Optimal. Leaf size=34 \[ \frac {5 x}{16}-\frac {1}{6} \sin ^5(x) \cos (x)-\frac {5}{24} \sin ^3(x) \cos (x)-\frac {5}{16} \sin (x) \cos (x) \]
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Rubi [A] time = 0.02, 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 = {2635, 8} \[ \frac {5 x}{16}-\frac {1}{6} \sin ^5(x) \cos (x)-\frac {5}{24} \sin ^3(x) \cos (x)-\frac {5}{16} \sin (x) \cos (x) \]
Antiderivative was successfully verified.
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Rule 8
Rule 2635
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 \[ \frac {5 x}{16}-\frac {15}{64} \sin (2 x)+\frac {3}{64} \sin (4 x)-\frac {1}{192} \sin (6 x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 25, normalized size = 0.74 \[ -\frac {1}{48} \, {\left (8 \, \cos \relax (x)^{5} - 26 \, \cos \relax (x)^{3} + 33 \, \cos \relax (x)\right )} \sin \relax (x) + \frac {5}{16} \, x \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.81, size = 22, normalized size = 0.65 \[ \frac {5}{16} \, x - \frac {1}{192} \, \sin \left (6 \, x\right ) + \frac {3}{64} \, \sin \left (4 \, x\right ) - \frac {15}{64} \, \sin \left (2 \, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 24, normalized size = 0.71 \[ \frac {5 x}{16}-\frac {\left (\sin ^{5}\relax (x )+\frac {5 \left (\sin ^{3}\relax (x )\right )}{4}+\frac {15 \sin \relax (x )}{8}\right ) \cos \relax (x )}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 24, normalized size = 0.71 \[ \frac {1}{48} \, \sin \left (2 \, x\right )^{3} + \frac {5}{16} \, x + \frac {3}{64} \, \sin \left (4 \, x\right ) - \frac {1}{4} \, \sin \left (2 \, x\right ) \]
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 \[ \frac {5\,x}{16}-\frac {15\,\sin \left (2\,x\right )}{64}+\frac {3\,\sin \left (4\,x\right )}{64}-\frac {\sin \left (6\,x\right )}{192} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.06, size = 36, normalized size = 1.06 \[ \frac {5 x}{16} - \frac {\sin ^{5}{\relax (x )} \cos {\relax (x )}}{6} - \frac {5 \sin ^{3}{\relax (x )} \cos {\relax (x )}}{24} - \frac {5 \sin {\relax (x )} \cos {\relax (x )}}{16} \]
Verification of antiderivative is not currently implemented for this CAS.
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