Integrand size = 14, antiderivative size = 41 \[ \int \frac {1}{\sqrt {3+3 \sin (e+f x)}} \, dx=-\frac {\sqrt {\frac {2}{3}} \text {arctanh}\left (\frac {\sqrt {\frac {3}{2}} \cos (e+f x)}{\sqrt {3+3 \sin (e+f x)}}\right )}{f} \]
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Time = 0.02 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.15, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2728, 212} \[ \int \frac {1}{\sqrt {3+3 \sin (e+f x)}} \, dx=-\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a \sin (e+f x)+a}}\right )}{\sqrt {a} f} \]
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Rule 212
Rule 2728
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \text {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,\frac {a \cos (e+f x)}{\sqrt {a+a \sin (e+f x)}}\right )}{f} \\ & = -\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a+a \sin (e+f x)}}\right )}{\sqrt {a} f} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.12 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.85 \[ \int \frac {1}{\sqrt {3+3 \sin (e+f x)}} \, dx=\frac {(2+2 i) (-1)^{3/4} \text {arctanh}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) (-1)^{3/4} \left (-1+\tan \left (\frac {1}{4} (e+f x)\right )\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )}{\sqrt {3} f \sqrt {1+\sin (e+f x)}} \]
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Time = 0.80 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.83
method | result | size |
default | \(-\frac {\left (\sin \left (f x +e \right )+1\right ) \sqrt {-a \left (\sin \left (f x +e \right )-1\right )}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-a \left (\sin \left (f x +e \right )-1\right )}\, \sqrt {2}}{2 \sqrt {a}}\right )}{\sqrt {a}\, \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}\) | \(75\) |
risch | \(\frac {2 i \left ({\mathrm e}^{i \left (f x +e \right )}+i\right ) \sqrt {2}\, {\mathrm e}^{-i \left (f x +e \right )}}{f \sqrt {-a \left (i {\mathrm e}^{2 i \left (f x +e \right )}-i-2 \,{\mathrm e}^{i \left (f x +e \right )}\right ) {\mathrm e}^{-i \left (f x +e \right )}}}-\frac {2 i \left ({\mathrm e}^{i \left (f x +e \right )}+i\right ) \left (\arctan \left (\frac {\sqrt {-i a \,{\mathrm e}^{i \left (f x +e \right )}}}{\sqrt {a}}\right ) a \sqrt {-i a \,{\mathrm e}^{i \left (f x +e \right )}}+a^{\frac {3}{2}}\right ) \sqrt {2}\, {\mathrm e}^{-i \left (f x +e \right )}}{f \,a^{\frac {3}{2}} \sqrt {-a \left (i {\mathrm e}^{2 i \left (f x +e \right )}-i-2 \,{\mathrm e}^{i \left (f x +e \right )}\right ) {\mathrm e}^{-i \left (f x +e \right )}}}\) | \(198\) |
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none
Time = 0.27 (sec) , antiderivative size = 167, normalized size of antiderivative = 4.07 \[ \int \frac {1}{\sqrt {3+3 \sin (e+f x)}} \, dx=\left [\frac {\sqrt {2} \log \left (-\frac {\cos \left (f x + e\right )^{2} - {\left (\cos \left (f x + e\right ) - 2\right )} \sin \left (f x + e\right ) - \frac {2 \, \sqrt {2} \sqrt {a \sin \left (f x + e\right ) + a} {\left (\cos \left (f x + e\right ) - \sin \left (f x + e\right ) + 1\right )}}{\sqrt {a}} + 3 \, \cos \left (f x + e\right ) + 2}{\cos \left (f x + e\right )^{2} - {\left (\cos \left (f x + e\right ) + 2\right )} \sin \left (f x + e\right ) - \cos \left (f x + e\right ) - 2}\right )}{2 \, \sqrt {a} f}, \frac {\sqrt {2} \sqrt {-\frac {1}{a}} \arctan \left (\frac {\sqrt {2} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-\frac {1}{a}}}{\cos \left (f x + e\right )}\right )}{f}\right ] \]
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\[ \int \frac {1}{\sqrt {3+3 \sin (e+f x)}} \, dx=\int \frac {1}{\sqrt {a \sin {\left (e + f x \right )} + a}}\, dx \]
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\[ \int \frac {1}{\sqrt {3+3 \sin (e+f x)}} \, dx=\int { \frac {1}{\sqrt {a \sin \left (f x + e\right ) + a}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 111 vs. \(2 (38) = 76\).
Time = 0.42 (sec) , antiderivative size = 111, normalized size of antiderivative = 2.71 \[ \int \frac {1}{\sqrt {3+3 \sin (e+f x)}} \, dx=\frac {\frac {\sqrt {2} \log \left ({\left | \frac {1}{\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )} + \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 2 \right |}\right )}{\sqrt {a} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} - \frac {\sqrt {2} \log \left ({\left | \frac {1}{\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )} + \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 2 \right |}\right )}{\sqrt {a} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{4 \, f} \]
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Time = 7.77 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.20 \[ \int \frac {1}{\sqrt {3+3 \sin (e+f x)}} \, dx=-\frac {\mathrm {F}\left (\frac {\pi }{4}-\frac {e}{2}-\frac {f\,x}{2}\middle |1\right )\,\sqrt {\frac {2\,\left (a+a\,\sin \left (e+f\,x\right )\right )}{a}}}{f\,\sqrt {a+a\,\sin \left (e+f\,x\right )}} \]
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