Optimal. Leaf size=15 \[ \frac {\cos (x)}{2}-\frac {1}{10} \cos (5 x) \]
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Rubi [A]
time = 0.00, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {4369}
\begin {gather*} \frac {\cos (x)}{2}-\frac {1}{10} \cos (5 x) \end {gather*}
Antiderivative was successfully verified.
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Rule 4369
Rubi steps
\begin {gather*} \begin {aligned} \text {Integral} &=\frac {\cos (x)}{2}-\frac {1}{10} \cos (5 x)\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.01, size = 15, normalized size = 1.00 \begin {gather*} \frac {\cos (x)}{2}-\frac {1}{10} \cos (5 x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 12, normalized size = 0.80
method | result | size |
default | \(\frac {\cos \left (x \right )}{2}-\frac {\cos \left (5 x \right )}{10}\) | \(12\) |
risch | \(\frac {\cos \left (x \right )}{2}-\frac {\cos \left (5 x \right )}{10}\) | \(12\) |
parallelrisch | \(\frac {\cos \left (x \right )}{2}-\frac {\cos \left (5 x \right )}{10}-\frac {2}{5}\) | \(13\) |
norman | \(\frac {-\frac {4 \left (\tan ^{2}\left (x \right )\right )}{5}-\frac {4 \left (\tan ^{2}\left (\frac {3 x}{2}\right )\right )}{5}+\frac {12 \tan \left (x \right ) \tan \left (\frac {3 x}{2}\right )}{5}}{\left (1+\tan ^{2}\left (x \right )\right ) \left (1+\tan ^{2}\left (\frac {3 x}{2}\right )\right )}\) | \(43\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.40, size = 11, normalized size = 0.73 \begin {gather*} -\frac {1}{10} \, \cos \left (5 \, x\right ) + \frac {1}{2} \, \cos \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.59, size = 13, normalized size = 0.87 \begin {gather*} -\frac {8}{5} \, \cos \left (x\right )^{5} + 2 \, \cos \left (x\right )^{3} \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. 26 vs.
\(2 (10) = 20\).
time = 0.14, size = 26, normalized size = 1.73 \begin {gather*} \frac {3 \sin {\left (2 x \right )} \sin {\left (3 x \right )}}{5} + \frac {2 \cos {\left (2 x \right )} \cos {\left (3 x \right )}}{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.50, size = 11, normalized size = 0.73 \begin {gather*} -\frac {1}{10} \, \cos \left (5 \, x\right ) + \frac {1}{2} \, \cos \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 13, normalized size = 0.87 \begin {gather*} 2\,{\cos \left (x\right )}^3-\frac {8\,{\cos \left (x\right )}^5}{5} \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 = 12, normalized size = 0.80 \begin {gather*} \frac {\left (3 \left (\sin ^{2}\left (x \right )\right )-1\right ) \cos \left (x \right )}{5} \end {gather*}
Warning: Unable to verify antiderivative.
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