Optimal. Leaf size=61 \[ \frac{2 \sqrt{3 \cos (c+d x)+2} \sqrt{\sec (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),\frac{6}{5}\right )}{\sqrt{5} d \sqrt{2 \sec (c+d x)+3}} \]
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Rubi [A] time = 0.0570815, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {3858, 2661} \[ \frac{2 \sqrt{3 \cos (c+d x)+2} \sqrt{\sec (c+d x)} F\left (\frac{1}{2} (c+d x)|\frac{6}{5}\right )}{\sqrt{5} d \sqrt{2 \sec (c+d x)+3}} \]
Antiderivative was successfully verified.
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Rule 3858
Rule 2661
Rubi steps
\begin{align*} \int \frac{\sqrt{\sec (c+d x)}}{\sqrt{3+2 \sec (c+d x)}} \, dx &=\frac{\left (\sqrt{2+3 \cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{2+3 \cos (c+d x)}} \, dx}{\sqrt{3+2 \sec (c+d x)}}\\ &=\frac{2 \sqrt{2+3 \cos (c+d x)} F\left (\frac{1}{2} (c+d x)|\frac{6}{5}\right ) \sqrt{\sec (c+d x)}}{\sqrt{5} d \sqrt{3+2 \sec (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.0572409, size = 61, normalized size = 1. \[ \frac{2 \sqrt{3 \cos (c+d x)+2} \sqrt{\sec (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),\frac{6}{5}\right )}{\sqrt{5} d \sqrt{2 \sec (c+d x)+3}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.215, size = 145, normalized size = 2.4 \begin{align*}{\frac{\sqrt{5} \left ( \sin \left ( dx+c \right ) \right ) ^{2}\cos \left ( dx+c \right ) \sqrt{10}\sqrt{2}}{5\,d \left ( 3\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}-\cos \left ( dx+c \right ) -2 \right ) }{\it EllipticF} \left ({\frac{ \left ( -1+\cos \left ( dx+c \right ) \right ) \sqrt{5}}{5\,\sin \left ( dx+c \right ) }},i\sqrt{5} \right ) \sqrt{ \left ( \cos \left ( dx+c \right ) \right ) ^{-1}}\sqrt{{\frac{2+3\,\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) }}}\sqrt{{\frac{2+3\,\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}\sqrt{ \left ( \cos \left ( dx+c \right ) +1 \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{\sec \left (d x + c\right )}}{\sqrt{2 \, \sec \left (d x + c\right ) + 3}}\,{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{\sec \left (d x + c\right )}}{\sqrt{2 \, \sec \left (d x + c\right ) + 3}}, 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{\sqrt{\sec{\left (c + d x \right )}}}{\sqrt{2 \sec{\left (c + d x \right )} + 3}}\, 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{\sqrt{\sec \left (d x + c\right )}}{\sqrt{2 \, \sec \left (d x + c\right ) + 3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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