Integrand size = 21, antiderivative size = 45 \[ \int (3+3 \sin (e+f x)) (c+d \sin (e+f x)) \, dx=\frac {3}{2} (2 c+d) x-\frac {3 (c+d) \cos (e+f x)}{f}-\frac {3 d \cos (e+f x) \sin (e+f x)}{2 f} \]
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Time = 0.02 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.07, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {2813} \[ \int (3+3 \sin (e+f x)) (c+d \sin (e+f x)) \, dx=-\frac {a (c+d) \cos (e+f x)}{f}+\frac {1}{2} a x (2 c+d)-\frac {a d \sin (e+f x) \cos (e+f x)}{2 f} \]
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Rule 2813
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} a (2 c+d) x-\frac {a (c+d) \cos (e+f x)}{f}-\frac {a d \cos (e+f x) \sin (e+f x)}{2 f} \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.98 \[ \int (3+3 \sin (e+f x)) (c+d \sin (e+f x)) \, dx=-\frac {3 (-2 (d e+2 c f x+d f x)+4 (c+d) \cos (e+f x)+d \sin (2 (e+f x)))}{4 f} \]
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Time = 0.54 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.98
method | result | size |
parallelrisch | \(\frac {\left (-\frac {d \sin \left (2 f x +2 e \right )}{4}+\left (-c -d \right ) \cos \left (f x +e \right )+f x c +\frac {d x f}{2}+c +d \right ) a}{f}\) | \(44\) |
parts | \(a c x -\frac {\left (a c +d a \right ) \cos \left (f x +e \right )}{f}+\frac {d a \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )}{f}\) | \(52\) |
risch | \(a c x +\frac {a x d}{2}-\frac {a \cos \left (f x +e \right ) c}{f}-\frac {a \cos \left (f x +e \right ) d}{f}-\frac {d a \sin \left (2 f x +2 e \right )}{4 f}\) | \(53\) |
derivativedivides | \(\frac {d a \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-a c \cos \left (f x +e \right )-d a \cos \left (f x +e \right )+a c \left (f x +e \right )}{f}\) | \(59\) |
default | \(\frac {d a \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-a c \cos \left (f x +e \right )-d a \cos \left (f x +e \right )+a c \left (f x +e \right )}{f}\) | \(59\) |
norman | \(\frac {\left (a c +\frac {1}{2} d a \right ) x +\left (a c +\frac {1}{2} d a \right ) x \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (2 a c +d a \right ) x \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\frac {\left (2 a c +2 d a \right ) \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}+\frac {d a \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}+\frac {2 \left (a c +d a \right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}-\frac {d a \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{f}}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{2}}\) | \(150\) |
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Time = 0.28 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.07 \[ \int (3+3 \sin (e+f x)) (c+d \sin (e+f x)) \, dx=-\frac {a d \cos \left (f x + e\right ) \sin \left (f x + e\right ) - {\left (2 \, a c + a d\right )} f x + 2 \, {\left (a c + a d\right )} \cos \left (f x + e\right )}{2 \, f} \]
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Leaf count of result is larger than twice the leaf count of optimal. 94 vs. \(2 (42) = 84\).
Time = 0.10 (sec) , antiderivative size = 94, normalized size of antiderivative = 2.09 \[ \int (3+3 \sin (e+f x)) (c+d \sin (e+f x)) \, dx=\begin {cases} a c x - \frac {a c \cos {\left (e + f x \right )}}{f} + \frac {a d x \sin ^{2}{\left (e + f x \right )}}{2} + \frac {a d x \cos ^{2}{\left (e + f x \right )}}{2} - \frac {a d \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{2 f} - \frac {a d \cos {\left (e + f x \right )}}{f} & \text {for}\: f \neq 0 \\x \left (c + d \sin {\left (e \right )}\right ) \left (a \sin {\left (e \right )} + a\right ) & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.27 \[ \int (3+3 \sin (e+f x)) (c+d \sin (e+f x)) \, dx=\frac {4 \, {\left (f x + e\right )} a c + {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} a d - 4 \, a c \cos \left (f x + e\right ) - 4 \, a d \cos \left (f x + e\right )}{4 \, f} \]
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Time = 0.47 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.16 \[ \int (3+3 \sin (e+f x)) (c+d \sin (e+f x)) \, dx=a c x + \frac {1}{2} \, a d x - \frac {a c \cos \left (f x + e\right )}{f} - \frac {a d \cos \left (f x + e\right )}{f} - \frac {a d \sin \left (2 \, f x + 2 \, e\right )}{4 \, f} \]
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Time = 7.10 (sec) , antiderivative size = 100, normalized size of antiderivative = 2.22 \[ \int (3+3 \sin (e+f x)) (c+d \sin (e+f x)) \, dx=a\,c\,x+\frac {a\,d\,x}{2}-\frac {-a\,d\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3+\left (2\,a\,c+2\,a\,d\right )\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+a\,d\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )+2\,a\,c+2\,a\,d}{f\,\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+1\right )} \]
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