Optimal. Leaf size=129 \[ \frac{4 \sqrt{2-3 \cos (c+d x)} \sqrt{\sec (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x+\pi ),\frac{6}{5}\right )}{3 \sqrt{5} d \sqrt{2 \sec (c+d x)-3}}-\frac{2 \sqrt{5} \sqrt{2 \sec (c+d x)-3} E\left (\frac{1}{2} (c+d x+\pi )|\frac{6}{5}\right )}{3 d \sqrt{2-3 \cos (c+d x)} \sqrt{\sec (c+d x)}} \]
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Rubi [A] time = 0.174113, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3862, 3856, 2654, 3858, 2662} \[ \frac{4 \sqrt{2-3 \cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\frac{1}{2} (c+d x+\pi )|\frac{6}{5}\right )}{3 \sqrt{5} d \sqrt{2 \sec (c+d x)-3}}-\frac{2 \sqrt{5} \sqrt{2 \sec (c+d x)-3} E\left (\frac{1}{2} (c+d x+\pi )|\frac{6}{5}\right )}{3 d \sqrt{2-3 \cos (c+d x)} \sqrt{\sec (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 3862
Rule 3856
Rule 2654
Rule 3858
Rule 2662
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{\sec (c+d x)} \sqrt{-3+2 \sec (c+d x)}} \, dx &=-\left (\frac{1}{3} \int \frac{\sqrt{-3+2 \sec (c+d x)}}{\sqrt{\sec (c+d x)}} \, dx\right )+\frac{2}{3} \int \frac{\sqrt{\sec (c+d x)}}{\sqrt{-3+2 \sec (c+d x)}} \, dx\\ &=\frac{\left (2 \sqrt{2-3 \cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{2-3 \cos (c+d x)}} \, dx}{3 \sqrt{-3+2 \sec (c+d x)}}-\frac{\sqrt{-3+2 \sec (c+d x)} \int \sqrt{2-3 \cos (c+d x)} \, dx}{3 \sqrt{2-3 \cos (c+d x)} \sqrt{\sec (c+d x)}}\\ &=\frac{4 \sqrt{2-3 \cos (c+d x)} F\left (\frac{1}{2} (c+\pi +d x)|\frac{6}{5}\right ) \sqrt{\sec (c+d x)}}{3 \sqrt{5} d \sqrt{-3+2 \sec (c+d x)}}-\frac{2 \sqrt{5} E\left (\frac{1}{2} (c+\pi +d x)|\frac{6}{5}\right ) \sqrt{-3+2 \sec (c+d x)}}{3 d \sqrt{2-3 \cos (c+d x)} \sqrt{\sec (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.077407, size = 72, normalized size = 0.56 \[ \frac{\sqrt{3 \cos (c+d x)-2} \sqrt{\sec (c+d x)} \left (4 \text{EllipticF}\left (\frac{1}{2} (c+d x),6\right )+2 E\left (\left .\frac{1}{2} (c+d x)\right |6\right )\right )}{3 d \sqrt{2 \sec (c+d x)-3}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.242, size = 370, normalized size = 2.9 \begin{align*} -{\frac{2}{3\,d\sin \left ( dx+c \right ) \left ( -2+3\,\cos \left ( dx+c \right ) \right ) } \left ( 3\,i\sin \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 ) }},i\sqrt{5} \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}}}-i\sin \left ( dx+c \right ) \cos \left ( dx+c \right ){\it EllipticE} \left ({\frac{i \left ( -1+\cos \left ( dx+c \right ) \right ) }{\sin \left ( dx+c \right ) }},i\sqrt{5} \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}}}+3\,i{\it EllipticF} \left ({\frac{i \left ( -1+\cos \left ( dx+c \right ) \right ) }{\sin \left ( dx+c \right ) }},i\sqrt{5} \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}}}\sin \left ( dx+c \right ) -i{\it EllipticE} \left ({\frac{i \left ( -1+\cos \left ( dx+c \right ) \right ) }{\sin \left ( dx+c \right ) }},i\sqrt{5} \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}}}\sin \left ( dx+c \right ) -3\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}+5\,\cos \left ( dx+c \right ) -2 \right ) \sqrt{-{\frac{-2+3\,\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) }}}{\frac{1}{\sqrt{ \left ( \cos \left ( dx+c \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{1}{\sqrt{2 \, \sec \left (d x + c\right ) - 3} \sqrt{\sec \left (d x + c\right )}}\,{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 )^{2} - 3 \, \sec \left (d x + c\right )}, 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{1}{\sqrt{2 \sec{\left (c + d x \right )} - 3} \sqrt{\sec{\left (c + d x \right )}}}\, 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{1}{\sqrt{2 \, \sec \left (d x + c\right ) - 3} \sqrt{\sec \left (d x + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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