Integrand size = 8, antiderivative size = 19 \[ \int \frac {1}{\sqrt {\sin (b x)}} \, dx=-\frac {2 \operatorname {EllipticF}\left (\frac {\pi }{4}-\frac {b x}{2},2\right )}{b} \]
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Time = 0.00 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2720} \[ \int \frac {1}{\sqrt {\sin (b x)}} \, dx=-\frac {2 \operatorname {EllipticF}\left (\frac {\pi }{4}-\frac {b x}{2},2\right )}{b} \]
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Rule 2720
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \operatorname {EllipticF}\left (\frac {\pi }{4}-\frac {b x}{2},2\right )}{b} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.11 \[ \int \frac {1}{\sqrt {\sin (b x)}} \, dx=-\frac {2 \operatorname {EllipticF}\left (\frac {1}{2} \left (\frac {\pi }{2}-b x\right ),2\right )}{b} \]
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Time = 0.07 (sec) , antiderivative size = 57, normalized size of antiderivative = 3.00
method | result | size |
default | \(\frac {\sqrt {\sin \left (b x \right )+1}\, \sqrt {-2 \sin \left (b x \right )+2}\, \sqrt {-\sin \left (b x \right )}\, F\left (\sqrt {\sin \left (b x \right )+1}, \frac {\sqrt {2}}{2}\right )}{\cos \left (b x \right ) \sqrt {\sin \left (b x \right )}\, b}\) | \(57\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.11 (sec) , antiderivative size = 47, normalized size of antiderivative = 2.47 \[ \int \frac {1}{\sqrt {\sin (b x)}} \, dx=\frac {\sqrt {2} \sqrt {-i} {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x\right ) + i \, \sin \left (b x\right )\right ) + \sqrt {2} \sqrt {i} {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x\right ) - i \, \sin \left (b x\right )\right )}{b} \]
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\[ \int \frac {1}{\sqrt {\sin (b x)}} \, dx=\int \frac {1}{\sqrt {\sin {\left (b x \right )}}}\, dx \]
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\[ \int \frac {1}{\sqrt {\sin (b x)}} \, dx=\int { \frac {1}{\sqrt {\sin \left (b x\right )}} \,d x } \]
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\[ \int \frac {1}{\sqrt {\sin (b x)}} \, dx=\int { \frac {1}{\sqrt {\sin \left (b x\right )}} \,d x } \]
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Time = 0.09 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int \frac {1}{\sqrt {\sin (b x)}} \, dx=-\frac {2\,\mathrm {F}\left (\frac {\pi }{4}-\frac {b\,x}{2}\middle |2\right )}{b} \]
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