Optimal. Leaf size=102 \[ \frac{2 \tan (e+f x) \sqrt{\frac{c+d \sec (e+f x)}{c+d}} \Pi \left (\frac{2 a}{a+b};\sin ^{-1}\left (\frac{\sqrt{1-\sec (e+f x)}}{\sqrt{2}}\right )|\frac{2 d}{c+d}\right )}{f (a+b) \sqrt{-\tan ^2(e+f x)} \sqrt{c+d \sec (e+f x)}} \]
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Rubi [A] time = 0.20217, antiderivative size = 102, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074, Rules used = {2829, 3973} \[ \frac{2 \tan (e+f x) \sqrt{\frac{c+d \sec (e+f x)}{c+d}} \Pi \left (\frac{2 a}{a+b};\sin ^{-1}\left (\frac{\sqrt{1-\sec (e+f x)}}{\sqrt{2}}\right )|\frac{2 d}{c+d}\right )}{f (a+b) \sqrt{-\tan ^2(e+f x)} \sqrt{c+d \sec (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 2829
Rule 3973
Rubi steps
\begin{align*} \int \frac{1}{(a+b \cos (e+f x)) \sqrt{c+d \sec (e+f x)}} \, dx &=\int \frac{\sec (e+f x)}{(b+a \sec (e+f x)) \sqrt{c+d \sec (e+f x)}} \, dx\\ &=\frac{2 \Pi \left (\frac{2 a}{a+b};\sin ^{-1}\left (\frac{\sqrt{1-\sec (e+f x)}}{\sqrt{2}}\right )|\frac{2 d}{c+d}\right ) \sqrt{\frac{c+d \sec (e+f x)}{c+d}} \tan (e+f x)}{(a+b) f \sqrt{c+d \sec (e+f x)} \sqrt{-\tan ^2(e+f x)}}\\ \end{align*}
Mathematica [A] time = 4.40852, size = 189, normalized size = 1.85 \[ -\frac{2 \sqrt{\sec (e+f x)} \sqrt{\sec (e+f x)+1} \sqrt{\cos (e+f x) \sec ^2\left (\frac{1}{2} (e+f x)\right )} \sqrt{\frac{c \cos (e+f x)+d}{(c+d) (\cos (e+f x)+1)}} \left ((a+b) F\left (\sin ^{-1}\left (\tan \left (\frac{1}{2} (e+f x)\right )\right )|\frac{c-d}{c+d}\right )+2 a \Pi \left (\frac{b-a}{a+b};-\sin ^{-1}\left (\tan \left (\frac{1}{2} (e+f x)\right )\right )|\frac{c-d}{c+d}\right )\right )}{f (a-b) (a+b) \sqrt{\sec ^2\left (\frac{1}{2} (e+f x)\right )} \sqrt{c+d \sec (e+f x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.214, size = 237, normalized size = 2.3 \begin{align*} -2\,{\frac{ \left ( \cos \left ( fx+e \right ) +1 \right ) ^{2} \left ( \cos \left ( fx+e \right ) -1 \right ) }{f \left ( a-b \right ) \left ( a+b \right ) \left ( d+c\cos \left ( fx+e \right ) \right ) \left ( \sin \left ( fx+e \right ) \right ) ^{2}}\sqrt{{\frac{d+c\cos \left ( fx+e \right ) }{\cos \left ( fx+e \right ) }}}\sqrt{{\frac{\cos \left ( fx+e \right ) }{\cos \left ( fx+e \right ) +1}}}\sqrt{{\frac{d+c\cos \left ( fx+e \right ) }{ \left ( c+d \right ) \left ( \cos \left ( fx+e \right ) +1 \right ) }}} \left ( a{\it EllipticF} \left ({\frac{\cos \left ( fx+e \right ) -1}{\sin \left ( fx+e \right ) }},\sqrt{{\frac{c-d}{c+d}}} \right ) +b{\it EllipticF} \left ({\frac{\cos \left ( fx+e \right ) -1}{\sin \left ( fx+e \right ) }},\sqrt{{\frac{c-d}{c+d}}} \right ) -2\,a{\it EllipticPi} \left ({\frac{\cos \left ( fx+e \right ) -1}{\sin \left ( fx+e \right ) }},-{\frac{a-b}{a+b}},\sqrt{{\frac{c-d}{c+d}}} \right ) \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{1}{{\left (b \cos \left (f x + e\right ) + a\right )} \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{d \sec \left (f x + e\right ) + c}}{b c \cos \left (f x + e\right ) + a c +{\left (b d \cos \left (f x + e\right ) + 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{1}{\left (a + b \cos{\left (e + f x \right )}\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{1}{{\left (b \cos \left (f x + e\right ) + a\right )} \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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