Integrand size = 10, antiderivative size = 41 \[ \int \frac {1}{\sqrt {\csc (a+b x)}} \, dx=\frac {2 \sqrt {\csc (a+b x)} E\left (\left .\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right )\right |2\right ) \sqrt {\sin (a+b x)}}{b} \]
[Out]
Time = 0.02 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3856, 2719} \[ \int \frac {1}{\sqrt {\csc (a+b x)}} \, dx=\frac {2 \sqrt {\sin (a+b x)} \sqrt {\csc (a+b x)} E\left (\left .\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right )\right |2\right )}{b} \]
[In]
[Out]
Rule 2719
Rule 3856
Rubi steps \begin{align*} \text {integral}& = \left (\sqrt {\csc (a+b x)} \sqrt {\sin (a+b x)}\right ) \int \sqrt {\sin (a+b x)} \, dx \\ & = \frac {2 \sqrt {\csc (a+b x)} E\left (\left .\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right )\right |2\right ) \sqrt {\sin (a+b x)}}{b} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.98 \[ \int \frac {1}{\sqrt {\csc (a+b x)}} \, dx=-\frac {2 \sqrt {\csc (a+b x)} E\left (\left .\frac {1}{4} (-2 a+\pi -2 b x)\right |2\right ) \sqrt {\sin (a+b x)}}{b} \]
[In]
[Out]
Time = 0.65 (sec) , antiderivative size = 91, normalized size of antiderivative = 2.22
method | result | size |
default | \(-\frac {\sqrt {\sin \left (x b +a \right )+1}\, \sqrt {-2 \sin \left (x b +a \right )+2}\, \sqrt {-\sin \left (x b +a \right )}\, \left (2 \operatorname {EllipticE}\left (\sqrt {\sin \left (x b +a \right )+1}, \frac {\sqrt {2}}{2}\right )-\operatorname {EllipticF}\left (\sqrt {\sin \left (x b +a \right )+1}, \frac {\sqrt {2}}{2}\right )\right )}{\cos \left (x b +a \right ) \sqrt {\sin \left (x b +a \right )}\, b}\) | \(91\) |
risch | \(-\frac {i \sqrt {2}}{b \sqrt {\frac {i {\mathrm e}^{i \left (x b +a \right )}}{{\mathrm e}^{2 i \left (x b +a \right )}-1}}}+\frac {i \left (-\frac {2 i \left (-i+i {\mathrm e}^{2 i \left (x b +a \right )}\right )}{\sqrt {{\mathrm e}^{i \left (x b +a \right )} \left (-i+i {\mathrm e}^{2 i \left (x b +a \right )}\right )}}-\frac {\sqrt {{\mathrm e}^{i \left (x b +a \right )}+1}\, \sqrt {-2 \,{\mathrm e}^{i \left (x b +a \right )}+2}\, \sqrt {-{\mathrm e}^{i \left (x b +a \right )}}\, \left (-2 \operatorname {EllipticE}\left (\sqrt {{\mathrm e}^{i \left (x b +a \right )}+1}, \frac {\sqrt {2}}{2}\right )+\operatorname {EllipticF}\left (\sqrt {{\mathrm e}^{i \left (x b +a \right )}+1}, \frac {\sqrt {2}}{2}\right )\right )}{\sqrt {i {\mathrm e}^{3 i \left (x b +a \right )}-i {\mathrm e}^{i \left (x b +a \right )}}}\right ) \sqrt {2}\, \sqrt {i {\mathrm e}^{i \left (x b +a \right )} \left ({\mathrm e}^{2 i \left (x b +a \right )}-1\right )}}{b \sqrt {\frac {i {\mathrm e}^{i \left (x b +a \right )}}{{\mathrm e}^{2 i \left (x b +a \right )}-1}}\, \left ({\mathrm e}^{2 i \left (x b +a \right )}-1\right )}\) | \(283\) |
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.08 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.34 \[ \int \frac {1}{\sqrt {\csc (a+b x)}} \, dx=\frac {\sqrt {2 i} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right )\right ) + \sqrt {-2 i} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )\right )}{b} \]
[In]
[Out]
\[ \int \frac {1}{\sqrt {\csc (a+b x)}} \, dx=\int \frac {1}{\sqrt {\csc {\left (a + b x \right )}}}\, dx \]
[In]
[Out]
\[ \int \frac {1}{\sqrt {\csc (a+b x)}} \, dx=\int { \frac {1}{\sqrt {\csc \left (b x + a\right )}} \,d x } \]
[In]
[Out]
\[ \int \frac {1}{\sqrt {\csc (a+b x)}} \, dx=\int { \frac {1}{\sqrt {\csc \left (b x + a\right )}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {1}{\sqrt {\csc (a+b x)}} \, dx=\int \frac {1}{\sqrt {\frac {1}{\sin \left (a+b\,x\right )}}} \,d x \]
[In]
[Out]