Optimal. Leaf size=25 \[ -\frac {1}{8} \cos (2 x)-\frac {1}{16} \cos (4 x)+\frac {1}{24} \cos (6 x) \]
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Rubi [A]
time = 0.02, antiderivative size = 25, 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, 2718}
\begin {gather*} -\frac {1}{8} \cos (2 x)-\frac {1}{16} \cos (4 x)+\frac {1}{24} \cos (6 x) \end {gather*}
Antiderivative was successfully verified.
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Rule 2718
Rule 4440
Rubi steps
\begin {align*} \int \sin (x) \sin (2 x) \sin (3 x) \, dx &=\int \left (\frac {1}{4} \sin (2 x)+\frac {1}{4} \sin (4 x)-\frac {1}{4} \sin (6 x)\right ) \, dx\\ &=\frac {1}{4} \int \sin (2 x) \, dx+\frac {1}{4} \int \sin (4 x) \, dx-\frac {1}{4} \int \sin (6 x) \, dx\\ &=-\frac {1}{8} \cos (2 x)-\frac {1}{16} \cos (4 x)+\frac {1}{24} \cos (6 x)\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 25, normalized size = 1.00 \begin {gather*} -\frac {1}{8} \cos (2 x)-\frac {1}{16} \cos (4 x)+\frac {1}{24} \cos (6 x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 20, normalized size = 0.80
method | result | size |
default | \(-\frac {\cos \left (2 x \right )}{8}-\frac {\cos \left (4 x \right )}{16}+\frac {\cos \left (6 x \right )}{24}\) | \(20\) |
risch | \(-\frac {\cos \left (2 x \right )}{8}-\frac {\cos \left (4 x \right )}{16}+\frac {\cos \left (6 x \right )}{24}\) | \(20\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 2.23, size = 19, normalized size = 0.76 \begin {gather*} \frac {1}{24} \, \cos \left (6 \, x\right ) - \frac {1}{16} \, \cos \left (4 \, x\right ) - \frac {1}{8} \, \cos \left (2 \, x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.66, size = 17, normalized size = 0.68 \begin {gather*} \frac {4}{3} \, \cos \left (x\right )^{6} - \frac {5}{2} \, \cos \left (x\right )^{4} + \cos \left (x\right )^{2} \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. 114 vs.
\(2 (19) = 38\).
time = 1.24, size = 114, normalized size = 4.56 \begin {gather*} \frac {x \sin {\left (x \right )} \sin {\left (2 x \right )} \sin {\left (3 x \right )}}{4} + \frac {x \sin {\left (x \right )} \cos {\left (2 x \right )} \cos {\left (3 x \right )}}{4} + \frac {x \sin {\left (2 x \right )} \cos {\left (x \right )} \cos {\left (3 x \right )}}{4} - \frac {x \sin {\left (3 x \right )} \cos {\left (x \right )} \cos {\left (2 x \right )}}{4} - \frac {5 \sin {\left (x \right )} \sin {\left (2 x \right )} \cos {\left (3 x \right )}}{24} - \frac {\sin {\left (2 x \right )} \sin {\left (3 x \right )} \cos {\left (x \right )}}{8} - \frac {\cos {\left (x \right )} \cos {\left (2 x \right )} \cos {\left (3 x \right )}}{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.83, size = 17, normalized size = 0.68 \begin {gather*} \frac {4}{3} \, \cos \left (x\right )^{6} - \frac {5}{2} \, \cos \left (x\right )^{4} + \cos \left (x\right )^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.17, size = 14, normalized size = 0.56 \begin {gather*} -\frac {{\sin \left (x\right )}^4\,\left (8\,{\sin \left (x\right )}^2-9\right )}{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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