Integrand size = 14, antiderivative size = 25 \[ \int \sqrt {3+3 \sin (e+f x)} \, dx=-\frac {6 \cos (e+f x)}{f \sqrt {3+3 \sin (e+f x)}} \]
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Time = 0.01 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.04, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {2725} \[ \int \sqrt {3+3 \sin (e+f x)} \, dx=-\frac {2 a \cos (e+f x)}{f \sqrt {a \sin (e+f x)+a}} \]
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Rule 2725
Rubi steps \begin{align*} \text {integral}& = -\frac {2 a \cos (e+f x)}{f \sqrt {a+a \sin (e+f x)}} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(68\) vs. \(2(25)=50\).
Time = 0.02 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.72 \[ \int \sqrt {3+3 \sin (e+f x)} \, dx=\frac {2 \sqrt {3} \left (-\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {1+\sin (e+f x)}}{f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )} \]
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Time = 0.70 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.72
method | result | size |
default | \(\frac {2 \left (\sin \left (f x +e \right )+1\right ) \left (\sin \left (f x +e \right )-1\right ) a}{\cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}\) | \(43\) |
risch | \(-\frac {i \sqrt {2}\, \sqrt {-a \left (-2-2 \sin \left (f x +e \right )\right )}\, \left ({\mathrm e}^{i \left (f x +e \right )}-i\right ) \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )}{\left ({\mathrm e}^{2 i \left (f x +e \right )}+2 i {\mathrm e}^{i \left (f x +e \right )}-1\right ) f}\) | \(74\) |
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Leaf count of result is larger than twice the leaf count of optimal. 50 vs. \(2 (24) = 48\).
Time = 0.26 (sec) , antiderivative size = 50, normalized size of antiderivative = 2.00 \[ \int \sqrt {3+3 \sin (e+f x)} \, dx=-\frac {2 \, \sqrt {a \sin \left (f x + e\right ) + a} {\left (\cos \left (f x + e\right ) - \sin \left (f x + e\right ) + 1\right )}}{f \cos \left (f x + e\right ) + f \sin \left (f x + e\right ) + f} \]
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\[ \int \sqrt {3+3 \sin (e+f x)} \, dx=\int \sqrt {a \sin {\left (e + f x \right )} + a}\, dx \]
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\[ \int \sqrt {3+3 \sin (e+f x)} \, dx=\int { \sqrt {a \sin \left (f x + e\right ) + a} \,d x } \]
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none
Time = 0.51 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.44 \[ \int \sqrt {3+3 \sin (e+f x)} \, dx=\frac {2 \, \sqrt {2} \sqrt {a} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{f} \]
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Time = 0.23 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.32 \[ \int \sqrt {3+3 \sin (e+f x)} \, dx=-\frac {2\,\cos \left (e+f\,x\right )\,\sqrt {a\,\left (\sin \left (e+f\,x\right )+1\right )}}{f\,\left (\sin \left (e+f\,x\right )+1\right )} \]
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