Integrand size = 22, antiderivative size = 40 \[ \int \frac {\cos ^2(a+b x)}{\sqrt {\sin (2 a+2 b x)}} \, dx=\frac {\operatorname {EllipticF}\left (a-\frac {\pi }{4}+b x,2\right )}{2 b}+\frac {\sqrt {\sin (2 a+2 b x)}}{2 b} \]
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Time = 0.05 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {4382, 2720} \[ \int \frac {\cos ^2(a+b x)}{\sqrt {\sin (2 a+2 b x)}} \, dx=\frac {\sqrt {\sin (2 a+2 b x)}}{2 b}+\frac {\operatorname {EllipticF}\left (a+b x-\frac {\pi }{4},2\right )}{2 b} \]
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Rule 2720
Rule 4382
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {\sin (2 a+2 b x)}}{2 b}+\frac {1}{2} \int \frac {1}{\sqrt {\sin (2 a+2 b x)}} \, dx \\ & = \frac {\operatorname {EllipticF}\left (a-\frac {\pi }{4}+b x,2\right )}{2 b}+\frac {\sqrt {\sin (2 a+2 b x)}}{2 b} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 1.56 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.35 \[ \int \frac {\cos ^2(a+b x)}{\sqrt {\sin (2 a+2 b x)}} \, dx=\frac {\left (1+\operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},-\tan ^2(a+b x)\right ) \sqrt {\sec ^2(a+b x)}\right ) \sqrt {\sin (2 (a+b x))}}{2 b} \]
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result has leaf size over 500,000. Avoiding possible recursion issues.
Time = 9.33 (sec) , antiderivative size = 67488705, normalized size of antiderivative = 1687217.62
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\[ \int \frac {\cos ^2(a+b x)}{\sqrt {\sin (2 a+2 b x)}} \, dx=\int { \frac {\cos \left (b x + a\right )^{2}}{\sqrt {\sin \left (2 \, b x + 2 \, a\right )}} \,d x } \]
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Timed out. \[ \int \frac {\cos ^2(a+b x)}{\sqrt {\sin (2 a+2 b x)}} \, dx=\text {Timed out} \]
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\[ \int \frac {\cos ^2(a+b x)}{\sqrt {\sin (2 a+2 b x)}} \, dx=\int { \frac {\cos \left (b x + a\right )^{2}}{\sqrt {\sin \left (2 \, b x + 2 \, a\right )}} \,d x } \]
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\[ \int \frac {\cos ^2(a+b x)}{\sqrt {\sin (2 a+2 b x)}} \, dx=\int { \frac {\cos \left (b x + a\right )^{2}}{\sqrt {\sin \left (2 \, b x + 2 \, a\right )}} \,d x } \]
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Timed out. \[ \int \frac {\cos ^2(a+b x)}{\sqrt {\sin (2 a+2 b x)}} \, dx=\int \frac {{\cos \left (a+b\,x\right )}^2}{\sqrt {\sin \left (2\,a+2\,b\,x\right )}} \,d x \]
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