3.3.68 \(\int \sin (x^{-n}) \, dx\) [268]

Optimal. Leaf size=43 \[ \frac {i x \left (-\operatorname {ExpIntegralE}\left (1+\frac {1}{n},-i x^{-n}\right )+\operatorname {ExpIntegralE}\left (1+\frac {1}{n},i x^{-n}\right )\right )}{2 n} \]

[Out]

1/2*I*x*(-Ei(1+1/n,-I/(x^n))+Ei(1+1/n,I/(x^n)))/n

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Rubi [A]
time = 0.01, antiderivative size = 77, normalized size of antiderivative = 1.79, number of steps used = 3, number of rules used = 2, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3446, 2239} \begin {gather*} \frac {i x \left (i x^{-n}\right )^{\frac {1}{n}} \Gamma \left (-\frac {1}{n},i x^{-n}\right )}{2 n}-\frac {i x \left (-i x^{-n}\right )^{\frac {1}{n}} \Gamma \left (-\frac {1}{n},-i x^{-n}\right )}{2 n} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sin[x^(-n)],x]

[Out]

((-1/2*I)*x*((-I)/x^n)^n^(-1)*Gamma[-n^(-1), (-I)/x^n])/n + ((I/2)*x*(I/x^n)^n^(-1)*Gamma[-n^(-1), I/x^n])/n

Rule 2239

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_)), x_Symbol] :> Simp[(-F^a)*(c + d*x)*(Gamma[1/n, (-b)*(c + d
*x)^n*Log[F]]/(d*n*((-b)*(c + d*x)^n*Log[F])^(1/n))), x] /; FreeQ[{F, a, b, c, d, n}, x] &&  !IntegerQ[2/n]

Rule 3446

Int[Sin[(c_.) + (d_.)*((e_.) + (f_.)*(x_))^(n_)], x_Symbol] :> Dist[I/2, Int[E^((-c)*I - d*I*(e + f*x)^n), x],
x] - Dist[I/2, Int[E^(c*I + d*I*(e + f*x)^n), x], x] /; FreeQ[{c, d, e, f, n}, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {Integral} &=\frac {1}{2} i \int e^{-i x^{-n}} \, dx-\frac {1}{2} i \int e^{i x^{-n}} \, dx\\ &=-\frac {i x \left (-i x^{-n}\right )^{\frac {1}{n}} \Gamma \left (-\frac {1}{n},-i x^{-n}\right )}{2 n}+\frac {i x \left (i x^{-n}\right )^{\frac {1}{n}} \Gamma \left (-\frac {1}{n},i x^{-n}\right )}{2 n}\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.02, size = 70, normalized size = 1.63 \begin {gather*} -\frac {i x \left (\left (-i x^{-n}\right )^{\frac {1}{n}} \Gamma \left (-\frac {1}{n},-i x^{-n}\right )-\left (i x^{-n}\right )^{\frac {1}{n}} \Gamma \left (-\frac {1}{n},i x^{-n}\right )\right )}{2 n} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sin[x^(-n)],x]

[Out]

((-1/2*I)*x*(((-I)/x^n)^n^(-1)*Gamma[-n^(-1), (-I)/x^n] - (I/x^n)^n^(-1)*Gamma[-n^(-1), I/x^n]))/n

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Maple [C] Result contains higher order function than in optimal. Order 5 vs. order 4.
time = 0.17, size = 47, normalized size = 1.09

method	result	size
meijerg	\(-\frac {x^{-n +1} {}_{1}^{}{\moversetsp {}{\mundersetsp {}{F_{2}^{}}}}\left (\frac {1}{2}-\frac {1}{2 n};\frac {3}{2},\frac {3}{2}-\frac {1}{2 n};-\frac {x^{-4 n} x^{2 n}}{4}\right )}{n -1}\)	\(47\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(1/(x^n)),x,method=_RETURNVERBOSE)

[Out]

-1/(n-1)*x^(-n+1)*hypergeom([1/2-1/2/n],[3/2,3/2-1/2/n],-1/4*(x^(-2*n))^2*x^(2*n))

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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(sin(1/(x^n)),x, algorithm="maxima")

[Out]

integrate(sin(1/(x^n)), x)

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Fricas [F]
time = 0.60, size = 8, normalized size = 0.19 \begin {gather*} {\rm integral}\left (\sin \left (\frac {1}{x^{n}}\right ), x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(1/(x^n)),x, algorithm="fricas")

[Out]

integral(sin(1/(x^n)), x)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (27) = 54\).
time = 0.80, size = 56, normalized size = 1.30 \begin {gather*} - \frac {x x^{- n} \Gamma \left (\frac {1}{2} - \frac {1}{2 n}\right ) {{}_{1}F_{2}\left (\begin {matrix} \frac {1}{2} - \frac {1}{2 n} \\ \frac {3}{2}, \frac {3}{2} - \frac {1}{2 n} \end {matrix}\middle | {- \frac {x^{- 2 n}}{4}} \right )}}{2 n \Gamma \left (\frac {3}{2} - \frac {1}{2 n}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(1/(x**n)),x)

[Out]

-x*gamma(1/2 - 1/(2*n))*hyper((1/2 - 1/(2*n),), (3/2, 3/2 - 1/(2*n)), -1/(4*x**(2*n)))/(2*n*x**n*gamma(3/2 - 1
/(2*n)))

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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(sin(1/(x^n)),x, algorithm="giac")

[Out]

integrate(sin(1/(x^n)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(1/x^n),x)

[Out]

int(sin(1/x^n), x)

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Chatgpt [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {not solved} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

int(sin(1/(x^n)),x)

[Out]

not solved

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