3.147 \(\int x^{-1-n} \sin (a+b x^n) \, dx\)

Optimal. Leaf size=46 \[ \frac {b \cos (a) \text {Ci}\left (b x^n\right )}{n}-\frac {b \sin (a) \text {Si}\left (b x^n\right )}{n}-\frac {x^{-n} \sin \left (a+b x^n\right )}{n} \]

[Out]

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

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Rubi [A]  time = 0.09, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {3379, 3297, 3303, 3299, 3302} \[ \frac {b \cos (a) \text {CosIntegral}\left (b x^n\right )}{n}-\frac {b \sin (a) \text {Si}\left (b x^n\right )}{n}-\frac {x^{-n} \sin \left (a+b x^n\right )}{n} \]

Antiderivative was successfully verified.

[In]

Int[x^(-1 - n)*Sin[a + b*x^n],x]

[Out]

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

Rule 3297

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] - Dist[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 3299

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

Rule 3302

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

Rule 3303

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[(c*f)/d + f*x]
/(c + d*x), x], x] + Dist[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 3379

Int[(x_)^(m_.)*((a_.) + (b_.)*Sin[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplif
y[(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[Simpl
ify[(m + 1)/n]] && (EqQ[p, 1] || EqQ[m, n - 1] || (IntegerQ[p] && GtQ[Simplify[(m + 1)/n], 0]))

Rubi steps

\begin {align*} \int x^{-1-n} \sin \left (a+b x^n\right ) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\sin (a+b x)}{x^2} \, dx,x,x^n\right )}{n}\\ &=-\frac {x^{-n} \sin \left (a+b x^n\right )}{n}+\frac {b \operatorname {Subst}\left (\int \frac {\cos (a+b x)}{x} \, dx,x,x^n\right )}{n}\\ &=-\frac {x^{-n} \sin \left (a+b x^n\right )}{n}+\frac {(b \cos (a)) \operatorname {Subst}\left (\int \frac {\cos (b x)}{x} \, dx,x,x^n\right )}{n}-\frac {(b \sin (a)) \operatorname {Subst}\left (\int \frac {\sin (b x)}{x} \, dx,x,x^n\right )}{n}\\ &=\frac {b \cos (a) \text {Ci}\left (b x^n\right )}{n}-\frac {x^{-n} \sin \left (a+b x^n\right )}{n}-\frac {b \sin (a) \text {Si}\left (b x^n\right )}{n}\\ \end {align*}

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Mathematica [A]  time = 0.08, size = 47, normalized size = 1.02 \[ \frac {x^{-n} \left (b \cos (a) x^n \text {Ci}\left (b x^n\right )-b \sin (a) x^n \text {Si}\left (b x^n\right )-\sin \left (a+b x^n\right )\right )}{n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.78, size = 62, normalized size = 1.35 \[ \frac {b x^{n} \cos \relax (a) \operatorname {Ci}\left (b x^{n}\right ) + b x^{n} \cos \relax (a) \operatorname {Ci}\left (-b x^{n}\right ) - 2 \, b x^{n} \sin \relax (a) \operatorname {Si}\left (b x^{n}\right ) - 2 \, \sin \left (b x^{n} + a\right )}{2 \, n x^{n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{-n - 1} \sin \left (b x^{n} + a\right )\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.03, size = 44, normalized size = 0.96 \[ \frac {b \left (-\frac {\sin \left (a +b \,x^{n}\right ) x^{-n}}{b}-\Si \left (b \,x^{n}\right ) \sin \relax (a )+\Ci \left (b \,x^{n}\right ) \cos \relax (a )\right )}{n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{-n - 1} \sin \left (b x^{n} + a\right )\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\sin \left (a+b\,x^n\right )}{x^{n+1}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(a + b*x^n)/x^(n + 1),x)

[Out]

int(sin(a + b*x^n)/x^(n + 1), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{- n - 1} \sin {\left (a + b x^{n} \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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