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 = 0.0167524, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 4, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, 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 2635
Rule 8
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*}
Mathematica [A] time = 0.0021813, 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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Maple [A] time = 0., size = 24, normalized size = 0.7 \begin{align*} -{\frac{\cos \left ( x \right ) }{6} \left ( \left ( \sin \left ( x \right ) \right ) ^{5}+{\frac{5\, \left ( \sin \left ( x \right ) \right ) ^{3}}{4}}+{\frac{15\,\sin \left ( x \right ) }{8}} \right ) }+{\frac{5\,x}{16}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.924352, size = 32, normalized size = 0.94 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.99774, size = 84, normalized size = 2.47 \begin{align*} -\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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.061398, size = 36, normalized size = 1.06 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.05327, size = 30, normalized size = 0.88 \begin{align*} \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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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