Integrand size = 12, antiderivative size = 43 \[ \int \frac {1}{\sqrt {c \csc (a+b x)}} \, dx=\frac {2 E\left (\left .\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right )\right |2\right )}{b \sqrt {c \csc (a+b x)} \sqrt {\sin (a+b x)}} \]
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Time = 0.02 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3856, 2719} \[ \int \frac {1}{\sqrt {c \csc (a+b x)}} \, dx=\frac {2 E\left (\left .\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right )\right |2\right )}{b \sqrt {\sin (a+b x)} \sqrt {c \csc (a+b x)}} \]
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Rule 2719
Rule 3856
Rubi steps \begin{align*} \text {integral}& = \frac {\int \sqrt {\sin (a+b x)} \, dx}{\sqrt {c \csc (a+b x)} \sqrt {\sin (a+b x)}} \\ & = \frac {2 E\left (\left .\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right )\right |2\right )}{b \sqrt {c \csc (a+b x)} \sqrt {\sin (a+b x)}} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.98 \[ \int \frac {1}{\sqrt {c \csc (a+b x)}} \, dx=-\frac {2 E\left (\left .\frac {1}{4} (-2 a+\pi -2 b x)\right |2\right )}{b \sqrt {c \csc (a+b x)} \sqrt {\sin (a+b x)}} \]
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Result contains complex when optimal does not.
Time = 3.92 (sec) , antiderivative size = 297, normalized size of antiderivative = 6.91
method | result | size |
risch | \(-\frac {i \sqrt {2}}{b \sqrt {\frac {i c \,{\mathrm e}^{i \left (x b +a \right )}}{{\mathrm e}^{2 i \left (x b +a \right )}-1}}}+\frac {i \left (-\frac {2 i \left (i c \,{\mathrm e}^{2 i \left (x b +a \right )}-i c \right )}{c \sqrt {{\mathrm e}^{i \left (x b +a \right )} \left (i c \,{\mathrm e}^{2 i \left (x b +a \right )}-i c \right )}}-\frac {\sqrt {{\mathrm e}^{i \left (x b +a \right )}+1}\, \sqrt {-2 \,{\mathrm e}^{i \left (x b +a \right )}+2}\, \sqrt {-{\mathrm e}^{i \left (x b +a \right )}}\, \left (-2 \operatorname {EllipticE}\left (\sqrt {{\mathrm e}^{i \left (x b +a \right )}+1}, \frac {\sqrt {2}}{2}\right )+\operatorname {EllipticF}\left (\sqrt {{\mathrm e}^{i \left (x b +a \right )}+1}, \frac {\sqrt {2}}{2}\right )\right )}{\sqrt {i c \,{\mathrm e}^{3 i \left (x b +a \right )}-i c \,{\mathrm e}^{i \left (x b +a \right )}}}\right ) \sqrt {2}\, \sqrt {i c \,{\mathrm e}^{i \left (x b +a \right )} \left ({\mathrm e}^{2 i \left (x b +a \right )}-1\right )}}{b \sqrt {\frac {i c \,{\mathrm e}^{i \left (x b +a \right )}}{{\mathrm e}^{2 i \left (x b +a \right )}-1}}\, \left ({\mathrm e}^{2 i \left (x b +a \right )}-1\right )}\) | \(297\) |
default | \(-\frac {\csc \left (x b +a \right ) \left (2 \cos \left (x b +a \right ) \sqrt {-i \left (i+\cot \left (x b +a \right )-\csc \left (x b +a \right )\right )}\, \sqrt {-i \left (i-\cot \left (x b +a \right )+\csc \left (x b +a \right )\right )}\, \sqrt {i \left (\csc \left (x b +a \right )-\cot \left (x b +a \right )\right )}\, \operatorname {EllipticE}\left (\sqrt {-i \left (i-\cot \left (x b +a \right )+\csc \left (x b +a \right )\right )}, \frac {\sqrt {2}}{2}\right )-\cos \left (x b +a \right ) \sqrt {-i \left (i+\cot \left (x b +a \right )-\csc \left (x b +a \right )\right )}\, \sqrt {-i \left (i-\cot \left (x b +a \right )+\csc \left (x b +a \right )\right )}\, \sqrt {i \left (\csc \left (x b +a \right )-\cot \left (x b +a \right )\right )}\, \operatorname {EllipticF}\left (\sqrt {-i \left (i-\cot \left (x b +a \right )+\csc \left (x b +a \right )\right )}, \frac {\sqrt {2}}{2}\right )+2 \sqrt {-i \left (i+\cot \left (x b +a \right )-\csc \left (x b +a \right )\right )}\, \sqrt {-i \left (i-\cot \left (x b +a \right )+\csc \left (x b +a \right )\right )}\, \sqrt {i \left (\csc \left (x b +a \right )-\cot \left (x b +a \right )\right )}\, \operatorname {EllipticE}\left (\sqrt {-i \left (i-\cot \left (x b +a \right )+\csc \left (x b +a \right )\right )}, \frac {\sqrt {2}}{2}\right )-\sqrt {-i \left (i+\cot \left (x b +a \right )-\csc \left (x b +a \right )\right )}\, \sqrt {-i \left (i-\cot \left (x b +a \right )+\csc \left (x b +a \right )\right )}\, \sqrt {i \left (\csc \left (x b +a \right )-\cot \left (x b +a \right )\right )}\, \operatorname {EllipticF}\left (\sqrt {-i \left (i-\cot \left (x b +a \right )+\csc \left (x b +a \right )\right )}, \frac {\sqrt {2}}{2}\right )+\cos \left (x b +a \right ) \sqrt {2}-\sqrt {2}\right ) \sqrt {2}}{b \sqrt {c \csc \left (x b +a \right )}}\) | \(429\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.08 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.44 \[ \int \frac {1}{\sqrt {c \csc (a+b x)}} \, dx=\frac {\sqrt {2 i \, c} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right )\right ) + \sqrt {-2 i \, c} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )\right )}{b c} \]
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\[ \int \frac {1}{\sqrt {c \csc (a+b x)}} \, dx=\int \frac {1}{\sqrt {c \csc {\left (a + b x \right )}}}\, dx \]
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\[ \int \frac {1}{\sqrt {c \csc (a+b x)}} \, dx=\int { \frac {1}{\sqrt {c \csc \left (b x + a\right )}} \,d x } \]
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\[ \int \frac {1}{\sqrt {c \csc (a+b x)}} \, dx=\int { \frac {1}{\sqrt {c \csc \left (b x + a\right )}} \,d x } \]
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Timed out. \[ \int \frac {1}{\sqrt {c \csc (a+b x)}} \, dx=\int \frac {1}{\sqrt {\frac {c}{\sin \left (a+b\,x\right )}}} \,d x \]
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