Integrand size = 21, antiderivative size = 77 \[ \int \frac {\csc ^3(e+f x)}{\sqrt {d \csc (e+f x)}} \, dx=-\frac {2 \cos (e+f x) (d \csc (e+f x))^{3/2}}{3 d^2 f}+\frac {2 \sqrt {d \csc (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),2\right ) \sqrt {\sin (e+f x)}}{3 d f} \]
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Time = 0.03 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {16, 3853, 3856, 2720} \[ \int \frac {\csc ^3(e+f x)}{\sqrt {d \csc (e+f x)}} \, dx=\frac {2 \sqrt {\sin (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),2\right ) \sqrt {d \csc (e+f x)}}{3 d f}-\frac {2 \cos (e+f x) (d \csc (e+f x))^{3/2}}{3 d^2 f} \]
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Rule 16
Rule 2720
Rule 3853
Rule 3856
Rubi steps \begin{align*} \text {integral}& = \frac {\int (d \csc (e+f x))^{5/2} \, dx}{d^3} \\ & = -\frac {2 \cos (e+f x) (d \csc (e+f x))^{3/2}}{3 d^2 f}+\frac {\int \sqrt {d \csc (e+f x)} \, dx}{3 d} \\ & = -\frac {2 \cos (e+f x) (d \csc (e+f x))^{3/2}}{3 d^2 f}+\frac {\left (\sqrt {d \csc (e+f x)} \sqrt {\sin (e+f x)}\right ) \int \frac {1}{\sqrt {\sin (e+f x)}} \, dx}{3 d} \\ & = -\frac {2 \cos (e+f x) (d \csc (e+f x))^{3/2}}{3 d^2 f}+\frac {2 \sqrt {d \csc (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),2\right ) \sqrt {\sin (e+f x)}}{3 d f} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.78 \[ \int \frac {\csc ^3(e+f x)}{\sqrt {d \csc (e+f x)}} \, dx=-\frac {2 \csc ^2(e+f x) \left (\cos (e+f x)+\operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),2\right ) \sin ^{\frac {3}{2}}(e+f x)\right )}{3 f \sqrt {d \csc (e+f x)}} \]
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Result contains complex when optimal does not.
Time = 0.96 (sec) , antiderivative size = 250, normalized size of antiderivative = 3.25
method | result | size |
default | \(\frac {\sqrt {2}\, \left (-i \sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, \sqrt {-i \left (i+\cot \left (f x +e \right )-\csc \left (f x +e \right )\right )}\, \sqrt {i \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, F\left (\sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}, \frac {\sqrt {2}}{2}\right ) \cos \left (f x +e \right ) \sin \left (f x +e \right )-i \sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, \sqrt {-i \left (i+\cot \left (f x +e \right )-\csc \left (f x +e \right )\right )}\, \sqrt {i \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, F\left (\sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}, \frac {\sqrt {2}}{2}\right ) \sin \left (f x +e \right )+\sqrt {2}\, \cos \left (f x +e \right )\right )}{3 f \sqrt {d \csc \left (f x +e \right )}\, \left (\cos ^{2}\left (f x +e \right )-1\right )}\) | \(250\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.29 \[ \int \frac {\csc ^3(e+f x)}{\sqrt {d \csc (e+f x)}} \, dx=\frac {-i \, \sqrt {2 i \, d} \sin \left (f x + e\right ) {\rm weierstrassPInverse}\left (4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right ) + i \, \sqrt {-2 i \, d} \sin \left (f x + e\right ) {\rm weierstrassPInverse}\left (4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right ) - 2 \, \sqrt {\frac {d}{\sin \left (f x + e\right )}} \cos \left (f x + e\right )}{3 \, d f \sin \left (f x + e\right )} \]
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\[ \int \frac {\csc ^3(e+f x)}{\sqrt {d \csc (e+f x)}} \, dx=\int \frac {\csc ^{3}{\left (e + f x \right )}}{\sqrt {d \csc {\left (e + f x \right )}}}\, dx \]
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\[ \int \frac {\csc ^3(e+f x)}{\sqrt {d \csc (e+f x)}} \, dx=\int { \frac {\csc \left (f x + e\right )^{3}}{\sqrt {d \csc \left (f x + e\right )}} \,d x } \]
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\[ \int \frac {\csc ^3(e+f x)}{\sqrt {d \csc (e+f x)}} \, dx=\int { \frac {\csc \left (f x + e\right )^{3}}{\sqrt {d \csc \left (f x + e\right )}} \,d x } \]
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Timed out. \[ \int \frac {\csc ^3(e+f x)}{\sqrt {d \csc (e+f x)}} \, dx=\int \frac {1}{{\sin \left (e+f\,x\right )}^3\,\sqrt {\frac {d}{\sin \left (e+f\,x\right )}}} \,d x \]
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