Optimal. Leaf size=21 \[ \frac {2}{3} \sin \left (\frac {3 x}{4}\right )-\frac {2}{5} \sin \left (\frac {5 x}{4}\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 21, 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 {2}{3} \sin \left (\frac {3 x}{4}\right )-\frac {2}{5} \sin \left (\frac {5 x}{4}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 4367
Rubi steps
\begin {align*} \int \sin \left (\frac {x}{4}\right ) \sin (x) \, dx &=\frac {2}{3} \sin \left (\frac {3 x}{4}\right )-\frac {2}{5} \sin \left (\frac {5 x}{4}\right )\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 21, normalized size = 1.00 \begin {gather*} \frac {2}{3} \sin \left (\frac {3 x}{4}\right )-\frac {2}{5} \sin \left (\frac {5 x}{4}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 14, normalized size = 0.67
method | result | size |
default | \(\frac {2 \sin \left (\frac {3 x}{4}\right )}{3}-\frac {2 \sin \left (\frac {5 x}{4}\right )}{5}\) | \(14\) |
risch | \(\frac {2 \sin \left (\frac {3 x}{4}\right )}{3}-\frac {2 \sin \left (\frac {5 x}{4}\right )}{5}\) | \(14\) |
norman | \(\frac {-\frac {8 \tan \left (\frac {x}{2}\right ) \left (\tan ^{2}\left (\frac {x}{8}\right )\right )}{15}+\frac {32 \left (\tan ^{2}\left (\frac {x}{2}\right )\right ) \tan \left (\frac {x}{8}\right )}{15}+\frac {8 \tan \left (\frac {x}{2}\right )}{15}-\frac {32 \tan \left (\frac {x}{8}\right )}{15}}{\left (1+\tan ^{2}\left (\frac {x}{8}\right )\right ) \left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )}\) | \(59\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 4.38, size = 13, normalized size = 0.62 \begin {gather*} -\frac {2}{5} \, \sin \left (\frac {5}{4} \, x\right ) + \frac {2}{3} \, \sin \left (\frac {3}{4} \, x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.67, size = 24, normalized size = 1.14 \begin {gather*} -\frac {16}{15} \, {\left (6 \, \cos \left (\frac {1}{4} \, x\right )^{4} - 7 \, \cos \left (\frac {1}{4} \, x\right )^{2} + 1\right )} \sin \left (\frac {1}{4} \, x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.13, size = 22, normalized size = 1.05 \begin {gather*} - \frac {16 \sin {\left (\frac {x}{4} \right )} \cos {\left (x \right )}}{15} + \frac {4 \sin {\left (x \right )} \cos {\left (\frac {x}{4} \right )}}{15} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.33, size = 17, normalized size = 0.81 \begin {gather*} -\frac {32}{5} \, \sin \left (\frac {1}{4} \, x\right )^{5} + \frac {16}{3} \, \sin \left (\frac {1}{4} \, x\right )^{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.17, size = 13, normalized size = 0.62 \begin {gather*} \frac {2\,\sin \left (\frac {3\,x}{4}\right )}{3}-\frac {2\,\sin \left (\frac {5\,x}{4}\right )}{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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