Optimal. Leaf size=30 \[ \frac {x}{4}+\frac {1}{8} \sin (2 x)+\frac {1}{16} \sin (4 x)+\frac {1}{24} \sin (6 x) \]
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Rubi [A]
time = 0.02, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {4440, 2717}
\begin {gather*} \frac {x}{4}+\frac {1}{8} \sin (2 x)+\frac {1}{16} \sin (4 x)+\frac {1}{24} \sin (6 x) \end {gather*}
Antiderivative was successfully verified.
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Rule 2717
Rule 4440
Rubi steps
\begin {align*} \int \cos (x) \cos (2 x) \cos (3 x) \, dx &=\int \left (\frac {1}{4}+\frac {1}{4} \cos (2 x)+\frac {1}{4} \cos (4 x)+\frac {1}{4} \cos (6 x)\right ) \, dx\\ &=\frac {x}{4}+\frac {1}{4} \int \cos (2 x) \, dx+\frac {1}{4} \int \cos (4 x) \, dx+\frac {1}{4} \int \cos (6 x) \, dx\\ &=\frac {x}{4}+\frac {1}{8} \sin (2 x)+\frac {1}{16} \sin (4 x)+\frac {1}{24} \sin (6 x)\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 30, normalized size = 1.00 \begin {gather*} \frac {x}{4}+\frac {1}{8} \sin (2 x)+\frac {1}{16} \sin (4 x)+\frac {1}{24} \sin (6 x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 23, normalized size = 0.77
method | result | size |
default | \(\frac {x}{4}+\frac {\sin \left (2 x \right )}{8}+\frac {\sin \left (4 x \right )}{16}+\frac {\sin \left (6 x \right )}{24}\) | \(23\) |
risch | \(\frac {x}{4}+\frac {\sin \left (2 x \right )}{8}+\frac {\sin \left (4 x \right )}{16}+\frac {\sin \left (6 x \right )}{24}\) | \(23\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 1.48, size = 22, normalized size = 0.73 \begin {gather*} \frac {1}{4} \, x + \frac {1}{24} \, \sin \left (6 \, x\right ) + \frac {1}{16} \, \sin \left (4 \, x\right ) + \frac {1}{8} \, \sin \left (2 \, x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.05, size = 25, normalized size = 0.83 \begin {gather*} \frac {1}{12} \, {\left (16 \, \cos \left (x\right )^{5} - 10 \, \cos \left (x\right )^{3} + 3 \, \cos \left (x\right )\right )} \sin \left (x\right ) + \frac {1}{4} \, x \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. 116 vs.
\(2 (22) = 44\).
time = 1.24, size = 116, normalized size = 3.87 \begin {gather*} - \frac {x \sin {\left (x \right )} \sin {\left (2 x \right )} \cos {\left (3 x \right )}}{4} + \frac {x \sin {\left (x \right )} \sin {\left (3 x \right )} \cos {\left (2 x \right )}}{4} + \frac {x \sin {\left (2 x \right )} \sin {\left (3 x \right )} \cos {\left (x \right )}}{4} + \frac {x \cos {\left (x \right )} \cos {\left (2 x \right )} \cos {\left (3 x \right )}}{4} + \frac {3 \sin {\left (x \right )} \sin {\left (2 x \right )} \sin {\left (3 x \right )}}{8} + \frac {\sin {\left (x \right )} \cos {\left (2 x \right )} \cos {\left (3 x \right )}}{3} + \frac {5 \sin {\left (2 x \right )} \cos {\left (x \right )} \cos {\left (3 x \right )}}{24} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.73, size = 22, normalized size = 0.73 \begin {gather*} \frac {1}{4} \, x + \frac {1}{24} \, \sin \left (6 \, x\right ) + \frac {1}{16} \, \sin \left (4 \, x\right ) + \frac {1}{8} \, \sin \left (2 \, x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.29, size = 22, normalized size = 0.73 \begin {gather*} \frac {x}{4}+\frac {\sin \left (2\,x\right )}{8}+\frac {\sin \left (4\,x\right )}{16}+\frac {\sin \left (6\,x\right )}{24} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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