Integrand size = 12, antiderivative size = 43 \[ \int \sqrt {c \csc (a+b x)} \, dx=\frac {2 \sqrt {c \csc (a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right ),2\right ) \sqrt {\sin (a+b x)}}{b} \]
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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, 2720} \[ \int \sqrt {c \csc (a+b x)} \, dx=\frac {2 \sqrt {\sin (a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right ),2\right ) \sqrt {c \csc (a+b x)}}{b} \]
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Rule 2720
Rule 3856
Rubi steps \begin{align*} \text {integral}& = \left (\sqrt {c \csc (a+b x)} \sqrt {\sin (a+b x)}\right ) \int \frac {1}{\sqrt {\sin (a+b x)}} \, dx \\ & = \frac {2 \sqrt {c \csc (a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right ),2\right ) \sqrt {\sin (a+b x)}}{b} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.98 \[ \int \sqrt {c \csc (a+b x)} \, dx=-\frac {2 \sqrt {c \csc (a+b x)} \operatorname {EllipticF}\left (\frac {1}{4} (-2 a+\pi -2 b x),2\right ) \sqrt {\sin (a+b x)}}{b} \]
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Result contains complex when optimal does not.
Time = 3.44 (sec) , antiderivative size = 120, normalized size of antiderivative = 2.79
method | result | size |
default | \(\frac {i \left (1+\cos \left (x b +a \right )\right ) \sqrt {2}\, \sqrt {c \csc \left (x b +a \right )}\, \sqrt {-i \left (i-\cot \left (x b +a \right )+\csc \left (x b +a \right )\right )}\, \sqrt {-i \left (i+\cot \left (x b +a \right )-\csc \left (x b +a \right )\right )}\, \sqrt {i \left (\csc \left (x b +a \right )-\cot \left (x b +a \right )\right )}\, \operatorname {EllipticF}\left (\sqrt {-i \left (i-\cot \left (x b +a \right )+\csc \left (x b +a \right )\right )}, \frac {\sqrt {2}}{2}\right )}{b}\) | \(120\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.28 \[ \int \sqrt {c \csc (a+b x)} \, dx=\frac {-i \, \sqrt {2 i \, c} {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right ) + i \, \sqrt {-2 i \, c} {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )}{b} \]
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\[ \int \sqrt {c \csc (a+b x)} \, dx=\int \sqrt {c \csc {\left (a + b x \right )}}\, dx \]
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\[ \int \sqrt {c \csc (a+b x)} \, dx=\int { \sqrt {c \csc \left (b x + a\right )} \,d x } \]
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\[ \int \sqrt {c \csc (a+b x)} \, dx=\int { \sqrt {c \csc \left (b x + a\right )} \,d x } \]
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Time = 22.32 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.47 \[ \int \sqrt {c \csc (a+b x)} \, dx=-\frac {2\,\sqrt {\sin \left (a+b\,x\right )}\,\mathrm {F}\left (\mathrm {asin}\left (\frac {\sqrt {2}\,\sqrt {1-\sin \left (a+b\,x\right )}}{2}\right )\middle |2\right )\,\sqrt {\frac {c}{\sin \left (a+b\,x\right )}}\,\sqrt {{\cos \left (a+b\,x\right )}^2}}{b\,\cos \left (a+b\,x\right )} \]
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