Optimal. Leaf size=44 \[ \frac{35 x}{128}-\frac{1}{8} \sin ^7(x) \cos (x)-\frac{7}{48} \sin ^5(x) \cos (x)-\frac{35}{192} \sin ^3(x) \cos (x)-\frac{35}{128} \sin (x) \cos (x) \]
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Rubi [A] time = 0.0214529, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 4, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {2635, 8} \[ \frac{35 x}{128}-\frac{1}{8} \sin ^7(x) \cos (x)-\frac{7}{48} \sin ^5(x) \cos (x)-\frac{35}{192} \sin ^3(x) \cos (x)-\frac{35}{128} \sin (x) \cos (x) \]
Antiderivative was successfully verified.
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Rule 2635
Rule 8
Rubi steps
\begin{align*} \int \sin ^8(x) \, dx &=-\frac{1}{8} \cos (x) \sin ^7(x)+\frac{7}{8} \int \sin ^6(x) \, dx\\ &=-\frac{7}{48} \cos (x) \sin ^5(x)-\frac{1}{8} \cos (x) \sin ^7(x)+\frac{35}{48} \int \sin ^4(x) \, dx\\ &=-\frac{35}{192} \cos (x) \sin ^3(x)-\frac{7}{48} \cos (x) \sin ^5(x)-\frac{1}{8} \cos (x) \sin ^7(x)+\frac{35}{64} \int \sin ^2(x) \, dx\\ &=-\frac{35}{128} \cos (x) \sin (x)-\frac{35}{192} \cos (x) \sin ^3(x)-\frac{7}{48} \cos (x) \sin ^5(x)-\frac{1}{8} \cos (x) \sin ^7(x)+\frac{35 \int 1 \, dx}{128}\\ &=\frac{35 x}{128}-\frac{35}{128} \cos (x) \sin (x)-\frac{35}{192} \cos (x) \sin ^3(x)-\frac{7}{48} \cos (x) \sin ^5(x)-\frac{1}{8} \cos (x) \sin ^7(x)\\ \end{align*}
Mathematica [A] time = 0.0025816, size = 38, normalized size = 0.86 \[ \frac{35 x}{128}-\frac{7}{32} \sin (2 x)+\frac{7}{128} \sin (4 x)-\frac{1}{96} \sin (6 x)+\frac{\sin (8 x)}{1024} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.034, size = 30, normalized size = 0.7 \begin{align*} -{\frac{\cos \left ( x \right ) }{8} \left ( \left ( \sin \left ( x \right ) \right ) ^{7}+{\frac{7\, \left ( \sin \left ( x \right ) \right ) ^{5}}{6}}+{\frac{35\, \left ( \sin \left ( x \right ) \right ) ^{3}}{24}}+{\frac{35\,\sin \left ( x \right ) }{16}} \right ) }+{\frac{35\,x}{128}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.940387, size = 41, normalized size = 0.93 \begin{align*} \frac{1}{24} \, \sin \left (2 \, x\right )^{3} + \frac{35}{128} \, x + \frac{1}{1024} \, \sin \left (8 \, x\right ) + \frac{7}{128} \, \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 = 1.69731, size = 111, normalized size = 2.52 \begin{align*} \frac{1}{384} \,{\left (48 \, \cos \left (x\right )^{7} - 200 \, \cos \left (x\right )^{5} + 326 \, \cos \left (x\right )^{3} - 279 \, \cos \left (x\right )\right )} \sin \left (x\right ) + \frac{35}{128} \, x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.059871, size = 48, normalized size = 1.09 \begin{align*} \frac{35 x}{128} - \frac{\sin ^{7}{\left (x \right )} \cos{\left (x \right )}}{8} - \frac{7 \sin ^{5}{\left (x \right )} \cos{\left (x \right )}}{48} - \frac{35 \sin ^{3}{\left (x \right )} \cos{\left (x \right )}}{192} - \frac{35 \sin{\left (x \right )} \cos{\left (x \right )}}{128} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.07199, size = 38, normalized size = 0.86 \begin{align*} \frac{35}{128} \, x + \frac{1}{1024} \, \sin \left (8 \, x\right ) - \frac{1}{96} \, \sin \left (6 \, x\right ) + \frac{7}{128} \, \sin \left (4 \, x\right ) - \frac{7}{32} \, \sin \left (2 \, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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