Integrand size = 21, antiderivative size = 74 \[ \int \csc ^2(e+f x) \sqrt {d \csc (e+f x)} \, dx=-\frac {2 \cos (e+f x) (d \csc (e+f x))^{3/2}}{3 d f}+\frac {2 \sqrt {d \csc (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),2\right ) \sqrt {\sin (e+f x)}}{3 f} \]
[Out]
Time = 0.03 (sec) , antiderivative size = 74, 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} \[ \int \csc ^2(e+f x) \sqrt {d \csc (e+f x)} \, dx=\frac {2 \sqrt {\sin (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),2\right ) \sqrt {d \csc (e+f x)}}{3 f}-\frac {2 \cos (e+f x) (d \csc (e+f x))^{3/2}}{3 d f} \]
[In]
[Out]
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^2} \\ & = -\frac {2 \cos (e+f x) (d \csc (e+f x))^{3/2}}{3 d f}+\frac {1}{3} \int \sqrt {d \csc (e+f x)} \, dx \\ & = -\frac {2 \cos (e+f x) (d \csc (e+f x))^{3/2}}{3 d f}+\frac {1}{3} \left (\sqrt {d \csc (e+f x)} \sqrt {\sin (e+f x)}\right ) \int \frac {1}{\sqrt {\sin (e+f x)}} \, dx \\ & = -\frac {2 \cos (e+f x) (d \csc (e+f x))^{3/2}}{3 d 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 f} \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.74 \[ \int \csc ^2(e+f x) \sqrt {d \csc (e+f x)} \, dx=-\frac {2 (d \csc (e+f x))^{3/2} \left (\cos (e+f x)+\operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),2\right ) \sin ^{\frac {3}{2}}(e+f x)\right )}{3 d f} \]
[In]
[Out]
Result contains complex when optimal does not.
Time = 0.85 (sec) , antiderivative size = 227, normalized size of antiderivative = 3.07
method | result | size |
default | \(\frac {\sqrt {2}\, \sqrt {d \csc \left (f x +e \right )}\, \left (i \sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, \sqrt {-i \left (i+\cot \left (f x +e \right )-\csc \left (f x +e \right )\right )}\, \sqrt {i \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, F\left (\sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}, \frac {\sqrt {2}}{2}\right ) \cos \left (f x +e \right )+i \sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, \sqrt {-i \left (i+\cot \left (f x +e \right )-\csc \left (f x +e \right )\right )}\, \sqrt {i \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, F\left (\sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}, \frac {\sqrt {2}}{2}\right )-\sqrt {2}\, \cot \left (f x +e \right )\right )}{3 f}\) | \(227\) |
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.12 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.30 \[ \int \csc ^2(e+f x) \sqrt {d \csc (e+f x)} \, dx=\frac {-i \, \sqrt {2 i \, d} \sin \left (f x + e\right ) {\rm weierstrassPInverse}\left (4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right ) + i \, \sqrt {-2 i \, d} \sin \left (f x + e\right ) {\rm weierstrassPInverse}\left (4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right ) - 2 \, \sqrt {\frac {d}{\sin \left (f x + e\right )}} \cos \left (f x + e\right )}{3 \, f \sin \left (f x + e\right )} \]
[In]
[Out]
\[ \int \csc ^2(e+f x) \sqrt {d \csc (e+f x)} \, dx=\int \sqrt {d \csc {\left (e + f x \right )}} \csc ^{2}{\left (e + f x \right )}\, dx \]
[In]
[Out]
\[ \int \csc ^2(e+f x) \sqrt {d \csc (e+f x)} \, dx=\int { \sqrt {d \csc \left (f x + e\right )} \csc \left (f x + e\right )^{2} \,d x } \]
[In]
[Out]
\[ \int \csc ^2(e+f x) \sqrt {d \csc (e+f x)} \, dx=\int { \sqrt {d \csc \left (f x + e\right )} \csc \left (f x + e\right )^{2} \,d x } \]
[In]
[Out]
Timed out. \[ \int \csc ^2(e+f x) \sqrt {d \csc (e+f x)} \, dx=\int \frac {\sqrt {\frac {d}{\sin \left (e+f\,x\right )}}}{{\sin \left (e+f\,x\right )}^2} \,d x \]
[In]
[Out]