Optimal. Leaf size=125 \[ -\frac{2 \cot (c+d x) \sqrt{-\frac{b (1-\sec (c+d x))}{a+b \sec (c+d x)}} \sqrt{\frac{b (\sec (c+d x)+1)}{a+b \sec (c+d x)}} (a+b \sec (c+d x)) \Pi \left (\frac{a}{a+b};\sin ^{-1}\left (\frac{\sqrt{a+b}}{\sqrt{a+b \sec (c+d x)}}\right )|\frac{a-b}{a+b}\right )}{d \sqrt{a+b}} \]
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Rubi [A] time = 0.0288398, antiderivative size = 125, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {3780} \[ -\frac{2 \cot (c+d x) \sqrt{-\frac{b (1-\sec (c+d x))}{a+b \sec (c+d x)}} \sqrt{\frac{b (\sec (c+d x)+1)}{a+b \sec (c+d x)}} (a+b \sec (c+d x)) \Pi \left (\frac{a}{a+b};\sin ^{-1}\left (\frac{\sqrt{a+b}}{\sqrt{a+b \sec (c+d x)}}\right )|\frac{a-b}{a+b}\right )}{d \sqrt{a+b}} \]
Antiderivative was successfully verified.
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Rule 3780
Rubi steps
\begin{align*} \int \sqrt{a+b \sec (c+d x)} \, dx &=-\frac{2 \cot (c+d x) \Pi \left (\frac{a}{a+b};\sin ^{-1}\left (\frac{\sqrt{a+b}}{\sqrt{a+b \sec (c+d x)}}\right )|\frac{a-b}{a+b}\right ) \sqrt{-\frac{b (1-\sec (c+d x))}{a+b \sec (c+d x)}} \sqrt{\frac{b (1+\sec (c+d x))}{a+b \sec (c+d x)}} (a+b \sec (c+d x))}{\sqrt{a+b} d}\\ \end{align*}
Mathematica [A] time = 1.56896, size = 153, normalized size = 1.22 \[ -\frac{4 \cos ^2\left (\frac{1}{2} (c+d x)\right ) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} \sqrt{a+b \sec (c+d x)} \left ((a-b) \text{EllipticF}\left (\sin ^{-1}\left (\tan \left (\frac{1}{2} (c+d x)\right )\right ),\frac{a-b}{a+b}\right )+2 a \Pi \left (-1;-\sin ^{-1}\left (\tan \left (\frac{1}{2} (c+d x)\right )\right )|\frac{a-b}{a+b}\right )\right )}{d (a \cos (c+d x)+b)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.265, size = 215, normalized size = 1.7 \begin{align*} -2\,{\frac{ \left ( \cos \left ( dx+c \right ) +1 \right ) ^{2} \left ( -1+\cos \left ( dx+c \right ) \right ) }{d \left ( b+a\cos \left ( dx+c \right ) \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{2}}\sqrt{{\frac{b+a\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) }}}\sqrt{{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}\sqrt{{\frac{b+a\cos \left ( dx+c \right ) }{ \left ( a+b \right ) \left ( \cos \left ( dx+c \right ) +1 \right ) }}} \left ({\it EllipticF} \left ({\frac{-1+\cos \left ( dx+c \right ) }{\sin \left ( dx+c \right ) }},\sqrt{{\frac{a-b}{a+b}}} \right ) a-{\it EllipticF} \left ({\frac{-1+\cos \left ( dx+c \right ) }{\sin \left ( dx+c \right ) }},\sqrt{{\frac{a-b}{a+b}}} \right ) b-2\,a{\it EllipticPi} \left ({\frac{-1+\cos \left ( dx+c \right ) }{\sin \left ( dx+c \right ) }},-1,\sqrt{{\frac{a-b}{a+b}}} \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 \sqrt{b \sec \left (d x + c\right ) + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 \sqrt{a + b \sec{\left (c + d 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 \sqrt{b \sec \left (d x + c\right ) + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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