Optimal. Leaf size=17 \[ \frac {1}{4} \sin (2 x)+\frac {1}{16} \sin (8 x) \]
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Rubi [A]
time = 0.01, antiderivative size = 17, 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 = {4368}
\begin {gather*} \frac {1}{4} \sin (2 x)+\frac {1}{16} \sin (8 x) \end {gather*}
Antiderivative was successfully verified.
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Rule 4368
Rubi steps
\begin {align*} \int \cos (3 x) \cos (5 x) \, dx &=\frac {1}{4} \sin (2 x)+\frac {1}{16} \sin (8 x)\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 17, normalized size = 1.00 \begin {gather*} \frac {1}{4} \sin (2 x)+\frac {1}{16} \sin (8 x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 14, normalized size = 0.82
method | result | size |
default | \(\frac {\sin \left (2 x \right )}{4}+\frac {\sin \left (8 x \right )}{16}\) | \(14\) |
risch | \(\frac {\sin \left (2 x \right )}{4}+\frac {\sin \left (8 x \right )}{16}\) | \(14\) |
norman | \(\frac {\frac {3 \tan \left (\frac {3 x}{2}\right ) \left (\tan ^{2}\left (\frac {5 x}{2}\right )\right )}{8}-\frac {5 \left (\tan ^{2}\left (\frac {3 x}{2}\right )\right ) \tan \left (\frac {5 x}{2}\right )}{8}-\frac {3 \tan \left (\frac {3 x}{2}\right )}{8}+\frac {5 \tan \left (\frac {5 x}{2}\right )}{8}}{\left (1+\tan ^{2}\left (\frac {3 x}{2}\right )\right ) \left (1+\tan ^{2}\left (\frac {5 x}{2}\right )\right )}\) | \(59\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 2.80, size = 13, normalized size = 0.76 \begin {gather*} \frac {1}{16} \, \sin \left (8 \, x\right ) + \frac {1}{4} \, \sin \left (2 \, x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.52, size = 22, normalized size = 1.29 \begin {gather*} {\left (8 \, \cos \left (x\right )^{7} - 12 \, \cos \left (x\right )^{5} + 5 \, \cos \left (x\right )^{3}\right )} \sin \left (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. 26 vs.
\(2 (12) = 24\).
time = 0.12, size = 26, normalized size = 1.53 \begin {gather*} - \frac {3 \sin {\left (3 x \right )} \cos {\left (5 x \right )}}{16} + \frac {5 \sin {\left (5 x \right )} \cos {\left (3 x \right )}}{16} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.12, size = 13, normalized size = 0.76 \begin {gather*} \frac {1}{16} \, \sin \left (8 \, x\right ) + \frac {1}{4} \, \sin \left (2 \, x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.06, size = 13, normalized size = 0.76 \begin {gather*} \frac {\sin \left (2\,x\right )}{4}+\frac {\sin \left (8\,x\right )}{16} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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