Integrand size = 23, antiderivative size = 51 \[ \int \frac {\sqrt {\sin (e+f x)}}{\sqrt {b \sec (e+f x)}} \, dx=\frac {E\left (\left .e-\frac {\pi }{4}+f x\right |2\right ) \sqrt {\sin (e+f x)}}{f \sqrt {b \sec (e+f x)} \sqrt {\sin (2 e+2 f x)}} \]
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Time = 0.05 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {2665, 2652, 2719} \[ \int \frac {\sqrt {\sin (e+f x)}}{\sqrt {b \sec (e+f x)}} \, dx=\frac {\sqrt {\sin (e+f x)} E\left (\left .e+f x-\frac {\pi }{4}\right |2\right )}{f \sqrt {\sin (2 e+2 f x)} \sqrt {b \sec (e+f x)}} \]
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Rule 2652
Rule 2665
Rule 2719
Rubi steps \begin{align*} \text {integral}& = \frac {\int \sqrt {b \cos (e+f x)} \sqrt {\sin (e+f x)} \, dx}{\sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)}} \\ & = \frac {\sqrt {\sin (e+f x)} \int \sqrt {\sin (2 e+2 f x)} \, dx}{\sqrt {b \sec (e+f x)} \sqrt {\sin (2 e+2 f x)}} \\ & = \frac {E\left (\left .e-\frac {\pi }{4}+f x\right |2\right ) \sqrt {\sin (e+f x)}}{f \sqrt {b \sec (e+f x)} \sqrt {\sin (2 e+2 f x)}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 1.15 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.18 \[ \int \frac {\sqrt {\sin (e+f x)}}{\sqrt {b \sec (e+f x)}} \, dx=-\frac {b \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{4},\frac {1}{2},\sec ^2(e+f x)\right ) \sqrt [4]{-\tan ^2(e+f x)}}{f (b \sec (e+f x))^{3/2} \sqrt {\sin (e+f x)}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(370\) vs. \(2(71)=142\).
Time = 0.84 (sec) , antiderivative size = 371, normalized size of antiderivative = 7.27
method | result | size |
default | \(-\frac {\sqrt {2}\, \left (2 \sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}\, \sqrt {\cot \left (f x +e \right )-\csc \left (f x +e \right )+1}\, \sqrt {\cot \left (f x +e \right )-\csc \left (f x +e \right )}\, E\left (\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}, \frac {\sqrt {2}}{2}\right )-\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}\, \sqrt {\cot \left (f x +e \right )-\csc \left (f x +e \right )+1}\, \sqrt {\cot \left (f x +e \right )-\csc \left (f x +e \right )}\, F\left (\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}, \frac {\sqrt {2}}{2}\right )+2 \sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}\, \sqrt {\cot \left (f x +e \right )-\csc \left (f x +e \right )+1}\, \sqrt {\cot \left (f x +e \right )-\csc \left (f x +e \right )}\, E\left (\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}, \frac {\sqrt {2}}{2}\right ) \sec \left (f x +e \right )-\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}\, \sqrt {\cot \left (f x +e \right )-\csc \left (f x +e \right )+1}\, \sqrt {\cot \left (f x +e \right )-\csc \left (f x +e \right )}\, F\left (\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}, \frac {\sqrt {2}}{2}\right ) \sec \left (f x +e \right )+\sqrt {2}\, \cos \left (f x +e \right )-\sqrt {2}\right )}{2 f \sqrt {b \sec \left (f x +e \right )}\, \sqrt {\sin \left (f x +e \right )}}\) | \(371\) |
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\[ \int \frac {\sqrt {\sin (e+f x)}}{\sqrt {b \sec (e+f x)}} \, dx=\int { \frac {\sqrt {\sin \left (f x + e\right )}}{\sqrt {b \sec \left (f x + e\right )}} \,d x } \]
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\[ \int \frac {\sqrt {\sin (e+f x)}}{\sqrt {b \sec (e+f x)}} \, dx=\int \frac {\sqrt {\sin {\left (e + f x \right )}}}{\sqrt {b \sec {\left (e + f x \right )}}}\, dx \]
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\[ \int \frac {\sqrt {\sin (e+f x)}}{\sqrt {b \sec (e+f x)}} \, dx=\int { \frac {\sqrt {\sin \left (f x + e\right )}}{\sqrt {b \sec \left (f x + e\right )}} \,d x } \]
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\[ \int \frac {\sqrt {\sin (e+f x)}}{\sqrt {b \sec (e+f x)}} \, dx=\int { \frac {\sqrt {\sin \left (f x + e\right )}}{\sqrt {b \sec \left (f x + e\right )}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {\sin (e+f x)}}{\sqrt {b \sec (e+f x)}} \, dx=\int \frac {\sqrt {\sin \left (e+f\,x\right )}}{\sqrt {\frac {b}{\cos \left (e+f\,x\right )}}} \,d x \]
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