Integrand size = 19, antiderivative size = 47 \[ \int \frac {\sec (e+f x)}{\sqrt {d \tan (e+f x)}} \, dx=\frac {\operatorname {EllipticF}\left (e-\frac {\pi }{4}+f x,2\right ) \sec (e+f x) \sqrt {\sin (2 e+2 f x)}}{f \sqrt {d \tan (e+f x)}} \]
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Time = 0.07 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2694, 2653, 2720} \[ \int \frac {\sec (e+f x)}{\sqrt {d \tan (e+f x)}} \, dx=\frac {\sqrt {\sin (2 e+2 f x)} \sec (e+f x) \operatorname {EllipticF}\left (e+f x-\frac {\pi }{4},2\right )}{f \sqrt {d \tan (e+f x)}} \]
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Rule 2653
Rule 2694
Rule 2720
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {\sin (e+f x)} \int \frac {1}{\sqrt {\cos (e+f x)} \sqrt {\sin (e+f x)}} \, dx}{\sqrt {\cos (e+f x)} \sqrt {d \tan (e+f x)}} \\ & = \frac {\left (\sec (e+f x) \sqrt {\sin (2 e+2 f x)}\right ) \int \frac {1}{\sqrt {\sin (2 e+2 f x)}} \, dx}{\sqrt {d \tan (e+f x)}} \\ & = \frac {\operatorname {EllipticF}\left (e-\frac {\pi }{4}+f x,2\right ) \sec (e+f x) \sqrt {\sin (2 e+2 f x)}}{f \sqrt {d \tan (e+f x)}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.18 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.64 \[ \int \frac {\sec (e+f x)}{\sqrt {d \tan (e+f x)}} \, dx=-\frac {2 \sqrt [4]{-1} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt [4]{-1} \sqrt {\tan (e+f x)}\right ),-1\right ) \sec ^3(e+f x) \sqrt {\tan (e+f x)}}{f \sqrt {d \tan (e+f x)} \left (1+\tan ^2(e+f x)\right )^{3/2}} \]
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Time = 0.97 (sec) , antiderivative size = 103, normalized size of antiderivative = 2.19
method | result | size |
default | \(\frac {F\left (\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}, \frac {\sqrt {2}}{2}\right ) \sqrt {\cot \left (f x +e \right )-\csc \left (f x +e \right )}\, \sqrt {\cot \left (f x +e \right )-\csc \left (f x +e \right )+1}\, \sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}\, \left (1+\sec \left (f x +e \right )\right ) \sqrt {2}}{f \sqrt {d \tan \left (f x +e \right )}}\) | \(103\) |
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Result contains complex when optimal does not.
Time = 0.09 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.21 \[ \int \frac {\sec (e+f x)}{\sqrt {d \tan (e+f x)}} \, dx=-\frac {\sqrt {i \, d} F(\arcsin \left (\cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\,|\,-1) + \sqrt {-i \, d} F(\arcsin \left (\cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\,|\,-1)}{d f} \]
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\[ \int \frac {\sec (e+f x)}{\sqrt {d \tan (e+f x)}} \, dx=\int \frac {\sec {\left (e + f x \right )}}{\sqrt {d \tan {\left (e + f x \right )}}}\, dx \]
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\[ \int \frac {\sec (e+f x)}{\sqrt {d \tan (e+f x)}} \, dx=\int { \frac {\sec \left (f x + e\right )}{\sqrt {d \tan \left (f x + e\right )}} \,d x } \]
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\[ \int \frac {\sec (e+f x)}{\sqrt {d \tan (e+f x)}} \, dx=\int { \frac {\sec \left (f x + e\right )}{\sqrt {d \tan \left (f x + e\right )}} \,d x } \]
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Timed out. \[ \int \frac {\sec (e+f x)}{\sqrt {d \tan (e+f x)}} \, dx=\int \frac {1}{\cos \left (e+f\,x\right )\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}} \,d x \]
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