Integrand size = 22, antiderivative size = 29 \[ \int (2 a+2 b \cos (d+e x)-2 a \sin (d+e x)) \, dx=2 a x+\frac {2 a \cos (d+e x)}{e}+\frac {2 b \sin (d+e x)}{e} \]
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Time = 0.02 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2717, 2718} \[ \int (2 a+2 b \cos (d+e x)-2 a \sin (d+e x)) \, dx=\frac {2 a \cos (d+e x)}{e}+2 a x+\frac {2 b \sin (d+e x)}{e} \]
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Rule 2717
Rule 2718
Rubi steps \begin{align*} \text {integral}& = 2 a x-(2 a) \int \sin (d+e x) \, dx+(2 b) \int \cos (d+e x) \, dx \\ & = 2 a x+\frac {2 a \cos (d+e x)}{e}+\frac {2 b \sin (d+e x)}{e} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.83 \[ \int (2 a+2 b \cos (d+e x)-2 a \sin (d+e x)) \, dx=2 a x+\frac {2 a \cos (d) \cos (e x)}{e}+\frac {2 b \cos (e x) \sin (d)}{e}+\frac {2 b \cos (d) \sin (e x)}{e}-\frac {2 a \sin (d) \sin (e x)}{e} \]
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Time = 0.43 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.03
method | result | size |
derivativedivides | \(\frac {2 \left (e x +d \right ) a +2 b \sin \left (e x +d \right )+2 a \cos \left (e x +d \right )}{e}\) | \(30\) |
default | \(2 a x +\frac {2 a \cos \left (e x +d \right )}{e}+\frac {2 b \sin \left (e x +d \right )}{e}\) | \(30\) |
risch | \(2 a x +\frac {2 a \cos \left (e x +d \right )}{e}+\frac {2 b \sin \left (e x +d \right )}{e}\) | \(30\) |
parts | \(2 a x +\frac {2 a \cos \left (e x +d \right )}{e}+\frac {2 b \sin \left (e x +d \right )}{e}\) | \(30\) |
parallelrisch | \(\frac {2 b \sin \left (e x +d \right )+2 a \cos \left (e x +d \right )-2 a}{e}+2 a x\) | \(32\) |
norman | \(\frac {2 a x +2 a x \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}+\frac {4 b \tan \left (\frac {e x}{2}+\frac {d}{2}\right )}{e}-\frac {4 a \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}}{e}}{1+\tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}}\) | \(69\) |
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Time = 0.24 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.90 \[ \int (2 a+2 b \cos (d+e x)-2 a \sin (d+e x)) \, dx=\frac {2 \, {\left (a e x + a \cos \left (e x + d\right ) + b \sin \left (e x + d\right )\right )}}{e} \]
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Time = 0.07 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.34 \[ \int (2 a+2 b \cos (d+e x)-2 a \sin (d+e x)) \, dx=2 a x - 2 a \left (\begin {cases} - \frac {\cos {\left (d + e x \right )}}{e} & \text {for}\: e \neq 0 \\x \sin {\left (d \right )} & \text {otherwise} \end {cases}\right ) + 2 b \left (\begin {cases} \frac {\sin {\left (d + e x \right )}}{e} & \text {for}\: e \neq 0 \\x \cos {\left (d \right )} & \text {otherwise} \end {cases}\right ) \]
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Time = 0.24 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int (2 a+2 b \cos (d+e x)-2 a \sin (d+e x)) \, dx=2 \, a x + \frac {2 \, a \cos \left (e x + d\right )}{e} + \frac {2 \, b \sin \left (e x + d\right )}{e} \]
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Time = 0.27 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int (2 a+2 b \cos (d+e x)-2 a \sin (d+e x)) \, dx=2 \, a x + \frac {2 \, a \cos \left (e x + d\right )}{e} + \frac {2 \, b \sin \left (e x + d\right )}{e} \]
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Time = 26.38 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int (2 a+2 b \cos (d+e x)-2 a \sin (d+e x)) \, dx=2\,a\,x+\frac {2\,a\,\cos \left (d+e\,x\right )}{e}+\frac {2\,b\,\sin \left (d+e\,x\right )}{e} \]
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