Integrand size = 10, antiderivative size = 16 \[ \int \frac {1}{\sqrt {\cos (a+b x)}} \, dx=\frac {2 \operatorname {EllipticF}\left (\frac {1}{2} (a+b x),2\right )}{b} \]
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Time = 0.01 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2720} \[ \int \frac {1}{\sqrt {\cos (a+b x)}} \, dx=\frac {2 \operatorname {EllipticF}\left (\frac {1}{2} (a+b x),2\right )}{b} \]
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Rule 2720
Rubi steps \begin{align*} \text {integral}& = \frac {2 \operatorname {EllipticF}\left (\frac {1}{2} (a+b x),2\right )}{b} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {\cos (a+b x)}} \, dx=\frac {2 \operatorname {EllipticF}\left (\frac {1}{2} (a+b x),2\right )}{b} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.25 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12
method | result | size |
default | \(\frac {2 \,\operatorname {am}^{-1}\left (\frac {b x}{2}+\frac {a}{2}| \sqrt {2}\right )}{b}\) | \(18\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 51, normalized size of antiderivative = 3.19 \[ \int \frac {1}{\sqrt {\cos (a+b x)}} \, dx=\frac {-i \, \sqrt {2} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right ) + i \, \sqrt {2} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )}{b} \]
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\[ \int \frac {1}{\sqrt {\cos (a+b x)}} \, dx=\int \frac {1}{\sqrt {\cos {\left (a + b x \right )}}}\, dx \]
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\[ \int \frac {1}{\sqrt {\cos (a+b x)}} \, dx=\int { \frac {1}{\sqrt {\cos \left (b x + a\right )}} \,d x } \]
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\[ \int \frac {1}{\sqrt {\cos (a+b x)}} \, dx=\int { \frac {1}{\sqrt {\cos \left (b x + a\right )}} \,d x } \]
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Time = 13.59 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.94 \[ \int \frac {1}{\sqrt {\cos (a+b x)}} \, dx=\frac {2\,\mathrm {F}\left (\frac {a}{2}+\frac {b\,x}{2}\middle |2\right )}{b} \]
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