Integrand size = 12, antiderivative size = 38 \[ \int \sqrt {b \sec (e+f x)} \, dx=\frac {2 \sqrt {\cos (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} (e+f x),2\right ) \sqrt {b \sec (e+f x)}}{f} \]
[Out]
Time = 0.02 (sec) , antiderivative size = 38, 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} \[ \int \sqrt {b \sec (e+f x)} \, dx=\frac {2 \sqrt {\cos (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} (e+f x),2\right ) \sqrt {b \sec (e+f x)}}{f} \]
[In]
[Out]
Rule 2720
Rule 3856
Rubi steps \begin{align*} \text {integral}& = \left (\sqrt {\cos (e+f x)} \sqrt {b \sec (e+f x)}\right ) \int \frac {1}{\sqrt {\cos (e+f x)}} \, dx \\ & = \frac {2 \sqrt {\cos (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} (e+f x),2\right ) \sqrt {b \sec (e+f x)}}{f} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00 \[ \int \sqrt {b \sec (e+f x)} \, dx=\frac {2 \sqrt {\cos (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} (e+f x),2\right ) \sqrt {b \sec (e+f x)}}{f} \]
[In]
[Out]
Result contains complex when optimal does not.
Time = 0.34 (sec) , antiderivative size = 77, normalized size of antiderivative = 2.03
method | result | size |
default | \(\frac {2 i \left (\cos \left (f x +e \right )+1\right ) \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, F\left (i \left (\cot \left (f x +e \right )-\csc \left (f x +e \right )\right ), i\right ) \sqrt {b \sec \left (f x +e \right )}}{f}\) | \(77\) |
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.50 \[ \int \sqrt {b \sec (e+f x)} \, dx=\frac {-i \, \sqrt {2} \sqrt {b} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right ) + i \, \sqrt {2} \sqrt {b} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )}{f} \]
[In]
[Out]
\[ \int \sqrt {b \sec (e+f x)} \, dx=\int \sqrt {b \sec {\left (e + f x \right )}}\, dx \]
[In]
[Out]
\[ \int \sqrt {b \sec (e+f x)} \, dx=\int { \sqrt {b \sec \left (f x + e\right )} \,d x } \]
[In]
[Out]
\[ \int \sqrt {b \sec (e+f x)} \, dx=\int { \sqrt {b \sec \left (f x + e\right )} \,d x } \]
[In]
[Out]
Time = 0.18 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.92 \[ \int \sqrt {b \sec (e+f x)} \, dx=\frac {2\,\sqrt {\cos \left (e+f\,x\right )}\,\sqrt {\frac {b}{\cos \left (e+f\,x\right )}}\,\mathrm {F}\left (\frac {e}{2}+\frac {f\,x}{2}\middle |2\right )}{f} \]
[In]
[Out]