Integrand size = 19, antiderivative size = 46 \[ \int \frac {\csc (e+f x)}{(d \csc (e+f x))^{3/2}} \, dx=\frac {2 E\left (\left .\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )\right |2\right )}{d f \sqrt {d \csc (e+f x)} \sqrt {\sin (e+f x)}} \]
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Time = 0.02 (sec) , antiderivative size = 46, 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} \[ \int \frac {\csc (e+f x)}{(d \csc (e+f x))^{3/2}} \, dx=\frac {2 E\left (\left .\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )\right |2\right )}{d f \sqrt {\sin (e+f x)} \sqrt {d \csc (e+f x)}} \]
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Rule 16
Rule 2719
Rule 3856
Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {1}{\sqrt {d \csc (e+f x)}} \, dx}{d} \\ & = \frac {\int \sqrt {\sin (e+f x)} \, dx}{d \sqrt {d \csc (e+f x)} \sqrt {\sin (e+f x)}} \\ & = \frac {2 E\left (\left .\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )\right |2\right )}{d f \sqrt {d \csc (e+f x)} \sqrt {\sin (e+f x)}} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.98 \[ \int \frac {\csc (e+f x)}{(d \csc (e+f x))^{3/2}} \, dx=-\frac {2 E\left (\left .\frac {1}{4} (-2 e+\pi -2 f x)\right |2\right )}{d f \sqrt {d \csc (e+f x)} \sqrt {\sin (e+f x)}} \]
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Result contains complex when optimal does not.
Time = 1.17 (sec) , antiderivative size = 303, normalized size of antiderivative = 6.59
method | result | size |
risch | \(-\frac {i \sqrt {2}}{f d \sqrt {\frac {i d \,{\mathrm e}^{i \left (f x +e \right )}}{{\mathrm e}^{2 i \left (f x +e \right )}-1}}}+\frac {i \left (-\frac {2 i \left (i d \,{\mathrm e}^{2 i \left (f x +e \right )}-i d \right )}{d \sqrt {{\mathrm e}^{i \left (f x +e \right )} \left (i d \,{\mathrm e}^{2 i \left (f x +e \right )}-i d \right )}}-\frac {\sqrt {{\mathrm e}^{i \left (f x +e \right )}+1}\, \sqrt {-2 \,{\mathrm e}^{i \left (f x +e \right )}+2}\, \sqrt {-{\mathrm e}^{i \left (f x +e \right )}}\, \left (-2 E\left (\sqrt {{\mathrm e}^{i \left (f x +e \right )}+1}, \frac {\sqrt {2}}{2}\right )+F\left (\sqrt {{\mathrm e}^{i \left (f x +e \right )}+1}, \frac {\sqrt {2}}{2}\right )\right )}{\sqrt {i d \,{\mathrm e}^{3 i \left (f x +e \right )}-i d \,{\mathrm e}^{i \left (f x +e \right )}}}\right ) \sqrt {2}\, \sqrt {i d \,{\mathrm e}^{i \left (f x +e \right )} \left ({\mathrm e}^{2 i \left (f x +e \right )}-1\right )}}{f d \sqrt {\frac {i d \,{\mathrm e}^{i \left (f x +e \right )}}{{\mathrm e}^{2 i \left (f x +e \right )}-1}}\, \left ({\mathrm e}^{2 i \left (f x +e \right )}-1\right )}\) | \(303\) |
default | \(-\frac {\sqrt {2}\, \left (2 \sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, \sqrt {-i \left (i+\cot \left (f x +e \right )-\csc \left (f x +e \right )\right )}\, \sqrt {-i \left (\cot \left (f x +e \right )-\csc \left (f x +e \right )\right )}\, E\left (\sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}, \frac {\sqrt {2}}{2}\right ) \cos \left (f x +e \right )-\sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, \sqrt {-i \left (i+\cot \left (f x +e \right )-\csc \left (f x +e \right )\right )}\, \sqrt {-i \left (\cot \left (f x +e \right )-\csc \left (f x +e \right )\right )}\, F\left (\sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}, \frac {\sqrt {2}}{2}\right ) \cos \left (f x +e \right )+2 \sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, \sqrt {-i \left (i+\cot \left (f x +e \right )-\csc \left (f x +e \right )\right )}\, \sqrt {-i \left (\cot \left (f x +e \right )-\csc \left (f x +e \right )\right )}\, E\left (\sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}, \frac {\sqrt {2}}{2}\right )-\sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, \sqrt {-i \left (i+\cot \left (f x +e \right )-\csc \left (f x +e \right )\right )}\, \sqrt {-i \left (\cot \left (f x +e \right )-\csc \left (f x +e \right )\right )}\, F\left (\sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}, \frac {\sqrt {2}}{2}\right )+\sqrt {2}\, \cos \left (f x +e \right )-\sqrt {2}\right ) \csc \left (f x +e \right )}{d f \sqrt {d \csc \left (f x +e \right )}}\) | \(432\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.35 \[ \int \frac {\csc (e+f x)}{(d \csc (e+f x))^{3/2}} \, dx=\frac {\sqrt {2 i \, d} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\right ) + \sqrt {-2 i \, d} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right )}{d^{2} f} \]
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\[ \int \frac {\csc (e+f x)}{(d \csc (e+f x))^{3/2}} \, dx=\int \frac {\csc {\left (e + f x \right )}}{\left (d \csc {\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {\csc (e+f x)}{(d \csc (e+f x))^{3/2}} \, dx=\int { \frac {\csc \left (f x + e\right )}{\left (d \csc \left (f x + e\right )\right )^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {\csc (e+f x)}{(d \csc (e+f x))^{3/2}} \, dx=\int { \frac {\csc \left (f x + e\right )}{\left (d \csc \left (f x + e\right )\right )^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\csc (e+f x)}{(d \csc (e+f x))^{3/2}} \, dx=\int \frac {1}{\sin \left (e+f\,x\right )\,{\left (\frac {d}{\sin \left (e+f\,x\right )}\right )}^{3/2}} \,d x \]
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