Integrand size = 19, antiderivative size = 44 \[ \int \sqrt {d \csc (e+f x)} \sin (e+f x) \, dx=\frac {2 d E\left (\left .\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )\right |2\right )}{f \sqrt {d \csc (e+f x)} \sqrt {\sin (e+f x)}} \]
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Time = 0.02 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {16, 3856, 2719} \[ \int \sqrt {d \csc (e+f x)} \sin (e+f x) \, dx=\frac {2 d E\left (\left .\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )\right |2\right )}{f \sqrt {\sin (e+f x)} \sqrt {d \csc (e+f x)}} \]
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Rule 16
Rule 2719
Rule 3856
Rubi steps \begin{align*} \text {integral}& = d \int \frac {1}{\sqrt {d \csc (e+f x)}} \, dx \\ & = \frac {d \int \sqrt {\sin (e+f x)} \, dx}{\sqrt {d \csc (e+f x)} \sqrt {\sin (e+f x)}} \\ & = \frac {2 d E\left (\left .\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )\right |2\right )}{f \sqrt {d \csc (e+f x)} \sqrt {\sin (e+f x)}} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.98 \[ \int \sqrt {d \csc (e+f x)} \sin (e+f x) \, dx=-\frac {2 d E\left (\left .\frac {1}{4} (-2 e+\pi -2 f x)\right |2\right )}{f \sqrt {d \csc (e+f x)} \sqrt {\sin (e+f x)}} \]
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Result contains complex when optimal does not.
Time = 0.95 (sec) , antiderivative size = 314, normalized size of antiderivative = 7.14
method | result | size |
risch | \(-\frac {\left ({\mathrm e}^{2 i \left (f x +e \right )}-1\right ) \sqrt {2}\, \sqrt {\frac {i d \,{\mathrm e}^{i \left (f x +e \right )}}{{\mathrm e}^{2 i \left (f x +e \right )}-1}}\, {\mathrm e}^{-i \left (f x +e \right )}}{f}+\frac {\left (-\frac {2 i \left (i d \,{\mathrm e}^{2 i \left (f x +e \right )}-i d \right )}{d \sqrt {{\mathrm e}^{i \left (f x +e \right )} \left (i d \,{\mathrm e}^{2 i \left (f x +e \right )}-i d \right )}}-\frac {\sqrt {{\mathrm e}^{i \left (f x +e \right )}+1}\, \sqrt {-2 \,{\mathrm e}^{i \left (f x +e \right )}+2}\, \sqrt {-{\mathrm e}^{i \left (f x +e \right )}}\, \left (-2 E\left (\sqrt {{\mathrm e}^{i \left (f x +e \right )}+1}, \frac {\sqrt {2}}{2}\right )+F\left (\sqrt {{\mathrm e}^{i \left (f x +e \right )}+1}, \frac {\sqrt {2}}{2}\right )\right )}{\sqrt {i d \,{\mathrm e}^{3 i \left (f x +e \right )}-i d \,{\mathrm e}^{i \left (f x +e \right )}}}\right ) \sqrt {2}\, \sqrt {\frac {i d \,{\mathrm e}^{i \left (f x +e \right )}}{{\mathrm e}^{2 i \left (f x +e \right )}-1}}\, \sqrt {i d \,{\mathrm e}^{i \left (f x +e \right )} \left ({\mathrm e}^{2 i \left (f x +e \right )}-1\right )}\, {\mathrm e}^{-i \left (f x +e \right )}}{f}\) | \(314\) |
default | \(-\frac {\sqrt {2}\, \sqrt {d \csc \left (f x +e \right )}\, \left (2 \sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, \sqrt {-i \left (i+\cot \left (f x +e \right )-\csc \left (f x +e \right )\right )}\, \sqrt {i \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, E\left (\sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}, \frac {\sqrt {2}}{2}\right ) \cos \left (f x +e \right )-\sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, \sqrt {-i \left (i+\cot \left (f x +e \right )-\csc \left (f x +e \right )\right )}\, \sqrt {i \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, F\left (\sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}, \frac {\sqrt {2}}{2}\right ) \cos \left (f x +e \right )+2 \sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, \sqrt {-i \left (i+\cot \left (f x +e \right )-\csc \left (f x +e \right )\right )}\, \sqrt {i \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, E\left (\sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}, \frac {\sqrt {2}}{2}\right )-\sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, \sqrt {-i \left (i+\cot \left (f x +e \right )-\csc \left (f x +e \right )\right )}\, \sqrt {i \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, F\left (\sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}, \frac {\sqrt {2}}{2}\right )+\sqrt {2}\, \cos \left (f x +e \right )-\sqrt {2}\right )}{f}\) | \(423\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.34 \[ \int \sqrt {d \csc (e+f x)} \sin (e+f x) \, dx=\frac {\sqrt {2 i \, d} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\right ) + \sqrt {-2 i \, d} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right )}{f} \]
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\[ \int \sqrt {d \csc (e+f x)} \sin (e+f x) \, dx=\int \sqrt {d \csc {\left (e + f x \right )}} \sin {\left (e + f x \right )}\, dx \]
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\[ \int \sqrt {d \csc (e+f x)} \sin (e+f x) \, dx=\int { \sqrt {d \csc \left (f x + e\right )} \sin \left (f x + e\right ) \,d x } \]
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\[ \int \sqrt {d \csc (e+f x)} \sin (e+f x) \, dx=\int { \sqrt {d \csc \left (f x + e\right )} \sin \left (f x + e\right ) \,d x } \]
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Timed out. \[ \int \sqrt {d \csc (e+f x)} \sin (e+f x) \, dx=\int \sin \left (e+f\,x\right )\,\sqrt {\frac {d}{\sin \left (e+f\,x\right )}} \,d x \]
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