Optimal. Leaf size=61 \[ \frac{2 \sqrt{2 \cos (c+d x)+3} \sqrt{\sec (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),\frac{4}{5}\right )}{\sqrt{5} d \sqrt{-3 \sec (c+d x)-2}} \]
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Rubi [A] time = 0.0700731, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {3858, 2663, 2661} \[ \frac{2 \sqrt{2 \cos (c+d x)+3} \sqrt{\sec (c+d x)} F\left (\frac{1}{2} (c+d x)|\frac{4}{5}\right )}{\sqrt{5} d \sqrt{-3 \sec (c+d x)-2}} \]
Antiderivative was successfully verified.
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Rule 3858
Rule 2663
Rule 2661
Rubi steps
\begin{align*} \int \frac{\sqrt{\sec (c+d x)}}{\sqrt{-2-3 \sec (c+d x)}} \, dx &=\frac{\left (\sqrt{-3-2 \cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{-3-2 \cos (c+d x)}} \, dx}{\sqrt{-2-3 \sec (c+d x)}}\\ &=\frac{\left (\sqrt{3+2 \cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\frac{3}{5}+\frac{2}{5} \cos (c+d x)}} \, dx}{\sqrt{5} \sqrt{-2-3 \sec (c+d x)}}\\ &=\frac{2 \sqrt{3+2 \cos (c+d x)} F\left (\frac{1}{2} (c+d x)|\frac{4}{5}\right ) \sqrt{\sec (c+d x)}}{\sqrt{5} d \sqrt{-2-3 \sec (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.0432494, size = 61, normalized size = 1. \[ \frac{2 \sqrt{2 \cos (c+d x)+3} \sqrt{\sec (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),\frac{4}{5}\right )}{\sqrt{5} d \sqrt{-3 \sec (c+d x)-2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.232, size = 139, normalized size = 2.3 \begin{align*}{\frac{{\frac{i}{5}} \left ( \sin \left ( dx+c \right ) \right ) ^{2}\cos \left ( dx+c \right ) \sqrt{10}\sqrt{2}}{d \left ( 2\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}+\cos \left ( dx+c \right ) -3 \right ) }\sqrt{ \left ( \cos \left ( dx+c \right ) \right ) ^{-1}}\sqrt{-{\frac{3+2\,\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) }}}{\it EllipticF} \left ({\frac{i \left ( -1+\cos \left ( dx+c \right ) \right ) }{\sin \left ( dx+c \right ) }},{\frac{\sqrt{5}}{5}} \right ) \sqrt{ \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}\sqrt{{\frac{3+2\,\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{-3 \, \sec \left (d x + c\right ) - 2}}\,{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{-3 \, \sec \left (d x + c\right ) - 2} \sqrt{\sec \left (d x + c\right )}}{3 \, \sec \left (d x + c\right ) + 2}, 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{- 3 \sec{\left (c + d x \right )} - 2}}\, 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{-3 \, \sec \left (d x + c\right ) - 2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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