Optimal. Leaf size=55 \[ \frac{2 \sqrt{-3 \cos (c+d x)-2} \sqrt{\sec (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x+\pi ),6\right )}{d \sqrt{-2 \sec (c+d x)-3}} \]
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Rubi [A] time = 0.0580804, antiderivative size = 55, 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, 2662} \[ \frac{2 \sqrt{-3 \cos (c+d x)-2} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x+\pi )\right |6\right )}{d \sqrt{-2 \sec (c+d x)-3}} \]
Antiderivative was successfully verified.
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Rule 3858
Rule 2662
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 (\left .\frac{1}{2} (c+\pi +d x)\right |6\right ) \sqrt{\sec (c+d x)}}{d \sqrt{-3-2 \sec (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.0466407, size = 61, normalized size = 1.11 \[ \frac{2 \sqrt{3 \cos (c+d x)+2} \sqrt{\sec (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),\frac{6}{5}\right )}{\sqrt{5} d \sqrt{-2 \sec (c+d x)-3}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.235, size = 142, normalized size = 2.6 \begin{align*}{\frac{{\frac{i}{5}} \left ( \sin \left ( dx+c \right ) \right ) ^{2}\cos \left ( dx+c \right ) \sqrt{2}\sqrt{10}}{d \left ( 3\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}-\cos \left ( dx+c \right ) -2 \right ) }{\it EllipticF} \left ({\frac{i \left ( -1+\cos \left ( dx+c \right ) \right ) }{\sin \left ( dx+c \right ) }},{\frac{i}{5}}\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{ \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}\sqrt{{\frac{2+3\,\cos \left ( dx+c \right ) }{\cos \left ( dx+c \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{-2 \, \sec \left (d x + c\right ) - 3} \sqrt{\sec \left (d x + c\right )}}{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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