Integrand size = 30, antiderivative size = 46 \[ \int \frac {1}{\sqrt {3+3 \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \, dx=\frac {\text {arctanh}(\sin (e+f x)) \cos (e+f x)}{f \sqrt {3+3 \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \]
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Time = 0.06 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2820, 3855} \[ \int \frac {1}{\sqrt {3+3 \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \, dx=\frac {\cos (e+f x) \text {arctanh}(\sin (e+f x))}{f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}} \]
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Rule 2820
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \frac {\cos (e+f x) \int \sec (e+f x) \, dx}{\sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \\ & = \frac {\text {arctanh}(\sin (e+f x)) \cos (e+f x)}{f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(112\) vs. \(2(46)=92\).
Time = 0.39 (sec) , antiderivative size = 112, normalized size of antiderivative = 2.43 \[ \int \frac {1}{\sqrt {3+3 \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \, dx=-\frac {\left (\log \left (1-\tan \left (\frac {1}{2} (e+f x)\right )\right )-\log \left (1+\tan \left (\frac {1}{2} (e+f x)\right )\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )}{\sqrt {3} f \sqrt {1+\sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \]
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Time = 4.79 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.61
method | result | size |
default | \(-\frac {\cos \left (f x +e \right ) \left (\ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )-1\right )-\ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1\right )\right )}{f \sqrt {-c \left (\sin \left (f x +e \right )-1\right )}\, \sqrt {a \left (\sin \left (f x +e \right )+1\right )}}\) | \(74\) |
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Time = 0.30 (sec) , antiderivative size = 160, normalized size of antiderivative = 3.48 \[ \int \frac {1}{\sqrt {3+3 \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \, dx=\left [\frac {\sqrt {a c} \log \left (-\frac {a c \cos \left (f x + e\right )^{3} - 2 \, a c \cos \left (f x + e\right ) - 2 \, \sqrt {a c} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c} \sin \left (f x + e\right )}{\cos \left (f x + e\right )^{3}}\right )}{2 \, a c f}, -\frac {\sqrt {-a c} \arctan \left (\frac {\sqrt {-a c} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}}{a c \cos \left (f x + e\right ) \sin \left (f x + e\right )}\right )}{a c f}\right ] \]
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\[ \int \frac {1}{\sqrt {3+3 \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \, dx=\int \frac {1}{\sqrt {a \left (\sin {\left (e + f x \right )} + 1\right )} \sqrt {- c \left (\sin {\left (e + f x \right )} - 1\right )}}\, dx \]
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\[ \int \frac {1}{\sqrt {3+3 \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \, dx=\int { \frac {1}{\sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}} \,d x } \]
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Exception generated. \[ \int \frac {1}{\sqrt {3+3 \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {1}{\sqrt {3+3 \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \, dx=\int \frac {1}{\sqrt {a+a\,\sin \left (e+f\,x\right )}\,\sqrt {c-c\,\sin \left (e+f\,x\right )}} \,d x \]
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