Integrand size = 7, antiderivative size = 35 \[ \int \cos (x) \sin (m x) \, dx=\frac {\cos ((1-m) x)}{2 (1-m)}-\frac {\cos ((1+m) x)}{2 (1+m)} \]
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Time = 0.04 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {4670, 2718} \[ \int \cos (x) \sin (m x) \, dx=\frac {\cos ((1-m) x)}{2 (1-m)}-\frac {\cos ((m+1) x)}{2 (m+1)} \]
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Rule 2718
Rule 4670
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {1}{2} \sin ((1-m) x)+\frac {1}{2} \sin ((1+m) x)\right ) \, dx \\ & = -\left (\frac {1}{2} \int \sin ((1-m) x) \, dx\right )+\frac {1}{2} \int \sin ((1+m) x) \, dx \\ & = \frac {\cos ((1-m) x)}{2 (1-m)}-\frac {\cos ((1+m) x)}{2 (1+m)} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.74 \[ \int \cos (x) \sin (m x) \, dx=\frac {m \cos (x) \cos (m x)+\sin (x) \sin (m x)}{1-m^2} \]
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Time = 0.41 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.80
method | result | size |
default | \(-\frac {\cos \left (x \left (-1+m \right )\right )}{2 \left (-1+m \right )}-\frac {\cos \left (\left (1+m \right ) x \right )}{2 \left (1+m \right )}\) | \(28\) |
risch | \(-\frac {\cos \left (x \left (-1+m \right )\right )}{2 \left (-1+m \right )}-\frac {\cos \left (\left (1+m \right ) x \right )}{2 \left (1+m \right )}\) | \(28\) |
parallelrisch | \(\frac {\left (-1-m \right ) \cos \left (x \left (-1+m \right )\right )+\left (1-m \right ) \cos \left (\left (1+m \right ) x \right )+2 m}{2 m^{2}-2}\) | \(39\) |
norman | \(\frac {\frac {2 m \tan \left (\frac {x}{2}\right )^{2}}{m^{2}-1}+\frac {2 m \tan \left (\frac {m x}{2}\right )^{2}}{m^{2}-1}-\frac {4 \tan \left (\frac {x}{2}\right ) \tan \left (\frac {m x}{2}\right )}{m^{2}-1}}{\left (1+\tan \left (\frac {x}{2}\right )^{2}\right ) \left (1+\tan \left (\frac {m x}{2}\right )^{2}\right )}\) | \(75\) |
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Time = 0.25 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.71 \[ \int \cos (x) \sin (m x) \, dx=-\frac {m \cos \left (m x\right ) \cos \left (x\right ) + \sin \left (m x\right ) \sin \left (x\right )}{m^{2} - 1} \]
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Time = 0.27 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.26 \[ \int \cos (x) \sin (m x) \, dx=\begin {cases} - \frac {\sin ^{2}{\left (x \right )}}{2} & \text {for}\: m = -1 \\\frac {\sin ^{2}{\left (x \right )}}{2} & \text {for}\: m = 1 \\- \frac {m \cos {\left (x \right )} \cos {\left (m x \right )}}{m^{2} - 1} - \frac {\sin {\left (x \right )} \sin {\left (m x \right )}}{m^{2} - 1} & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.77 \[ \int \cos (x) \sin (m x) \, dx=-\frac {\cos \left ({\left (m + 1\right )} x\right )}{2 \, {\left (m + 1\right )}} - \frac {\cos \left ({\left (m - 1\right )} x\right )}{2 \, {\left (m - 1\right )}} \]
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Time = 0.26 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.83 \[ \int \cos (x) \sin (m x) \, dx=-\frac {\cos \left (m x + x\right )}{2 \, {\left (m + 1\right )}} - \frac {\cos \left (m x - x\right )}{2 \, {\left (m - 1\right )}} \]
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Time = 27.35 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.63 \[ \int \cos (x) \sin (m x) \, dx=\left \{\begin {array}{cl} \frac {{\sin \left (x\right )}^2}{2} & \text {\ if\ \ }m=1\\ \frac {{\cos \left (x\right )}^2}{2} & \text {\ if\ \ }m=-1\\ -\frac {\cos \left (x\,\left (m-1\right )\right )}{2\,m-2}-\frac {\cos \left (x\,\left (m+1\right )\right )}{2\,m+2} & \text {\ if\ \ }m\neq -1\wedge m\neq 1 \end {array}\right . \]
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