Optimal. Leaf size=17 \[ \frac {1}{6} \sin (3 x)+\frac {1}{10} \sin (5 x) \]
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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 = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {4368}
\begin {gather*} \frac {1}{6} \sin (3 x)+\frac {1}{10} \sin (5 x) \end {gather*}
Antiderivative was successfully verified.
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Rule 4368
Rubi steps
\begin {align*} \int \cos (x) \cos (4 x) \, dx &=\frac {1}{6} \sin (3 x)+\frac {1}{10} \sin (5 x)\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 17, normalized size = 1.00 \begin {gather*} \frac {1}{6} \sin (3 x)+\frac {1}{10} \sin (5 x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 14, normalized size = 0.82
method | result | size |
default | \(\frac {\sin \left (3 x \right )}{6}+\frac {\sin \left (5 x \right )}{10}\) | \(14\) |
risch | \(\frac {\sin \left (3 x \right )}{6}+\frac {\sin \left (5 x \right )}{10}\) | \(14\) |
norman | \(\frac {-\frac {8 \tan \left (2 x \right ) \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{15}+\frac {2 \left (\tan ^{2}\left (2 x \right )\right ) \tan \left (\frac {x}{2}\right )}{15}+\frac {8 \tan \left (2 x \right )}{15}-\frac {2 \tan \left (\frac {x}{2}\right )}{15}}{\left (1+\tan ^{2}\left (\frac {x}{2}\right )\right ) \left (1+\tan ^{2}\left (2 x \right )\right )}\) | \(59\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 1.20, size = 13, normalized size = 0.76 \begin {gather*} \frac {1}{10} \, \sin \left (5 \, x\right ) + \frac {1}{6} \, \sin \left (3 \, x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.08, size = 18, normalized size = 1.06 \begin {gather*} \frac {1}{15} \, {\left (24 \, \cos \left (x\right )^{4} - 8 \, \cos \left (x\right )^{2} - 1\right )} \sin \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.12, size = 20, normalized size = 1.18 \begin {gather*} - \frac {\sin {\left (x \right )} \cos {\left (4 x \right )}}{15} + \frac {4 \sin {\left (4 x \right )} \cos {\left (x \right )}}{15} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.93, size = 13, normalized size = 0.76 \begin {gather*} \frac {1}{10} \, \sin \left (5 \, x\right ) + \frac {1}{6} \, \sin \left (3 \, x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.19, size = 13, normalized size = 0.76 \begin {gather*} \frac {\sin \left (3\,x\right )}{6}+\frac {\sin \left (5\,x\right )}{10} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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