3.82 \(\int x^{-1-2 n} \cos (a+b x^n) \, dx\)

Optimal. Leaf size=78 \[ -\frac{b^2 \cos (a) \text{CosIntegral}\left (b x^n\right )}{2 n}+\frac{b^2 \sin (a) \text{Si}\left (b x^n\right )}{2 n}+\frac{b x^{-n} \sin \left (a+b x^n\right )}{2 n}-\frac{x^{-2 n} \cos \left (a+b x^n\right )}{2 n} \]

[Out]

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

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Rubi [A]  time = 0.111042, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312, Rules used = {3380, 3297, 3303, 3299, 3302} \[ -\frac{b^2 \cos (a) \text{CosIntegral}\left (b x^n\right )}{2 n}+\frac{b^2 \sin (a) \text{Si}\left (b x^n\right )}{2 n}+\frac{b x^{-n} \sin \left (a+b x^n\right )}{2 n}-\frac{x^{-2 n} \cos \left (a+b x^n\right )}{2 n} \]

Antiderivative was successfully verified.

[In]

Int[x^(-1 - 2*n)*Cos[a + b*x^n],x]

[Out]

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

Rule 3380

Int[((a_.) + Cos[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplif
y[(m + 1)/n] - 1)*(a + b*Cos[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]))

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 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 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]

Rubi steps

\begin{align*} \int x^{-1-2 n} \cos \left (a+b x^n\right ) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\cos (a+b x)}{x^3} \, dx,x,x^n\right )}{n}\\ &=-\frac{x^{-2 n} \cos \left (a+b x^n\right )}{2 n}-\frac{b \operatorname{Subst}\left (\int \frac{\sin (a+b x)}{x^2} \, dx,x,x^n\right )}{2 n}\\ &=-\frac{x^{-2 n} \cos \left (a+b x^n\right )}{2 n}+\frac{b x^{-n} \sin \left (a+b x^n\right )}{2 n}-\frac{b^2 \operatorname{Subst}\left (\int \frac{\cos (a+b x)}{x} \, dx,x,x^n\right )}{2 n}\\ &=-\frac{x^{-2 n} \cos \left (a+b x^n\right )}{2 n}+\frac{b x^{-n} \sin \left (a+b x^n\right )}{2 n}-\frac{\left (b^2 \cos (a)\right ) \operatorname{Subst}\left (\int \frac{\cos (b x)}{x} \, dx,x,x^n\right )}{2 n}+\frac{\left (b^2 \sin (a)\right ) \operatorname{Subst}\left (\int \frac{\sin (b x)}{x} \, dx,x,x^n\right )}{2 n}\\ &=-\frac{x^{-2 n} \cos \left (a+b x^n\right )}{2 n}-\frac{b^2 \cos (a) \text{Ci}\left (b x^n\right )}{2 n}+\frac{b x^{-n} \sin \left (a+b x^n\right )}{2 n}+\frac{b^2 \sin (a) \text{Si}\left (b x^n\right )}{2 n}\\ \end{align*}

Mathematica [A]  time = 0.129637, size = 70, normalized size = 0.9 \[ -\frac{x^{-2 n} \left (b^2 \cos (a) x^{2 n} \text{CosIntegral}\left (b x^n\right )-b^2 \sin (a) x^{2 n} \text{Si}\left (b x^n\right )-b x^n \sin \left (a+b x^n\right )+\cos \left (a+b x^n\right )\right )}{2 n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.037, size = 65, normalized size = 0.8 \begin{align*}{\frac{{b}^{2}}{n} \left ( -{\frac{\cos \left ( a+b{x}^{n} \right ) }{2\, \left ({x}^{n} \right ) ^{2}{b}^{2}}}+{\frac{\sin \left ( a+b{x}^{n} \right ) }{2\,b{x}^{n}}}+{\frac{{\it Si} \left ( b{x}^{n} \right ) \sin \left ( a \right ) }{2}}-{\frac{{\it Ci} \left ( b{x}^{n} \right ) \cos \left ( a \right ) }{2}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{-2 \, n - 1} \cos \left (b x^{n} + a\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.6691, size = 254, normalized size = 3.26 \begin{align*} -\frac{b^{2} x^{2 \, n} \cos \left (a\right ) \operatorname{Ci}\left (b x^{n}\right ) + b^{2} x^{2 \, n} \cos \left (a\right ) \operatorname{Ci}\left (-b x^{n}\right ) - 2 \, b^{2} x^{2 \, n} \sin \left (a\right ) \operatorname{Si}\left (b x^{n}\right ) - 2 \, b x^{n} \sin \left (b x^{n} + a\right ) + 2 \, \cos \left (b x^{n} + a\right )}{4 \, n x^{2 \, n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(b^2*x^(2*n)*cos(a)*cos_integral(b*x^n) + b^2*x^(2*n)*cos(a)*cos_integral(-b*x^n) - 2*b^2*x^(2*n)*sin(a)*
sin_integral(b*x^n) - 2*b*x^n*sin(b*x^n + a) + 2*cos(b*x^n + a))/(n*x^(2*n))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{-2 \, n - 1} \cos \left (b x^{n} + a\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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