Optimal. Leaf size=54 \[ \frac{2 \sqrt{3-2 \cos (c+d x)} \sqrt{\sec (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),-4\right )}{d \sqrt{3 \sec (c+d x)-2}} \]
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Rubi [A] time = 0.0566, antiderivative size = 54, 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} \[ \frac{2 \sqrt{3-2 \cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |-4\right )}{d \sqrt{3 \sec (c+d x)-2}} \]
Antiderivative was successfully verified.
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Rule 3858
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{2 \sqrt{3-2 \cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |-4\right ) \sqrt{\sec (c+d x)}}{d \sqrt{-2+3 \sec (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.0561017, size = 54, normalized size = 1. \[ \frac{2 \sqrt{3-2 \cos (c+d x)} \sqrt{\sec (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),-4\right )}{d \sqrt{3 \sec (c+d x)-2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.234, size = 137, normalized size = 2.5 \begin{align*}{\frac{i \left ( \sin \left ( dx+c \right ) \right ) ^{2}\cos \left ( dx+c \right ) \sqrt{2}}{d \left ( 2\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}-5\,\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 ) }},\sqrt{5} \right ) \sqrt{ \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}\sqrt{-2\,{\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{\sec \left (d x + c\right )}}{\sqrt{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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