Optimal. Leaf size=192 \[ \frac{2 \sqrt{a+b} \cot (e+f x) (c+d \sec (e+f x)) \sqrt{\frac{(b c-a d) (1-\sec (e+f x))}{(a+b) (c+d \sec (e+f x))}} \sqrt{-\frac{(b c-a d) (\sec (e+f x)+1)}{(a-b) (c+d \sec (e+f x))}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{c+d} \sqrt{a+b \sec (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sec (e+f x)}}\right ),\frac{(a+b) (c-d)}{(a-b) (c+d)}\right )}{f \sqrt{c+d} (b c-a d)} \]
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Rubi [A] time = 0.168126, antiderivative size = 192, 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} \[ \frac{2 \sqrt{a+b} \cot (e+f x) (c+d \sec (e+f x)) \sqrt{\frac{(b c-a d) (1-\sec (e+f x))}{(a+b) (c+d \sec (e+f x))}} \sqrt{-\frac{(b c-a d) (\sec (e+f x)+1)}{(a-b) (c+d \sec (e+f x))}} F\left (\sin ^{-1}\left (\frac{\sqrt{c+d} \sqrt{a+b \sec (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sec (e+f x)}}\right )|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right )}{f \sqrt{c+d} (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 3984
Rubi steps
\begin{align*} \int \frac{\sec (e+f x)}{\sqrt{a+b \sec (e+f x)} \sqrt{c+d \sec (e+f x)}} \, dx &=\frac{2 \sqrt{a+b} \cot (e+f x) F\left (\sin ^{-1}\left (\frac{\sqrt{c+d} \sqrt{a+b \sec (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sec (e+f x)}}\right )|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right ) \sqrt{\frac{(b c-a d) (1-\sec (e+f x))}{(a+b) (c+d \sec (e+f x))}} \sqrt{-\frac{(b c-a d) (1+\sec (e+f x))}{(a-b) (c+d \sec (e+f x))}} (c+d \sec (e+f x))}{\sqrt{c+d} (b c-a d) f}\\ \end{align*}
Mathematica [A] time = 3.48568, size = 233, normalized size = 1.21 \[ \frac{4 \sin ^2\left (\frac{1}{2} (e+f x)\right ) \csc (e+f x) \sqrt{a+b \sec (e+f x)} \sqrt{\frac{(c+d) \cot ^2\left (\frac{1}{2} (e+f x)\right )}{c-d}} \sqrt{\frac{(a+b) \csc ^2\left (\frac{1}{2} (e+f x)\right ) (c \cos (e+f x)+d)}{a d-b c}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{(a+b) \csc ^2\left (\frac{1}{2} (e+f x)\right ) (c \cos (e+f x)+d)}{a d-b c}}}{\sqrt{2}}\right ),\frac{2 (b c-a d)}{(a+b) (c-d)}\right )}{f (a+b) \sqrt{c+d \sec (e+f x)} \sqrt{\frac{(c+d) \csc ^2\left (\frac{1}{2} (e+f x)\right ) (a \cos (e+f x)+b)}{b c-a d}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.389, size = 219, normalized size = 1.1 \begin{align*} 2\,{\frac{ \left ( \sin \left ( fx+e \right ) \right ) ^{2}\cos \left ( fx+e \right ) }{f \left ( -1+\cos \left ( fx+e \right ) \right ) \left ( d+c\cos \left ( fx+e \right ) \right ) \left ( a\cos \left ( fx+e \right ) +b \right ) }\sqrt{{\frac{a\cos \left ( fx+e \right ) +b}{\cos \left ( fx+e \right ) }}}\sqrt{{\frac{d+c\cos \left ( fx+e \right ) }{\cos \left ( fx+e \right ) }}}\sqrt{{\frac{d+c\cos \left ( fx+e \right ) }{ \left ( c+d \right ) \left ( 1+\cos \left ( fx+e \right ) \right ) }}}\sqrt{{\frac{a\cos \left ( fx+e \right ) +b}{ \left ( a+b \right ) \left ( 1+\cos \left ( fx+e \right ) \right ) }}}{\it EllipticF} \left ({\frac{-1+\cos \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }\sqrt{{\frac{a-b}{a+b}}}},\sqrt{{\frac{ \left ( c-d \right ) \left ( a+b \right ) }{ \left ( a-b \right ) \left ( c+d \right ) }}} \right ){\frac{1}{\sqrt{{\frac{a-b}{a+b}}}}}} \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{b \sec \left (f x + e\right ) + a} \sqrt{d \sec \left (f x + e\right ) + c}}\,{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{b \sec \left (f x + e\right ) + a} \sqrt{d \sec \left (f x + e\right ) + c} \sec \left (f x + e\right )}{b d \sec \left (f x + e\right )^{2} + a c +{\left (b c + a d\right )} \sec \left (f x + e\right )}, 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{a + b \sec{\left (e + f x \right )}} \sqrt{c + d \sec{\left (e + f x \right )}}}\, 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{b \sec \left (f x + e\right ) + a} \sqrt{d \sec \left (f x + e\right ) + c}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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