} + 8 b^{2} x^{2} \tan ^{2}{\left (\frac {a}{2} + \frac {b}{2 x} \right )} + 4 b^{2} x^{2}} & \text {for}\: b \neq 0 \\- \frac {\sin ^{2}{\left (a \right )}}{2 x^{2}} & \text {otherwise} \end {cases} \] Input:
integrate(sin(a+b/x)**2/x**3,x)
Output:
Piecewise((-b**2*tan(a/2 + b/(2*x))**4/(4*b**2*x**2*tan(a/2 + b/(2*x))**4 + 8*b**2*x**2*tan(a/2 + b/(2*x))**2 + 4*b**2*x**2) - 2*b**2*tan(a/2 + b/(2 *x))**2/(4*b**2*x**2*tan(a/2 + b/(2*x))**4 + 8*b**2*x**2*tan(a/2 + b/(2*x) )**2 + 4*b**2*x**2) - b**2/(4*b**2*x**2*tan(a/2 + b/(2*x))**4 + 8*b**2*x** 2*tan(a/2 + b/(2*x))**2 + 4*b**2*x**2) - 4*b*x*tan(a/2 + b/(2*x))**3/(4*b* *2*x**2*tan(a/2 + b/(2*x))**4 + 8*b**2*x**2*tan(a/2 + b/(2*x))**2 + 4*b**2 *x**2) + 4*b*x*tan(a/2 + b/(2*x))/(4*b**2*x**2*tan(a/2 + b/(2*x))**4 + 8*b **2*x**2*tan(a/2 + b/(2*x))**2 + 4*b**2*x**2) - 4*x**2*tan(a/2 + b/(2*x))* *2/(4*b**2*x**2*tan(a/2 + b/(2*x))**4 + 8*b**2*x**2*tan(a/2 + b/(2*x))**2 + 4*b**2*x**2), Ne(b, 0)), (-sin(a)**2/(2*x**2), True))
Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.07 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.33 \[ \int \frac {\sin ^2\left (a+\frac {b}{x}\right )}{x^3} \, dx=\frac {{\left ({\left (\Gamma \left (2, \frac {2 i \, b}{x}\right ) + \Gamma \left (2, -\frac {2 i \, b}{x}\right )\right )} \cos \left (2 \, a\right ) - {\left (i \, \Gamma \left (2, \frac {2 i \, b}{x}\right ) - i \, \Gamma \left (2, -\frac {2 i \, b}{x}\right )\right )} \sin \left (2 \, a\right )\right )} x^{2} - 4 \, b^{2}}{16 \, b^{2} x^{2}} \] Input:
integrate(sin(a+b/x)^2/x^3,x, algorithm="maxima")
Output:
1/16*(((gamma(2, 2*I*b/x) + gamma(2, -2*I*b/x))*cos(2*a) - (I*gamma(2, 2*I *b/x) - I*gamma(2, -2*I*b/x))*sin(2*a))*x^2 - 4*b^2)/(b^2*x^2)
Time = 0.16 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.51 \[ \int \frac {\sin ^2\left (a+\frac {b}{x}\right )}{x^3} \, dx=-\frac {2 \, a \sin \left (\frac {2 \, {\left (a x + b\right )}}{x}\right ) - \frac {4 \, {\left (a x + b\right )} a}{x} - \frac {2 \, {\left (a x + b\right )} \sin \left (\frac {2 \, {\left (a x + b\right )}}{x}\right )}{x} + \frac {2 \, {\left (a x + b\right )}^{2}}{x^{2}} - \cos \left (\frac {2 \, {\left (a x + b\right )}}{x}\right )}{8 \, b^{2}} \] Input:
integrate(sin(a+b/x)^2/x^3,x, algorithm="giac")
Output:
-1/8*(2*a*sin(2*(a*x + b)/x) - 4*(a*x + b)*a/x - 2*(a*x + b)*sin(2*(a*x + b)/x)/x + 2*(a*x + b)^2/x^2 - cos(2*(a*x + b)/x))/b^2
Time = 42.67 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.80 \[ \int \frac {\sin ^2\left (a+\frac {b}{x}\right )}{x^3} \, dx=\frac {\cos \left (2\,a+\frac {2\,b}{x}\right )}{8\,b^2}-\frac {1}{4\,x^2}+\frac {\sin \left (2\,a+\frac {2\,b}{x}\right )}{4\,b\,x} \] Input:
int(sin(a + b/x)^2/x^3,x)
Output:
cos(2*a + (2*b)/x)/(8*b^2) - 1/(4*x^2) + sin(2*a + (2*b)/x)/(4*b*x)
\[ \int \frac {\sin ^2\left (a+\frac {b}{x}\right )}{x^3} \, dx=\int \frac {\sin \left (\frac {a x +b}{x}\right )^{2}}{x^{3}}d x \] Input:
int(sin(a+b/x)^2/x^3,x)
Output:
int(sin((a*x + b)/x)**2/x**3,x)