Optimal. Leaf size=34 \[ \frac {5 x}{16}-\frac {5}{16} \cos (x) \sin (x)-\frac {5}{24} \cos (x) \sin ^3(x)-\frac {1}{6} \cos (x) \sin ^5(x) \]
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Rubi [A]
time = 0.01, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 2, integrand size = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2715, 8}
\begin {gather*} \frac {5 x}{16}-\frac {1}{6} \sin ^5(x) \cos (x)-\frac {5}{24} \sin ^3(x) \cos (x)-\frac {5}{16} \sin (x) \cos (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 2715
Rubi steps
\begin {align*} \int \sin ^6(x) \, dx &=-\frac {1}{6} \cos (x) \sin ^5(x)+\frac {5}{6} \int \sin ^4(x) \, dx\\ &=-\frac {5}{24} \cos (x) \sin ^3(x)-\frac {1}{6} \cos (x) \sin ^5(x)+\frac {5}{8} \int \sin ^2(x) \, dx\\ &=-\frac {5}{16} \cos (x) \sin (x)-\frac {5}{24} \cos (x) \sin ^3(x)-\frac {1}{6} \cos (x) \sin ^5(x)+\frac {5 \int 1 \, dx}{16}\\ &=\frac {5 x}{16}-\frac {5}{16} \cos (x) \sin (x)-\frac {5}{24} \cos (x) \sin ^3(x)-\frac {1}{6} \cos (x) \sin ^5(x)\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 30, normalized size = 0.88 \begin {gather*} \frac {5 x}{16}-\frac {15}{64} \sin (2 x)+\frac {3}{64} \sin (4 x)-\frac {1}{192} \sin (6 x) \end {gather*}
Antiderivative was successfully verified.
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Mathics [A]
time = 1.95, size = 32, normalized size = 0.94 \begin {gather*} \frac {5 x}{16}-\frac {5 \text {Sin}\left [2 x\right ]}{24}-\frac {\left (1-\text {Cos}\left [2 x\right ]\right )^2 \text {Sin}\left [2 x\right ]}{48}+\frac {5 \text {Sin}\left [4 x\right ]}{192} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.00, size = 24, normalized size = 0.71
method | result | size |
risch | \(\frac {5 x}{16}-\frac {\sin \left (6 x \right )}{192}+\frac {3 \sin \left (4 x \right )}{64}-\frac {15 \sin \left (2 x \right )}{64}\) | \(23\) |
default | \(-\frac {\left (\sin ^{5}\left (x \right )+\frac {5 \left (\sin ^{3}\left (x \right )\right )}{4}+\frac {15 \sin \left (x \right )}{8}\right ) \cos \left (x \right )}{6}+\frac {5 x}{16}\) | \(24\) |
norman | \(\frac {\frac {5 x}{16}-\frac {85 \left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{24}-\frac {33 \left (\tan ^{5}\left (\frac {x}{2}\right )\right )}{4}+\frac {33 \left (\tan ^{7}\left (\frac {x}{2}\right )\right )}{4}+\frac {85 \left (\tan ^{9}\left (\frac {x}{2}\right )\right )}{24}+\frac {5 \left (\tan ^{11}\left (\frac {x}{2}\right )\right )}{8}+\frac {15 x \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{8}+\frac {75 x \left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{16}+\frac {25 x \left (\tan ^{6}\left (\frac {x}{2}\right )\right )}{4}+\frac {75 x \left (\tan ^{8}\left (\frac {x}{2}\right )\right )}{16}+\frac {15 x \left (\tan ^{10}\left (\frac {x}{2}\right )\right )}{8}+\frac {5 x \left (\tan ^{12}\left (\frac {x}{2}\right )\right )}{16}-\frac {5 \tan \left (\frac {x}{2}\right )}{8}}{\left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )^{6}}\) | \(116\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 24, normalized size = 0.71 \begin {gather*} \frac {1}{48} \, \sin \left (2 \, x\right )^{3} + \frac {5}{16} \, x + \frac {3}{64} \, \sin \left (4 \, x\right ) - \frac {1}{4} \, \sin \left (2 \, x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.32, size = 25, normalized size = 0.74 \begin {gather*} -\frac {1}{48} \, {\left (8 \, \cos \left (x\right )^{5} - 26 \, \cos \left (x\right )^{3} + 33 \, \cos \left (x\right )\right )} \sin \left (x\right ) + \frac {5}{16} \, x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.03, size = 36, normalized size = 1.06 \begin {gather*} \frac {5 x}{16} - \frac {\sin ^{5}{\left (x \right )} \cos {\left (x \right )}}{6} - \frac {5 \sin ^{3}{\left (x \right )} \cos {\left (x \right )}}{24} - \frac {5 \sin {\left (x \right )} \cos {\left (x \right )}}{16} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 32, normalized size = 0.94 \begin {gather*} \frac {5}{16} x-\frac {15}{64} \sin \left (2 x\right )+\frac {3}{64} \sin \left (4 x\right )-\frac {\sin \left (6 x\right )}{192} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 22, normalized size = 0.65 \begin {gather*} \frac {5\,x}{16}-\frac {15\,\sin \left (2\,x\right )}{64}+\frac {3\,\sin \left (4\,x\right )}{64}-\frac {\sin \left (6\,x\right )}{192} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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