\(\int \frac {\sin ^2(a+\frac {b}{x})}{x^5} \, dx\) [118]

Optimal result

Integrand size = 14, antiderivative size = 107 \[ \int \frac {\sin ^2\left (a+\frac {b}{x}\right )}{x^5} \, dx=-\frac {1}{8 x^4}+\frac {3}{8 b^2 x^2}+\frac {\cos \left (a+\frac {b}{x}\right ) \sin \left (a+\frac {b}{x}\right )}{2 b x^3}-\frac {3 \cos \left (a+\frac {b}{x}\right ) \sin \left (a+\frac {b}{x}\right )}{4 b^3 x}+\frac {3 \sin ^2\left (a+\frac {b}{x}\right )}{8 b^4}-\frac {3 \sin ^2\left (a+\frac {b}{x}\right )}{4 b^2 x^2} \] Output:

-1/8/x^4+3/8/b^2/x^2+1/2*cos(a+b/x)*sin(a+b/x)/b/x^3-3/4*cos(a+b/x)*sin(a+ 
b/x)/b^3/x+3/8*sin(a+b/x)^2/b^4-3/4*sin(a+b/x)^2/b^2/x^2
 

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.61 \[ \int \frac {\sin ^2\left (a+\frac {b}{x}\right )}{x^5} \, dx=-\frac {3 \left (-2 b^2 x^2+x^4\right ) \cos \left (2 \left (a+\frac {b}{x}\right )\right )+2 b \left (b^3+\left (-2 b^2 x+3 x^3\right ) \sin \left (2 \left (a+\frac {b}{x}\right )\right )\right )}{16 b^4 x^4} \] Input:

Integrate[Sin[a + b/x]^2/x^5,x]
 

Output:

-1/16*(3*(-2*b^2*x^2 + x^4)*Cos[2*(a + b/x)] + 2*b*(b^3 + (-2*b^2*x + 3*x^ 
3)*Sin[2*(a + b/x)]))/(b^4*x^4)
 

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.05, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3860, 3042, 3792, 15, 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.

\(\displaystyle \int \frac {\sin ^2\left (a+\frac {b}{x}\right )}{x^5} \, dx\)

\(\Big \downarrow \) 3860

\(\displaystyle -\int \frac {\sin ^2\left (a+\frac {b}{x}\right )}{x^3}d\frac {1}{x}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\int \frac {\sin \left (a+\frac {b}{x}\right )^2}{x^3}d\frac {1}{x}\)

\(\Big \downarrow \) 3792

\(\displaystyle \frac {3 \int \frac {\sin ^2\left (a+\frac {b}{x}\right )}{x}d\frac {1}{x}}{2 b^2}-\frac {1}{2} \int \frac {1}{x^3}d\frac {1}{x}-\frac {3 \sin ^2\left (a+\frac {b}{x}\right )}{4 b^2 x^2}+\frac {\sin \left (a+\frac {b}{x}\right ) \cos \left (a+\frac {b}{x}\right )}{2 b x^3}\)

\(\Big \downarrow \) 15

\(\displaystyle \frac {3 \int \frac {\sin ^2\left (a+\frac {b}{x}\right )}{x}d\frac {1}{x}}{2 b^2}-\frac {3 \sin ^2\left (a+\frac {b}{x}\right )}{4 b^2 x^2}+\frac {\sin \left (a+\frac {b}{x}\right ) \cos \left (a+\frac {b}{x}\right )}{2 b x^3}-\frac {1}{8 x^4}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {3 \int \frac {\sin \left (a+\frac {b}{x}\right )^2}{x}d\frac {1}{x}}{2 b^2}-\frac {3 \sin ^2\left (a+\frac {b}{x}\right )}{4 b^2 x^2}+\frac {\sin \left (a+\frac {b}{x}\right ) \cos \left (a+\frac {b}{x}\right )}{2 b x^3}-\frac {1}{8 x^4}\)

\(\Big \downarrow \) 3791

\(\displaystyle \frac {3 \left (\frac {1}{2} \int \frac {1}{x}d\frac {1}{x}+\frac {\sin ^2\left (a+\frac {b}{x}\right )}{4 b^2}-\frac {\sin \left (a+\frac {b}{x}\right ) \cos \left (a+\frac {b}{x}\right )}{2 b x}\right )}{2 b^2}-\frac {3 \sin ^2\left (a+\frac {b}{x}\right )}{4 b^2 x^2}+\frac {\sin \left (a+\frac {b}{x}\right ) \cos \left (a+\frac {b}{x}\right )}{2 b x^3}-\frac {1}{8 x^4}\)

\(\Big \downarrow \) 15

\(\displaystyle -\frac {3 \sin ^2\left (a+\frac {b}{x}\right )}{4 b^2 x^2}+\frac {3 \left (\frac {\sin ^2\left (a+\frac {b}{x}\right )}{4 b^2}-\frac {\sin \left (a+\frac {b}{x}\right ) \cos \left (a+\frac {b}{x}\right )}{2 b x}+\frac {1}{4 x^2}\right )}{2 b^2}+\frac {\sin \left (a+\frac {b}{x}\right ) \cos \left (a+\frac {b}{x}\right )}{2 b x^3}-\frac {1}{8 x^4}\)

Input:

Int[Sin[a + b/x]^2/x^5,x]
 

Output:

-1/8*1/x^4 + (Cos[a + b/x]*Sin[a + b/x])/(2*b*x^3) - (3*Sin[a + b/x]^2)/(4 
*b^2*x^2) + (3*(1/(4*x^2) - (Cos[a + b/x]*Sin[a + b/x])/(2*b*x) + Sin[a + 
b/x]^2/(4*b^2)))/(2*b^2)
 

Defintions of rubi rules used

rule 15

Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ 
{a, m}, x] && NeQ[m, -1]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3791

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]
 

rule 3792

Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbo 
l] :> Simp[d*m*(c + d*x)^(m - 1)*((b*Sin[e + f*x])^n/(f^2*n^2)), x] + (-Sim 
p[b*(c + d*x)^m*Cos[e + f*x]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x] + Simp[b^ 
2*((n - 1)/n)   Int[(c + d*x)^m*(b*Sin[e + f*x])^(n - 2), x], x] - Simp[d^2 
*m*((m - 1)/(f^2*n^2))   Int[(c + d*x)^(m - 2)*(b*Sin[e + f*x])^n, x], x]) 
/; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 1]
 

rule 3860

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]))
 

Maple [A] (verified)

Time = 1.45 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.63

method	result	size
risch	\(-\frac {1}{8 x^{4}}+\frac {3 \left (2 b^{2}-x^{2}\right ) \cos \left (\frac {2 a x +2 b}{x}\right )}{16 x^{2} b^{4}}+\frac {\left (2 b^{2}-3 x^{2}\right ) \sin \left (\frac {2 a x +2 b}{x}\right )}{8 b^{3} x^{3}}\)	\(67\)
parallelrisch	\(\frac {\left (6 x^{2} b^{2}-3 x^{4}\right ) \cos \left (\frac {2 a x +2 b}{x}\right )+\left (4 b^{3} x -6 b \,x^{3}\right ) \sin \left (\frac {2 a x +2 b}{x}\right )-2 b^{4}+3 x^{4}}{16 x^{4} b^{4}}\)	\(75\)
orering	\(-\frac {\left (2 b^{4}+25 x^{2} b^{2}-63 x^{4}\right ) \sin \left (a +\frac {b}{x}\right )^{2}}{8 x^{4} b^{4}}-\frac {x^{4} \left (8 b^{2}-21 x^{2}\right ) \left (-\frac {2 \sin \left (a +\frac {b}{x}\right ) b \cos \left (a +\frac {b}{x}\right )}{x^{7}}-\frac {5 \sin \left (a +\frac {b}{x}\right )^{2}}{x^{6}}\right )}{8 b^{4}}-\frac {\left (b^{2}-3 x^{2}\right ) x^{5} \left (\frac {2 b^{2} \cos \left (a +\frac {b}{x}\right )^{2}}{x^{9}}+\frac {24 \sin \left (a +\frac {b}{x}\right ) b \cos \left (a +\frac {b}{x}\right )}{x^{8}}-\frac {2 \sin \left (a +\frac {b}{x}\right )^{2} b^{2}}{x^{9}}+\frac {30 \sin \left (a +\frac {b}{x}\right )^{2}}{x^{7}}\right )}{16 b^{4}}\)	\(187\)
norman	\(\frac {-\frac {1}{8}+\frac {x \tan \left (\frac {a}{2}+\frac {b}{2 x}\right )}{b}-\frac {\tan \left (\frac {a}{2}+\frac {b}{2 x}\right )^{2}}{4}-\frac {\tan \left (\frac {a}{2}+\frac {b}{2 x}\right )^{4}}{8}+\frac {3 x^{2}}{8 b^{2}}-\frac {3 x^{3} \tan \left (\frac {a}{2}+\frac {b}{2 x}\right )}{2 b^{3}}+\frac {3 x^{3} \tan \left (\frac {a}{2}+\frac {b}{2 x}\right )^{3}}{2 b^{3}}-\frac {9 x^{2} \tan \left (\frac {a}{2}+\frac {b}{2 x}\right )^{2}}{4 b^{2}}+\frac {3 x^{2} \tan \left (\frac {a}{2}+\frac {b}{2 x}\right )^{4}}{8 b^{2}}-\frac {x \tan \left (\frac {a}{2}+\frac {b}{2 x}\right )^{3}}{b}+\frac {3 x^{4} \tan \left (\frac {a}{2}+\frac {b}{2 x}\right )^{2}}{2 b^{4}}}{\left (1+\tan \left (\frac {a}{2}+\frac {b}{2 x}\right )^{2}\right )^{2} x^{4}}\)	\(200\)
derivativedivides	\(-\frac {-a^{3} \left (-\frac {\cos \left (a +\frac {b}{x}\right ) \sin \left (a +\frac {b}{x}\right )}{2}+\frac {a}{2}+\frac {b}{2 x}\right )+3 a^{2} \left (\left (a +\frac {b}{x}\right ) \left (-\frac {\cos \left (a +\frac {b}{x}\right ) \sin \left (a +\frac {b}{x}\right )}{2}+\frac {a}{2}+\frac {b}{2 x}\right )-\frac {\left (a +\frac {b}{x}\right )^{2}}{4}+\frac {\sin \left (a +\frac {b}{x}\right )^{2}}{4}\right )-3 a \left (\left (a +\frac {b}{x}\right )^{2} \left (-\frac {\cos \left (a +\frac {b}{x}\right ) \sin \left (a +\frac {b}{x}\right )}{2}+\frac {a}{2}+\frac {b}{2 x}\right )-\frac {\left (a +\frac {b}{x}\right ) \cos \left (a +\frac {b}{x}\right )^{2}}{2}+\frac {\cos \left (a +\frac {b}{x}\right ) \sin \left (a +\frac {b}{x}\right )}{4}+\frac {b}{4 x}+\frac {a}{4}-\frac {\left (a +\frac {b}{x}\right )^{3}}{3}\right )+\left (a +\frac {b}{x}\right )^{3} \left (-\frac {\cos \left (a +\frac {b}{x}\right ) \sin \left (a +\frac {b}{x}\right )}{2}+\frac {a}{2}+\frac {b}{2 x}\right )-\frac {3 \left (a +\frac {b}{x}\right )^{2} \cos \left (a +\frac {b}{x}\right )^{2}}{4}+\frac {3 \left (a +\frac {b}{x}\right ) \left (\frac {\cos \left (a +\frac {b}{x}\right ) \sin \left (a +\frac {b}{x}\right )}{2}+\frac {b}{2 x}+\frac {a}{2}\right )}{2}-\frac {3 \left (a +\frac {b}{x}\right )^{2}}{8}-\frac {3 \sin \left (a +\frac {b}{x}\right )^{2}}{8}-\frac {3 \left (a +\frac {b}{x}\right )^{4}}{8}}{b^{4}}\)	\(334\)
default	\(-\frac {-a^{3} \left (-\frac {\cos \left (a +\frac {b}{x}\right ) \sin \left (a +\frac {b}{x}\right )}{2}+\frac {a}{2}+\frac {b}{2 x}\right )+3 a^{2} \left (\left (a +\frac {b}{x}\right ) \left (-\frac {\cos \left (a +\frac {b}{x}\right ) \sin \left (a +\frac {b}{x}\right )}{2}+\frac {a}{2}+\frac {b}{2 x}\right )-\frac {\left (a +\frac {b}{x}\right )^{2}}{4}+\frac {\sin \left (a +\frac {b}{x}\right )^{2}}{4}\right )-3 a \left (\left (a +\frac {b}{x}\right )^{2} \left (-\frac {\cos \left (a +\frac {b}{x}\right ) \sin \left (a +\frac {b}{x}\right )}{2}+\frac {a}{2}+\frac {b}{2 x}\right )-\frac {\left (a +\frac {b}{x}\right ) \cos \left (a +\frac {b}{x}\right )^{2}}{2}+\frac {\cos \left (a +\frac {b}{x}\right ) \sin \left (a +\frac {b}{x}\right )}{4}+\frac {b}{4 x}+\frac {a}{4}-\frac {\left (a +\frac {b}{x}\right )^{3}}{3}\right )+\left (a +\frac {b}{x}\right )^{3} \left (-\frac {\cos \left (a +\frac {b}{x}\right ) \sin \left (a +\frac {b}{x}\right )}{2}+\frac {a}{2}+\frac {b}{2 x}\right )-\frac {3 \left (a +\frac {b}{x}\right )^{2} \cos \left (a +\frac {b}{x}\right )^{2}}{4}+\frac {3 \left (a +\frac {b}{x}\right ) \left (\frac {\cos \left (a +\frac {b}{x}\right ) \sin \left (a +\frac {b}{x}\right )}{2}+\frac {b}{2 x}+\frac {a}{2}\right )}{2}-\frac {3 \left (a +\frac {b}{x}\right )^{2}}{8}-\frac {3 \sin \left (a +\frac {b}{x}\right )^{2}}{8}-\frac {3 \left (a +\frac {b}{x}\right )^{4}}{8}}{b^{4}}\)	\(334\)

Input:

int(sin(a+b/x)^2/x^5,x,method=_RETURNVERBOSE)
 

Output:

-1/8/x^4+3/16/x^2*(2*b^2-x^2)/b^4*cos(2*(a*x+b)/x)+1/8/b^3*(2*b^2-3*x^2)/x 
^3*sin(2*(a*x+b)/x)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.84 \[ \int \frac {\sin ^2\left (a+\frac {b}{x}\right )}{x^5} \, dx=-\frac {2 \, b^{4} + 6 \, b^{2} x^{2} - 3 \, x^{4} - 6 \, {\left (2 \, b^{2} x^{2} - x^{4}\right )} \cos \left (\frac {a x + b}{x}\right )^{2} - 4 \, {\left (2 \, b^{3} x - 3 \, b x^{3}\right )} \cos \left (\frac {a x + b}{x}\right ) \sin \left (\frac {a x + b}{x}\right )}{16 \, b^{4} x^{4}} \] Input:

integrate(sin(a+b/x)^2/x^5,x, algorithm="fricas")
 

Output:

-1/16*(2*b^4 + 6*b^2*x^2 - 3*x^4 - 6*(2*b^2*x^2 - x^4)*cos((a*x + b)/x)^2 
- 4*(2*b^3*x - 3*b*x^3)*cos((a*x + b)/x)*sin((a*x + b)/x))/(b^4*x^4)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 726 vs. \(2 (92) = 184\).

Time = 2.42 (sec) , antiderivative size = 726, normalized size of antiderivative = 6.79 \[ \int \frac {\sin ^2\left (a+\frac {b}{x}\right )}{x^5} \, dx =\text {Too large to display} \] Input:

integrate(sin(a+b/x)**2/x**5,x)
 

Output:

Piecewise((-b**4*tan(a/2 + b/(2*x))**4/(8*b**4*x**4*tan(a/2 + b/(2*x))**4 
+ 16*b**4*x**4*tan(a/2 + b/(2*x))**2 + 8*b**4*x**4) - 2*b**4*tan(a/2 + b/( 
2*x))**2/(8*b**4*x**4*tan(a/2 + b/(2*x))**4 + 16*b**4*x**4*tan(a/2 + b/(2* 
x))**2 + 8*b**4*x**4) - b**4/(8*b**4*x**4*tan(a/2 + b/(2*x))**4 + 16*b**4* 
x**4*tan(a/2 + b/(2*x))**2 + 8*b**4*x**4) - 8*b**3*x*tan(a/2 + b/(2*x))**3 
/(8*b**4*x**4*tan(a/2 + b/(2*x))**4 + 16*b**4*x**4*tan(a/2 + b/(2*x))**2 + 
 8*b**4*x**4) + 8*b**3*x*tan(a/2 + b/(2*x))/(8*b**4*x**4*tan(a/2 + b/(2*x) 
)**4 + 16*b**4*x**4*tan(a/2 + b/(2*x))**2 + 8*b**4*x**4) + 3*b**2*x**2*tan 
(a/2 + b/(2*x))**4/(8*b**4*x**4*tan(a/2 + b/(2*x))**4 + 16*b**4*x**4*tan(a 
/2 + b/(2*x))**2 + 8*b**4*x**4) - 18*b**2*x**2*tan(a/2 + b/(2*x))**2/(8*b* 
*4*x**4*tan(a/2 + b/(2*x))**4 + 16*b**4*x**4*tan(a/2 + b/(2*x))**2 + 8*b** 
4*x**4) + 3*b**2*x**2/(8*b**4*x**4*tan(a/2 + b/(2*x))**4 + 16*b**4*x**4*ta 
n(a/2 + b/(2*x))**2 + 8*b**4*x**4) + 12*b*x**3*tan(a/2 + b/(2*x))**3/(8*b* 
*4*x**4*tan(a/2 + b/(2*x))**4 + 16*b**4*x**4*tan(a/2 + b/(2*x))**2 + 8*b** 
4*x**4) - 12*b*x**3*tan(a/2 + b/(2*x))/(8*b**4*x**4*tan(a/2 + b/(2*x))**4 
+ 16*b**4*x**4*tan(a/2 + b/(2*x))**2 + 8*b**4*x**4) + 12*x**4*tan(a/2 + b/ 
(2*x))**2/(8*b**4*x**4*tan(a/2 + b/(2*x))**4 + 16*b**4*x**4*tan(a/2 + b/(2 
*x))**2 + 8*b**4*x**4), Ne(b, 0)), (-sin(a)**2/(4*x**4), True))
 

Maxima [C] (verification not implemented)

Result contains higher order function than in optimal. Order 4 vs. order 3.

Time = 0.08 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.64 \[ \int \frac {\sin ^2\left (a+\frac {b}{x}\right )}{x^5} \, dx=-\frac {{\left ({\left (\Gamma \left (4, \frac {2 i \, b}{x}\right ) + \Gamma \left (4, -\frac {2 i \, b}{x}\right )\right )} \cos \left (2 \, a\right ) - {\left (i \, \Gamma \left (4, \frac {2 i \, b}{x}\right ) - i \, \Gamma \left (4, -\frac {2 i \, b}{x}\right )\right )} \sin \left (2 \, a\right )\right )} x^{4} + 8 \, b^{4}}{64 \, b^{4} x^{4}} \] Input:

integrate(sin(a+b/x)^2/x^5,x, algorithm="maxima")
 

Output:

-1/64*(((gamma(4, 2*I*b/x) + gamma(4, -2*I*b/x))*cos(2*a) - (I*gamma(4, 2* 
I*b/x) - I*gamma(4, -2*I*b/x))*sin(2*a))*x^4 + 8*b^4)/(b^4*x^4)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 255 vs. \(2 (95) = 190\).

Time = 0.12 (sec) , antiderivative size = 255, normalized size of antiderivative = 2.38 \[ \int \frac {\sin ^2\left (a+\frac {b}{x}\right )}{x^5} \, dx=-\frac {4 \, a^{3} \sin \left (\frac {2 \, {\left (a x + b\right )}}{x}\right ) - \frac {8 \, {\left (a x + b\right )} a^{3}}{x} - 6 \, a^{2} \cos \left (\frac {2 \, {\left (a x + b\right )}}{x}\right ) - \frac {12 \, {\left (a x + b\right )} a^{2} \sin \left (\frac {2 \, {\left (a x + b\right )}}{x}\right )}{x} + \frac {12 \, {\left (a x + b\right )}^{2} a^{2}}{x^{2}} + \frac {12 \, {\left (a x + b\right )} a \cos \left (\frac {2 \, {\left (a x + b\right )}}{x}\right )}{x} - 6 \, a \sin \left (\frac {2 \, {\left (a x + b\right )}}{x}\right ) + \frac {12 \, {\left (a x + b\right )}^{2} a \sin \left (\frac {2 \, {\left (a x + b\right )}}{x}\right )}{x^{2}} - \frac {8 \, {\left (a x + b\right )}^{3} a}{x^{3}} - \frac {6 \, {\left (a x + b\right )}^{2} \cos \left (\frac {2 \, {\left (a x + b\right )}}{x}\right )}{x^{2}} - \frac {4 \, {\left (a x + b\right )}^{3} \sin \left (\frac {2 \, {\left (a x + b\right )}}{x}\right )}{x^{3}} + \frac {6 \, {\left (a x + b\right )} \sin \left (\frac {2 \, {\left (a x + b\right )}}{x}\right )}{x} + \frac {2 \, {\left (a x + b\right )}^{4}}{x^{4}} + 3 \, \cos \left (\frac {2 \, {\left (a x + b\right )}}{x}\right )}{16 \, b^{4}} \] Input:

integrate(sin(a+b/x)^2/x^5,x, algorithm="giac")
 

Output:

-1/16*(4*a^3*sin(2*(a*x + b)/x) - 8*(a*x + b)*a^3/x - 6*a^2*cos(2*(a*x + b 
)/x) - 12*(a*x + b)*a^2*sin(2*(a*x + b)/x)/x + 12*(a*x + b)^2*a^2/x^2 + 12 
*(a*x + b)*a*cos(2*(a*x + b)/x)/x - 6*a*sin(2*(a*x + b)/x) + 12*(a*x + b)^ 
2*a*sin(2*(a*x + b)/x)/x^2 - 8*(a*x + b)^3*a/x^3 - 6*(a*x + b)^2*cos(2*(a* 
x + b)/x)/x^2 - 4*(a*x + b)^3*sin(2*(a*x + b)/x)/x^3 + 6*(a*x + b)*sin(2*( 
a*x + b)/x)/x + 2*(a*x + b)^4/x^4 + 3*cos(2*(a*x + b)/x))/b^4
 

Mupad [B] (verification not implemented)

Time = 43.20 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.79 \[ \int \frac {\sin ^2\left (a+\frac {b}{x}\right )}{x^5} \, dx=-\frac {3\,\cos \left (2\,a+\frac {2\,b}{x}\right )}{16\,b^4}-\frac {\frac {b^4}{8}-\frac {3\,b^2\,x^2\,\cos \left (2\,a+\frac {2\,b}{x}\right )}{8}+\frac {3\,b\,x^3\,\sin \left (2\,a+\frac {2\,b}{x}\right )}{8}-\frac {b^3\,x\,\sin \left (2\,a+\frac {2\,b}{x}\right )}{4}}{b^4\,x^4} \] Input:

int(sin(a + b/x)^2/x^5,x)
 

Output:

- (3*cos(2*a + (2*b)/x))/(16*b^4) - (b^4/8 - (3*b^2*x^2*cos(2*a + (2*b)/x) 
)/8 + (3*b*x^3*sin(2*a + (2*b)/x))/8 - (b^3*x*sin(2*a + (2*b)/x))/4)/(b^4* 
x^4)
 

Reduce [F]

\[ \int \frac {\sin ^2\left (a+\frac {b}{x}\right )}{x^5} \, dx=\int \frac {\sin \left (\frac {a x +b}{x}\right )^{2}}{x^{5}}d x \] Input:

int(sin(a+b/x)^2/x^5,x)
 

Output:

int(sin((a*x + b)/x)**2/x**5,x)