Integrand size = 25, antiderivative size = 53 \[ \int \sqrt {d \cos (a+b x)} \sqrt {c \sin (a+b x)} \, dx=\frac {\sqrt {d \cos (a+b x)} E\left (\left .a-\frac {\pi }{4}+b x\right |2\right ) \sqrt {c \sin (a+b x)}}{b \sqrt {\sin (2 a+2 b x)}} \]
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Time = 0.03 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {2652, 2719} \[ \int \sqrt {d \cos (a+b x)} \sqrt {c \sin (a+b x)} \, dx=\frac {E\left (\left .a+b x-\frac {\pi }{4}\right |2\right ) \sqrt {c \sin (a+b x)} \sqrt {d \cos (a+b x)}}{b \sqrt {\sin (2 a+2 b x)}} \]
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Rule 2652
Rule 2719
Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {d \cos (a+b x)} \sqrt {c \sin (a+b x)}\right ) \int \sqrt {\sin (2 a+2 b x)} \, dx}{\sqrt {\sin (2 a+2 b x)}} \\ & = \frac {\sqrt {d \cos (a+b x)} E\left (\left .a-\frac {\pi }{4}+b x\right |2\right ) \sqrt {c \sin (a+b x)}}{b \sqrt {\sin (2 a+2 b x)}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.04 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.26 \[ \int \sqrt {d \cos (a+b x)} \sqrt {c \sin (a+b x)} \, dx=\frac {2 \sqrt {d \cos (a+b x)} \sqrt [4]{\cos ^2(a+b x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {3}{4},\frac {7}{4},\sin ^2(a+b x)\right ) \sqrt {c \sin (a+b x)} \tan (a+b x)}{3 b} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(392\) vs. \(2(73)=146\).
Time = 0.26 (sec) , antiderivative size = 393, normalized size of antiderivative = 7.42
method | result | size |
default | \(-\frac {\sqrt {2}\, \left (2 \sqrt {-\cot \left (b x +a \right )+\csc \left (b x +a \right )+1}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )+1}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )}\, E\left (\sqrt {-\cot \left (b x +a \right )+\csc \left (b x +a \right )+1}, \frac {\sqrt {2}}{2}\right ) \cos \left (b x +a \right )-\sqrt {-\cot \left (b x +a \right )+\csc \left (b x +a \right )+1}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )+1}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )}\, F\left (\sqrt {-\cot \left (b x +a \right )+\csc \left (b x +a \right )+1}, \frac {\sqrt {2}}{2}\right ) \cos \left (b x +a \right )+2 \sqrt {-\cot \left (b x +a \right )+\csc \left (b x +a \right )+1}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )+1}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )}\, E\left (\sqrt {-\cot \left (b x +a \right )+\csc \left (b x +a \right )+1}, \frac {\sqrt {2}}{2}\right )-\sqrt {-\cot \left (b x +a \right )+\csc \left (b x +a \right )+1}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )+1}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )}\, F\left (\sqrt {-\cot \left (b x +a \right )+\csc \left (b x +a \right )+1}, \frac {\sqrt {2}}{2}\right )+\sqrt {2}\, \left (\cos ^{2}\left (b x +a \right )\right )-\sqrt {2}\, \cos \left (b x +a \right )\right ) \sqrt {d \cos \left (b x +a \right )}\, \sqrt {c \sin \left (b x +a \right )}\, \sec \left (b x +a \right ) \csc \left (b x +a \right )}{2 b}\) | \(393\) |
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\[ \int \sqrt {d \cos (a+b x)} \sqrt {c \sin (a+b x)} \, dx=\int { \sqrt {d \cos \left (b x + a\right )} \sqrt {c \sin \left (b x + a\right )} \,d x } \]
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\[ \int \sqrt {d \cos (a+b x)} \sqrt {c \sin (a+b x)} \, dx=\int \sqrt {c \sin {\left (a + b x \right )}} \sqrt {d \cos {\left (a + b x \right )}}\, dx \]
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\[ \int \sqrt {d \cos (a+b x)} \sqrt {c \sin (a+b x)} \, dx=\int { \sqrt {d \cos \left (b x + a\right )} \sqrt {c \sin \left (b x + a\right )} \,d x } \]
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\[ \int \sqrt {d \cos (a+b x)} \sqrt {c \sin (a+b x)} \, dx=\int { \sqrt {d \cos \left (b x + a\right )} \sqrt {c \sin \left (b x + a\right )} \,d x } \]
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Timed out. \[ \int \sqrt {d \cos (a+b x)} \sqrt {c \sin (a+b x)} \, dx=\int \sqrt {d\,\cos \left (a+b\,x\right )}\,\sqrt {c\,\sin \left (a+b\,x\right )} \,d x \]
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