Optimal. Leaf size=17 \[ \frac {1}{6} \sin (3 x)-\frac {1}{18} \sin (9 x) \]
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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 = {4367}
\begin {gather*} \frac {1}{6} \sin (3 x)-\frac {1}{18} \sin (9 x) \end {gather*}
Antiderivative was successfully verified.
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Rule 4367
Rubi steps
\begin {align*} \int \sin (3 x) \sin (6 x) \, dx &=\frac {1}{6} \sin (3 x)-\frac {1}{18} \sin (9 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}{18} \sin (9 x) \end {gather*}
Antiderivative was successfully verified.
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Mathics [A]
time = 1.93, size = 8, normalized size = 0.47 \begin {gather*} \frac {2 \text {Sin}\left [3 x\right ]^3}{9} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 9, normalized size = 0.53
method | result | size |
derivativedivides | \(\frac {2 \left (\sin ^{3}\left (3 x \right )\right )}{9}\) | \(9\) |
default | \(\frac {2 \left (\sin ^{3}\left (3 x \right )\right )}{9}\) | \(9\) |
risch | \(\frac {\sin \left (3 x \right )}{6}-\frac {\sin \left (9 x \right )}{18}\) | \(14\) |
norman | \(\frac {-\frac {2 \tan \left (3 x \right ) \left (\tan ^{2}\left (\frac {3 x}{2}\right )\right )}{9}+\frac {4 \left (\tan ^{2}\left (3 x \right )\right ) \tan \left (\frac {3 x}{2}\right )}{9}+\frac {2 \tan \left (3 x \right )}{9}-\frac {4 \tan \left (\frac {3 x}{2}\right )}{9}}{\left (1+\tan ^{2}\left (\frac {3 x}{2}\right )\right ) \left (1+\tan ^{2}\left (3 x \right )\right )}\) | \(59\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 13, normalized size = 0.76 \begin {gather*} -\frac {1}{18} \, \sin \left (9 \, 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 = 0.34, size = 14, normalized size = 0.82 \begin {gather*} -\frac {2}{9} \, {\left (\cos \left (3 \, x\right )^{2} - 1\right )} \sin \left (3 \, x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.14, size = 24, normalized size = 1.41 \begin {gather*} - \frac {2 \sin {\left (3 x \right )} \cos {\left (6 x \right )}}{9} + \frac {\sin {\left (6 x \right )} \cos {\left (3 x \right )}}{9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 12, normalized size = 0.71 \begin {gather*} \frac {2 \sin ^{3}\left (3 x\right )}{3\cdot 3} \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 (3\,x\right )}{6}-\frac {\sin \left (9\,x\right )}{18} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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