Optimal. Leaf size=170 \[ \frac{2 g (b c-a d) \sqrt{g \sec (e+f x)} \sqrt{\frac{c \cos (e+f x)+d}{c+d}} \Pi \left (\frac{2 a}{a+b};\frac{1}{2} (e+f x)|\frac{2 c}{c+d}\right )}{b f (a+b) \sqrt{c+d \sec (e+f x)}}+\frac{2 d g \sqrt{g \sec (e+f x)} \sqrt{\frac{c \cos (e+f x)+d}{c+d}} \Pi \left (2;\frac{1}{2} (e+f x)|\frac{2 c}{c+d}\right )}{b f \sqrt{c+d \sec (e+f x)}} \]
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Rubi [A] time = 0.848721, antiderivative size = 170, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.128, Rules used = {3971, 3859, 2807, 2805, 3975} \[ \frac{2 g (b c-a d) \sqrt{g \sec (e+f x)} \sqrt{\frac{c \cos (e+f x)+d}{c+d}} \Pi \left (\frac{2 a}{a+b};\frac{1}{2} (e+f x)|\frac{2 c}{c+d}\right )}{b f (a+b) \sqrt{c+d \sec (e+f x)}}+\frac{2 d g \sqrt{g \sec (e+f x)} \sqrt{\frac{c \cos (e+f x)+d}{c+d}} \Pi \left (2;\frac{1}{2} (e+f x)|\frac{2 c}{c+d}\right )}{b f \sqrt{c+d \sec (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 3971
Rule 3859
Rule 2807
Rule 2805
Rule 3975
Rubi steps
\begin{align*} \int \frac{(g \sec (e+f x))^{3/2} \sqrt{c+d \sec (e+f x)}}{a+b \sec (e+f x)} \, dx &=\frac{d \int \frac{(g \sec (e+f x))^{3/2}}{\sqrt{c+d \sec (e+f x)}} \, dx}{b}-\frac{(-b c+a d) \int \frac{(g \sec (e+f x))^{3/2}}{(a+b \sec (e+f x)) \sqrt{c+d \sec (e+f x)}} \, dx}{b}\\ &=\frac{\left (d g \sqrt{d+c \cos (e+f x)} \sqrt{g \sec (e+f x)}\right ) \int \frac{\sec (e+f x)}{\sqrt{d+c \cos (e+f x)}} \, dx}{b \sqrt{c+d \sec (e+f x)}}-\frac{\left ((-b c+a d) g \sqrt{d+c \cos (e+f x)} \sqrt{g \sec (e+f x)}\right ) \int \frac{1}{(b+a \cos (e+f x)) \sqrt{d+c \cos (e+f x)}} \, dx}{b \sqrt{c+d \sec (e+f x)}}\\ &=\frac{\left (d g \sqrt{\frac{d+c \cos (e+f x)}{c+d}} \sqrt{g \sec (e+f x)}\right ) \int \frac{\sec (e+f x)}{\sqrt{\frac{d}{c+d}+\frac{c \cos (e+f x)}{c+d}}} \, dx}{b \sqrt{c+d \sec (e+f x)}}-\frac{\left ((-b c+a d) g \sqrt{\frac{d+c \cos (e+f x)}{c+d}} \sqrt{g \sec (e+f x)}\right ) \int \frac{1}{(b+a \cos (e+f x)) \sqrt{\frac{d}{c+d}+\frac{c \cos (e+f x)}{c+d}}} \, dx}{b \sqrt{c+d \sec (e+f x)}}\\ &=\frac{2 d g \sqrt{\frac{d+c \cos (e+f x)}{c+d}} \Pi \left (2;\frac{1}{2} (e+f x)|\frac{2 c}{c+d}\right ) \sqrt{g \sec (e+f x)}}{b f \sqrt{c+d \sec (e+f x)}}+\frac{2 (b c-a d) g \sqrt{\frac{d+c \cos (e+f x)}{c+d}} \Pi \left (\frac{2 a}{a+b};\frac{1}{2} (e+f x)|\frac{2 c}{c+d}\right ) \sqrt{g \sec (e+f x)}}{b (a+b) f \sqrt{c+d \sec (e+f x)}}\\ \end{align*}
Mathematica [C] time = 4.02838, size = 223, normalized size = 1.31 \[ -\frac{2 i g \cot (e+f x) \sqrt{g \sec (e+f x)} \sqrt{-\frac{c (\cos (e+f x)-1)}{c+d}} \sqrt{\frac{c (\cos (e+f x)+1)}{c-d}} \sqrt{c+d \sec (e+f x)} \left (\Pi \left (1-\frac{c}{d};i \sinh ^{-1}\left (\sqrt{\frac{1}{c-d}} \sqrt{d+c \cos (e+f x)}\right )|\frac{d-c}{c+d}\right )-\Pi \left (\frac{a (d-c)}{a d-b c};i \sinh ^{-1}\left (\sqrt{\frac{1}{c-d}} \sqrt{d+c \cos (e+f x)}\right )|\frac{d-c}{c+d}\right )\right )}{b f \sqrt{\frac{1}{c-d}} \sqrt{c \cos (e+f x)+d}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.393, size = 465, normalized size = 2.7 \begin{align*}{\frac{-2\,i \left ( \cos \left ( fx+e \right ) \right ) ^{2}}{fb \left ( a+b \right ) \left ( a-b \right ) \left ( d+c\cos \left ( fx+e \right ) \right ) }\sqrt{{\frac{d+c\cos \left ( fx+e \right ) }{ \left ( c+d \right ) \left ( 1+\cos \left ( fx+e \right ) \right ) }}} \left ({\it EllipticF} \left ({\frac{i \left ( -1+\cos \left ( fx+e \right ) \right ) }{\sin \left ( fx+e \right ) }},\sqrt{-{\frac{c-d}{c+d}}} \right ) abc-{\it EllipticF} \left ({\frac{i \left ( -1+\cos \left ( fx+e \right ) \right ) }{\sin \left ( fx+e \right ) }},\sqrt{-{\frac{c-d}{c+d}}} \right ) abd+{\it EllipticF} \left ({\frac{i \left ( -1+\cos \left ( fx+e \right ) \right ) }{\sin \left ( fx+e \right ) }},\sqrt{-{\frac{c-d}{c+d}}} \right ){b}^{2}c-{\it EllipticF} \left ({\frac{i \left ( -1+\cos \left ( fx+e \right ) \right ) }{\sin \left ( fx+e \right ) }},\sqrt{-{\frac{c-d}{c+d}}} \right ){b}^{2}d+2\,{\it EllipticPi} \left ({\frac{i \left ( -1+\cos \left ( fx+e \right ) \right ) }{\sin \left ( fx+e \right ) }},-{\frac{a-b}{a+b}},i\sqrt{{\frac{c-d}{c+d}}} \right ){a}^{2}d-2\,{\it EllipticPi} \left ({\frac{i \left ( -1+\cos \left ( fx+e \right ) \right ) }{\sin \left ( fx+e \right ) }},-{\frac{a-b}{a+b}},i\sqrt{{\frac{c-d}{c+d}}} \right ) abc-2\,{\it EllipticPi} \left ({\frac{i \left ( -1+\cos \left ( fx+e \right ) \right ) }{\sin \left ( fx+e \right ) }},-1,i\sqrt{{\frac{c-d}{c+d}}} \right ){a}^{2}d+2\,{\it EllipticPi} \left ({\frac{i \left ( -1+\cos \left ( fx+e \right ) \right ) }{\sin \left ( fx+e \right ) }},-1,i\sqrt{{\frac{c-d}{c+d}}} \right ){b}^{2}d \right ) \sqrt{{\frac{d+c\cos \left ( fx+e \right ) }{\cos \left ( fx+e \right ) }}} \left ({\frac{g}{\cos \left ( fx+e \right ) }} \right ) ^{{\frac{3}{2}}}{\frac{1}{\sqrt{ \left ( 1+\cos \left ( fx+e \right ) \right ) ^{-1}}}}} \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{\sqrt{d \sec \left (f x + e\right ) + c} \left (g \sec \left (f x + e\right )\right )^{\frac{3}{2}}}{b \sec \left (f x + e\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(-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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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{d \sec \left (f x + e\right ) + c} \left (g \sec \left (f x + e\right )\right )^{\frac{3}{2}}}{b \sec \left (f x + e\right ) + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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