Integrand size = 4, antiderivative size = 44 \[ \int \sin ^8(x) \, dx=\frac {35 x}{128}-\frac {35}{128} \cos (x) \sin (x)-\frac {35}{192} \cos (x) \sin ^3(x)-\frac {7}{48} \cos (x) \sin ^5(x)-\frac {1}{8} \cos (x) \sin ^7(x) \]
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Time = 0.02 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2715, 8} \[ \int \sin ^8(x) \, dx=\frac {35 x}{128}-\frac {1}{8} \sin ^7(x) \cos (x)-\frac {7}{48} \sin ^5(x) \cos (x)-\frac {35}{192} \sin ^3(x) \cos (x)-\frac {35}{128} \sin (x) \cos (x) \]
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Rule 8
Rule 2715
Rubi steps \begin{align*} \text {integral}& = -\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*}
Time = 0.01 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.86 \[ \int \sin ^8(x) \, dx=\frac {35 x}{128}-\frac {7}{32} \sin (2 x)+\frac {7}{128} \sin (4 x)-\frac {1}{96} \sin (6 x)+\frac {\sin (8 x)}{1024} \]
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Time = 0.45 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.66
method | result | size |
risch | \(\frac {35 x}{128}+\frac {\sin \left (8 x \right )}{1024}-\frac {\sin \left (6 x \right )}{96}+\frac {7 \sin \left (4 x \right )}{128}-\frac {7 \sin \left (2 x \right )}{32}\) | \(29\) |
parallelrisch | \(\frac {35 x}{128}+\frac {\sin \left (8 x \right )}{1024}-\frac {\sin \left (6 x \right )}{96}+\frac {7 \sin \left (4 x \right )}{128}-\frac {7 \sin \left (2 x \right )}{32}\) | \(29\) |
default | \(-\frac {\left (\sin ^{7}\left (x \right )+\frac {7 \left (\sin ^{5}\left (x \right )\right )}{6}+\frac {35 \left (\sin ^{3}\left (x \right )\right )}{24}+\frac {35 \sin \left (x \right )}{16}\right ) \cos \left (x \right )}{8}+\frac {35 x}{128}\) | \(30\) |
norman | \(\frac {\frac {35 x}{128}-\frac {805 \left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{192}-\frac {2681 \left (\tan ^{5}\left (\frac {x}{2}\right )\right )}{192}-\frac {5053 \left (\tan ^{7}\left (\frac {x}{2}\right )\right )}{192}+\frac {5053 \left (\tan ^{9}\left (\frac {x}{2}\right )\right )}{192}+\frac {2681 \left (\tan ^{11}\left (\frac {x}{2}\right )\right )}{192}+\frac {805 \left (\tan ^{13}\left (\frac {x}{2}\right )\right )}{192}+\frac {35 \left (\tan ^{15}\left (\frac {x}{2}\right )\right )}{64}+\frac {35 x \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{16}+\frac {245 x \left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{32}+\frac {245 x \left (\tan ^{6}\left (\frac {x}{2}\right )\right )}{16}+\frac {1225 x \left (\tan ^{8}\left (\frac {x}{2}\right )\right )}{64}+\frac {245 x \left (\tan ^{10}\left (\frac {x}{2}\right )\right )}{16}+\frac {245 x \left (\tan ^{12}\left (\frac {x}{2}\right )\right )}{32}+\frac {35 x \left (\tan ^{14}\left (\frac {x}{2}\right )\right )}{16}+\frac {35 x \left (\tan ^{16}\left (\frac {x}{2}\right )\right )}{128}-\frac {35 \tan \left (\frac {x}{2}\right )}{64}}{\left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )^{8}}\) | \(150\) |
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Time = 0.25 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.70 \[ \int \sin ^8(x) \, dx=\frac {1}{384} \, {\left (48 \, \cos \left (x\right )^{7} - 200 \, \cos \left (x\right )^{5} + 326 \, \cos \left (x\right )^{3} - 279 \, \cos \left (x\right )\right )} \sin \left (x\right ) + \frac {35}{128} \, x \]
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Time = 0.02 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.09 \[ \int \sin ^8(x) \, dx=\frac {35 x}{128} - \frac {\sin ^{7}{\left (x \right )} \cos {\left (x \right )}}{8} - \frac {7 \sin ^{5}{\left (x \right )} \cos {\left (x \right )}}{48} - \frac {35 \sin ^{3}{\left (x \right )} \cos {\left (x \right )}}{192} - \frac {35 \sin {\left (x \right )} \cos {\left (x \right )}}{128} \]
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Time = 0.21 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.68 \[ \int \sin ^8(x) \, dx=\frac {1}{24} \, \sin \left (2 \, x\right )^{3} + \frac {35}{128} \, x + \frac {1}{1024} \, \sin \left (8 \, x\right ) + \frac {7}{128} \, \sin \left (4 \, x\right ) - \frac {1}{4} \, \sin \left (2 \, x\right ) \]
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Time = 0.28 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.64 \[ \int \sin ^8(x) \, dx=\frac {35}{128} \, x + \frac {1}{1024} \, \sin \left (8 \, x\right ) - \frac {1}{96} \, \sin \left (6 \, x\right ) + \frac {7}{128} \, \sin \left (4 \, x\right ) - \frac {7}{32} \, \sin \left (2 \, x\right ) \]
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Time = 0.03 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.64 \[ \int \sin ^8(x) \, dx=\frac {35\,x}{128}-\frac {7\,\sin \left (2\,x\right )}{32}+\frac {7\,\sin \left (4\,x\right )}{128}-\frac {\sin \left (6\,x\right )}{96}+\frac {\sin \left (8\,x\right )}{1024} \]
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