Optimal. Leaf size=125 \[ \frac{2 i \cot (e+f x) \sqrt{\frac{1-\sec (e+f x)}{3 \sec (e+f x)+2}} \sqrt{\frac{\sec (e+f x)+1}{3 \sec (e+f x)+2}} (3 \sec (e+f x)+2) \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{5} \sqrt{4-5 \sec (e+f x)}}{\sqrt{3 \sec (e+f x)+2}}\right ),\frac{1}{45}\right )}{3 \sqrt{5} f} \]
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Rubi [A] time = 0.119524, antiderivative size = 125, 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} \[ \frac{2 i \cot (e+f x) \sqrt{\frac{1-\sec (e+f x)}{3 \sec (e+f x)+2}} \sqrt{\frac{\sec (e+f x)+1}{3 \sec (e+f x)+2}} (3 \sec (e+f x)+2) F\left (i \sinh ^{-1}\left (\frac{\sqrt{5} \sqrt{4-5 \sec (e+f x)}}{\sqrt{3 \sec (e+f x)+2}}\right )|\frac{1}{45}\right )}{3 \sqrt{5} f} \]
Antiderivative was successfully verified.
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Rule 3984
Rubi steps
\begin{align*} \int \frac{\sec (e+f x)}{\sqrt{4-5 \sec (e+f x)} \sqrt{2+3 \sec (e+f x)}} \, dx &=\frac{2 i \cot (e+f x) F\left (i \sinh ^{-1}\left (\frac{\sqrt{5} \sqrt{4-5 \sec (e+f x)}}{\sqrt{2+3 \sec (e+f x)}}\right )|\frac{1}{45}\right ) \sqrt{\frac{1-\sec (e+f x)}{2+3 \sec (e+f x)}} \sqrt{\frac{1+\sec (e+f x)}{2+3 \sec (e+f x)}} (2+3 \sec (e+f x))}{3 \sqrt{5} f}\\ \end{align*}
Mathematica [A] time = 0.444602, size = 176, normalized size = 1.41 \[ -\frac{4 \sin ^4\left (\frac{1}{2} (e+f x)\right ) \sqrt{-\cot ^2\left (\frac{1}{2} (e+f x)\right )} \csc (e+f x) \sec (e+f x) \sqrt{-(2 \cos (e+f x)+3) \csc ^2\left (\frac{1}{2} (e+f x)\right )} \sqrt{-(4 \cos (e+f x)-5) \csc ^2\left (\frac{1}{2} (e+f x)\right )} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{5}{22}} \sqrt{\frac{4 \cos (e+f x)-5}{\cos (e+f x)-1}}\right ),\frac{44}{45}\right )}{3 \sqrt{5} f \sqrt{4-5 \sec (e+f x)} \sqrt{3 \sec (e+f x)+2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.309, size = 170, normalized size = 1.4 \begin{align*}{\frac{-{\frac{i}{15}} \left ( \sin \left ( fx+e \right ) \right ) ^{2}\cos \left ( fx+e \right ) \sqrt{10}}{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{4\,\cos \left ( fx+e \right ) -5}{\cos \left ( fx+e \right ) }}}\sqrt{{\frac{2\,\cos \left ( fx+e \right ) +3}{\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}{1+\cos \left ( fx+e \right ) }}}{\it EllipticF} \left ({\frac{3\,i \left ( -1+\cos \left ( fx+e \right ) \right ) }{\sin \left ( fx+e \right ) }},{\frac{\sqrt{5}}{15}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sec \left (f x + e\right )}{\sqrt{3 \, \sec \left (f x + e\right ) + 2} \sqrt{-5 \, \sec \left (f x + e\right ) + 4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{3 \, \sec \left (f x + e\right ) + 2} \sqrt{-5 \, \sec \left (f x + e\right ) + 4} \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.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sec{\left (e + f x \right )}}{\sqrt{4 - 5 \sec{\left (e + f x \right )}} \sqrt{3 \sec{\left (e + f x \right )} + 2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sec \left (f x + e\right )}{\sqrt{3 \, \sec \left (f x + e\right ) + 2} \sqrt{-5 \, \sec \left (f x + e\right ) + 4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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