3.1.79 \(\int x^{-1-n} \cos (a+b x^n) \, dx\) [79]

Optimal. Leaf size=47 \[ -\frac {x^{-n} \cos \left (a+b x^n\right )}{n}-\frac {b \text {CosIntegral}\left (b x^n\right ) \sin (a)}{n}-\frac {b \cos (a) \text {Si}\left (b x^n\right )}{n} \]

[Out]

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

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Rubi [A]
time = 0.06, antiderivative size = 47, 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 = {3461, 3378, 3384, 3380, 3383} \begin {gather*} -\frac {b \sin (a) \text {CosIntegral}\left (b x^n\right )}{n}-\frac {b \cos (a) \text {Si}\left (b x^n\right )}{n}-\frac {x^{-n} \cos \left (a+b x^n\right )}{n} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^(-1 - n)*Cos[a + b*x^n],x]

[Out]

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

Rule 3378

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 3380

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 3383

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 3384

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 3461

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]))

Rubi steps

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

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Mathematica [A]
time = 0.10, size = 45, normalized size = 0.96 \begin {gather*} -\frac {x^{-n} \left (\cos \left (a+b x^n\right )+b x^n \text {CosIntegral}\left (b x^n\right ) \sin (a)+b x^n \cos (a) \text {Si}\left (b x^n\right )\right )}{n} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.14, size = 45, normalized size = 0.96

method	result	size
default	\(\frac {b \left (-\frac {\cos \left (a +b \,x^{n}\right ) x^{-n}}{b}-\sinIntegral \left (b \,x^{n}\right ) \cos \left (a \right )-\cosineIntegral \left (b \,x^{n}\right ) \sin \left (a \right )\right )}{n}\)	\(45\)
risch	\(\frac {b \,{\mathrm e}^{-i a} \pi \,\mathrm {csgn}\left (b \,x^{n}\right )}{2 n}-\frac {b \,{\mathrm e}^{-i a} \sinIntegral \left (b \,x^{n}\right )}{n}+\frac {i b \,{\mathrm e}^{-i a} \expIntegral \left (1, -i b \,x^{n}\right )}{2 n}-\frac {i b \,{\mathrm e}^{i a} \expIntegral \left (1, -i b \,x^{n}\right )}{2 n}-\frac {\cos \left (a +b \,x^{n}\right ) x^{-n}}{n}\)	\(97\)
meijerg	error in int/gproduct: numeric exception: division by zero\	N/A

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.38, size = 62, normalized size = 1.32 \begin {gather*} -\frac {b x^{n} \operatorname {Ci}\left (b x^{n}\right ) \sin \left (a\right ) + b x^{n} \operatorname {Ci}\left (-b x^{n}\right ) \sin \left (a\right ) + 2 \, b x^{n} \cos \left (a\right ) \operatorname {Si}\left (b x^{n}\right ) + 2 \, \cos \left (b x^{n} + a\right )}{2 \, n x^{n}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{- n - 1} \cos {\left (a + b x^{n} \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {\cos \left (a+b\,x^n\right )}{x^{n+1}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(a + b*x^n)/x^(n + 1),x)

[Out]

int(cos(a + b*x^n)/x^(n + 1), x)

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