Optimal. Leaf size=43 \[ \frac {i x \left (-\operatorname {ExpIntegralE}\left (1+\frac {1}{n},-i x^{-n}\right )+\operatorname {ExpIntegralE}\left (1+\frac {1}{n},i x^{-n}\right )\right )}{2 n} \]
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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.
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Rule 2239
Rule 3446
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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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