Integrand size = 19, antiderivative size = 68 \[ \int \csc (e+f x) \sqrt {d \csc (e+f x)} \, dx=-\frac {2 \cos (e+f x) \sqrt {d \csc (e+f x)}}{f}-\frac {2 d E\left (\left .\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )\right |2\right )}{f \sqrt {d \csc (e+f x)} \sqrt {\sin (e+f x)}} \]
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Time = 0.03 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {16, 3853, 3856, 2719} \[ \int \csc (e+f x) \sqrt {d \csc (e+f x)} \, dx=-\frac {2 \cos (e+f x) \sqrt {d \csc (e+f x)}}{f}-\frac {2 d E\left (\left .\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )\right |2\right )}{f \sqrt {\sin (e+f x)} \sqrt {d \csc (e+f x)}} \]
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Rule 16
Rule 2719
Rule 3853
Rule 3856
Rubi steps \begin{align*} \text {integral}& = \frac {\int (d \csc (e+f x))^{3/2} \, dx}{d} \\ & = -\frac {2 \cos (e+f x) \sqrt {d \csc (e+f x)}}{f}-d \int \frac {1}{\sqrt {d \csc (e+f x)}} \, dx \\ & = -\frac {2 \cos (e+f x) \sqrt {d \csc (e+f x)}}{f}-\frac {d \int \sqrt {\sin (e+f x)} \, dx}{\sqrt {d \csc (e+f x)} \sqrt {\sin (e+f x)}} \\ & = -\frac {2 \cos (e+f x) \sqrt {d \csc (e+f x)}}{f}-\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 = 0.11 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.84 \[ \int \csc (e+f x) \sqrt {d \csc (e+f x)} \, dx=\frac {(d \csc (e+f x))^{3/2} \left (2 E\left (\left .\frac {1}{4} (-2 e+\pi -2 f x)\right |2\right ) \sin ^{\frac {3}{2}}(e+f x)-\sin (2 (e+f x))\right )}{d f} \]
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Result contains complex when optimal does not.
Time = 0.92 (sec) , antiderivative size = 409, normalized size of antiderivative = 6.01
method | result | size |
default | \(-\frac {\sqrt {2}\, \sqrt {d \csc \left (f x +e \right )}\, \left (-2 \sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, \sqrt {-i \left (i+\cot \left (f x +e \right )-\csc \left (f x +e \right )\right )}\, \sqrt {i \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, E\left (\sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}, \frac {\sqrt {2}}{2}\right ) \cos \left (f x +e \right )+\sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, \sqrt {-i \left (i+\cot \left (f x +e \right )-\csc \left (f x +e \right )\right )}\, \sqrt {i \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, F\left (\sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}, \frac {\sqrt {2}}{2}\right ) \cos \left (f x +e \right )-2 \sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, \sqrt {-i \left (i+\cot \left (f x +e \right )-\csc \left (f x +e \right )\right )}\, \sqrt {i \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, E\left (\sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}, \frac {\sqrt {2}}{2}\right )+\sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, \sqrt {-i \left (i+\cot \left (f x +e \right )-\csc \left (f x +e \right )\right )}\, \sqrt {i \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, F\left (\sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}, \frac {\sqrt {2}}{2}\right )+\sqrt {2}\right )}{f}\) | \(409\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.18 \[ \int \csc (e+f x) \sqrt {d \csc (e+f x)} \, dx=-\frac {2 \, \sqrt {\frac {d}{\sin \left (f x + e\right )}} \cos \left (f x + e\right ) + \sqrt {2 i \, d} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\right ) + \sqrt {-2 i \, d} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right )}{f} \]
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\[ \int \csc (e+f x) \sqrt {d \csc (e+f x)} \, dx=\int \sqrt {d \csc {\left (e + f x \right )}} \csc {\left (e + f x \right )}\, dx \]
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\[ \int \csc (e+f x) \sqrt {d \csc (e+f x)} \, dx=\int { \sqrt {d \csc \left (f x + e\right )} \csc \left (f x + e\right ) \,d x } \]
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\[ \int \csc (e+f x) \sqrt {d \csc (e+f x)} \, dx=\int { \sqrt {d \csc \left (f x + e\right )} \csc \left (f x + e\right ) \,d x } \]
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Timed out. \[ \int \csc (e+f x) \sqrt {d \csc (e+f x)} \, dx=\int \frac {\sqrt {\frac {d}{\sin \left (e+f\,x\right )}}}{\sin \left (e+f\,x\right )} \,d x \]
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