Integrand size = 12, antiderivative size = 36 \[ \int (3+3 \sin (e+f x))^2 \, dx=\frac {27 x}{2}-\frac {18 \cos (e+f x)}{f}-\frac {9 \cos (e+f x) \sin (e+f x)}{2 f} \]
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Time = 0.01 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.25, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {2723} \[ \int (3+3 \sin (e+f x))^2 \, dx=-\frac {2 a^2 \cos (e+f x)}{f}-\frac {a^2 \sin (e+f x) \cos (e+f x)}{2 f}+\frac {3 a^2 x}{2} \]
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Rule 2723
Rubi steps \begin{align*} \text {integral}& = \frac {3 a^2 x}{2}-\frac {2 a^2 \cos (e+f x)}{f}-\frac {a^2 \cos (e+f x) \sin (e+f x)}{2 f} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.89 \[ \int (3+3 \sin (e+f x))^2 \, dx=-\frac {9 (-2 (e+3 f x)+8 \cos (e+f x)+\sin (2 (e+f x)))}{4 f} \]
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Time = 0.50 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.89
method | result | size |
parallelrisch | \(-\frac {a^{2} \left (-6 f x +8 \cos \left (f x +e \right )+\sin \left (2 f x +2 e \right )-8\right )}{4 f}\) | \(32\) |
risch | \(\frac {3 a^{2} x}{2}-\frac {2 a^{2} \cos \left (f x +e \right )}{f}-\frac {a^{2} \sin \left (2 f x +2 e \right )}{4 f}\) | \(39\) |
parts | \(a^{2} x +\frac {a^{2} \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )}{f}-\frac {2 a^{2} \cos \left (f x +e \right )}{f}\) | \(50\) |
derivativedivides | \(\frac {a^{2} \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-2 \cos \left (f x +e \right ) a^{2}+\left (f x +e \right ) a^{2}}{f}\) | \(52\) |
default | \(\frac {a^{2} \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-2 \cos \left (f x +e \right ) a^{2}+\left (f x +e \right ) a^{2}}{f}\) | \(52\) |
norman | \(\frac {\frac {a^{2} \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}+\frac {4 a^{2} \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}+\frac {3 a^{2} x}{2}-\frac {a^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{f}+3 a^{2} x \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\frac {3 a^{2} x \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2}+\frac {4 a^{2} \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{2}}\) | \(131\) |
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Time = 0.28 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.14 \[ \int (3+3 \sin (e+f x))^2 \, dx=\frac {3 \, a^{2} f x - a^{2} \cos \left (f x + e\right ) \sin \left (f x + e\right ) - 4 \, a^{2} \cos \left (f x + e\right )}{2 \, f} \]
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Time = 0.10 (sec) , antiderivative size = 78, normalized size of antiderivative = 2.17 \[ \int (3+3 \sin (e+f x))^2 \, dx=\begin {cases} \frac {a^{2} x \sin ^{2}{\left (e + f x \right )}}{2} + \frac {a^{2} x \cos ^{2}{\left (e + f x \right )}}{2} + a^{2} x - \frac {a^{2} \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{2 f} - \frac {2 a^{2} \cos {\left (e + f x \right )}}{f} & \text {for}\: f \neq 0 \\x \left (a \sin {\left (e \right )} + a\right )^{2} & \text {otherwise} \end {cases} \]
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Timed out. \[ \int (3+3 \sin (e+f x))^2 \, dx=\text {Timed out} \]
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Time = 0.44 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.06 \[ \int (3+3 \sin (e+f x))^2 \, dx=\frac {3}{2} \, a^{2} x - \frac {2 \, a^{2} \cos \left (f x + e\right )}{f} - \frac {a^{2} \sin \left (2 \, f x + 2 \, e\right )}{4 \, f} \]
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Time = 6.91 (sec) , antiderivative size = 123, normalized size of antiderivative = 3.42 \[ \int (3+3 \sin (e+f x))^2 \, dx=\frac {3\,a^2\,x}{2}-\frac {a^2\,\left (\frac {3\,e}{2}+\frac {3\,f\,x}{2}\right )-a^2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3-a^2\,\left (\frac {3\,e}{2}+\frac {3\,f\,x}{2}-4\right )+a^2\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (2\,a^2\,\left (\frac {3\,e}{2}+\frac {3\,f\,x}{2}\right )-a^2\,\left (3\,e+3\,f\,x-4\right )\right )}{f\,{\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+1\right )}^2} \]
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