Integrand size = 4, antiderivative size = 24 \[ \int \sin ^4(x) \, dx=\frac {3 x}{8}-\frac {3}{8} \cos (x) \sin (x)-\frac {1}{4} \cos (x) \sin ^3(x) \]
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Time = 0.01 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2715, 8} \[ \int \sin ^4(x) \, dx=\frac {3 x}{8}-\frac {1}{4} \sin ^3(x) \cos (x)-\frac {3}{8} \sin (x) \cos (x) \]
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Rule 8
Rule 2715
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{4} \cos (x) \sin ^3(x)+\frac {3}{4} \int \sin ^2(x) \, dx \\ & = -\frac {3}{8} \cos (x) \sin (x)-\frac {1}{4} \cos (x) \sin ^3(x)+\frac {3 \int 1 \, dx}{8} \\ & = \frac {3 x}{8}-\frac {3}{8} \cos (x) \sin (x)-\frac {1}{4} \cos (x) \sin ^3(x) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \sin ^4(x) \, dx=\frac {3 x}{8}-\frac {1}{4} \sin (2 x)+\frac {1}{32} \sin (4 x) \]
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Time = 0.06 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.71
method | result | size |
risch | \(\frac {3 x}{8}+\frac {\sin \left (4 x \right )}{32}-\frac {\sin \left (2 x \right )}{4}\) | \(17\) |
parallelrisch | \(\frac {3 x}{8}+\frac {\sin \left (4 x \right )}{32}-\frac {\sin \left (2 x \right )}{4}\) | \(17\) |
default | \(-\frac {\left (\sin ^{3}\left (x \right )+\frac {3 \sin \left (x \right )}{2}\right ) \cos \left (x \right )}{4}+\frac {3 x}{8}\) | \(18\) |
norman | \(\frac {\frac {3 x}{8}-\frac {11 \left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{4}+\frac {11 \left (\tan ^{5}\left (\frac {x}{2}\right )\right )}{4}+\frac {3 \left (\tan ^{7}\left (\frac {x}{2}\right )\right )}{4}+\frac {3 x \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{2}+\frac {9 x \left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{4}+\frac {3 x \left (\tan ^{6}\left (\frac {x}{2}\right )\right )}{2}+\frac {3 x \left (\tan ^{8}\left (\frac {x}{2}\right )\right )}{8}-\frac {3 \tan \left (\frac {x}{2}\right )}{4}}{\left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )^{4}}\) | \(82\) |
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Time = 0.24 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.79 \[ \int \sin ^4(x) \, dx=\frac {1}{8} \, {\left (2 \, \cos \left (x\right )^{3} - 5 \, \cos \left (x\right )\right )} \sin \left (x\right ) + \frac {3}{8} \, x \]
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Time = 0.02 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \sin ^4(x) \, dx=\frac {3 x}{8} - \frac {\sin ^{3}{\left (x \right )} \cos {\left (x \right )}}{4} - \frac {3 \sin {\left (x \right )} \cos {\left (x \right )}}{8} \]
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Time = 0.19 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.67 \[ \int \sin ^4(x) \, dx=\frac {3}{8} \, x + \frac {1}{32} \, \sin \left (4 \, x\right ) - \frac {1}{4} \, \sin \left (2 \, x\right ) \]
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Time = 0.27 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.67 \[ \int \sin ^4(x) \, dx=\frac {3}{8} \, x + \frac {1}{32} \, \sin \left (4 \, x\right ) - \frac {1}{4} \, \sin \left (2 \, x\right ) \]
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Time = 0.04 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.67 \[ \int \sin ^4(x) \, dx=\frac {3\,x}{8}-\frac {\sin \left (2\,x\right )}{4}+\frac {\sin \left (4\,x\right )}{32} \]
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