\(\int x^{-1-2 n} \sin (a+b x^n) \, dx\) [150]

Optimal result

Integrand size = 16, antiderivative size = 78 \[ \int x^{-1-2 n} \sin \left (a+b x^n\right ) \, dx=-\frac {b x^{-n} \cos \left (a+b x^n\right )}{2 n}-\frac {b^2 \operatorname {CosIntegral}\left (b x^n\right ) \sin (a)}{2 n}-\frac {x^{-2 n} \sin \left (a+b x^n\right )}{2 n}-\frac {b^2 \cos (a) \text {Si}\left (b x^n\right )}{2 n} \] Output:

-1/2*b*cos(a+b*x^n)/n/(x^n)-1/2*b^2*Ci(b*x^n)*sin(a)/n-1/2*sin(a+b*x^n)/n/ 
(x^(2*n))-1/2*b^2*cos(a)*Si(b*x^n)/n
 

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.87 \[ \int x^{-1-2 n} \sin \left (a+b x^n\right ) \, dx=-\frac {x^{-2 n} \left (b x^n \cos \left (a+b x^n\right )+b^2 x^{2 n} \operatorname {CosIntegral}\left (b x^n\right ) \sin (a)+\sin \left (a+b x^n\right )+b^2 x^{2 n} \cos (a) \text {Si}\left (b x^n\right )\right )}{2 n} \] Input:

Integrate[x^(-1 - 2*n)*Sin[a + b*x^n],x]
 

Output:

-1/2*(b*x^n*Cos[a + b*x^n] + b^2*x^(2*n)*CosIntegral[b*x^n]*Sin[a] + Sin[a 
 + b*x^n] + b^2*x^(2*n)*Cos[a]*SinIntegral[b*x^n])/(n*x^(2*n))
 

Rubi [A] (verified)

Time = 0.53 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.83, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.688, Rules used = {3860, 3042, 3778, 3042, 3778, 25, 3042, 3784, 3042, 3780, 3783}

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.

\(\displaystyle \int x^{-2 n-1} \sin \left (a+b x^n\right ) \, dx\)

\(\Big \downarrow \) 3860

\(\displaystyle \frac {\int x^{-3 n} \sin \left (b x^n+a\right )dx^n}{n}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int x^{-3 n} \sin \left (b x^n+a\right )dx^n}{n}\)

\(\Big \downarrow \) 3778

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

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {1}{2} b \int x^{-2 n} \sin \left (b x^n+a+\frac {\pi }{2}\right )dx^n-\frac {1}{2} x^{-2 n} \sin \left (a+b x^n\right )}{n}\)

\(\Big \downarrow \) 3778

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3784

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

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {1}{2} b \left (x^{-n} \left (-\cos \left (a+b x^n\right )\right )-b \left (\sin (a) \int x^{-n} \sin \left (b x^n+\frac {\pi }{2}\right )dx^n+\cos (a) \int x^{-n} \sin \left (b x^n\right )dx^n\right )\right )-\frac {1}{2} x^{-2 n} \sin \left (a+b x^n\right )}{n}\)

\(\Big \downarrow \) 3780

\(\displaystyle \frac {\frac {1}{2} b \left (x^{-n} \left (-\cos \left (a+b x^n\right )\right )-b \left (\sin (a) \int x^{-n} \sin \left (b x^n+\frac {\pi }{2}\right )dx^n+\cos (a) \text {Si}\left (b x^n\right )\right )\right )-\frac {1}{2} x^{-2 n} \sin \left (a+b x^n\right )}{n}\)

\(\Big \downarrow \) 3783

\(\displaystyle \frac {\frac {1}{2} b \left (x^{-n} \left (-\cos \left (a+b x^n\right )\right )-b \left (\sin (a) \operatorname {CosIntegral}\left (b x^n\right )+\cos (a) \text {Si}\left (b x^n\right )\right )\right )-\frac {1}{2} x^{-2 n} \sin \left (a+b x^n\right )}{n}\)

Input:

Int[x^(-1 - 2*n)*Sin[a + b*x^n],x]
 

Output:

(-1/2*Sin[a + b*x^n]/x^(2*n) + (b*(-(Cos[a + b*x^n]/x^n) - b*(CosIntegral[ 
b*x^n]*Sin[a] + Cos[a]*SinIntegral[b*x^n])))/2)/n
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 3042

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

rule 3778

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[(c 
 + d*x)^(m + 1)*(Sin[e + f*x]/(d*(m + 1))), x] - Simp[f/(d*(m + 1))   Int[( 
c + d*x)^(m + 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && LtQ[m, - 
1]
 

rule 3780

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinInte 
gral[e + f*x]/d, x] /; FreeQ[{c, d, e, f}, x] && EqQ[d*e - c*f, 0]
 

rule 3783

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CosInte 
gral[e - Pi/2 + f*x]/d, x] /; FreeQ[{c, d, e, f}, x] && EqQ[d*(e - Pi/2) - 
c*f, 0]
 

rule 3784

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[Cos[(d* 
e - c*f)/d]   Int[Sin[c*(f/d) + f*x]/(c + d*x), x], x] + Simp[Sin[(d*e - c* 
f)/d]   Int[Cos[c*(f/d) + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f}, x] 
&& NeQ[d*e - c*f, 0]
 

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 = 0.84 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.83

method	result	size
default	\(\frac {b^{2} \left (-\frac {\sin \left (a +b \,x^{n}\right ) x^{-2 n}}{2 b^{2}}-\frac {\cos \left (a +b \,x^{n}\right ) x^{-n}}{2 b}-\frac {\operatorname {Si}\left (b \,x^{n}\right ) \cos \left (a \right )}{2}-\frac {\operatorname {Ci}\left (b \,x^{n}\right ) \sin \left (a \right )}{2}\right )}{n}\)	\(65\)
risch	\(-\frac {\left (-b^{2} {\mathrm e}^{-i a} \pi \,\operatorname {csgn}\left (b \,x^{n}\right ) x^{2 n}-i b^{2} {\mathrm e}^{-i a} \operatorname {expIntegral}_{1}\left (-i b \,x^{n}\right ) x^{2 n}+i b^{2} {\mathrm e}^{i a} \operatorname {expIntegral}_{1}\left (-i b \,x^{n}\right ) x^{2 n}+2 b^{2} {\mathrm e}^{-i a} \operatorname {Si}\left (b \,x^{n}\right ) x^{2 n}+2 x^{n} \cos \left (a +b \,x^{n}\right ) b +2 \sin \left (a +b \,x^{n}\right )\right ) x^{-2 n}}{4 n}\)	\(129\)
meijerg	\(\frac {b^{2} \sqrt {\pi }\, \left (-\frac {x^{2 \left (\frac {-1-2 n}{2 n}+\frac {1}{2 n}\right ) n} 2^{-\frac {-1-2 n}{n}-\frac {1}{n}}}{\sqrt {\pi }\, b^{2}}+\frac {\left (-1\right )^{-\frac {-1-2 n}{2 n}-\frac {1}{2 n}} \left (-\Psi \left (1-\frac {-1-2 n}{2 n}-\frac {1}{2 n}\right )-\Psi \left (\frac {1}{2}-\frac {-1-2 n}{2 n}-\frac {1}{2 n}\right )+2 n \ln \left (x \right )-2 \ln \left (2\right )+\ln \left (b^{2}\right )\right ) \sqrt {2}\, 2^{-\frac {-1-2 n}{n}-\frac {1}{n}-\frac {1}{2}}}{2 \sqrt {\pi }\, \Gamma \left (-\frac {-1-2 n}{n}-\frac {1}{n}\right )}-\frac {\left (-1\right )^{-\frac {-1-2 n}{2 n}-\frac {1}{2 n}} 2^{1-\frac {-1-2 n}{n}-\frac {1}{n}} x^{-2 n} \left (-\frac {9 x^{2 n} b^{2}}{2}+3\right )}{\sqrt {\pi }\, b^{2} \Gamma \left (2-\frac {-1-2 n}{n}-\frac {1}{n}\right )}-\frac {3 \left (-1\right )^{-\frac {-1-2 n}{2 n}-\frac {1}{2 n}} 2^{1-\frac {-1-2 n}{n}-\frac {1}{n}} \gamma }{\sqrt {\pi }\, \Gamma \left (2-\frac {-1-2 n}{n}-\frac {1}{n}\right )}-\frac {3 \left (-1\right )^{-\frac {-1-2 n}{2 n}-\frac {1}{2 n}} 2^{1-\frac {-1-2 n}{n}-\frac {1}{n}} \ln \left (2\right )}{\sqrt {\pi }\, \Gamma \left (2-\frac {-1-2 n}{n}-\frac {1}{n}\right )}-\frac {3 \left (-1\right )^{-\frac {-1-2 n}{2 n}-\frac {1}{2 n}} 2^{1-\frac {-1-2 n}{n}-\frac {1}{n}} \ln \left (\frac {b \,x^{n}}{2}\right )}{\sqrt {\pi }\, \Gamma \left (2-\frac {-1-2 n}{n}-\frac {1}{n}\right )}+\frac {3 \,2^{1-\frac {-1-2 n}{n}-\frac {1}{n}} \left (-1\right )^{-\frac {-1-2 n}{2 n}-\frac {1}{2 n}} x^{-2 n} \cos \left (b \,x^{n}\right )}{\sqrt {\pi }\, b^{2} \Gamma \left (2-\frac {-1-2 n}{n}-\frac {1}{n}\right )}-\frac {3 \left (-1\right )^{-\frac {-1-2 n}{2 n}-\frac {1}{2 n}} 2^{1-\frac {-1-2 n}{n}-\frac {1}{n}} x^{-n} \sin \left (b \,x^{n}\right )}{\sqrt {\pi }\, b \Gamma \left (2-\frac {-1-2 n}{n}-\frac {1}{n}\right )}+\frac {3 \,2^{1-\frac {-1-2 n}{n}-\frac {1}{n}} \left (-1\right )^{-\frac {-1-2 n}{2 n}-\frac {1}{2 n}} \operatorname {Ci}\left (b \,x^{n}\right )}{\sqrt {\pi }\, \Gamma \left (2-\frac {-1-2 n}{n}-\frac {1}{n}\right )}\right ) \sin \left (a \right )}{8 n}+\frac {b^{2} \sqrt {\pi }\, \left (-\frac {4 x^{-n} \cos \left (b \,x^{n}\right )}{\sqrt {\pi }\, b}-\frac {4 x^{-2 n} \sin \left (b \,x^{n}\right )}{\sqrt {\pi }\, b^{2}}-\frac {4 \,\operatorname {Si}\left (b \,x^{n}\right )}{\sqrt {\pi }}\right ) \cos \left (a \right )}{8 n}\)	\(761\)

Input:

int(x^(-1-2*n)*sin(a+b*x^n),x,method=_RETURNVERBOSE)
 

Output:

1/n*b^2*(-1/2*sin(a+b*x^n)/b^2/(x^n)^2-1/2*cos(a+b*x^n)/b/(x^n)-1/2*Si(b*x 
^n)*cos(a)-1/2*Ci(b*x^n)*sin(a))
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.87 \[ \int x^{-1-2 n} \sin \left (a+b x^n\right ) \, dx=-\frac {b^{2} x^{2 \, n} \operatorname {Ci}\left (b x^{n}\right ) \sin \left (a\right ) + b^{2} x^{2 \, n} \cos \left (a\right ) \operatorname {Si}\left (b x^{n}\right ) + b x^{n} \cos \left (b x^{n} + a\right ) + \sin \left (b x^{n} + a\right )}{2 \, n x^{2 \, n}} \] Input:

integrate(x^(-1-2*n)*sin(a+b*x^n),x, algorithm="fricas")
 

Output:

-1/2*(b^2*x^(2*n)*cos_integral(b*x^n)*sin(a) + b^2*x^(2*n)*cos(a)*sin_inte 
gral(b*x^n) + b*x^n*cos(b*x^n + a) + sin(b*x^n + a))/(n*x^(2*n))
 

Sympy [F]

\[ \int x^{-1-2 n} \sin \left (a+b x^n\right ) \, dx=\int x^{- 2 n - 1} \sin {\left (a + b x^{n} \right )}\, dx \] Input:

integrate(x**(-1-2*n)*sin(a+b*x**n),x)
 

Output:

Integral(x**(-2*n - 1)*sin(a + b*x**n), x)
 

Maxima [F]

\[ \int x^{-1-2 n} \sin \left (a+b x^n\right ) \, dx=\int { x^{-2 \, n - 1} \sin \left (b x^{n} + a\right ) \,d x } \] Input:

integrate(x^(-1-2*n)*sin(a+b*x^n),x, algorithm="maxima")
 

Output:

integrate(x^(-2*n - 1)*sin(b*x^n + a), x)
 

Giac [F]

\[ \int x^{-1-2 n} \sin \left (a+b x^n\right ) \, dx=\int { x^{-2 \, n - 1} \sin \left (b x^{n} + a\right ) \,d x } \] Input:

integrate(x^(-1-2*n)*sin(a+b*x^n),x, algorithm="giac")
 

Output:

integrate(x^(-2*n - 1)*sin(b*x^n + a), x)
 

Mupad [F(-1)]

Timed out. \[ \int x^{-1-2 n} \sin \left (a+b x^n\right ) \, dx=\int \frac {\sin \left (a+b\,x^n\right )}{x^{2\,n+1}} \,d x \] Input:

int(sin(a + b*x^n)/x^(2*n + 1),x)
 

Output:

int(sin(a + b*x^n)/x^(2*n + 1), x)
 

Reduce [F]

\[ \int x^{-1-2 n} \sin \left (a+b x^n\right ) \, dx=\frac {-x^{n} \cos \left (x^{n} b +a \right ) b -x^{2 n} \left (\int \frac {\sin \left (x^{n} b +a \right )}{x}d x \right ) b^{2} n -\sin \left (x^{n} b +a \right )}{2 x^{2 n} n} \] Input:

int(x^(-1-2*n)*sin(a+b*x^n),x)
 

Output:

( - (x**n*cos(x**n*b + a)*b + x**(2*n)*int(sin(x**n*b + a)/x,x)*b**2*n + s 
in(x**n*b + a)))/(2*x**(2*n)*n)