Optimal. Leaf size=27 \[ \frac {1}{8} (3-x)^8+\frac {x^2}{2}-\cos (3-x) \]
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Rubi [A]
time = 0.00, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2718}
\begin {gather*} \frac {x^2}{2}+\frac {1}{8} (3-x)^8-\cos (3-x) \end {gather*}
Antiderivative was successfully verified.
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Rule 2718
Rubi steps
\begin {gather*} \begin {aligned} \text {Integral} &=\frac {1}{8} (3-x)^8+\frac {x^2}{2}-\int \sin (3-x) \, dx\\ &=\frac {1}{8} (3-x)^8+\frac {x^2}{2}-\cos (3-x)\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.01, size = 29, normalized size = 1.07 \begin {gather*} \frac {1}{8} (-3+x)^8+\frac {x^2}{2}-\cos (3) \cos (x)-\sin (3) \sin (x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 20, normalized size = 0.74
method | result | size |
default | \(\frac {x^{2}}{2}+\frac {\left (-3+x \right )^{8}}{8}-\cos \left (-3+x \right )\) | \(20\) |
derivativedivides | \(-9+3 x +\frac {\left (-3+x \right )^{2}}{2}+\frac {\left (-3+x \right )^{8}}{8}-\cos \left (-3+x \right )\) | \(26\) |
parts | \(-2187 x +2552 x^{2}-1701 x^{3}+\frac {2835 x^{4}}{4}-189 x^{5}+\frac {63 x^{6}}{2}-3 x^{7}+\frac {x^{8}}{8}-\cos \left (-3+x \right )\) | \(46\) |
risch | \(2552 x^{2}+\frac {x^{8}}{8}-3 x^{7}+\frac {63 x^{6}}{2}-189 x^{5}+\frac {2835 x^{4}}{4}-1701 x^{3}-2187 x +\frac {6561}{8}-\cos \left (-3+x \right )\) | \(47\) |
parallelrisch | \(\frac {x^{8}}{8}-3 x^{7}+\frac {63 x^{6}}{2}-189 x^{5}+\frac {2835 x^{4}}{4}-1701 x^{3}+2552 x^{2}-2187 x -1-\cos \left (-3+x \right )\) | \(47\) |
norman | \(\frac {-2187 x +2552 x^{2}-1701 x^{3}+\frac {2835 x^{4}}{4}-189 x^{5}+\frac {63 x^{6}}{2}-3 x^{7}+\frac {x^{8}}{8}-2187 x \left (\tan ^{2}\left (-\frac {3}{2}+\frac {x}{2}\right )\right )+2552 x^{2} \left (\tan ^{2}\left (-\frac {3}{2}+\frac {x}{2}\right )\right )-1701 x^{3} \left (\tan ^{2}\left (-\frac {3}{2}+\frac {x}{2}\right )\right )+\frac {2835 x^{4} \left (\tan ^{2}\left (-\frac {3}{2}+\frac {x}{2}\right )\right )}{4}-189 x^{5} \left (\tan ^{2}\left (-\frac {3}{2}+\frac {x}{2}\right )\right )+\frac {63 x^{6} \left (\tan ^{2}\left (-\frac {3}{2}+\frac {x}{2}\right )\right )}{2}-3 x^{7} \left (\tan ^{2}\left (-\frac {3}{2}+\frac {x}{2}\right )\right )+\frac {x^{8} \left (\tan ^{2}\left (-\frac {3}{2}+\frac {x}{2}\right )\right )}{8}-2}{1+\tan ^{2}\left (-\frac {3}{2}+\frac {x}{2}\right )}\) | \(156\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.33, size = 19, normalized size = 0.70 \begin {gather*} \frac {1}{8} \, {\left (x - 3\right )}^{8} + \frac {1}{2} \, x^{2} - \cos \left (x - 3\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 45 vs.
\(2 (19) = 38\).
time = 0.62, size = 45, normalized size = 1.67 \begin {gather*} \frac {1}{8} \, x^{8} - 3 \, x^{7} + \frac {63}{2} \, x^{6} - 189 \, x^{5} + \frac {2835}{4} \, x^{4} - 1701 \, x^{3} + 2552 \, x^{2} - 2187 \, x - \cos \left (x - 3\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.05, size = 15, normalized size = 0.56 \begin {gather*} \frac {x^{2}}{2} + \frac {\left (x - 3\right )^{8}}{8} - \cos {\left (x - 3 \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.43, size = 19, normalized size = 0.70 \begin {gather*} \frac {1}{8} \, {\left (x - 3\right )}^{8} + \frac {1}{2} \, x^{2} - \cos \left (x - 3\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.20, size = 45, normalized size = 1.67 \begin {gather*} 2552\,x^2-\cos \left (x-3\right )-2187\,x-1701\,x^3+\frac {2835\,x^4}{4}-189\,x^5+\frac {63\,x^6}{2}-3\,x^7+\frac {x^8}{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Chatgpt [F] Failed to verify
time = 1.00, size = 17, normalized size = 0.63 \begin {gather*} \frac {\left (x -3\right )^{8}}{8}+\frac {x^{2}}{2}+\cos \left (x -3\right ) \end {gather*}
Warning: Unable to verify antiderivative.
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