Optimal. Leaf size=110 \[ \frac{2 \cot (e+f x) (4-5 \sec (e+f x)) \sqrt{\frac{1-\sec (e+f x)}{4-5 \sec (e+f x)}} \sqrt{\frac{\sec (e+f x)+1}{4-5 \sec (e+f x)}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{3 \sec (e+f x)+2}}{\sqrt{5} \sqrt{5 \sec (e+f x)-4}}\right ),45\right )}{f} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.12409, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.029, Rules used = {3984} \[ \frac{2 \cot (e+f x) (4-5 \sec (e+f x)) \sqrt{\frac{1-\sec (e+f x)}{4-5 \sec (e+f x)}} \sqrt{\frac{\sec (e+f x)+1}{4-5 \sec (e+f x)}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{3 \sec (e+f x)+2}}{\sqrt{5} \sqrt{5 \sec (e+f x)-4}}\right )\right |45\right )}{f} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3984
Rubi steps
\begin{align*} \int \frac{\sec (e+f x)}{\sqrt{2+3 \sec (e+f x)} \sqrt{-4+5 \sec (e+f x)}} \, dx &=\frac{2 \cot (e+f x) F\left (\left .\sin ^{-1}\left (\frac{\sqrt{2+3 \sec (e+f x)}}{\sqrt{5} \sqrt{-4+5 \sec (e+f x)}}\right )\right |45\right ) (4-5 \sec (e+f x)) \sqrt{\frac{1-\sec (e+f x)}{4-5 \sec (e+f x)}} \sqrt{\frac{1+\sec (e+f x)}{4-5 \sec (e+f x)}}}{f}\\ \end{align*}
Mathematica [A] time = 1.75091, size = 176, normalized size = 1.6 \[ -\frac{4 \sin ^4\left (\frac{1}{2} (e+f x)\right ) \sqrt{-\cot ^2\left (\frac{1}{2} (e+f x)\right )} \csc (e+f x) \sec (e+f x) \sqrt{-(2 \cos (e+f x)+3) \csc ^2\left (\frac{1}{2} (e+f x)\right )} \sqrt{-(4 \cos (e+f x)-5) \csc ^2\left (\frac{1}{2} (e+f x)\right )} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{5}{22}} \sqrt{\frac{4 \cos (e+f x)-5}{\cos (e+f x)-1}}\right ),\frac{44}{45}\right )}{3 \sqrt{5} f \sqrt{3 \sec (e+f x)+2} \sqrt{5 \sec (e+f x)-4}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.339, size = 177, normalized size = 1.6 \begin{align*}{\frac{{\frac{i}{5}}\sqrt{5}\sqrt{10}\cos \left ( fx+e \right ) \left ( \sin \left ( fx+e \right ) \right ) ^{2}}{f \left ( 8\, \left ( \cos \left ( fx+e \right ) \right ) ^{3}-6\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}-17\,\cos \left ( fx+e \right ) +15 \right ) }\sqrt{{\frac{2\,\cos \left ( fx+e \right ) +3}{1+\cos \left ( fx+e \right ) }}}\sqrt{-2\,{\frac{4\,\cos \left ( fx+e \right ) -5}{1+\cos \left ( fx+e \right ) }}}\sqrt{{\frac{2\,\cos \left ( fx+e \right ) +3}{\cos \left ( fx+e \right ) }}}\sqrt{-{\frac{4\,\cos \left ( fx+e \right ) -5}{\cos \left ( fx+e \right ) }}}{\it EllipticF} \left ({\frac{{\frac{i}{5}} \left ( -1+\cos \left ( fx+e \right ) \right ) \sqrt{5}}{\sin \left ( fx+e \right ) }},3\,\sqrt{5} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sec \left (f x + e\right )}{\sqrt{5 \, \sec \left (f x + e\right ) - 4} \sqrt{3 \, \sec \left (f x + e\right ) + 2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{5 \, \sec \left (f x + e\right ) - 4} \sqrt{3 \, \sec \left (f x + e\right ) + 2} \sec \left (f x + e\right )}{15 \, \sec \left (f x + e\right )^{2} - 2 \, \sec \left (f x + e\right ) - 8}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sec{\left (e + f x \right )}}{\sqrt{3 \sec{\left (e + f x \right )} + 2} \sqrt{5 \sec{\left (e + f x \right )} - 4}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sec \left (f x + e\right )}{\sqrt{5 \, \sec \left (f x + e\right ) - 4} \sqrt{3 \, \sec \left (f x + e\right ) + 2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]